Topological and geometrical aspects of band theory

Let us consider a single-particle Hamiltonian H having translation invariance under a Bravais lattice (e.g. a tight-binding Hamiltonian). Most of this section is valid in any space dimension D, but at the end we will focus on space dimension D = 2 which is special for topology as we have seen in the TLS section. A parameter-dependent Bloch Hamiltonian is defined via a unitary transformation as:

$$\begin{equation} H({\boldsymbol{k}}) = e^{- i {\boldsymbol{k}} \cdot {\boldsymbol{r}}} H e^{i {\boldsymbol{k}} \cdot {\boldsymbol{r}}}, \end{equation} \tag{ 40 }$$

where ${\boldsymbol{r}}$ is the position operator and ${\boldsymbol{k}}$ is a parameter with the dimension of a wave vector. In the following, we will refer to it as the canonical Bloch Hamiltonian or simply the Bloch Hamiltonian. Bloch's theorem allows us to diagonalize the Hamiltonian as:

$$\begin{equation} H |\psi_{n,{\boldsymbol{k}}}\rangle = E_n({\boldsymbol{k}}) |\psi_{n,{\boldsymbol{k}}}\rangle, \end{equation} \tag{ 41 }$$

where ${\boldsymbol{k}}$ is the crystal momentum and n is a discrete band index. The Bloch eigenvectors are:

$$\begin{equation} |\psi_{n,{\boldsymbol{k}}}\rangle = e^{i {\boldsymbol{k}} \cdot {\boldsymbol{r}}} |u_{n}({\boldsymbol{k}})\rangle, \end{equation} \tag{ 42 }$$

where in coordinate representation the 'cell-periodic Bloch state' $u_{n,{\boldsymbol{k}}}({\boldsymbol{r}}) = \langle{{\boldsymbol{r}} | u_{n} ({\boldsymbol{k}}) }\rangle$ obeys:

$$\begin{equation} u_{n,{\boldsymbol{k}}}({\boldsymbol{r}}+{\boldsymbol{R}}) = u_{n,{\boldsymbol{k}}}({\boldsymbol{r}}), \end{equation} \tag{ 43 }$$

namely it is periodic in real space with the periodicity of the unit cell (${\boldsymbol{R}}$ belongs to the Bravais lattice). In the following, in order to emphasize the fact that the Bloch wavevector ${\boldsymbol{k}}$ now plays the role of a parameter (unlike the band index n which remains a quantum number), we have chosen to write the cell-periodic Bloch eigenvector as $| u_{n} ({\boldsymbol{k}}) \rangle$ rather than $| u_{n,{\boldsymbol{k}}} \rangle$. Let us now examine the ${\boldsymbol{k}}$-dependence of $| u_{n} ({\boldsymbol{k}}) \rangle$ in the reciprocal space. One must have $|\psi_{n,{\boldsymbol{k}}+{\boldsymbol{G}}}\rangle \propto |\psi_{n,{\boldsymbol{k}}}\rangle$ up to a global phase, which shows that the crystal momentum ${\boldsymbol{k}}$ can be restricted to the first BZ. A common choice of phase is to ask that $|\psi_{n,{\boldsymbol{k}}+{\boldsymbol{G}}}\rangle = |\psi_{n,{\boldsymbol{k}}}\rangle$ so that:

$$\begin{equation} |u_{n}({\boldsymbol{k}}+{\boldsymbol{G}})\rangle = e^{-i {\boldsymbol{G}} \cdot {\boldsymbol{r}}} |u_{n}({\boldsymbol{k}})\rangle, \end{equation} \tag{ 44 }$$

which is known as the 'periodic gauge choice' [52]. It does not fully fix the gauge but restricts the possible gauge choices. This choice is not always possible: in 2D there is a famous obstruction to it (known as a non-zero Chern number), that we discuss below. The cell-periodic Bloch eigenvectors $|u_{n}({\boldsymbol{k}})\rangle$ will be the main players in the following ⁴ . In general, they do not have the periodicity of the reciprocal lattice, see equation (44). However, the energy bands do have the periodicity of the reciprocal lattice $E_n({\boldsymbol{k}}+{\boldsymbol{G}}) = E_n({\boldsymbol{k}})$.

The reason for performing the unitary transformation (40) is that we want a parameter-dependent Hamiltonian in order to be able to separate two different dynamics: that associated with changing the wave vector ${\boldsymbol{k}}$ (slow) and that associated with changing the band index n (fast). The goal is to obtain an effective description for the dynamics of an electron restricted to a single band (this will be done by projecting on a single band) nevertheless taking into account the coupling to other bands (this will occur via an emergent gauge field). To make this discussion more concrete, let us consider a one-dimensional tight-binding model (but with several bands) with hopping amplitude t, lattice spacing a and crystal size L. The timescale for the intra-band dynamics can be estimated as ∼ћL/(ta) (because the spacing between allowed k values is Δk = 2π/L and the typical band velocity is ta/ћ so that the typical energy change ΔE ∼ taΔk ∼ ta/L), and that for the inter-band dynamics as ∼ћ/bandgap ∼ћ/t. For a macroscopic crystal $L\gg a$, the first timescale is much larger than the second. The idea that the emergent (Berry) gauge structure generally appears via a separation of two timescales such that the effective dynamics of the slow (or 'heavy') degrees of freedom gets modified by the integration over the fast (or 'light') degrees of freedom is well explained in [74]. Paraphrasing Michael Berry, the reaction of the fast degrees of freedom ('light system') onto the slow degrees of freedom ('heavy system') occurs via the appearance of an emergent gauge field, as we will see.

The $|u_{n}({\boldsymbol{k}})\rangle$'s are the parameter-dependent eigenvectors of the Bloch Hamiltonian:

$$\begin{equation} H({\boldsymbol{k}}) |u_{n}({\boldsymbol{k}})\rangle = E_n({\boldsymbol{k}}) |u_{n}({\boldsymbol{k}})\rangle. \end{equation} \tag{ 45 }$$

This last equation appears very similar to (41) but is actually quite different. Whereas the set $\{|\psi_{n,{\boldsymbol{k}}}\rangle,n,{\boldsymbol{k}}\}$ forms an orthonormal basis in Hilbert space, it is not so for $\{|u_{n}({\boldsymbol{k}})\rangle,n,{\boldsymbol{k}}\}$. Indeed $|u_{n}({\boldsymbol{k}})\rangle$ and $|u_{n}({\boldsymbol{k}}^{^{\prime}})\rangle$ are eigenvectors of two different Hamiltonians $H({\boldsymbol{k}})$ and $H({\boldsymbol{k}}^{^{\prime}})$ and therefore need not be orthogonal. However $\langle u_n({\boldsymbol{k}})|u_{n^{^{\prime}}}({\boldsymbol{k}})\rangle = \delta_{n,n^{^{\prime}}}$ as $|u_{n}({\boldsymbol{k}})\rangle$ and $|u_{n^{^{\prime}}}({\boldsymbol{k}})\rangle$ are eigenvectors of the same Hamiltonian $H({\boldsymbol{k}})$. Their overlap

$$\begin{equation} \langle u_n({\boldsymbol{k}})|u_n({\boldsymbol{k}}^{^{\prime}})\rangle \neq \delta_{{\boldsymbol{k}},{\boldsymbol{k}}^{^{\prime}}} \end{equation} \tag{ 46 }$$

is a non-zero complex number in general. Note that, precisely at this point, we have restricted the discussion to a single band (the nth band). This is the moment where we stop discussing the dynamics in the full Hilbert space and project on a single band of interest. The corresponding fiber F is a one dimensional complex vector space, i.e. essentially a U(1) phase, while the first BZ torus T^D plays the role of the base space B.

When ${\boldsymbol{k}}^{^{\prime}} = {\boldsymbol{k}} +d {\boldsymbol{k}}$ is close to ${\boldsymbol{k}}$, one may study the deviation of this overlap (46) from unity and define the evolution (i) of its phase and (ii) of its norm:

(i) The evolution of its phase, by expanding at first order in $\boldsymbol{dk}$,

$$\begin{equation} \langle u_n({\boldsymbol{k}})|u_n({\boldsymbol{k}} +\boldsymbol{dk})\rangle\approx \langle u_n({\boldsymbol{k}})|(1+\boldsymbol{dk} \cdot \boldsymbol{\nabla}_{{\boldsymbol{k}}})|u_n({\boldsymbol{k}})\rangle \approx e^{-i \boldsymbol{dk} \cdot \boldsymbol{A}_n({\boldsymbol{k}})}, \end{equation} \tag{ 47 }$$

defines the Berry connection:

$$\begin{equation} \mathbf{A}_n({\boldsymbol{k}}) = i\langle u_n({\boldsymbol{k}})|\boldsymbol{\nabla}_{{\boldsymbol{k}}} u_n({\boldsymbol{k}})\rangle, \end{equation} \tag{ 48 }$$

which is the Bloch version of the TLS formula equation (23). This quantity is called $\boldsymbol{\mathfrak{X}}_{nn}({\boldsymbol{k}})$ in [7] and is related to the projected position operator. The Berry curvature is given by the curl of the Berry connection:

$$\begin{equation} F_{ij}^{ n}({\boldsymbol{k}}) = i\langle \partial_i u_n|\partial_j u_n\rangle +\textrm{c.c.}, \end{equation} \tag{ 49 }$$

which is the Bloch counterpart of equation (28) for the TLS. The Berry curvature $\mathbf{F}_n({\boldsymbol{k}}) = \textrm{curl } \mathbf{A}_n({\boldsymbol{k}})$ is called $\boldsymbol{\Omega}_{n}({\boldsymbol{k}})$ in [7]. The geometry of the fiber bundle is described by the Berry connection, curvature and phase, while its topology is characterized by the Chern number (see below). In the context of band theory, this was first recognized by Thouless et al [12, 75] and Simon and coworkers [15, 76] who underlined the relation with Berry's contribution.

(ii) The evolution of the norm of this overlap (46) defines another geometric quantity, known as the quantum metric, obtained by introducing a distance (squared) in the Hilbert space:

$$\begin{equation} ds^2 = 1-|\langle u_n({\boldsymbol{k}})|u_n({\boldsymbol{k}} +\boldsymbol{dk})\rangle|^2\approx \sum_{i,j} g_{ij}^{ n}({\boldsymbol{k}}) dk_i dk_j. \end{equation} \tag{ 50 }$$

This distance in projective Hilbert space was introduced by Provost and Vallée [77]. Expanding at second order in $\boldsymbol{dk}$, the quantum metric is obtained as:

$$\begin{equation} g_{ij}^{ n}({\boldsymbol{k}}) = \textrm{Re} \langle \partial_i u_n|(\unicode{x1D7D9}-|u_n \rangle\langle u_n|)|\partial_j u_n\rangle, \end{equation} \tag{ 51 }$$

where $\partial_i$ is a short-hand notation for $\partial_{k_i}$ and $\unicode{x1D7D9}$ is the identity. The quantum metric and the Berry curvature are the real and imaginary part of a more general object called the quantum geometric tensor:

$$\begin{equation} T_{ij}^{ n}({\boldsymbol{k}}) = \langle \partial_i u_n|(\unicode{x1D7D9}-|u_n \rangle\langle u_n|)|\partial_j u_n\rangle. \end{equation} \tag{ 52 }$$

Indeed $g_{ij}^{ n} = \textrm{Re} T_{ij}^{ n}$ and $F_{ij}^{ n} = -2 \textrm{Im} T_{ij}^{ n}$. For more information on the quantum metric, see the article by Berry in [68]. The quantum metric appears in some physical quantities such as the magnetic orbital susceptibility [78] or the superfluid weight [79]. It is also useful to define localization in an insulator [53] and gives a measure of the minimal wavepacket spreading for a Bloch electron [80].

There is a gauge freedom in the cell-periodic Bloch states seen as functions of the parameter ${\boldsymbol{k}}$. Indeed, provided the phase $\varphi({\boldsymbol{k}})$ is a smooth enough function of ${\boldsymbol{k}}$,

$$\begin{equation} |\tilde{u}_n({\boldsymbol{k}})\rangle = e^{i\varphi({\boldsymbol{k}})}|u_n({\boldsymbol{k}})\rangle, \end{equation} \tag{ 53 }$$

is also a valid choice for the cell-periodic Bloch states. We refer to it as a Berry gauge transformation to distinguish it from a real-space electromagnetic gauge transformation. In this gauge transformation $|u_n({\boldsymbol{k}})\rangle \to |\tilde{u}_n({\boldsymbol{k}})\rangle$, the Berry connection gets modified as:

$$\begin{equation} \tilde{\mathbf{A}}_n = i\langle \tilde{u}_n |\boldsymbol{\nabla}_{{\boldsymbol{k}}} \tilde{u}_n\rangle = \mathbf{A}_n-\boldsymbol{\nabla}_{{\boldsymbol{k}}} \varphi, \end{equation} \tag{ 54 }$$

and is therefore not gauge-invariant just like the vector potential in electromagnetism. However, the Berry curvature is gauge-invariant since:

$$\begin{equation} \tilde{\mathbf{F}}_n = \textrm{curl } \tilde{\mathbf{A}}_n = \textrm{curl } \mathbf{A}_n + 0 = \mathbf{F}_n, \end{equation} \tag{ 55 }$$

just like the magnetic field. More generally, the quantum geometric tensor, i.e. both the Berry curvature and the quantum metric, is gauge-invariant and therefore measurable. A map of the Berry curvature and of the quantum metric in the whole BZ has been measured in artificial crystals, see e.g. [81, 82].

The above expression may give the impression that the Berry curvature in the nth band only depends on a single band. This is actually not the case. To show that the Berry curvature is related to virtual transitions between bands (at fixed ${\boldsymbol{k}}$), it is useful to rewrite it using perturbation theory to express $|\partial_j u_n\rangle$ as a function of $|u_{n^{^{\prime}}}\rangle$ with $n^{^{\prime}}\neq n$:

$$\begin{equation} F_{ij}^{ n}({\boldsymbol{k}}) = i\sum_{n^{^{\prime}}\neq n }\frac{\langle u_n|\partial_i H({\boldsymbol{k}}) | u_{n^{^{\prime}}}\rangle \langle u_{n^{^{\prime}}}|\partial_j H({\boldsymbol{k}}) | u_{n}\rangle}{(E_{n^{^{\prime}}}-E_n)^2} +\textrm{c.c.} \end{equation} \tag{ 56 }$$

This expression has the flavor of second-order perturbation theory and clearly shows that the Berry curvature in the $n^{\textrm{th}}$ band is the result of virtual transitions (see figure 4) to other bands and that the inter-band coupling is related to the velocity operator $\partial_i H({\boldsymbol{k}})$. As a consequence, it is obvious that if there is a single band in the model, these effects are totally absent. Also, the Berry curvature is well-defined only when there is a gap between bands (at fixed ${\boldsymbol{k}}$, i.e. $E_{n^{^{\prime}}}({\boldsymbol{k}})\neq E_n({\boldsymbol{k}})$) and becomes large when this gap becomes small. The above expression makes it obvious that the Berry curvature is gauge-independent (as for each bra $\langle u_{n^{^{\prime}}}|$, it involves the corresponding ket $| u_{n^{^{\prime}}}\rangle$), in contrast to the Berry connection which is gauge-dependent (as it involves a bra $\langle u_{n}|$ but a different ket $|\boldsymbol{\nabla}_{\boldsymbol{k}} u_{n}\rangle$). The Berry curvature also has the periodicity of the reciprocal Bravais lattice $F_{ij}^{ n}({\boldsymbol{k}}+{\boldsymbol{G}}) = F_{ij}^{ n}({\boldsymbol{k}})$, although this is not the case of the $|u_n ({\boldsymbol{k}})\rangle$'s. A consequence of equation (56) is that the sum over all bands of the Berry curvature at a given ${\boldsymbol{k}}$ point vanishes:

$$\begin{equation} \sum_n F_{ij}^{ n}({\boldsymbol{k}}) = 0. \end{equation} \tag{ 57 }$$

Time-reversal symmetry implies:

$$\begin{equation} F_{ij}^{ n}(-{\boldsymbol{k}}) = - F_{ij}^{ n}({\boldsymbol{k}}), \end{equation} \tag{ 58 }$$

while inversion symmetry imposes:

$$\begin{equation} F_{ij}^{ n}(-{\boldsymbol{k}}) = F_{ij}^{ n}({\boldsymbol{k}}). \end{equation} \tag{ 59 }$$

Hence, if both symmetries are present, the Berry curvature vanishes everywhere in the BZ.

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In general, the Bloch Hamiltonian $H({\boldsymbol{k}})$ does not have the periodicity of the reciprocal lattice, even if the spectrum $E_n({\boldsymbol{k}})$ always has it. The reason is that the definition of the Bloch Hamiltonian (40) involves the Hamiltonian H and the position operator ${\boldsymbol{r}}$. The latter depends on the position of every site in the crystal, including sites within the unit cell (e.g. for a lattice with a basis). The distance between sites within the unit cell need not have a special relationship (i.e. need not be commensurable) with the Bravais lattice vectors. Therefore, in general, the Bloch Hamiltonian need not be a periodic function of ${\boldsymbol{k}}$ [83]. In section 5, we will see on the example of the SSH chain that the Bloch Hamiltonian is periodic but with a double periodicity compared to the reciprocal lattice. In section 6, we will see another example: the honeycomb lattice, for which the Bloch Hamiltonian has a triple periodicity compared to the reciprocal lattice [83].

An alternative 'periodic Bloch Hamiltonian' is often defined as follows:

$$\begin{equation} \mathcal{H}({\boldsymbol{k}}) = e^{- i {\boldsymbol{k}} \cdot {\boldsymbol{R}}} H e^{i {\boldsymbol{k}} \cdot {\boldsymbol{R}}}. \end{equation} \tag{ 60 }$$

It depends only on the position operator for the unit cell ${\boldsymbol{R}}$ (i.e. the position on the Bravais lattice) and not on the full position operator ${\boldsymbol{r}}$. We will write ${\boldsymbol{r}} = {\boldsymbol{R}}+\boldsymbol{\delta}$, where $\boldsymbol{\delta}$ is the position operator within the unit cell. The canonical and the periodic Bloch Hamiltonian have the same energy bands $E_n({\boldsymbol{k}})$. This alternative Bloch Hamiltonian is periodic with the reciprocal lattice $\mathcal{H}({\boldsymbol{k}}+{\boldsymbol{G}}) = \mathcal{H}({\boldsymbol{k}})$ unlike the Bloch Hamiltonian $H({\boldsymbol{k}}+{\boldsymbol{G}}) = e^{-i{\boldsymbol{k}} \cdot \boldsymbol{\delta}}H({\boldsymbol{k}})e^{i{\boldsymbol{k}} \cdot \boldsymbol{\delta}}$, which is not, in general. Note also that the canonical Bloch Hamiltonian is unique, while the periodic Bloch Hamiltonian depends on the choice of the unit cell. One should therefore better speak of a, rather than the, periodic Bloch Hamiltonian. This issue of canonical versus periodic Bloch Hamiltonian is crucial in the case of a lattice with a basis (e.g. the honeycomb lattice or the SSH chain). In the literature, it is sometimes known as basis I versus basis II, see [83–86]. The unique (canonical) Bloch Hamiltonian $H({\boldsymbol{k}})$ being that in basis II and the various possible periodic Bloch Hamiltonians $\mathcal{H}({\boldsymbol{k}})$ belonging to basis I.

The eigenvectors of a periodic Bloch Hamiltonian are not the cell-periodic Bloch eigenvectors $|u_n({\boldsymbol{k}})\rangle$. In order to distinguish them, we call them $|v_n({\boldsymbol{k}})\rangle = e^{-i{\boldsymbol{k}} \cdot {\boldsymbol{R}}}|\psi_{n {\boldsymbol{k}}}\rangle = e^{i{\boldsymbol{k}} \cdot \boldsymbol{\delta}}|u_n({\boldsymbol{k}})\rangle$, where $\boldsymbol{\delta} = {\boldsymbol{r}}-{\boldsymbol{R}}$ is the position operator within the unit cell. They also satisfy:

$$\begin{equation} \mathcal{H}({\boldsymbol{k}}) |v_{n}({\boldsymbol{k}})\rangle = E_n({\boldsymbol{k}}) |v_{n}({\boldsymbol{k}})\rangle, \end{equation} \tag{ 61 }$$

and the $v_{n, {\boldsymbol{k}}}({\boldsymbol{r}})$'s have the periodicity of the Bravais lattice. The $|v_{n}({\boldsymbol{k}})\rangle$'s have the periodicity of the reciprocal lattice (at least under the 'periodic gauge choice'): $|v_{n}({\boldsymbol{k}}+{\boldsymbol{G}})\rangle = |v_{n}({\boldsymbol{k}})\rangle$.

Let us restrict for a moment to a tight-binding model with Hamiltonian $H = \sum_{i,j} t_{ij} |i\rangle \langle j|$ and position operator ${\boldsymbol{r}} = \sum_i {\boldsymbol{r}}_i |i \rangle \langle i|$. The Hamiltonian stores the information about the connectivity between orbitals, that are assumed to form a complete orthogonal set: $\langle i |j \rangle = \delta_{i,j}$ and $\sum_i | i \rangle \langle i | = \unicode{x1D7D9}$. The position operator contains the information on the position of the orbitals in space. The canonical Bloch Hamiltonian should be used to compute geometrical quantities that crucially depend on the spatial location (or spatial embedding, as Haldane calls it) of the orbitals defining the model (i.e. Berry curvature, quantum metric, etc). The periodic Bloch Hamiltonian is more convenient in computing topological invariants such as winding numbers, Chern numbers, symmetry-based indicators (such as that of Fu and Kane [32]) etc. To be on the safe side, it is always better to work with the canonical Bloch Hamiltonian. As the periodic Bloch Hamiltonian is blind to the exact location of orbitals within the unit cell, the two Bloch Hamiltonians are not equivalent when there is a lattice with a basis (e.g. graphene or the SSH chain). The canonical Bloch Hamiltonian incorporates more information about the spatial location of orbitals than the periodic Bloch Hamiltonian. Colloquially speaking, the canonical Bloch Hamiltonian $H({\boldsymbol{k}})$ knows the connectivity contained in the tight-binding Hamiltonian H and the complete position operator ${\boldsymbol{r}} = {\boldsymbol{R}}+\boldsymbol{\delta}$. In constrast, the periodic Bloch Hamiltonian $\mathcal{H}({\boldsymbol{k}})$ only knows H and the Bravais lattice position operator ${\boldsymbol{R}}$ but is unaware of the position operator within the unit cell $\boldsymbol{\delta}$.

In conclusion of this section, we wish to emphasize that there is no need of introducing Berry phases. One could study the complete quantum mechanical problem of an electron in a band structure in the presence of external fields, without ever projecting on a given subset of bands. The appearance of Berry phase effects is only related to the approximate treatment of restricting to a subset of bands and asking for an effective description within this subspace. In practice, solving the full quantum mechanical problem is often un-doable analytically (but may be done numerically) and one ends up asking for an analytically-tractable effective description restricted to a subset of bands. In such a case, Berry phase effects necessarily appear. For example, the orbital magnetic susceptibility or the electric polarization, which are usually discussed in terms of Berry phases, can be studied using only the energy spectrum numerically-computed in a finite magnetic field (Hofstadter butterfly) [87] or in a finite electric field (Wannier–Stark ladder) [88].

If an electron in the nth band performs a closed (and contractible) orbit $\mathcal{C}$ in ${\boldsymbol{k}}$-space under the influence of a force, it will acquire a geometric phase due to the encircled Berry flux in addition to the dynamical phase. This extra phase is known as the Berry phase [14]:

$$\begin{equation} \Gamma_n(\mathcal{C}) = \oint_\mathcal{C} \boldsymbol{dk}\cdot\boldsymbol{A}_n({\boldsymbol{k}}) = \int_\mathcal{S} d^2k \, F_{xy}^{ n}({\boldsymbol{k}}) \,\,[2\pi], \end{equation} \tag{ 62 }$$

where $\partial\mathcal{S} = \mathcal{C}$ (here we assumed that the electron is moving in the xy plane). The Berry phase is defined modulo 2π. Using Stokes' theorem in order to go from the expression involving the connection to that involving the curvature, one needs to assume that the connection is well-defined over the whole patch $\mathcal{S}$. When expressed in terms of the Berry curvature, it is obvious that the Berry phase is gauge-invariant. This Berry phase is a dual of the Aharonov–Bohm phase [89] in the sense that it is acquired in ${\boldsymbol{k}}$-space (rather than real space) and due to the Berry curvature (rather than to the magnetic field). In the case of a closed cyclotron orbit performed under an applied magnetic field, the electron wave function actually acquires both an Aharonov–Bohm phase (in real space) and a Berry phase (in reciprocal space). See, for example, the discussion of semi-classical quantization of cyclotron orbits for Bloch electrons in [85]. The Berry phase factor $W(\mathcal{C}) = \exp[i\Gamma_n(\mathcal{C})]$ is sometimes called an abelian Wilson loop (see section 4.5).

If the Berry phase is computed over a non-contractible loop $\mathcal{P}$ of the BZ torus, then it is known as a Zak phase [19]:

$$\begin{equation} Z_n(\mathcal{P}) = \int_\mathcal{P} \boldsymbol{dk}\cdot\boldsymbol{A}_n({\boldsymbol{k}}) \,\,[2\pi]. \end{equation} \tag{ 63 }$$

Stokes' theorem can no longer be used to relate it to a Berry curvature and it is not obvious that the Zak phase is gauge-invariant. Here there is no smooth gauge for the Berry connection over the complete path $\mathcal{P}$. Actually, the Zak phase is only gauge-invariant provided the 'periodic gauge choice' condition is imposed [19, 52]. This is an example of open-path geometrical phase, as clearly discussed by Resta [52]. Indeed, the final ket $|u_n({\boldsymbol{k}}_f)\rangle$ in the path is not the same as the initial one $|u_n({\boldsymbol{k}}_i)\rangle$ because the canonical Bloch Hamiltonian is not in general periodic with the first BZ (see the discussion about the canonical Bloch Hamiltonian versus a periodic Bloch Hamiltonian). It is possible to impose a definite phase relation between $|u_n({\boldsymbol{k}}_f)\rangle$ and $|u_n({\boldsymbol{k}}_i)\rangle$ in order to render the Zak phase gauge independent. This phase relation involves the position operator and is known as the 'periodic gauge choice':

$$\begin{equation} |u_n({\boldsymbol{k}}_f)\rangle = |u_n({\boldsymbol{k}}_i+{\boldsymbol{G}})\rangle = e^{-i{\boldsymbol{G}}\cdot {\boldsymbol{r}}}|u_n({\boldsymbol{k}}_i)\rangle. \end{equation} \tag{ 64 }$$

It does not completely fix the gauge, but only restricts possible gauge choices. As a consequence of this periodic gauge choice, the Zak phase depends explicitly on the position operator ${\boldsymbol{r}}$ and therefore on the choice of position origin [90]. The Zak phase is therefore best thought as being a particular position within the unit cell known as the Wannier center. The Zak phase is especially relevant in one dimension, where the BZ is a circle (see section 5). If the base space (here the BZ) was not a torus but a sphere or a manifold with a trivial first homotopy group (i.e. no non-contractible loop), then there would be no sense in defining a Zak phase and only the Berry phase would be defined. The Zak phase factor $W(\mathcal{P}) = \exp([i Z_n(\mathcal{P})]$ is sometimes called an abelian Wilson-Zak loop.

The Zak phase should be clearly distinguished from the Berry phase. The former can not be expressed in terms of the Berry curvature and is closely related to the position operator. A convenient habit is to think of the Zak phase as the Wannier center. In contrast, the Berry phase measures the Berry flux across a patch in BZ, has a minor dependence on the position operator (see the above discussion about basis I versus basis II) and does not depend on the choice of position origin. A word of caution to the reader: in many papers, a winding number is mistakenly called a Zak phase.

Here we give a minimal introduction to Wannier functions, which is a whole subject in its own, see [54, 80]. They were introduced long ago [91] as the Fourier transform of Bloch states in a given band:

$$\begin{equation} w_{n,{\boldsymbol{R}}} ({\boldsymbol{r}}) = \int_\textrm{BZ} \frac{d {\boldsymbol{k}}}{(2\pi)^D} \, e^{-i{\boldsymbol{k}} \cdot {\boldsymbol{R}}} \psi_{n,{\boldsymbol{k}}} ({\boldsymbol{r}}) = \int_\textrm{BZ} \frac{d {\boldsymbol{k}}}{(2\pi)^D} \, e^{-i{\boldsymbol{k}} \cdot ({\boldsymbol{R}}-{\boldsymbol{r}})} u_{n,{\boldsymbol{k}}} ({\boldsymbol{r}}) \,. \end{equation} \tag{ 65 }$$

In 1D, they can also be defined as eigenvectors of the projected position operator onto a given band [92]. There is one Wannier function per unit cell and per band. The Wannier states form an orthonormalized basis $\langle w_{n,{\boldsymbol{R}}}|w_{n^{^{\prime}},{\boldsymbol{R}}^{^{\prime}}}\rangle = \delta_{n,n^{^{\prime}}}\delta({\boldsymbol{R}}-{\boldsymbol{R}}^{^{\prime}})$. In a given band, one may concentrate on $w_{n}({\boldsymbol{r}}) = w_{n,\mathbf{0}}({\boldsymbol{r}})$ as other Wannier functions in the same band are obtained by translation by a Bravais lattice vector $w_{n,{\boldsymbol{R}}} ({\boldsymbol{r}}) = w_{n} ({\boldsymbol{r}}-{\boldsymbol{R}})$. Physically, Wannier functions are the solid-state equivalent of atomic or molecular orbitals and are sometimes called Wannier orbitals. As compared to Bloch states, they are better localized in real space but they are not energy eigenvectors. There is a certain gauge-freedom in their definition (related to the Berry-gauge freedom in the definition of the cell-periodic Bloch states). Two important characterization of Wannier functions are:

• Their average position defined as:
  $$\begin{equation} \langle {\boldsymbol{r}}_{n,{\boldsymbol{R}}} \rangle = \int d {\boldsymbol{r}} \, {\boldsymbol{r}} \, |w_{n,{\boldsymbol{R}}}({\boldsymbol{r}})|^2 = \langle {\boldsymbol{r}}_n \rangle + {\boldsymbol{R}}, \end{equation} \tag{ 66 }$$
  which is gauge-invariant and known as the Wannier (or band) center. One is usually mainly interested in $\langle {\boldsymbol{r}}_{n} \rangle$ which is the Wannier center modulo a Bravais lattice vector ${\boldsymbol{R}}$. The Wannier center is related to the electric polarization [20–22] (see section 5.2.4).
• Their localization, which depends on the chosen gauge. The issue of exponential localization (or not) of Wannier functions has a long history starting with Kohn [93] (see e.g. [94] for a discussion of its relation to singularities in Bloch states). Roughly speaking, in a trivial insulator, Wannier functions can be exponentially localized [95]; in a Chern insulator, they are only algebraically localized [96, 97]; and in a metal, they are delocalized. In particular, Thouless has shown that a non-zero Chern number implies an obstruction in finding an exponentially-localized Wannier function [96]. One may characterize the localization by defining the spread or extension $\langle {\boldsymbol{r}}_n^2 \rangle-\langle {\boldsymbol{r}}_n \rangle^2$ of a Wannier function. Playing with the gauge freedom, it is possible to define so-called maximally-localized Wannier functions (MLWF) [80]. The gauge-invariant part of the spread of a MLWF is related to the quantum metric [53, 54]. The obstruction in finding exponential-localized Wannier functions respecting a given symmetry will turn out to play an important role in the definition of SPT insulators.

In space dimension two, the integral of the Berry curvature over the whole BZ torus T² is quantized:

$$\begin{equation} C_n = \frac{1}{2\pi} \int_{T^2} d^2 k F_{xy}^{ n}({\boldsymbol{k}}) \in \mathbb{Z}. \end{equation} \tag{ 67 }$$

This integer is called the Chern number and tells whether the corresponding Bloch bundle is twisted or not. For time-reversal-invariant materials, the Chern number is always zero, being the integral of an odd function of ${\boldsymbol{k}}$ over the BZ. Therefore breaking time-reversal symmetry is a necessary condition to obtain a band with non-zero Chern number, but it is not sufficient, as we will see in the Haldane model. The Chern number can be seen as an obstruction to having a well-defined connection over the whole BZ. Indeed, if a well-defined Berry connection exists over the whole BZ, then the Berry phase computed over the null path should be equal to the total Berry flux across the BZ via Stokes' theorem and should therefore vanish, i.e. C_n  = 0. Therefore C_n ≠ 0 means that there is no well-defined connection over the whole BZ.

An alternative view of the Chern number is the following. A band contact point (or degeneracy) acts as a magnetic monopole in parameter space (a Berry monopole) [14]. Berry calls it a diabolical point because of its diabolo shape [40]. It is also known as a conical intersection. The Chern number counts the number of such degeneracies which are enclosed by the BZ torus, i.e. which are inside the torus. The precise meaning of inside is the following. The band structure has no band degeneracy on the surface of the BZ torus, otherwise the bands would not be well separated, there would be no gap, and the Chern number would not be well-defined. Therefore the band degeneracies that we are talking about actually occur not on the surface of the BZ torus, but really inside a toroid or solid torus [15]. It means that one should imagine extending the 2D $(k_x,k_y)$ model with a third dimension, that we call k_z even it does not correspond to a spatial direction but to some parameter of the Bloch Hamiltonian (such as a hopping amplitude or an on-site energy) that upon tuning creates such a degeneracy. The torus spanned by $(k_x,k_y)$ is now filled inside into a toroid spanned by $(k_x,k_y,k_z)$. We will give a concrete example later in this review when discussing 3D Weyl points in section 8.2. The latter is a contact point between two bands in 3D reciprocal space which is very analogous to the Dirac magnetic monopole.

An equivalent of the Chern number can also be defined at finite temperature [98]. It requires extending the notion of a geometric phase to mixed (i.e. not pure) states and is known as the Uhlmann phase.

In order to discuss the Berry phase effects in the semi-classical equations of motion for a Bloch electron, we first recall the standard equations (see, e.g. [6, 99]) which were obtained in the 1930s by Bloch [3], Peierls [4], Jones and Zener [5]. For simplicity, we consider the motion of a spinless electron restricted to a non-degenerate band and under the influence of external electromagnetic fields (they can be slightly inhomogeneous but here independent of time). The latter are responsible for the dynamics and also for possible transitions between bands. We assume that the electron stays in a given band (adiabatic following, semi-classical approximation): there are no inter-band transitions. This means that the external fields are sufficiently small and that the gap between the bands are sufficiently large. The coupled equations of motion for an electron with average (or center of mass) wave vector ${\boldsymbol{k}}$ and position ${\boldsymbol{r}}_c$ in the nth band are:

$$\begin{align} \hbar \dot{{\boldsymbol{k}}}& = -\boldsymbol{\nabla}_{{\boldsymbol{r}}_c}\widetilde{E}_n -\dot{{\boldsymbol{r}}}_c \times e{\boldsymbol{B}}({\boldsymbol{r}}_c) = -e\left[{\boldsymbol{E}}({\boldsymbol{r}}_c)+ \dot{{\boldsymbol{r}}}_c \times {\boldsymbol{B}}({\boldsymbol{r}}_c)\right] \nonumber\\ \dot{{\boldsymbol{r}}}_c& = \frac{1}{\hbar}\boldsymbol{\nabla}_{{\boldsymbol{k}}}\widetilde{E}_n = \frac{1}{\hbar}\boldsymbol{\nabla}_{{\boldsymbol{k}}} E_n, \end{align} \tag{ 68 }$$

where the semi-classical energy:

$$\begin{equation} \widetilde{E}_n ({\boldsymbol{k}},{\boldsymbol{r}}_c) = E_n({\boldsymbol{k}})-e A_0 ({\boldsymbol{r}}_c), \, \end{equation} \tag{ 69 }$$

is the sum of the band energy and the potential energy of a charge −e in the external electrostatic potential $A_0({\boldsymbol{r}})$. The first equation of the system (68) looks like Newton's equation for a particle with gauge-invariant momentum $\hbar {\boldsymbol{k}}$ and electric charge −e under the influence of the Coulomb and Lorentz forces in an electric field ${\boldsymbol{E}} = -\boldsymbol{\nabla}_{{\boldsymbol{r}}} A_0({\boldsymbol{r}})$ and in a magnetic field ${\boldsymbol{B}} = \boldsymbol{\nabla}_{\boldsymbol{r}}\times \boldsymbol{A} ({\boldsymbol{r}})$. The second equation in (68) is the statement that the electron velocity $\dot{{\boldsymbol{r}}}_c$ is given by the group velocity of the band dispersion relation $E_n({\boldsymbol{k}})$.

It is possible to requantize the above equations and obtain an effective one-band quantum Hamiltonian describing the electron in the nth band:

$$\begin{equation} H_n^{\textrm{eff}} = \widetilde{E}_n ({\boldsymbol{k}},{\boldsymbol{r}}_c) = E_n({\boldsymbol{k}})-e A_0({\boldsymbol{r}}_c), \end{equation} \tag{ 70 }$$

with,

$$\begin{align} {\boldsymbol{k}}& = {\boldsymbol{q}}+\frac{e}{\hbar}\boldsymbol{A}({\boldsymbol{r}}_c) \to -i\boldsymbol{\nabla}_{{\boldsymbol{r}}_c}+\frac{e}{\hbar}\boldsymbol{A}({\boldsymbol{r}}_c)\nonumber\\ {\boldsymbol{r}}_c & = {\boldsymbol{x}}, \end{align} \tag{ 71 }$$

where $\hbar {\boldsymbol{q}}$ and ${\boldsymbol{x}}$ are the canonical momentum and position operators, and $\hbar {\boldsymbol{k}}$ is the (electromagnetic) gauge-invariant momentum operator. This is known as the Peierls substitution [100]. To summarize the 'Peierls strategy', one diagonalizes the Bloch Hamiltonian $H({\boldsymbol{k}})$ in the absence of external fields, projects on a given band to obtain an effective Hamiltonian $H_n^\textrm{eff} = E_n({\boldsymbol{k}})$, then introduces external fields in the effective Hamiltonian $H_n^\textrm{eff} = E_n({\boldsymbol{q}}+\frac{e}{\hbar}\boldsymbol{A})-e A_0({\boldsymbol{r}}_c)$ and eventually requantizes ${\boldsymbol{q}}\to -i\boldsymbol{\nabla}_{{\boldsymbol{r}}_c}$. It is then obvious that inter-band transitions induced by the external fields are neglected in this process.

These equations were able to explain and predict many transport phenomena occurring in crystals (e.g. Bloch oscillations, Hall effect, conduction by holes, cyclotron motion). However, in the 1950s and 1960s, it became clear that these equations were not complete because the electron dynamics is actually influenced by the possibility of virtual transitions to other bands driven by the external fields [7]. In other words, while it is possible to render real (Landau–Zener) inter-band transitions vanishingly small (by having small external fields and large gaps), it is not possible to forbid virtual inter-band transitions. In the 'Peierls strategy', in order to describe the effective behavior in a given band, one relies only on the band's dispersion relation $E_n({\boldsymbol{k}})$ (obtained in the absence of external fields) and on no other information (e.g. such as the cell-periodic Bloch states $|u_{n^{^{\prime}}}({\boldsymbol{k}})\rangle$). Berry phase effects are essentially the statement that cell-periodic Bloch states do play a role in the effective dynamics, as we now discuss.

The complete (at first order in the external fields) equations of motion were obtained in final form by Qian Niu and coworkers [24, 26, 27] using a wave packet approach. The derivation is quite tedious and we do not reproduce it here. For an alternative Hamiltonian approach (i.e. without wave packets), see [101]. The semi-classical equations of motion for an electron wave packet built from states in the nth band and having average wave vector ${\boldsymbol{k}}$ and position ${\boldsymbol{r}}_c$ are:

$$\begin{align} \dot{{\boldsymbol{k}}}& = -\frac{1}{\hbar}\boldsymbol{\nabla}_{{\boldsymbol{r}}_c}\widetilde{E}_n -\dot{{\boldsymbol{r}}}_c \times \frac{e}{\hbar}{\boldsymbol{B}}({\boldsymbol{r}}_c) \nonumber\\ \dot{{\boldsymbol{r}}}_c& = \frac{1}{\hbar}\boldsymbol{\nabla}_{{\boldsymbol{k}}}\widetilde{E}_n -\dot{{\boldsymbol{k}}} \times \boldsymbol{F}_n({\boldsymbol{k}}), \end{align} \tag{ 72 }$$

where the semi-classical energy is:

$$\begin{equation} \widetilde{E}_n ({\boldsymbol{k}},{\boldsymbol{r}}_c) = E_n({\boldsymbol{k}})-e A_0({\boldsymbol{r}}_c)-\boldsymbol{m}_n({\boldsymbol{k}})\cdot {\boldsymbol{B}}({\boldsymbol{r}}_c). \end{equation} \tag{ 73 }$$

At second order, extra terms appear, some of which are discussed in [102]. Compared to the standard equations (68) and (69), equations (72) and (73) contain two extra contributions (there is also a third hidden modification, related to the phase-space measure and that we discuss below):

One $-\dot{{\boldsymbol{k}}} \times \boldsymbol{{\boldsymbol{F}}}_n$ is called the anomalous velocity and depends on the Berry curvature ${\boldsymbol{F}}_n({\boldsymbol{k}}) = \boldsymbol{\nabla}_{{\boldsymbol{k}}} \times \boldsymbol{A}_n$. It was first found by Karplus and Luttinger in a particular case [8]. It is a dual of the Lorentz magnetic force (in ${\boldsymbol{k}}$-space the Berry curvature is the analogous of a magnetic field and $\dot{{\boldsymbol{k}}}$ is the analogous of a velocity) and is at the origin, for example, of the IQHE (see below).

The other extra term $-\boldsymbol{m}_n\cdot {\boldsymbol{B}}$, first obtained by Kohn [9], is a kind of Zeeman effect and involves another geometrical object (not previously introduced) called the orbital magnetic moment $\boldsymbol{m}_n({\boldsymbol{k}}) = -\frac{e}{2}\langle {\boldsymbol{r}} \times$ $(\boldsymbol{v} - \langle \boldsymbol{v} \rangle) \rangle$, where $\boldsymbol{v}$ is the velocity operator and the average is taken over a wave packet restricted to the nth band [24]. It has an expression similar to (56) for the Berry curvature:

$$\begin{equation} {\boldsymbol{m}}_n({\boldsymbol{k}}) = i\frac{e}{2\hbar} \sum_{n^{^{\prime}}\neq n }\frac{\langle u_n|\boldsymbol{\nabla}_{{\boldsymbol{k}}} H({\boldsymbol{k}}) | u_{n^{^{\prime}}}\rangle \times \langle u_{n^{^{\prime}}}|\boldsymbol{\nabla}_{{\boldsymbol{k}}} H({\boldsymbol{k}}) | u_{n}\rangle}{E_{n^{^{\prime}}}-E_n}. \end{equation} \tag{ 74 }$$

This Zeeman-like effect is emergent as the electron considered here is spinless. Its emergence is similar to that of the Zeeman effect in the Pauli equation (with the famous g = 2 factor) when taking the low-energy limit of the 3D Dirac equation and projecting on the positive energy states [7]. Niu and co-workers [24] provide the following picture of this emerging orbital magnetic moment. When restricted to a finite energy band, an electron wavepacket has a minimum size due to the uncertainty principle. This minimum size is similar to the Compton wavelength for the Dirac electron as discussed by Blount [7]. This means that the electronic charge is now spread over a finite volume and there is the possibility of self-rotation as in the Uhlenbeck and Goudsmit picture of the rotating electron as a mechanism for the appearance of the intrinsic magnetic moment. The orbital magnetic moment is a crucial ingredient in the 'modern theory of orbital magnetism', see [24, 25] for review.

It is also possible to generalize the Peierls substitution in order to obtain an effective one-band quantum Hamiltonian including the Berry phase corrections [24]. As before the effective Hamiltonian is given by the semi-classical energy:

$$\begin{equation} H_n^\textrm{eff} = \widetilde{E}_n ({\boldsymbol{k}},{\boldsymbol{r}}_c) = E_n({\boldsymbol{k}})-e A_0({\boldsymbol{r}}_c)-\boldsymbol{m}_n({\boldsymbol{k}})\cdot {\boldsymbol{B}}({\boldsymbol{r}}_c), \end{equation} \tag{ 75 }$$

upon identifying ${\boldsymbol{k}}$ and ${\boldsymbol{r}}_c$ with operators. However, the canonical quantization is not easy, because ${\boldsymbol{r}}_c$ and $\hbar{\boldsymbol{k}}$ are not canonical position and momentum: their Poisson bracket is not standard. We consider three cases.

• (a)  
  In the particular case in which the Berry curvature vanishes, one uses the standard Peierls substitution (71).
• (b)  
  In the particular case where the magnetic field vanishes, these operators are given by a kind of dual of the Peierls substitution:

  $$\begin{align} {\boldsymbol{k}} & = {\boldsymbol{q}} \nonumber\\ {\boldsymbol{r}}_c & = {\boldsymbol{x}} + \boldsymbol{A}_n({\boldsymbol{k}})\to i \boldsymbol{\nabla}_{{\boldsymbol{k}}}+\boldsymbol{A}_n({\boldsymbol{k}}), \end{align} \tag{ 76 }$$

  where $\hbar {\boldsymbol{q}}$ and ${\boldsymbol{x}}$ are the canonical momentum and position operators, and ${\boldsymbol{r}}_c$ is the (Berry) gauge-invariant position operator. It can be shown that ${\boldsymbol{r}}_c = \boldsymbol{A}_n({\boldsymbol{k}}) + {\boldsymbol{x}}$ is also the position operator projected onto the nth band. (In the crystal momentum representation [7], the complete position operator ${\boldsymbol{r}}$ also has matrix elements between different bands, that are given by: $\boldsymbol{A}_{n,n^{^{\prime}}}({\boldsymbol{k}}) = \langle u_n|i \boldsymbol{\nabla}_{\boldsymbol{k}} u_{n^{^{\prime}}}\rangle$ so that ${\boldsymbol{r}}_{n,n^{^{\prime}}} = \delta_{n,n^{^{\prime}}} i \boldsymbol{\nabla}_{\boldsymbol{k}} + \boldsymbol{A}_{n,n^{^{\prime}}}({\boldsymbol{k}}) = \delta_{n,n^{^{\prime}}}{\boldsymbol{r}}_c + (1-\delta_{n,n^{^{\prime}}})\boldsymbol{A}_{n,n^{^{\prime}}}({\boldsymbol{k}})$. This is the reason for distinguishing between the position operator ${\boldsymbol{r}}$ and the projected position operator ${\boldsymbol{r}}_c$.) Its average over all wave vectors ${\boldsymbol{k}}$ in the BZ $\langle{\boldsymbol{r}}_c\rangle = \langle {\boldsymbol{r}}_{n} \rangle +{\boldsymbol{R}}$ is equal to the Wannier center $\langle {\boldsymbol{r}}_{n} \rangle = \langle \boldsymbol{A}_n\rangle$ modulo a Bravais lattice vector ${\boldsymbol{R}}$. Roughly speaking, the Wannier center $\langle {\boldsymbol{r}}_{n} \rangle$ is the electron position within the unit cell and ${\boldsymbol{R}}$ is the position of the unit cell. The Wannier center plays an important role in the 'modern theory of electric polarization' [20–22], as we discuss below.
• (c)  
  In the general case, when both the magnetic field and the Berry curvature are non-zero, it can be shown that the operator identification [24, 101] is:

  $$\begin{align} {\boldsymbol{k}} & = {\boldsymbol{q}}+\frac{e}{\hbar}\boldsymbol{A} ({\boldsymbol{r}}_c)+e{\boldsymbol{B}}({\boldsymbol{r}}_c)\times \boldsymbol{A}_n({\boldsymbol{k}})\nonumber\\ {\boldsymbol{r}}_c & = {\boldsymbol{x}} + \boldsymbol{A}_n({\boldsymbol{k}}), \end{align} \tag{ 77 }$$

  where $(\hbar {\boldsymbol{q}},{\boldsymbol{x}})$ are the canonical momentum and position operators. Note that this is not simply (71) together with (76).

There is a third modification of the electron dynamics that is not apparent in the above equations of motion ((72) and (73)), but which is important. It is a consequence of the fact that the wave packet momentum $\hbar{\boldsymbol{k}}$ and position ${\boldsymbol{r}}_c$ are gauge-invariant (both under an electromagnetic gauge transformation and under a Berry gauge transformation) but are not canonical. Their Poisson bracket (and the corresponding quantum commutator) is not the usual one but is modified by the Berry curvature and the magnetic field. Therefore the volume occupied by a state in the $({\boldsymbol{r}}_c,{\boldsymbol{k}})$ phase-space is no longer (2π)^D . In order to take this fact into account, one should modify the phase-space measure as follows [24, 103]:

$$\begin{equation} \frac{d {\boldsymbol{r}}_c d {\boldsymbol{k}}}{(2\pi)^D} \to \frac{d {\boldsymbol{r}}_c d {\boldsymbol{k}}}{(2\pi)^D}\left(1+\frac{e}{\hbar}{\boldsymbol{B}}\cdot {\boldsymbol{F}}\right). \end{equation} \tag{ 78 }$$

This modified phase-space density simply comes from the Jacobian in the transformation from the canonical (${\boldsymbol{x}}$,$\hbar{\boldsymbol{q}}$) to the gauge-invariant (${\boldsymbol{r}}_c$,$\hbar{\boldsymbol{k}}$) position and momentum variables [101] as given in equation (77):

$$\begin{equation} d {\boldsymbol{x}} d {\boldsymbol{q}} = d {\boldsymbol{r}}_c d {\boldsymbol{k}} \left(1+\frac{e}{\hbar}{\boldsymbol{B}}\cdot {\boldsymbol{F}}\right). \end{equation} \tag{ 79 }$$

The modified phase-space density affects thermodynamic quantities such as the orbital magnetization and susceptibility [24, 25].

As a side remark, it is easy to recover the behavior of the Berry curvature under time-reversal (58) and space inversion (59) from the anomalous velocity in equation (72). Under time reversal, the velocity, momentum and time-derivative change sign so that $\dot{{\boldsymbol{k}}}$ does not change, which means that $\boldsymbol{F}_n$ must change sign. Therefore $\boldsymbol{F}_n({\boldsymbol{k}})\to -\boldsymbol{F}_n(-{\boldsymbol{k}})$ under time-reversal and if it is a symmetry then $\boldsymbol{F}_n({\boldsymbol{k}}) = -\boldsymbol{F}_n(-{\boldsymbol{k}})$. Under space inversion, the velocity and momentum change sign so that $\dot{{\boldsymbol{k}}}$ changes sign as well and the curvature must remain. Therefore $\boldsymbol{F}_n({\boldsymbol{k}})\to \boldsymbol{F}_n(-{\boldsymbol{k}})$ under inversion and if it is a symmetry then $\boldsymbol{F}_n({\boldsymbol{k}}) = \boldsymbol{F}_n(-{\boldsymbol{k}})$. From the Zeeman-like term $-{\boldsymbol{m}}_n \cdot {\boldsymbol{B}}$, the same considerations apply to the orbital magnetic moment (74): time-reversal symmetry makes it odd ${\boldsymbol{m}}_n(-{\boldsymbol{k}}) = -{\boldsymbol{m}}_n({\boldsymbol{k}})$ and inversion symmetry makes it even ${\boldsymbol{m}}_n(-{\boldsymbol{k}}) = {\boldsymbol{m}}_n({\boldsymbol{k}})$. If both symmetries are present, the Berry curvature and the orbital magnetic moment vanish at every ${\boldsymbol{k}}$.

In summary, at first order in the external fields, the equations of motion of a spinless Bloch electron restricted to a single band are given by equation (72) with the effective energy (73) and the relation (77) between gauge-invariant and canonical momentum and position. The extra terms (as compared to the standard equations) vanish in the presence of both time-reversal and inversion symmetries. Compared to the 'Peierls scheme', here the restriction to an effective single-band description is done in the presence of external fields and virtual transitions to other bands are allowed, which lead to geometric forces of reaction [74]. Only real (Landau–Zener) inter-band transitions are neglected.

Here, we specialize to space dimension D = 2 and time-reversal breaking systems. In their landmark paper, Thouless–Kohmoto–Nightingale–den Nijs (TKNN) have shown that the Hall conductivity of a band insulator is quantized because it is related to a topological number: the sum of the Chern numbers of the occupied bands [12]. They originally considered the case of a two-dimensional electron gas in an applied perpendicular magnetic field. Here, we will consider a slightly different version—the so-called QAHE—by assuming that time-reversal symmetry is broken but no external magnetic field is applied. We therefore consider both translation invariance under a Bravais lattice and time-reversal breaking. We will use the semi-classical equations of motion for an electron restricted to a given band and show that a filled band may nevertheless carry a quantized Hall current.

Consider the semi-classical equations of motion (72) for a single electron in a given band n in 2D in the presence of an applied electric field but no applied magnetic field. They read:

$$\begin{align} \dot{{\boldsymbol{k}}}& = -\frac{e}{\hbar}{\boldsymbol{E}} \nonumber\\ \dot{{\boldsymbol{r}}}_c& = \frac{1}{\hbar}\boldsymbol{\nabla}_{{\boldsymbol{k}}}E_n({\boldsymbol{k}}) -\dot{{\boldsymbol{k}}} \times {\boldsymbol{F}}_n = \frac{1}{\hbar}\boldsymbol{\nabla}_{{\boldsymbol{k}}}E_n({\boldsymbol{k}}) + \frac{e}{\hbar}{\boldsymbol{E}} \times {\boldsymbol{F}}_n. \end{align} \tag{ 80 }$$

The first term in the above velocity is the familiar group velocity, while the second term is the anomalous velocity of Karplus and Luttinger [8]. We can easily obtain the average velocity of an electron and deduce the electric current carried by a filled band as:

$$\begin{equation} {\boldsymbol{j}}_n = (-e) \int_{T^2}\frac{d^2 {\boldsymbol{k}}}{(2\pi)^2} \dot{{\boldsymbol{r}}}_c = (-e) \int_{T^2}\frac{d^2 {\boldsymbol{k}}}{(2\pi)^2}\left[\frac{1}{\hbar}\boldsymbol{\nabla}_{{\boldsymbol{k}}}E_n({\boldsymbol{k}}) +\frac{e}{\hbar}{\boldsymbol{E}} \times {\boldsymbol{F}}_n\right]. \end{equation} \tag{ 81 }$$

The first term (group velocity) vanishes after integration over a filled band as is well known ('a filled band does not conduct electricity'). However, the second term (anomalous velocity) need not vanish and involves the Chern number C_n of the nth band:

$$\begin{equation} {\boldsymbol{j}}_n = -\frac{e^2}{h}{\boldsymbol{E}} \times \int_{T^2}\frac{d^2 {\boldsymbol{k}}}{2\pi} {\boldsymbol{F}}_n = -\frac{e^2}{h}C_n {\boldsymbol{E}} \times {\boldsymbol{e}}_z . \end{equation} \tag{ 82 }$$

This means that, provided the Chern number is non-zero, a filled band can nevertheless have a Hall current, which is quantized in units of e²/h, as the Chern number is an integer. For a band insulator with several filled bands (those with $n\leqslant n_F$), one finds that the Hall conductivity is given by:

$$\begin{equation} \sigma_{xy} = -\frac{e^2}{h} \sum_{n\leqslant n_F} C_n = -\frac{e^2}{h} n_H. \end{equation} \tag{ 83 }$$

The topological invariant n_H characterizing the band insulator is called the Hall (or TKNN) number and is given by the sum of the Chern numbers of the filled bands. The Chern number C_n characterizes a band n, whereas the Hall number n_H is attached to a band gap and characterizes a band insulator (gap labeling). At this point, we have not yet shown that a band insulator with a non-zero Hall number exists. From symmetry arguments, we know that if we do not break time-reversal symmetry, then C_n  = 0 for all bands and therefore σ_xy  = 0. Below, we will discuss the Haldane model, which is precisely a band model that breaks time-reversal symmetry and is able to produce a non-zero Chern number for a band (note that breaking time-reversal is a necessary but not sufficient condition).

Thouless has also shown how the Chern number quantizes adiabatic pumping in a 1D band insulator [75]. Actually the 2D quantum Hall system can exactly be mapped on a time-periodic 1D system.

Another point to note is that from the perspective of the bulk, the Hall current is carried by all the filled bands. However, in a finite sample with edges, the quantized Hall current is carried by gapless and chiral edge modes as in the usual QHE [104]. In the familiar QHE, the bulk invariant n_H is just the number of filled Landau levels. And the number of chiral gapless modes per edge is also equal to n_H . This is a first example of bulk-edge correspondence between a topological invariant in the bulk (number of filled Landau levels) and a topological invariant on an edge (the number of chiral gapless edge modes).

When they occur in insulators, Berry curvature effects are quantized/topological because one integrates over the whole BZ (topological numbers). That is for example the case of the QAHE that occurs in Chern insulators. A typical representative is the Haldane model of graphene [16], that we will treat in detail in section 6.3. Another example is the quantum spin Hall effect (QSHE) that occurs in time-reversal invariant topological insulators with strong intrinsic SOC, see e.g. Kane and Mele's model [28, 29]. Figure 5 summarizes three kinds of quantized Hall effects.

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We have seen that Berry phase effects, such as the existence of the anomalous velocity, lead to robust quantized responses in insulators. However, the anomalous velocity also exists in doped semiconductors and metals where it induces observable but unquantized effects, such as the anomalous Hall effect [106] in time-reversal breaking compounds (ferromagnetic semiconductors or metals), or the spin Hall effect [107] in time-reversal invariant materials with strong-SOC. These effects are still due to geometric quantities such as the Berry curvature but which are not integrated over the complete BZ, but over a finite portion of the BZ delimited by the Fermi surface, and are therefore not topological but merely geometrical. The anomalous Hall effect [106] and the spin Hall effect [107] are fundamental in the field of spintronics and have important applications. The inverse spin Hall effect is a very convenient method to measure a spin current by a charge signal.

Even more recently the nonlinear electromagnetic responses of non-centrosymmetric crystals have been reinterpreted in the perspective of Berry curvature properties [108–113]. The photogalvanic effect, the second-harmonic generation or the frequency difference generation are captured by various frequency dependencies of the nonlinear second-order susceptibility tensor. At low frequency of the driving field, the rectified current is determined by the scattering time and the intrinsic Berry curvature dipole, which is a measure of the average gradient of Berry curvature of the occupied states [109, 112, 113]. Provided inversion symmetry is broken, this intraband effect is present even in time-reversal invariant materials and has been coined nonlinear Hall effect (or nonlinear anomalous Hall effect depending on the authors). Nonlinear Hall or optical effects are also present in 3D materials, and are especially strong in recently discovered Weyl semimetals. Experimentally the nonlinear Hall effect has been measured in the few-layer transition metal dichalcogenides (TMDC) WTe₂ [114, 115] and also in 3D Dirac or Weyl semimetals [116]. At higher frequencies, the DC rectified current is an inter-band effect known as the shift current. Interestingly, the frequency integral of the rectified second order conductivity is a purely geometrical quantity that depends neither on the scattering time nor on the band dispersion, but is solely determined by the Berry curvature dipole [113]. In principle, those nonlinear response susceptibilities are not quantized in the metallic regime, but in some particular circumstances they also might be quantized [117].

In the absence of disorder, translation invariance allows one to diagonalize the Hamiltonian with respect to the cell index n. The field operators c_l (n) in real space are expanded over operators in reciprocal space c_l (k) as:

$$\begin{equation} c_l (n) = \frac{1}{\sqrt{N}} \sum_{k} e^{i k n} c_{l}(k), \end{equation} \tag{ 89 }$$

where l = A, B is the sublattice index, k the crystal momentum, and N the number of unit cells. The unit cell is chosen to have a unit length a = 1 and we choose units such that ћ = 1. After substitution of equation (89) in equation (88), the Hamiltonian becomes:

$$\begin{equation} H = \sum_k c_l^\dagger(k) \mathcal{H}_{lm}(k) \, c_m (k), \end{equation} \tag{ 90 }$$

where momentum k is restricted to the 1D BZ [−π, π] and the Einstein summation over repeated sublattice indexes (l, m) is implied. The SSH Hamiltonian which is a 2N × 2N matrix in real space, becomes a 2 × 2 matrix $\mathcal{H}(k)$ for each value of k. Note that the exponential factors in equation (89) contain only the cell number n, without any mention about the real position of the sites within each cell. As a result, the Bloch Hamiltonian $\mathcal{H}(k)$ is 2π-periodic. This periodic Bloch Hamiltonian can be written as a linear combination of the Pauli matrices σ_x and σ_y only:

$$\begin{equation} \mathcal{H}(k) = d_x(k) \sigma_x + d_y(k) \sigma_y, \end{equation} \tag{ 91 }$$

with real coefficients:

$$\begin{equation} d_x(k) = v + w \cos{k } \quad \quad \textrm{and} \quad \quad d_y(k) = w \sin{k }. \end{equation} \tag{ 92 }$$

The Pauli matrices above act on the sublattice index (A, B) and not on real electronic spin (we assume spinless electrons here). We call $t = (v+w)/2$ the average hopping amplitude, take units such that t = 1 and call δ = v − w the difference in hopping amplitudes. As already discussed, a periodic Bloch Hamiltonian depends on the choice of unit cell. Making the other choice is equivalent to δ → −δ.

For comparison and later use, one defines also the canonical Bloch Hamiltonian (or simply the Bloch Hamiltonian) H(k). It is obtained by using the following unitary transformation for diagonalizing H:

$$\begin{equation} c_l (n) = \frac{1}{\sqrt{N}} \sum_{k} e^{i k \, x_{nl}} c_{l}(k), \end{equation} \tag{ 93 }$$

where x_nl (x_nl  = n + 0 if l=A and $x_{nl} = n+1/2$ if l=B) represents the exact position of the type-l site of the cell with number n. Note that one should have used a different notation for the operators c_l (k) in order to distinguish (89) and (93). After substitution of equation (93) in equation (88), one obtains the Bloch Hamiltonian:

$$\begin{equation} H(k) = (v+w)\cos(k/2) \sigma_x + (v-w) \sin(k/2) \sigma_y = 2\cos(k/2) \sigma_x + \delta \sin(k/2) \sigma_y. \end{equation} \tag{ 94 }$$

Since the exact position of sites is involved in the unitary transformation, the Bloch Hamiltonian H(k) does not depend on the choice of unit cell, but its periodicity in reciprocal space is altered. Indeed, note that $\mathcal{H}(k+2\pi) = \mathcal{H}(k)$ but $H(k+2\pi) = -H(k)$ such that H(k + 4π) = H(k). The Bloch Hamiltonian has a doubled periodicity related to the fact that the distance between the two sites within the unit cell is half of the unit cell size. Later, we will encounter a similar effect in graphene. The Bloch Hamiltonian seems to depend on the sign of δ. Actually $H_{-\delta}(k)$ can be mapped back to $H_{\delta}(k)$ by a unitary transform which exchanges A and B sites: $\sigma_x H_{-\delta}(k) \sigma_x = H_{\delta}(x)$. This relabeling transformation is a do-nothing transformation. One could therefore restrict the study of the infinite SSH chain to $\delta \geqslant 0$.

In both representations, we have obtained that d_x is an even and d_y an odd function of k by explicit calculation, but this is also a consequence of the symmetries of the SSH model. Like in any two-band model, the physics is encoded in the functions d_x (k), and d_y (k), which are similar to the components of the external magnetic field in the TLS case (section 3), and determine the geometry of the spinors. Note the absence of a $d_z(k) \sigma_z$ term in the SSH model. We will study the effect of such a term within the RM model in section 5.5.

The electronic band structure of the infinite chain is obtained by diagonalizing equation (91) or equation (94), which leads to the dispersion:

$$\begin{equation} E_{\pm}(k) = \pm |{\boldsymbol{d}}(k)| = \pm \sqrt{v^{2} + w^{2} + 2 v w \cos(k)} = \pm \sqrt{4 \cos^2(k/2)+\delta^2 \sin^2(k/2)}. \end{equation} \tag{ 95 }$$

In the general case δ ≠ 0, the conduction band ($E_{+}(k)\gt 0$) and the valence band ($E_{-}(k)\lt 0$) are separated by a gap, and the chain is therefore insulating at half-filling, see figure 7(left). One may also define an azimuthal angle (along the equator of the Bloch sphere) ϕ_k reflecting the geometrical/topological properties of the Bloch Hamiltonian beyond the energy spectrum:

$$\begin{equation} H(k) = E_+(k)(\cos \varphi_k \sigma_x + \sin \varphi_k \sigma_y)\, \textrm{and } \mathcal{H}(k) = E_+(k)(\cos \phi_k \sigma_x + \sin \phi_k \sigma_y). \end{equation} \tag{ 96 }$$

In order to clearly distinguish them, we call ϕ_k the azimuthal angle for the canonical Bloch Hamiltonian H(k) and φ_k that for a periodic Bloch Hamiltonian $\mathcal{H}(k)$. They are both plotted in figure 8.

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One may wonder if there are exceptional situations where the spectrum equation (95) becomes gapless. The condition for a gap closing between these two bands requires all coefficients d_i (k) (for i = x, y) in equation (92) to vanish simultaneously at the same point k of the BZ. One has two constraints and only one variable k, which is a typical level crossing/repulsion situation in quantum mechanics. Nevertheless, here it leads to $d_x = v + w \cos(k) = 0$ and d_y  = w sin(k) = 0. Therefore, considering solely positive v and w, the gap closing arises for:

$$\begin{equation} k = \pi, \quad \delta = v-w = 0. \end{equation} \tag{ 97 }$$

The SSH model becomes gapless when δ = 0, which in fact means that the chain is no longer dimerized. Indeed a monoatomic chain is known to lead to a single metallic band E(k) = 2cos(k/2). It can be described by a single band in its natural BZ [−2π, 2π]. Here the two bands in figure 7(middle) correspond to the folding of this metallic band in the BZ of the SSH model [−π, π] (see figure 7(right)).

Going beyond the energy level description, one considers now the evolution of the stationary states of H(k) when k runs over the circular BZ. For each k, the Bloch Hamiltonian has exactly the structure of the two-level Hamiltonian of section 3.1 with d_z  = 0 and therefore θ_k  = π/2. The spinors (cell-periodic Bloch states) are located along the equator of the Bloch sphere and read:

$$\begin{equation} |{u_+(k)}\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ e^{i \varphi_k} \end{pmatrix} \quad \textrm{and} \quad |{u_-(k)}\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ - e^{i \varphi_k} \end{pmatrix}, \end{equation} \tag{ 98 }$$

which are well defined for all k. The ket $|{u_+(k)}\rangle$ represents the excited state associated with the positive energy E(k) (conduction band), while the eigenstate $|{u_-(k)}\rangle$ is the ground state with negative energy −E(k) (valence band).

The symmetries of the SSH model translate into constraints fulfilled by its Bloch Hamiltonian. It turns out that, for the SSH model (but this is not a general property), these constraints are the same whether expressed on the canonical or a periodic Bloch Hamiltonian.

The time-reversal symmetry $\mathcal{T}$ for spinless fermions is expressed by:

$$\begin{equation} \mathcal{T}: \, \, \mathcal{K} \mathcal{H}(k) \mathcal{K} = \mathcal{H}(k)^{\,*} = \mathcal{H}(-k), \end{equation} \tag{ 99 }$$

where $\mathcal{K}$ means complex conjugation. This invariance is valid because the hopping parameters v and w are real. The chiral symmetry reads:

$$\begin{equation} \mathcal{S}: \, \, \sigma_z \mathcal{H}(k) \sigma_z = - \mathcal{H}(k). \end{equation} \tag{ 100 }$$

The chiral symmetry is equivalent to the absence of σ_z in $\mathcal{H}(k)$. The chiral symmetry, also called sublattice symmetry, $\mathcal{S}$ involves only $\mathcal{H}(k)$, and not a relation between periodic Bloch Hamiltonians $\mathcal{H}(k)$ and $\mathcal{H}(-k)$ at opposite quasimomenta k and −k. For each state $|{\Psi_k}\rangle$ with energy E, there is a state $\sigma_z |{\Psi_k}\rangle$ with opposite energy −E, which explains the electron/hole symmetry between the conduction and valence bands equation (95). The RM term Δσ_z would break this chiral symmetry as it does not respect bipartiteness. The charge conjugation appears to be a combination of time-reversal, which involves complex conjugation, and sublattice symmetry which involves flipping the sign of energy $\mathcal{C} = \mathcal{S}\mathcal{T}$:

$$\begin{equation} \mathcal{C}: \, \, \sigma_z \mathcal{H}(k)^{\,*} \sigma_z = - \mathcal{H}(-k). \end{equation} \tag{ 101 }$$

The SSH model is also a condensed matter realization of the quantum field theory concept of charge conjugation. In particle physics, charge conjugation is the interchange of particles and antiparticles. In polyacetylene, there are no genuine positrons of course but the electronic states splits into electron and hole states. In a half-filled chain, hole excitations can propagate with the same parameters of the electron excitation, except for the opposite electric charge. Formally it corresponds to a map between the positive energy solutions and the negative energy solutions of the model.

Finally, the SSH model has an additional symmetry which is the invariance under space inversion, which implies the exchange of A and B isospin index, $\mathcal{I}:c_A \rightarrow c_B$ and $\mathcal{I}:c_B \rightarrow c_A$, as the inversion center is mid-bond and not on-site. The inversion symmetry reads:

$$\begin{equation} \mathcal{I}: \, \, \sigma_x \mathcal{H}(k) \sigma_x = \mathcal{H}(-k). \end{equation} \tag{ 102 }$$

In the ten-fold classification of topological insulators [41, 42, 128, 129], it would therefore appear that the SSH model belongs to the BDI class of symmetry (i.e. time-reversal symmetry TRS with $\mathcal{T}^{\,2} = +1$, particle-hole symmetry PHS with $\mathcal{C}^2 = +1$ and therefore chiral or sublattice symmetry SLS $\mathcal{S} = 1$) which is characterized by a $\mathbb{Z}$ topological index (winding number). Actually, we will see that the bulk winding number has no precise meaning here and that the SSH model is best described as an inversion-symmetric 1D band insulator characterized by a $\mathbb{Z}_2$ invariant.

For each k, the periodic Bloch Hamiltonian $\mathcal{H}(k)$ (and hence its two eigenstates) is represented by the unit vector ${\hat{\mathbf{d}}}(k) = \mathbf{d}(k)/|\mathbf{d}(k)|$ on the Bloch sphere S². Besides explaining the electron/hole symmetry between the conduction and valence bands equation (95), the chiral symmetry also constrains the tip of this unit vector ${\hat{\mathbf{d}}}(k)$ to stay along the equator of the Bloch sphere when k runs over the BZ. The 1D BZ has the structure of the circle S¹ since its extremities k = ±π represent the same state. The mapping from the circle (BZ) to the circle (equator) allows one to define a winding number n_w , which is related to the first homotopy group of the circle $\Pi_1(S^1) = \mathbb{Z}$. The relative integer n_w is the winding number of $\mathbf{d}(k)$ around the origin. For our initial choice of unit-cell convention, when $w\lt v$, the origin is outside the circle traced by $\mathbf{d}(k)$, so n_w  = 0 (see figure 9(left)). In contrast, when $w\gt v$, the origin is inside the circle and $\mathbf{d}(k)$ winds exactly once around the origin: n_w  = 1 (see figure 9(right)). This winding number n_w is protected by the chiral symmetry of the SSH chain, namely $\sigma_z \mathcal{H}(k) \sigma_z = -\mathcal{H}(k)$ (this periodic Bloch Hamiltonian $\mathcal{H}(k)$ anticommutes with σ_z ).

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There is an ambiguity in the assignment of n_w  = 0 or n_w  = 1 to one phase or the other, namely to a definite sign of the parameter v − w. For a genuinely infinite system, one cannot determine whether the system is in a trivial or non-trivial phase based on n_w if one is not aware of the chosen unit cell. The only unambiguous statement is that this winding number changes by ±1 when going from one dimerization to the other. This ambiguity is lifted for finite chains, because it is then possible to state whether the chain starts (or ends) by a weak or a strong bond and there is a natural choice of unit cell. There is a relation between this winding property and the presence of protected zero modes at an edge or a domain wall [130, 131]. It is an example of the Jackiw and Rebbi mechanism (see below).

In contrast, the staggered on-site potential Δσ_z of the RM model breaks explicitly the chiral symmetry, and the tip of ${\hat{\mathbf{d}}}(k)$ is now free to follow any loop on the whole Bloch sphere. Since all such unconstrained loops can be smoothly deformed and reduced to a point, there is no possibility to define a winding number for the RM model (at least when inversion symmetry is not imposed, see below).

In summary, the winding number is useful in discussing open chains with edges or domain walls and the presence of protected edge modes. However, as a bulk invariant, it is meaningless as it depends on the unit cell choice.

A more interesting quantity is the Zak phase [19]. It is a kind of Berry phase defined for a given band but computed along a non-contractible path in the BZ thanks to its torus shape. In 1D, it reads:

$$\begin{equation} Z_n = \int_{-\pi}^\pi dk \, A_n(k) = \int_{-\pi}^\pi dk \, \langle u_n(k)|i\partial_k u_n(k)\rangle, \end{equation} \tag{ 103 }$$

for the nth band. This bulk quantity is measurable (and was actually measured [132]) and related to the electronic contribution to the polarization [23]. The Zak phase should be computed using the canonical Bloch Hamiltonian H(k) (and not the periodic one $\mathcal{H}(k)$), as it is related to the projected position operator $x_c = i\nabla_k + A_n(k)$. It is best thought as being an average electron position (known as the Wannier or band center) within the unit cell [19]:

$$\begin{equation} \langle x_-\rangle = \int_0^1 dx |w_n(x)|^2 x = \frac{Z_-}{2\pi}, \end{equation} \tag{ 104 }$$

modulo a = 1, where

$$\begin{equation} w_n(x) = \int_{-\pi}^\pi dk e\, ^{i k x} u_{n k}(x) = \int_{-\pi}^\pi dk \, \psi_{n k}(x), \end{equation} \tag{ 105 }$$

is the Wannier function of the nth band (here positioned in the unit cell at R = 0) [80]. Because the Wannier center is a position, it continuously depends on the choice of position origin. Therefore the Zak phase continuously depends on the choice of position origin. Also, it is actually an open-path geometrical phase and therefore becomes gauge-invariant only upon imposing a definite phase relation between initial and final states [19, 52]. Indeed, on the non-contractible path from k =−π to k =+π, the final state $|u_n (\pi)\rangle$ is not the same as the initial state $|u_n (-\pi)\rangle$: the path is open in Hilbert space, although it is closed in parameter space, i.e. in the BZ. In order to give a gauge-invariant quantity, the Zak phase should be computed under the following restriction called the 'periodic gauge choice' [19, 52] $|u_n (k+G)\rangle = e^{-iG x} |u_n (k)\rangle$, where x is the position operator, i.e. $|u_n (\pi)\rangle = e^{-i 2\pi x} |u_n (-\pi)\rangle$. The Zak phase being position origin-dependent, computed along a non-contractible path and an open-path geometrical phase subject to a periodic gauge condition, it is truly different from a Berry phase. The two should be carefully distinguished.

The 1D Berry connection defined from the Bloch spinor $|{u_-(k)}\rangle$ (ground state or valence band) reads:

$$\begin{equation} A_- (k) = i \langle{u_-(k) \mid \partial_k u_-(k)}\rangle = \frac{1}{2} \frac{d \varphi_k}{dk}, \end{equation} \tag{ 106 }$$

and can be easily integrated along a non-contractible closed path in the BZ to give the Zak phase:

$$\begin{equation} Z_- = \int_\textrm{BZ} A_- (k) dk = \frac{1}{2} \int_{k = -\pi}^{k = \pi} d \varphi_k\, = \frac{1}{2} \left( \varphi_\pi - \varphi_{-\pi} \right) = \frac{\pi}{2} \textrm{sign}\frac{w-v}{w+v} = \pm \frac{\pi}{2}. \end{equation} \tag{ 107 }$$

The Zak phase is also related to the electronic contribution to the polarization:

$$\begin{equation} P_\textrm{el} = -e \langle x_-\rangle = -e \frac{Z_-}{2\pi} = \mp \frac{e}{4}, \end{equation} \tag{ 108 }$$

modulo the electron charge e [23]. In the following, we take units such that e = 1 and it therefore seems that the polarization quantum [133], i.e. the modulo in the electric polarization, is P_q  = 1. But this is not correct. Actually, the only meaningful electric polarization is defined for a charge-neutral crystal and should therefore include the contribution of the ions:

$$\begin{equation} P_\textrm{tot} = P_\textrm{el}+P_\textrm{ions}. \end{equation} \tag{ 109 }$$

In order to have a charge-neutral crystal, we assume that each ion (A or B) carries a +1/2 charge. As another consequence of the presence of the ions (or of the rigid lattice of sites), it will turn out (see section 5.5) that the polarization is actually defined modulo P_q  = 1/2 and not P_q  = 1.

In conclusion of these short sections on the winding number and the Zak phase, we remark the following. The winding number is very often mistaken for the Zak phase. They should be clearly distinguished. The winding number is computed using a periodic Bloch Hamiltonian, whereas the Zak phase should be computed using the canonical Bloch Hamiltonian, as it is related to the position operator (see the above discussion). On the one hand, the winding number depends on the choice of unit cell and is therefore only well-defined when the choice of unit cell is fixed, e.g. because of a boundary [131] or at a domain wall between two dimerizations. On the other hand, the Zak phase requires using the periodic gauge choice, it continuously depends on the position origin (it is best thought as being proportional to the Wannier center) and it is evaluated along a non-contractible loop in BZ.

However, there is one way in defining a proper 1D topological crystalline insulator [136], i.e. a band insulator whose topology is protected by a point group symmetry. We have already noted that the SSH chain has an inversion symmetry and that the inversion center is mid-bond (at equal distance between two sites A and B) and not on-site. Another type of 1D two-band insulator with inversion symmetry is a regular tight-binding chain (as the SSH chain with δ = 0) but with staggered on-site energies E_A  = Δ and E_B  =−Δ. When half-filled, this gapped two-band model realizes a band insulator of the CDW type. It has inversion symmetry but the inversion center is now on-site (rather than mid-bond).

In order to realize a phase transition between these two types of inversion-symmetric insulators, we consider the RM model [123] having both the SSH dimerization δ and the CDW staggered on-site energy Δ. The (canonical) Bloch Hamiltonian reads:

$$\begin{equation} H(k) = 2\cos(k/2) \sigma_x + \delta \sin(k/2) \sigma_y + \Delta \sigma_z . \end{equation} \tag{ 117 }$$

Following [21, 24, 88], we introduce an angle $\theta_\textrm{RM}$ such that $\cos \theta_\textrm{RM} = \Delta/M$ and $\sin \theta_\textrm{RM} = \delta /M$, where $M = \sqrt{\Delta^2+\delta^2}\geqslant 0$, in order to parametrize the RM model. However, instead of considering the full range of parameters, we restrict them such that the model has inversion symmetry. On-site inversion symmetry means:

$$\begin{equation} H(k)\to H(-k) = 2\cos(k/2) \sigma_x - \delta \sin(k/2) \sigma_y + \Delta \sigma_z = H(k), \end{equation} \tag{ 118 }$$

which only occurs if δ = 0 and mid-bond inversion symmetry means:

$$\begin{equation} H(k)\to \sigma_x H(-k)\sigma_x = 2\cos(k/2) \sigma_x + \delta \sin(k/2) \sigma_y - \Delta \sigma_z = H(k), \end{equation} \tag{ 119 }$$

which is possible only if Δ = 0. Therefore, we choose (δ, Δ) to be (≠ 0, 0) i.e. $\theta_\textrm{RM} = \pi/2$ or 3π/2 (SSH); or (0, 0) (gapless); or (0, ≠ 0) i.e. $\theta_\textrm{RM} = 0$ or π (CDW). Inversion symmetry imposes that only two out of the three Pauli matrices appear simultaneously in the Bloch Hamiltonian, which means that this model actually has a chiral symmetry. Indeed, when Δ = 0, H(k) anticommutes with σ_z and when δ = 0, it anticommutes with σ_y (in the latter case, chiral symmetry is no longer related to the two sublattices and bipartiteness). This means that when k spans the BZ, the Bloch Hamiltonian is restricted to move on a great circle of the Bloch sphere. When Δ = 0, this great circle is the equator (xy plane), and when δ = 0, it is a meridian (xz plane). We also introduce the dimensionless parameter λ such that Δ = λMΘ(λ) and $\delta = -\lambda M \Theta(-\lambda)$, where Θ(x) is the Heaviside step function. When $\lambda\lt 0$, the model is an SSH chain with positive δ and when $\lambda\gt 0$ it is a CDW chain with positive Δ. When λ = 0 it is gapless. When λ varies from negative to positive, a phase transition occurs between two different types of band insulators having inversion symmetry. At the transition, the gap closes and the low-energy physics is described by a massless Dirac equation (see figure 13).

Both phases (SSH and CDW) are Dirac insulators. Indeed at low energy, they are both described by a massive Dirac equation. In the SSH case, the Dirac Hamiltonian is given in (110). In the CDW case, it is:

$$\begin{eqnarray} &&\mathcal{H}(k = \pi + q) \approx -q \sigma_y + \Delta \sigma_z. \end{eqnarray} \tag{ 120 }$$

Beware that the Dirac Hamiltonian is here obtained from linearizing the periodic (and not the canonical) Bloch Hamiltonian $\mathcal{H}(k)$. The Dirac mass is not the same in the two cases: δσ_x versus Δσ_z . However, a crucial point is that, in each phase, only two Pauli matrices appear, otherwise this would not be Dirac matrices satisfying the Clifford algebra in 1D. It is interesting to note that the Jackiw–Rebbi mechanism therefore applies to both phases as well with either a domain wall in δ(x) or in Δ(x). There is a subtlety involved here in the fact that the Jackiw–Rebbi mechanism strictly applies only in the continuum limit in which a → 0 and $t\to \infty$ such that the velocity ta remains finite. In the tight-binding model, there may be surprises as found in the corresponding 2D case (boron nitride) [137].

Let us now discuss the different phases by computing the electric polarization in the bulk. We first recall that, as we are considering spinless electrons and a half filled two-band chain, it means that there is a single electron per unit cell carrying a charge $-e = -1$. In order to ensure electric neutrality, we assume that the ions A and B carry each a charge +1/2 (they are cations). It is crucial to consider a neutral system, otherwise the electric polarization has no meaning. In addition, the complete polarization $P_\textrm{tot}$ is defined modulo P_q  = 1/2 (see e.g. the pedagogical discussion about the polarization quantum P_q in [138]). This can be seen as follows. Consider the CDW situation with the electron localized on A sites. The total charge is therefore $+1/2-1 = -1/2$ on A sites and +1/2 on B sites. Now, if we move every electron by a distance of 1/2 to the right we obtain a state in which every A site has a charge +1/2 and every B site a charge −1/2. But up to a translation by half a unit cell, these two states are identical (they correspond to the RM model with θ_RM = 0 and π). Because we have translated a single electron of charge −1 by a distance of 1/2 in order to be back on the same state, it means that the polarization is defined modulo a quantum of $1\times 1/2 = 1/2$. A subtle issue is the fact that the electronic contribution to the polarization reads $P_\textrm{el} = -e \langle x_- \rangle = -\frac{Z_-}{2\pi}$ (see below), which is obviously defined modulo 1 as Z₋ is a phase. However, the complete polarization $P_\textrm{tot}$ is defined modulo P_q  = 1/2. Next, we remark that under space inversion, the total polarization $P_\textrm{tot}\to -P_\textrm{tot}$ and because inversion is a symmetry it means that $-P_\textrm{tot} = P_\textrm{tot}$ modulo P_q . Therefore

$$\begin{equation} P_\textrm{tot} = 0\, \textrm{or } P_q/2\, \textrm{modulo}\, P_q. \end{equation} \tag{ 121 }$$

in a inversion-symmetric band insulator. This is a $\mathbb{Z}_2$ invariant.

The total electric polarization $P_\textrm{tot}$ has two contributions: that of ions and that of electrons. The former is computed classically as $P_\textrm{ions} = \sum_j q_j x_j = (x_A+x_B)/2 = \bar{x}$, where $q_j = +1/2$ is the ion charge. $P_\textrm{ions}$ depends on the average position $\bar{x}$ of ions within the unit cell. This contribution can be made to vanish by properly choosing the position origin on a bond, half way between A and B (this is an inversion center for the SSH chain), see figure 10. The electronic contribution can be easily computed thanks to the prescription given by King-Smith, Vanderbilt and Resta [23]: do as if the electron charge was entirely localized at the Wannier center of the occupied band $P_\textrm{el} = -\langle x_-\rangle = -Z_- /(2\pi)$. In the SSH phase, the Wannier centers are localized on the bonds, half-way between A and B sites (see figure 11). The sign of δ only decides whether these Wannier centers are to the left or to the right of A sites (they are on the strong bonds). For such a chain, the total electric polarization (including the electrons and the ions) is $P_\textrm{tot} = 0$ modulo 1/2. The Zak phase is 0, i.e. $\langle x_-\rangle = 0$, or π, i.e. $\langle x_-\rangle = 1/2$ (depending on the sign of δ and when computed from the inversion center, which is on a bond, can be either chosen to the left or to the right of an A site). In the CDW phase, the Wannier centers are located on-sites. The sign of Δ only decides whether they are located on an A site (if $\Delta\lt 0$) or a B site (if $\Delta\gt 0$). The total electric polarization here is $P_\textrm{tot} = 1/4$ modulo 1/2. The Zak phase is also 0 or π when computed from the inversion center which is now on-site. The value 0 or π depends on the sign of Δ but also on whether one has chosen the inversion center to be on an A or a B site. If one fixes the same position origin to compute the Zak phase in both the SSH and the CDW phases, then it is 0 or π in the SSH phase and ±π/2 in the CDW phase. Figure 12 presents the electric polarization for the complete RM model. When θ_RM is not 0, π/2, π or 3π/2, this model breaks inversion symmetry and the polarization is not constrained to 0 or 1/4.

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In a one-dimensional crystal with inversion symmetry, there are two special points in each unit cell that are known as Wyckoff positions. In our case, they are either on-site (A or B) or at mid-distance between two sites (to the right of A or to its left). We find that the Wannier center can only occupy one of these two positions and as a consequence the electric polarization is quantized: either P_q /2 or 0 modulo P_q . This is the fundamental reason for having two classes of 1D insulators protected by inversion symmetry: CDW has $P_\textrm{tot} = P_q/2$ and SSH has $P_\textrm{tot} = 0$ modulo P_q .

We give two different answers before concluding.

5.5.3.1. First answer

The vacuum is a band insulator with inversion symmetry and vanishing polarization. We take it as a definition of a trivial insulator. On the one hand, the SSH chain has a vanishing bulk polarization and is smoothly connected to a molecular insulator (this is obvious in the limit $|\delta| = 2$ in which it describes uncoupled non-polar dimers). It is therefore a trivial insulator. On the other hand, the CDW chain has a non-vanishing bulk polarization $P_\textrm{tot} = P_q/2\, [P_q]$ and is smoothly connected to an ionic insulator (a one-dimensional version of rock salt Na⁺ Cl⁻ obtained in the limit $t = (v+w)/2\to 0$). It is therefore a SPT insulator. The $\mathbb{Z}_2$ invariant ν can be taken to be:

$$\begin{equation} \nu = -\frac{P_\textrm{tot}}{P_q/2} = 4(\langle x_-\rangle-\bar{x}) = \frac{Z_- -2\pi \bar{x}}{\pi/2} \,\, [2]. \end{equation} \tag{ 122 }$$

This characterization by a $\mathbb{Z}_2$ invariant is consistent with that based on a topological θ-term in the effective gauge theory [128, 139], that we here recall. The long-wavelength electromagnetic response of a dielectric material in one spatial dimension is described by the following Lagrangian density:

$$\begin{equation} \mathcal{L} = -\frac{\epsilon}{4}F^{\mu \nu} F_{\mu \nu} -e\frac{\theta}{4\pi}\epsilon^{\mu\nu}F_{\mu \nu} = \epsilon\frac{E_x^2}{2}+P_\textrm{tot} E_x, \end{equation} \tag{ 123 }$$

where E_x is the electric field and $F^{\mu\nu}$ is the field tensor with µ = 0, 1 = t, x. There are two material parameters that distinguish the dielectric material (here a 1D band insulator) from the vacuum: the familiar dielectric constant —related to the electric susceptibility χ by  = 1 + χ—and the less familiar angle θ. The latter is related to the total (spontaneous) electric polarization by $P_\textrm{tot} = -P_q \theta/(2\pi)$ modulo P_q . The θ angle has only two possible values θ = 0 (trivial) or π (non-trivial) when the band insulator respects both time-reversal and inversion symmetries, which is the case here. It appears that $P_\textrm{tot}(\textrm{CDW}) = P_q/2\neq 0\, [P_q]$ (topological, θ = π) and $P_\textrm{tot}(\textrm{SSH}) = 0\, [P_q]$ (trivial, θ = 0). The $\mathbb{Z}_2$ invariant is:

$$\begin{equation} \nu = \frac{\theta}{\pi} \, [2]. \end{equation} \tag{ 124 }$$

5.5.3.2. Second answer

5.5.3.3. Conclusion

Imagine tuning a transition between the SSH and the CDW phases. We start with $\delta\gt 0$ and Δ = 0 (i.e. $\lambda\lt 0$), then diminish the dimerization until we reach δ = 0 maintaining Δ = 0 (i.e. λ = 0) and then we increase Δ while maintaining δ = 0 (i.e. $\lambda\gt 0$). In such a case, the inversion symmetry is preserved all the way. Then, we necessarily have to close the gap at λ = 0, a point at which the electron is delocalized over the whole crystal and the Wannier center is ill defined. Actually, in the process the Wannier center jumps from on-site to mid-bond exactly when λ = 0. See the phase diagram in figure 13.

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In summary, the RM model with inversion symmetry connects the SSH model to a CDW model. It allows one to study the phase transition between the two phases of a 1D band insulator with inversion symmetry. The non-trivial phase is a SPT insulator—rather than an intrinsic topological insulator. As the symmetry in question is the space inversion (and not an external symmetry such as time-reversal or particle-hole), it is known as a topological crystalline insulator [136]. It is characterized by a $\mathbb{Z}_2$ topological invariant given in (122) and built from the bulk electric polarization. The CDW chain is an ionic insulator (inversion-protected topological insulator), whereas the SSH chain is a molecular insulator (trivial).

If space inversion is broken (which happens in the RM model whenever $\theta_\textrm{RM}\neq 0$ modulo π/2), the system has a single gapped phase and is a trivial band insulator. Indeed, one can continuously go from the CDW to the SSH phase without closing the gap by breaking inversion symmetry. Its electric polarization is unquantized (neither 0 nor P_q /2, see figure 12) and the Wannier center is not at a special position.

Graphene is the one-atom thick layer of carbon atoms arranged with the honeycomb lattice structure, made of two interpenetrating triangular sublattices, respectively denoted A and B (figure 14). Each carbon atom has six electrons: five core electrons filling the inner shells (2 electrons in the 1s orbital and 3 electrons in the covalent sp² bonds) while a single valence electron fills the p_z orbital perpendicular to the plane. As in polyacetylene, the p_z orbitals lead to π bands that are well 'isolated' from filled lower bands and empty higher bands. Much of the physics of graphene is related to those two-dimensional π bands that are accurately described by the following tight-binding Hamiltonian [38]:

$$\begin{equation} H_0 = t \sum_{{\boldsymbol{r}}_A} \sum_{\alpha = 1}^{3} c_B^\dagger({\boldsymbol{r}}_A+\boldsymbol{\delta}_\alpha) c_A({\boldsymbol{r}}_A) + \textrm{h.c.} , \end{equation} \tag{ 125 }$$

where t ≃ −2.7 eV is the hopping amplitude between the p_z orbitals of two adjacent carbon atoms. The operator $c_l({\boldsymbol{r}})$ destroys an electron in the p_z orbital at site ${\boldsymbol{r}}$, with l = A, B indicating the sublattice. The sum over ${\boldsymbol{r}}_A$ runs over the A-sites which form a triangular Bravais lattice generated by the basis vectors (figure 14):

$$\begin{equation} \boldsymbol{a}_{1} = \sqrt{3} a \, \boldsymbol{e}_x , \, \qquad \boldsymbol{a}_2 = \frac{a}{2} \left( \sqrt{3} \boldsymbol{e}_x + 3 \boldsymbol{e}_y \right) , \end{equation} \tag{ 126 }$$

where a = 0.142 nm is the length of the carbon–carbon bond. The vectors $\boldsymbol{\delta}_{\alpha}$ defined by:

$$\begin{equation} \boldsymbol{\delta}_{1,2} = \frac{a}{2} \left( \pm \sqrt{3} \boldsymbol{e}_x + \boldsymbol{e}_y \right) , \, \qquad \boldsymbol{\delta}_{3} = - a \, \boldsymbol{e}_y, \end{equation} \tag{ 127 }$$

connect any A-site to its three B-type nearest neighbors (figure 14). The hopping matrix elements between next-nearest neighbors (NNNs) can be safely neglected, being roughly ten times smaller than the main hopping t [142, 143]. Sites from distinct sublattices, A and B, are crystallographically different by the orientations of the three attached bonds. In graphene, all sites are occupied by identical carbon atoms, while in hexagonal boron nitride (h-BN) the boron atoms are distributed on one sublattice while the nitrogen atoms lie on the other one, see section 6.2.

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The graphene lattice can be seen as a 2D version of the polyacetylene chain obtained by replacing the C–H bonds by C–C bonds to another chain. Although the tight-binding models of graphene and polyacetylene look quite similar, there are also important differences. First, in graphene each p_z orbital is coupled to 3 neighboring p_z orbitals, instead of two neighbors for polyacetylene. Second, there is no Peierls instability in graphene, although dimerization effects have been studied: for a magnetic-field induced Peierls instability see [157] and for a Kékulé type of distortion see [158]. Finally, and more fundamentally, the quasi-momentum spans a 2D BZ in graphene, which is crucial for topology, since a 2D compact manifold is a necessary condition for a Chern number to be defined.

Owing to translation invariance, the two-dimensional quasi-momentum ${\boldsymbol{k}} = (k_x,k_y)$ is a good quantum number. In order to diagonalize the Hamiltonian equation (125), we expand the field operator $c_l({\boldsymbol{r}})$ as a sum of Fourier modes:

$$\begin{equation} c_l({\boldsymbol{r}}) = \frac{1}{\sqrt{N}} \sum_{{\boldsymbol{k}}} e^{i {\boldsymbol{k}} \cdot {\boldsymbol{r}}} c_l({\boldsymbol{k}}), \end{equation} \tag{ 128 }$$

where l = A, B is the sublattice index and N is the total number of unit cells. Note that the exponential phase factors contain the exact location of the atomic sites. After substitution of equation (128), the Hamiltonian equation (125) becomes diagonal in momentum and reads:

$$\begin{equation} H_0 = \sum_{\boldsymbol{k}} c_l^\dagger({\boldsymbol{k}}) [H_0({\boldsymbol{k}})]_{lm} \, c_m ({\boldsymbol{k}}), \end{equation} \tag{ 129 }$$

where ${\boldsymbol{k}}$ is restricted to the first BZ. The (canonical) Bloch Hamiltonian $H_0({\boldsymbol{k}})$, which acts on the sublattice isospin, is given by:

$$\begin{equation} H_0({\boldsymbol{k}}) = d_x({\boldsymbol{k}}) \sigma_x + d_y({\boldsymbol{k}}) \sigma_y = |{\boldsymbol{d}}({\boldsymbol{k}})|(\cos\varphi_{\boldsymbol{k}} \sigma_x +\sin\varphi_{\boldsymbol{k}} \sigma_y), \end{equation} \tag{ 130 }$$

since only off-diagonal hopping amplitudes are included in the model defined by equation (125). The phase $\varphi_{\boldsymbol{k}}$ is the azimuthal angle along the equator of the Bloch sphere. The real functions $d_x({\boldsymbol{k}})$ and $d_y({\boldsymbol{k}})$ are defined by:

$$\begin{equation} d_x ({\boldsymbol{k}}) = t \sum_{\alpha = 1}^{3} \cos( {\boldsymbol{k}} \cdot \boldsymbol{\delta}_\alpha) \quad \textrm{and} \quad d_y ({\boldsymbol{k}}) = t \sum_{\alpha = 1}^{3} \sin( {\boldsymbol{k}} \cdot \boldsymbol{\delta}_\alpha), \end{equation} \tag{ 131 }$$

over the whole BZ. The functions $d_x ({\boldsymbol{k}})$ and $d_y({\boldsymbol{k}})$ are respectively even and odd in momentum reversal ${\boldsymbol{k}} \rightarrow -{\boldsymbol{k}}$, which is related to time-reversal and inversion symmetries of pristine graphene. The electronic energy spectrum is given by the length of the vector ${\boldsymbol{d}} = (d_x,d_y)$:

$$\begin{equation} E({\boldsymbol{k}}) = \pm |{\boldsymbol{d}}({\boldsymbol{k}})| = \pm \sqrt{d_x^2 ({\boldsymbol{k}})+d_y^2 ({\boldsymbol{k}})}, \end{equation} \tag{ 132 }$$

which describes a valence band (minus sign) and a conduction band (plus sign) that are symmetric with respect to E = 0. The zero energy corresponds to the common energy of the p_z atomic orbitals on sublattices A and B. The corresponding wave functions are the spinors:

$$\begin{equation} |{u_+({\boldsymbol{k}})}\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ e^{i\varphi_{\boldsymbol{k}}} \end{pmatrix}, \quad \textrm{and} \quad |{u_- ({\boldsymbol{k}})}\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ - e^{i\varphi_{\boldsymbol{k}}} \end{pmatrix}, \end{equation} \tag{ 133 }$$

in the upper and lower bands respectively. These are the same as for the SSH model albeit the angle $\varphi_{\boldsymbol{k}}$ is now a function of a 2D quasimomentum given by $\varphi_{\boldsymbol{k}} = \textrm{arg}(k_x+i k_y)$. The Bloch Hamiltonian does not have the periodicity of the reciprocal lattice. The reason is that its definition involves the distance between sites within the unit cell. For the honeycomb lattice, the distance between nearest-neighbor A and B sites is actually one-third of a lattice vector. For example, $\boldsymbol{\delta}_3 = (\boldsymbol{a}_1-2\boldsymbol{a}_2)/3$. This translates into the fact that the Bloch Hamiltonian has a triple periodicity, as can be seen by plotting the phase $\varphi_{\boldsymbol{k}}$ (see figure 15(left)) [83]. This fact has measurable consequences which have been observed by quantum tomography using cold atoms in an optical lattice, see [159]. If instead of using the canonical Bloch Hamiltonian (130), one uses a periodic Bloch Hamiltonian:

$$\begin{equation} \mathcal{H}_0({\boldsymbol{k}}) = |{\boldsymbol{d}}({\boldsymbol{k}})|(\cos\phi_{\boldsymbol{k}} \sigma_x +\sin\phi_{\boldsymbol{k}} \sigma_y), \end{equation} \tag{ 134 }$$

then the azimuthal angle is different and called $\phi_{\boldsymbol{k}}$. The latter has the periodicity of the hexagonal BZ and is related to $\varphi_{\boldsymbol{k}}$ by $\phi_{\boldsymbol{k}} = \varphi_{\boldsymbol{k}}-{\boldsymbol{k}}\cdot \boldsymbol{\delta}_{3}$ (see figure 15(right)), where $-\boldsymbol{\delta}_{3}$ is the position of a B site with respect to an A site within the unit cell (see figure 14(left)).

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The valence and conduction bands touch at isolated points of the BZ obtained by solving the equations $d_x({\boldsymbol{k}}) = d_y({\boldsymbol{k}}) = 0$. There are only two inequivalent solutions located at:

$$\begin{equation} {\boldsymbol{k}} = \pm {\boldsymbol{K}} = \pm \frac{4 \pi}{3\sqrt{3}a} \, \boldsymbol{e}_x, \end{equation} \tag{ 135 }$$

and called the Dirac points. Any other solutions of the equation ${\boldsymbol{d}}({\boldsymbol{k}}) = 0$ can be linked by a reciprocal lattice vector to one of these two solutions, and therefore would describe the same physical state. In the SSH chain, the band touching conditions were also obtained by cancelling two functions d_x and d_y , but there was only a single momentum component k, so it needed a fine tuning of hopping parameters to close the gap, namely $\delta = v-w = 0$. In graphene, the momentum ${\boldsymbol{k}}$ runs over a 2D manifold for graphene, so there is no need for fine tuning of the Hamiltonian parameters and one generically gets Dirac points. Even for anisotropic graphene, where the hoppings $t_\alpha$ along the bonding directions $\boldsymbol{\delta}_\alpha$ might differ, there will be always some isolated points ${\boldsymbol{k}} = (k_x,k_y)$ satisfying the 2 equations: $d_x({\boldsymbol{k}} ) = d_y({\boldsymbol{k}} ) = 0$ [160].

We consider now the low-energy theory for the single-particle states near the Dirac points [39], see equation (135). The momenta are written as ${\boldsymbol{k}} = \pm {\boldsymbol{K}} + {\boldsymbol{q}}$ where ${\boldsymbol{q}} = q_x {\boldsymbol{e}}_x + q_y {\boldsymbol{e}}_y$ is a small momentum deviation from the Dirac points, namely $|{\boldsymbol{q}}|a \ll 1$. The annihilation operators for these states are relabeled as $c_{A\,\,\pm\,\,{\boldsymbol{K}}} ({\boldsymbol{q}}) = c_{A}(\pm{\boldsymbol{K}}+{\boldsymbol{q}})$ to indicate the valley they belong to. Expanding to first order in momenta, the Bloch Hamiltonian describing the low-energy excitations in the valley near $\xi {\boldsymbol{K}}$ (ξ = ±1) reads:

$$\begin{equation} H_0(\xi {\boldsymbol{K}} + {\boldsymbol{q}}) = H_0^{(\xi)}({\boldsymbol{q}}) = v_F \begin{pmatrix} 0 & \xi q_x -i q_y \\ \xi q_x + i q_y & 0 \end{pmatrix} = v_F(q_x \xi \sigma_x+q_y \sigma_y) , \end{equation} \tag{ 136 }$$

where $v_F = -3 at/2 \simeq 10^6$ m s⁻¹≃c/300 is the Fermi velocity. The Fermi velocity is roughly the bandwidth t divided by the BZ size 1/a (when ћ = 1). The Hamiltonian can also be written $H_0^{(\xi)}({\boldsymbol{q}}) = v_{ij} q_i \sigma_j$ with i, j = x, y = 1, 2 here. Therefore, near each Dirac point, one obtains a 2D Weyl Hamiltonian describing massless fermions carrying a sublattice isospin coupled to their momentum. The two species of massless Dirac fermions are attached to a given valley, labelled by ξ. The chirality (or winding) of each Dirac point is given by $\chi_\xi = \textrm{sign } \textrm{det} (v_{ij}) = \xi$. The low-energy dispersion of those fermions is valley independent and reads:

$$\begin{equation} E_\pm ({\boldsymbol{q}}) = \pm v_F |{\boldsymbol{q}}| , \end{equation} \tag{ 137 }$$

which is typical of a relativistic massless particle with velocity of light replaced by v_F . This linearized dispersion is reminiscent of the 1D fermions obtained in the SSH chain near k = π at the gap closure point $m = v-w = 0$. It also has a 3D counterpart in Weyl and Dirac semimetals [161]. As a general reference on the distinction between Dirac (i.e. complex and having both chiralities), Weyl (i.e. complex and chiral) and Majorana (i.e. real and achiral) fermions, we recommend [162].

Nevertheless we would like to emphasize the differences between the Dirac (and Weyl) equation in the contexts of graphene and particle physics, respectively. In high-energy physics, the Dirac equation comes from Lorentz-invariance and very general considerations related to special relativity and quantum mechanics. Then, in 3+1 space-time dimensions, the minimal objects to satisfy Dirac equation are bispinors combining the spin and particle/antiparticle degrees of freedom. In graphene, the origin of the Dirac physics is totally different. As we have seen, the spinors originate from a ${\boldsymbol{k}} .{\boldsymbol{p}}$ expansion around special points of a particular band structure. Hence in graphene, there is no fundamental issue with the negative energy states that are just the valence band states (these states are in fact bounded from below by the bottom of the valence band). Finally the emergent Lorentz invariance of equation (137) is only valid near the Dirac point, namely for wave vectors ${\boldsymbol{q}}$ located in a disk whose radius is far smaller than the inverse lattice spacing 1/a, whereas Lorentz invariance applies in the whole Minkowski space-time. Finally the 4 components of the spinors are associated to the sublattice isospin (instead of real spin), and to the valley index (instead of particle/antiparticle label).

Graphene is invariant under space inversions with respect to particular points of the lattice, which are the centers of the hexagons of carbon atoms and the centers of the carbon–carbon bonds. In the absence of any magnetic field or impurities, graphene is also time-reversal invariant and therefore all hopping parameters are real numbers.

It is rather difficult to gap out the Dirac points between two bands of spinless fermions. Indeed, there are only four possible types of perturbations which mathematically corresponds to the identity and the three Pauli matrices. A scalar perturbation (proportional to identity in sublattice isospin space) will just shift both bands in energy without separating them. Moreover perturbations in σ_x or σ_y would shift the position of the band touching to another location in the BZ without removing the degeneracy. Only a perturbation acting as σ_z could gap out the Dirac points, but we are now going to explain that such a term is forbidden by the time-reversal and inversion symmetry of graphene, and more generally in any two-band model which is invariant under those two symmetries. Let us consider a generic two-band model in 2D, described by the Bloch Hamiltonian $H({\boldsymbol{k}}) = {\boldsymbol{d}}({\boldsymbol{k}})\cdot \boldsymbol{\sigma}$, and with sublattice isospin ($\mathcal{T}^{\,2} = +1$). The time-reversal $\mathcal{T}$ symmetry condition reads:

$$\begin{equation} H({\boldsymbol{k}}) = H^{\,*} (-{\boldsymbol{k}}) \implies \quad d_x({\boldsymbol{k}}) = d_x(-{\boldsymbol{k}}) , \quad d_y({\boldsymbol{k}}) = -d_y(-{\boldsymbol{k}}) \quad \textrm{and } \quad d_z({\boldsymbol{k}}) = d_z(-{\boldsymbol{k}}), \end{equation} \tag{ 138 }$$

which is indeed satisfied by graphene and h-BN.

The inversion $\mathcal{I}$ symmetry condition can be written:

$$\begin{equation} \sigma_x H({\boldsymbol{k}}) \sigma_x = H(-{\boldsymbol{k}}) \implies \quad d_x({\boldsymbol{k}}) = d_x(-{\boldsymbol{k}}) , \quad d_y({\boldsymbol{k}}) = -d_y(-{\boldsymbol{k}}) \quad \textrm{and } \quad d_z({\boldsymbol{k}}) = -d_z(-{\boldsymbol{k}}), \end{equation} \tag{ 139 }$$

which is satisfied by graphene but violated by h-BN.

Finally the combined $\mathcal{I}\mathcal{T}$ symmetry leads to:

$$\begin{equation} \sigma_x H^{\,*}({\boldsymbol{k}}) \sigma_x = H({\boldsymbol{k}}) \implies \quad d_x({\boldsymbol{k}}) = d_x({\boldsymbol{k}}) ,\quad d_y({\boldsymbol{k}}) = d_y({\boldsymbol{k}}) \quad \textrm{and } \quad d_z({\boldsymbol{k}}) = -d_z({\boldsymbol{k}}). \end{equation} \tag{ 140 }$$

Therefore, the presence of both $\mathcal{T}$ and $\mathcal{I}$ enforces the function $d_z({\boldsymbol{k}})$ to be odd and even in ${\boldsymbol{k}}$, which means $d_z({\boldsymbol{k}}) = 0$ for all ${\boldsymbol{k}}$. The third line shows that the symmetry $\mathcal{I}\mathcal{T}$ is enough to protect the existence of a robust contact, even if $\mathcal{T}$ and $\mathcal{I}$ were separately broken. Then a gap closing requires only the simultaneous cancellation of two functions $d_x({\boldsymbol{k}}) = d_y({\boldsymbol{k}}) = 0$ at some ${\boldsymbol{k}}$ which can be varied in a 2D manifold. This is why isolated solutions are expected as it happens in graphene. The existence of such isolated solutions of ${\boldsymbol{d}}({\boldsymbol{k}}) = 0$, preventing the system to become gapped, is robust even if some crystal symmetries are lost and more hopping amplitudes are added. For instance additional second-neighbor hopping will break the electron/hole symmetry discussed above, but will not affect the existence of Dirac points. Other perturbations, like an anisotropic deformations on one type of bond, only shift the Dirac points and modify the conical dispersion around them [160, 163, 164]. The touching points are protected by more fundamental symmetries, namely space inversion and time-reversal symmetries. Breaking at least one of these symmetries usually results in a gap opening at the Dirac points, as we will see below with noncentrosymmetric h-BN, and in the time-reversal breaking Haldane model.

To compute the Chern number, one can use equation (56) and inject:

$$\begin{equation} \partial_i H({\boldsymbol{k}}) = \sigma_a \partial_i d_a({\boldsymbol{k}}) \quad \textrm{and} \quad (E_{n^{^{\prime}}} -E_n)^2 = 4 d^2, \end{equation} \tag{ 158 }$$

where summation over repeated index a = x, y, z is implied. The Berry curvature is given by:

$$\begin{align} F_{xy}^{ n}({\boldsymbol{k}})& = \frac{i}{4d^2} \langle u_n|\partial_x H({\boldsymbol{k}}) \partial_y H({\boldsymbol{k}}) | u_{n}\rangle +\textrm{c.c.} = -\frac{1}{2d^2} \epsilon_{abc} \langle u_n| \sigma_c | u_{n}\rangle \partial_x d_a({\boldsymbol{k}}) \partial_y d_b({\boldsymbol{k}}), \end{align} \tag{ 159 }$$

$$\begin{align} & = \frac{1}{2d^3} \epsilon_{abc} d_c \partial_x d_a({\boldsymbol{k}}) \partial_y d_b({\boldsymbol{k}}), \end{align} \tag{ 160 }$$

where the last line holds for the valence band because then $\langle u_n| \sigma_c | u_{n}\rangle = -d_c/d$. The Berry curvature is opposite for the conduction band. Then we can express the Chern number of the valence band n =− under the following form:

$$\begin{equation} C_- = \frac{1}{2\pi} \int_{T^2} d^2 {\boldsymbol{k}} \, \, F_{xy}^-({\boldsymbol{k}}) = \frac{1}{4 \pi} \int d^2 {\boldsymbol{k}} \left( \frac{\partial {\boldsymbol{d}} }{\partial k_x} \times \frac{\partial {\boldsymbol{d}} }{\partial k_y} \right) . \, \, \frac{\hat{\boldsymbol{d}} }{d^2} = \frac{1}{4 \pi} \int d\Omega \,\, \in \mathbb{Z}, \end{equation} \tag{ 161 }$$

which is the Chern number of the fiber bundle. This formula is useful because it expresses the Chern number of a band as an integral over the parameter vector components with no explicit mention of spinors, unlike equation (49), nor velocity operator averages, unlike equation (56). It has a simple geometric interpretation, which explains why it has to be an integer. Let us consider the image $\mathcal{M}$ of the whole BZ through the map ${\boldsymbol{k}} \rightarrow {\boldsymbol{d}}({\boldsymbol{k}})$ (figure 19). The number C_n counts how many times the closed manifold $\mathcal{M}$ wraps around the origin, because it is the global solid angle Ω from which this image manifold is seen from the origin divided by 4π. If the origin is outside the image manifold $\mathcal{M}$ then the integral vanishes and C_n  = 0. If the origin is inside the image manifold, then the integrated is a multiple of 4π, and therefore C_n is a finite integer (see figure 18 right). To change C_n it is necessary to change the parameter of the bulk Hamiltonian $\hat{\boldsymbol{d}}({\boldsymbol{k}})$ in such a way that the bulk gap closes. Below, we will see that an alternative view on a Chern insulator is to consider it as a topological texture in reciprocal space. The corresponding invariant will be shown to be the skyrmion number (which is still another name for the Chern number, the Pontryagin index, the wrapping number, etc). An alternative formula using only the unit vector $\hat{\boldsymbol{d}}$ is:

$$\begin{equation} C_n = \frac{1}{4 \pi} \int d^2 {\boldsymbol{k}} \left( \frac{\partial \hat{\boldsymbol{d}} ({\boldsymbol{k}})}{\partial k_x} \times \frac{\partial \hat{\boldsymbol{d}}({\boldsymbol{k}}) }{\partial k_y} \right) \cdot \,\, \hat{\boldsymbol{d}} \end{equation} \tag{ 162 }$$

which is the wrapping number of the map ${\boldsymbol{k}} \rightarrow \hat{\boldsymbol{d}} ({\boldsymbol{k}}) = {\boldsymbol{d}}({\boldsymbol{k}})/|{\boldsymbol{d}}({\boldsymbol{k}})|$ between the BZ (torus T²) and the unit sphere (S²). In accordance with general classifications, C_n is a single number that characterizes the general structure of wave functions globally in ${\boldsymbol{k}}$-space. In section 4.4.3, we have seen that this topological number is observable and measures the charge Hall conductance of insulators in units of e²/h. In the following subsection, we evaluate C_n for the Haldane model in its different insulating phases.

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Finally, we can relate those Chern number of Bloch bands to expressions for Chern numbers in the TLS case. By integrating over the BZ and injecting the spinor components equation (144), one gets the Chern number:

$$\begin{equation} C = \frac{1}{2 \pi} \int_\textrm{BZ} d{\boldsymbol{k}} \,\,\, \frac{\sin \theta_{\boldsymbol{k}}}{2} \left( \frac{\partial \theta_{\boldsymbol{k}}}{\partial k_x} \frac{\partial \phi_{\boldsymbol{k}}}{\partial k_y} - \frac{\partial \phi_{\boldsymbol{k}}}{\partial k_x} \frac{\partial \theta_{\boldsymbol{k}}}{\partial k_y} \right) = \frac{1}{2 \pi} \int_\textrm{BZ'} d\theta d\varphi \,\,\, \frac{\sin \theta}{2}, \end{equation} \tag{ 163 }$$

where $\textrm{BZ}^{^{\prime}}$ is the image of the BZ on the Bloch sphere, or equivalently the central projection of the closed manifold $\mathcal{M}$ onto the unit sphere. The last equality simply follows from the change of integration variables between the ${\boldsymbol{k}}$ of the BZ and the polar angles on the Bloch sphere, the quantity between comes being the Jacobian of this transformation. This is the flux of a radial Berry curvature through the unit sphere.

Equation (162) expresses the Chern number as a winding number of the map ${\boldsymbol{k}} \rightarrow \hat{\boldsymbol{d}} ({\boldsymbol{k}}) = {\boldsymbol{d}}({\boldsymbol{k}})/|{\boldsymbol{d}}({\boldsymbol{k}})|$, which is the composition of the map ${\boldsymbol{k}} \rightarrow {\boldsymbol{d}}({\boldsymbol{k}})$ followed by the central projection π to the unit sphere. It is possible to transform this integral expression as a discrete sum by using the Brouwer degree of the map. The Brouwer degree reads [166]:

$$\begin{equation} C = \sum_{y \in Y_z} \, \, \, \sum_{{\boldsymbol{k}} \in d^{-1}(y)} \textrm{sign} \left[ \left( \partial_{k_x} {\boldsymbol{d}} \times \partial_{k_y} {\boldsymbol{d}} \right) \cdot {\boldsymbol{n}} \right], \end{equation} \tag{ 164 }$$

where z is an arbitrary point on S², ${\boldsymbol{n}}$ the unit vector towards z, and $Y_z = \mathcal{M} \cap \pi^{-1}(z)$. The procedure for evaluating C consists in choosing a point z on S² and then determining the intersections of $\mathcal{M}$ with the ray originating from (0, 0, 0) to z. These intersections typically form a discrete set Y_z of isolated points of $\mathbb{R}^3$. For our purpose, it is very useful to choose z to be on one of the σ axis, let us say z is the north pole of the Bloch sphere. Then we have to sum in equation (164) over points that satisfy $d_x({\boldsymbol{k}}) = d_y({\boldsymbol{k}}) = 0$.

Let us treat the case of the Haldane model to be more specific. Then $d_x({\boldsymbol{k}}) = d_y({\boldsymbol{k}})$ leads to the Dirac points of pristine graphene ${\boldsymbol{k}} = \pm {\boldsymbol{K}}$, but note that we are considering the gapped Haldane model, so that importantly $d_z(\pm {\boldsymbol{K}})$ is non zero. Near those points the Bloch Hamiltonians read:

$$\begin{equation} H(\xi {\boldsymbol{K}} + {\boldsymbol{q}}) = H^{(\xi)}({\boldsymbol{q}}) = v_F \left( \xi q_x \sigma_x + q_y \sigma_y \right) + d_z(\xi {\boldsymbol{K}}) \sigma_z, \end{equation} \tag{ 165 }$$

${\boldsymbol{d}}_\xi = (\xi v_F q_x , v_F q_y,d_z(\xi {\boldsymbol{K}}))$. At this point, one needs to consider different cases depending on the signs of $d_z(\pm {\boldsymbol{K}})$. Let us first assume that d_z is positive at both Dirac points $\pm {\boldsymbol{K}}$. Then the set Y_z consists in two points and the Chern number is:

$$\begin{equation} C = \sum_{\xi = \pm 1} \textrm{sign} \left[ \left( \partial_{q_x} {\boldsymbol{d}}_\xi \times \partial_{q_y} {\boldsymbol{d}}_\xi \right) \cdot {\boldsymbol{e}}_z \right] = \sum_{\xi = \pm 1} \textrm{sign}(\xi v_F^2) = 0, \end{equation} \tag{ 166 }$$

reflecting that the local orientations of the two sides of the surface $\mathcal{M}$ are opposite. If d_z is negative at both Dirac points $\pm {\boldsymbol{K}}$, then set Y_z  = and immediately C = 0. Therefore one finds that the valence band is trivial in the h-BN model and also in regions of the Haldane phase diagram where the masses have the same sign at the Dirac point.

We now consider the case with opposite signs of the masses, for instance $d_z({\boldsymbol{K}})\gt 0$ and $d_z(-{\boldsymbol{K}})\lt 0$. In such a case, the ray z intersects the surface $\mathcal{M}$ only once and the Chern number is simply:

$$\begin{equation} C = \sum_{\xi = + 1} \textrm{sign} \left[ \left( \partial_{q_x} {\boldsymbol{d}}_\xi \times \partial_{q_y} {\boldsymbol{d}}_\xi \right) \cdot {\boldsymbol{e}}_z \right] = \textrm{sign}(v_F^2) = 1. \end{equation} \tag{ 167 }$$

For the opposite choice of signs, $d_z({\boldsymbol{K}})\lt 0$ and $d_z(-{\boldsymbol{K}})\gt 0$, the intersection Y_z also reduces to a single point but which maps back to valley ξ =−1 in the BZ. Hence C =−1. The above formulas can be symmetrized by considering the whole axis z instead of the ray z, leading to:

$$\begin{equation} C = - \frac{1}{2} \sum_{\xi = \pm 1} \xi\, \textrm{sign} (M_\xi) = \frac{1}{2} \left[ \textrm{sign}(M_-) - \textrm{sign}(M_+) \right], \end{equation} \tag{ 168 }$$

where $M_\xi = d_z(\xi {\boldsymbol{K}})$.

Finally, the above formula has been obtained in a more general way in [166] which is valid for any two-band insulator and reads:

$$\begin{equation} C = \frac{1}{2}\sum_{{\boldsymbol{k}}_D} \, \chi ({\boldsymbol{k}}_D)\, \textrm{sign } [d_z({\boldsymbol{k}}_D)], \end{equation} \tag{ 169 }$$

where $\chi({\boldsymbol{k}}_D) = \pm 1$ denotes the chirality or winding number of the Dirac point located at ${\boldsymbol{k}}_D$ (for example, in graphene, this chirality is $\chi_\xi = \textrm{sign det}(v_{ij}) = \xi$) and $\textrm{sign } [d_z({\boldsymbol{k}}_D)] = \pm 1$ is the sign of the Dirac fermion mass. Lattice Dirac fermions always occur in pairs (fermion doubling theorem), which makes the Chern number an integer. Each massive Dirac fermion contributes ±1/2 to the Chern number. However, it is not always easy to identify all massive Dirac fermions from the energy spectrum alone. Indeed, there are often cases with spectator Dirac fermions of very large mass that are totally undetectable in the energy spectrum. Let us illustrate that in the case of two bands with $H({\boldsymbol{k}}) = {\boldsymbol{d}}({\boldsymbol{k}}) \cdot \boldsymbol{\sigma}$. The way to identify Dirac fermions correctly is to find points in reciprocal space at which two (out of three) components of ${\boldsymbol{d}}({\boldsymbol{k}})$ vanish (e.g. $d_x({\boldsymbol{k}}_D) = d_y({\boldsymbol{k}}_D) = 0$). The third component $d_z({\boldsymbol{k}}_D)$ giving the mass of the Dirac fermion (it can not be zero otherwise the system would be gapless at ${\boldsymbol{k}}_D$). The choice of components is arbitrary: it does not matter which two components vanish and which plays the role of the finite mass. Spectator fermions are discussed in [16] and also in section 8.2 of [18].

Let us now consider the example of graphene in presence of some inversion breaking and time-reversal breaking terms. So the mass matrix is $(M_S - 3 \sqrt{3} t_2 \sin (\phi) \xi ) \sigma_z$ implying that the gap can close for $M_S = \xi 3 \sqrt{3} t_2 \sin \phi$ in one valley (labelled by ξ = ±1). This equality signals a one-electron topological quantum transition separating an QAH insulator and a trivial atomic insulator. Finally we would like to make a comment on the terminology. This type of phase transition is purely a change between two one-electron Hamiltonians. It has in particular nothing to do with TO defined by Wen [167, 168]. In particular the transition discussed here is not a transition between two TOs. It is rather a transition between two band-insulators having distinct topological invariants (which characterize the winding of one-electron wave functions). More on this distinction in the conclusion, see section 9.

When proposing his model, Haldane had a two-fold goal [16]. One was to find a QHE without Landau levels, i.e. as a band effect, breaking time-reversal symmetry but preserving translation symmetry. This goal was achieved. His other goal was to have a condensed matter realization of the parity anomaly known to occur in the 2D massless Dirac equation. One manifestation of this anomaly (i.e. a symmetry which exists classically but does not survive quantization) is the fact that, in a perpendicular magnetic field, there should be a half-quantized QHE. However, in the Haldane model on the honeycomb lattice, Dirac fermions always occur in pairs even if one member of the pair has a zero mass and the other has a very large mass (of the order of the bandwidth). Therefore, no half-quantized Hall effect occurs in the Haldane model and the parity anomaly of one Dirac fermion (say the massless one in one valley) is compensated by the other Dirac fermion (say the spectator one in the other valley). Hence, from this perspective the second goal was not achieved. Achieving this parity anomaly requires a further trick, namely to produce a single 2D massless Dirac equation at the surface of a 3D topological insulator [32–34].

A well-known topological insulator, the Haldane model [16], has been discussed in detail in this review (see section 6.3). This is the telltale example and the representative of a larger class of topological insulators, called Chern insulators or QAH insulators. Its phase diagram shows several gapped phases separated by lines where the gap closes and identified by a specific value of the Chern number C₋ of the occupied valence band (figure 18, left). The Chern number of a band is a topological invariant which measures a global property of the Bloch wave functions of this band over the whole BZ. It can only be defined in the presence of a finite bulk gap. Each gapped phase is robust in the sense that modifying continuously the parameters of the model will not change the Chern number. The only way to jump from an integer value of the Chern number to another one is to cross a gap closing. At the gap closure, C₋ is ill-defined as can be seen from the formula (161) where the denominator d vanishes, or in more geometric terms, the manifold $\mathcal M$ crosses the origin which is the source of the Berry curvature flux (figure 18, right). Note that the total Chern number, including the two bands of the Haldane model (the occupied valence and the empty conduction band) is always zero: $C = C_- + C_+ = 0$. Each band can separately carries both zero Chern number (trivial phase), or opposite C_n = ±1 (topological phase). More generally, a Chern insulator is a two-dimensional band insulator that breaks time-reversal symmetry and that has a non-vanishing total Chern number of the occupied bands at zero temperature (it may have much more bands than the Haldane model). The QH insulator is a close relative where the translational invariance is broken by the magnetic field and replaced by the Landau level structure in the continuum, and eventually magnetic subbands and Hofstadter spectrum in presence of a periodic potential.

We now turn to the characterization of topological insulating phases in terms of macroscopic observables, like the transverse Hall conductance which is a bulk response function odd with respect to time-reversal. Via the TKNN formula (83), the Hall conductance is directly proportional to the TKNN number with a prefactor e²/h depending only on fundamental constants. There is a subtle distinction here between the TKNN number and Chern numbers. A Chern number characterizes a band (or a group of bands) whereas a TKNN number characterizes a gap. The TKNN number is simply the sum of the Chern numbers of the occupied bands below the gap in which the Fermi level is assumed to be. This explains why each insulating phase is distinguished by a robust and quantized Hall conductance σ_xy  = 0 or $\sigma_{xy} = \pm e^2/h$. Translation symmetry is very useful in order to relate the robust observable (here the Hall conductance) and the Bloch wave functions. Nevertheless it is not essential and the Hall quantization survives disorder. One may even envision the existence of topological invariants that do not rely on Bloch states, as in quasicrystals (see for example [169–172] for review) or in amorphous solids [173, 174]. In order to study such cases, one needs generalizations of the Chern number beyond reciprocal space. This was pioneered by Niu et al [175] with the sensitivity of the many-body ground-state to twisted boundary conditions. There are other possibilities such as the Bott index [176] or the local Chern marker [177].

So far, we have only discussed a bulk characterization of the gapped phases and stated that topological phases are separated from trivial phases by a gap closing transition, where the Chern number and the Hall conductance encounter a finite jump. This is also true for a transition between two topological phases with different Chern numbers. Note that the vacuum is considered to be a trivial band insulator. This fact has very important consequences for the physics at the boundary of a topological insulator, or at the interface between distinct topological phases. At such an interface, the parameters of the model have to be tuned in such a way that the Chern number jumps by an integer number, which implies a region where the gap closes. The gap closing makes possible the presence of gapless conducting states at the interface between topologically-distinct phases. In addition, the existence of chiral gapless modes is guaranteed by the bulk-edge correspondence and can also be understood as following from the Jackiw–Rebbi mechanism (see section 5.4). Bulk-edge correspondence states that the number of gapless chiral modes per edge is equal to the jump of the bulk topological invariant [178]. These boundary modes have unique properties, meaning that they typically cannot be realized in a lattice system. For Chern insulators, the 1D edge states are chiral, i.e. they circulate only in one direction around the insulating bulk (see figure 5). The chiral nature of motion indicates the time-reversal breaking. Such chiral fermions can only emerge at the boundary at a non trivial insulator: there are no 1D chain or wire with unidirectional electrons. In the QHE, the direction of motion is determined by the orientation of the magnetic field, while in the Haldane model it is determined by the internal inhomogeneous magnetic field. These modes are protected from backscattering by impurities or disorder because there is no available state for the carrier to back scatter on the same edge and tunneling to the opposite edge is exponentially forbidden.

From the above discussion, we can formulate a first definition of a topological insulator. It is a bulk band insulator, that is characterized by a non-zero integer, called a (bulk) topological invariant, and a robust quantized bulk response. To change this bulk topological invariant it is necessary to close the bulk gap. The edges of a topological insulator necessarily host gapless modes that are robust because they are chiral and therefore immune to disorder. In addition there is a relation between the number of these edge modes and the bulk topological invariant: this is the so-called bulk-edge correspondence. The case of the Chern insulator is particular in that it requires no symmetry for its protection and is truly robust to any perturbation, as long as it does not close the bulk gap. It is intrinsically topological.

It was long believed that broken time-reversal symmetry and space dimension two were a mandatory setting for the appearance of topological phases. In 2005, Kane and Mele [28, 29] uncovered a new type of topological insulator where time-reversal plays the role of a protective symmetry for metallic edges, and allows one to draw a topological distinction between two insulating states. They introduced a model of spinful fermions in graphene consisting in two independent copies of the Haldane model respectively attached to each spin projection, each copy being the time-reversed of the other, e.g. with Chern number $C_\uparrow = +1$ for spin-up fermions and $C_\downarrow = -1$ for spin down fermions. As a result the overall Kane–Mele model is time-reversal invariant and describes graphene with a very particular intrinsic SO coupling conserving the spin projection S_z . The total Chern number of the occupied bands is $C = C_\uparrow+C_\downarrow = 0$ (no QHE), but there is a $\mathbb{Z}_2$ topological invariant $\nu = (C_\uparrow-C_\downarrow)/2 \, \, [2]$, which distinguishes only two classes: trivial (ν = 0) and non-trivial (ν = 1).

Because the S_z -conserving Kane–Mele model consists in two independent copies of a Haldane model with opposite Chern numbers, it has counterpropagating edge states with a full spin-momentum locking property: spin up propagates in only one direction, and spin-down propagates in the opposite direction. This spin-momentum locking insures that backscattering by a non-magnetic impurity (i.e. an impurity that respects time-reversal symmetry) is not possible. One could expect that the gapless helical edge states are an artefact of this S_z conserving juxtaposition of two chiral edge states, and that any spin mixing perturbation would mix the two spins and gap the edge states. In fact the gapless character of the helical edge states is robust again both staggered potential perturbations and Rashba SO perturbations (mixing spin up and down projections) provided those perturbations do not close completely the bulk gap. With those time-reversal symmetric perturbations, the only way to get rid of the helical edge state is to close the bulk gap. In contrast, a time-reversal breaking perturbation (like a Zeeman effect due to transverse magnetic field in x or y directions) would immediately spoil the metallic character of the helical edge. The crossing of the left and right movers branches is protected by Kramers' theorem as $\mathcal{T}^{\,2} = -1$. In principle, it is possible to construct models with a larger number of pairs of counter-propagating modes. If the number of Kramers' pairs is even, then disorder may eventually gap out all the edge states. In contrast, for an odd number of Kramers' pairs there will always be a gapless pair left as long a the bulk gap is finite and time-reversal symmetry is obeyed. This expresses that the bulk $\mathbb{Z}_2$ index is the parity of the number of Kramers' pairs of edge states. Intrinsic SOC, at the heart of the Kane and Mele proposal, is too small in graphene for the effect to be measurable. However, QSHIs have been realized experimentally in HgTe/CdTe quantum wells following the theoretical proposal by Bernevig, Hughes and Zhang, see [18] for review.

In contrast to the Chern insulator (QHE), that only exists in two space dimension [179], this type of time-reversal symmetric insulator has a three-dimensional counterpart known as a strong topological insulator [32–34]. The strong topological insulator is also classified by a $\mathbb{Z}_2$ index and also has a topological metal on its surface. It is a single massless two-dimensional Dirac fermions that is spin-momentum locked. It appears similar to graphene, except that it has a single (instead of four) Dirac cone. Indeed, the real spin is actually the one appearing in the Dirac equation (and not a sublattice pseudo-spin as in graphene) and there is also no valley degeneracy. Such a strange metal can only exist as a surface mode of a higher dimensional system and has a true parity anomaly and a half-quantized QHE.

In conclusion, the $\mathbb{Z}_2$ topological index and the helical edge states are both protected by time-reversal symmetry. There is no smooth path (in the space of Hamiltonians) connecting the ν = 1 and ν = 0 phase while simultaneously keeping the bulk gap open and respecting the time-reversal invariance. To change the $\mathbb{Z}_2$ topological index, it is necessary either to use time-reversal breaking perturbations or to close the bulk gap. In 2D, the odd and even phases of the Kane–Mele model can be characterized by a bulk quantized response which is a spin Hall conductance. Nevertheless, this only holds for the S_z conserving models, and it is difficult to build a macroscopic observable that could play the role of the Hall conductance for Chern insulators. The bulk characterizations of the 2D or 3D time-reversal invariant phases relies on $\mathbb{Z}_2$ pumps [180, 181]. The 3D strong topological insulators have a specific orbital magnetoelectric polarizability described by axion electrodynamics (with a topological θ-term similar to the one discussed in the 1D context, see equation (123)) that can play such a role [182].

The ideas of Kane and Mele have been extended to extraordinary symmetries such as time-reversal, particle-hole (charge conjugation) and chiral and to the concept of SPT insulators. There is a general classification of topological insulators (and superconductors) for non-interacting fermions that is known as the ten-fold way periodic table [41, 42, 128, 129]. Depending on space dimension and on three extraordinary symmetries resulting in ten Altland-Zirnbauer classes, it indicates whether there is a single type of band insulator (only trivial), two types of band insulators ($\mathbb{Z}_2$ index separating trivial from non-trivial band insulators) or as many band insulators as relative integers $\mathbb{Z}$. Kitaev has realized that, as a function of space dimension D, there is a form of regularity or pattern in this table related to real and to complex K-theory and known as Bott periodicity [42]. Such topological insulators that are protected by a extraordinary symmetry are also called strong topological insulators [183] as they resist the absence of crystalline symmetries that we study next.

So far we have discussed topological band insulators protected by time-reversal symmetry, which is a fundamental and generic symmetry of space-time. As initiated by Liang Fu [136], it is also possible to use spatial symmetries (e.g. rotations or reflections) of the crystal itself to protect and maintain topological distinctions between various insulating phases. Such phases have been coined topological crystalline insulators (TCI) and have also been classified (see e.g. [184, 185]).

We have already discussed a very simple example, namely an inversion-symmetric insulator in 1D. In section 5.5, we have seen that provided inversion symmetry is maintained, it is impossible to connect the SSH insulating chain (at δ ≠ 0 and Δ = 0) to the CDW insulator (at δ = 0 and Δ ≠ 0). Both phases are characterized by a distinct robust observable, which is the electric polarization. It is vanishing in the SSH phase and quantized in the CDW phase.

This simple example of a quantized electric polarization in 1D has been generalized to quantized electric multipole insulators [120] leading to the concept of higher order topological insulators (HOTI) [186]. These latter systems are insulators that do not possess gapless surface states but rather gapless hinge states, which are gapless states living on the boundary of a boundary. For example, a 2D second order topological insulator known as the quadrupole insulator does not host 1D gapless edge states but possesses 0D gapless corner states for a square patch protected by mirror symmetries. In other words, the surface of this second order topological insulator is not a metal but is itself a 1D topological insulator. HOTI escape the simple bulk-boundary correspondence between a D-dimensional insulator and D−1 metal but rather feature a correspondence to D−2 or D−3 gapless states. They only exist under the protection of point-group symmetries.

A topological insulator is a band insulator (it has a bulk gap) characterized by a bulk topological invariant computed from its cell-periodic Bloch states (e.g. a Chern number or a $\mathbb{Z}_2$ invariant). This invariant is typically related to a quantized electromagnetic response (e.g. a quantized Hall conductivity or a quantized electric polarization). In order to change this bulk topological invariant it is necessary to close the bulk gap or to break the protecting symmetry. In a finite sample with a surface that does not break the protecting symmetry, a topological insulator host gapless surface (or hinge) states that have some form of robustness towards perturbations that respect the corresponding symmetry.

Bloch functions and Wannier functions are two dual possible representations of the electronic states of a crystal. Intrinsic topological insulators (Chern insulator and QH states) are characterized by obstructions in the realization of either of these representations. First, we explain the obstruction for the Bloch states in reciprocal space, and second the corresponding veto for exponentially-localized Wannier functions in real space.

A band with a finite Chern number implies an obstruction in finding a unique gauge for the cell-periodic Bloch states which would be smooth over the whole BZ. Let us consider the explicit case of the Haldane model and use the Bloch sphere representation to visualize the spinors (cell-periodic Bloch states) of the valence band. In the trivial phase, $d_z({\boldsymbol{k}})$ being always positive it is possible to pick the (n)-gauge which has no singularity except at the south pole. In the topological phase, $d_z({\boldsymbol{k}})$ changes sign somewhere in the BZ, and the spinor explores both north and south poles: it is not possible to use a single gauge and one has to use the Wu-Yang construction with overlapping (n) and (s) regions. From this perspective, going from a trivial to a topological phase requires a band inversion (in one of the valley for the Haldane model) which necessarily occurs via a gap closing. In other words, there is an obstruction to smoothly or adiabatically connect a topological insulator to an atomic insulator. Note that the vacuum is considered to be a trivial (atomic) band insulator. This obstruction is also a well-known property of the QH states [189]. Reciprocally, without this obstruction, it would be possible to apply Stokes' theorem to the whole BZ and to obtain that the flux of the Berry curvature vanishes in contradiction with the hypothesis of finite Chern number.

Let us now turn to the Wannier function representation. Wannier functions are the Fourier transforms of Bloch states that, in conventional insulators, are usually exponentially-localized around specific locations of the solid, the Wannier centers. The Wannier functions are not unique, because there is a gauge freedom to define the Bloch functions. Changing, locally in ${\boldsymbol{k}}$, the U(1) phases of the Bloch functions results in changing the spread of each Wannier function, while keeping the Wannier center of the band (or isolated group of bands) invariant. Marzari and Vanderbilt [190] devised a procedure to compute the maximally localized Wannier functions of a given lattice model, see [80] for a review. Going back to the Haldane model, Thonhauser and Vanderbilt have shown that this procedure fails at the topological transition, where the spread of the Wannier functions diverges [97]. Indeed, at the transition, Wannier functions are completely delocalized, as in any metallic gapless state. For a band with finite Chern number, it is not possible to build a basis of exponentially-localized Wannier functions, meaning that they are only algebraically localized [96, 97]. This is in fact a general property shared by all Chern insulators [95] and by QH insulators [96].

Finally, this impossibility to represent a topological phase in terms of localized basis functions shows that the quantum degrees of freedom are not stored locally in such phases. As a plausible consequence, the non-triviality of such phases should be evidenced by their sensitivity to twists in the boundary conditions. For the QHE, Niu et al [175] gave a real-space expression for the Chern number. This requires putting the system on a real space torus and twisting the boundary conditions by inserting fluxes in the inequivalent non-contractible loops of the torus. The space of boundary conditions replaces the BZ. The beauty of this formulation is that translation invariance is no longer needed. The topological invariant can be defined directly for the many-body ground-state (instead of 'band by band') and is valid also for a disordered (and in some cases interacting) system.

The idea of obstruction is also relevant to investigate SPT phases, but here the obstruction is based on the fact that one enforces a protective symmetry. Without such a constraint, the obstruction would generally be lifted.

For concreteness, we consider the case where the symmetry is time-reversal and first discuss real space. According to [95], it is always possible to construct a basis of exponentially-localized Wannier functions for time-reversal invariant insulators both in the even (trivial) and odd (topological) phases. This is true but it comes always at a price, which is that one cannot maintain the time-reversal invariance of the exponentially-localized Wannier functions in the topological phase [191, 192]. In a topological insulator, there is an obstruction to finding symmetric and exponentially-localized Wannier functions [140, 188].

In reciprocal space, the obstruction is manifest in the fact that it becomes impossible to find a smooth gauge for the whole BZ that respects time-reversal symmetry. Although it is possible to find a smooth gauge over the whole BZ (because the Chern number vanishes), it necessarily breaks time-reversal symmetry. This is nicely reviewed in [57].

We start by considering the simplest possible 1D monoatomic chain and search for topological insulator phases protected by inversion symmetry (the simplest point-group symmetry). The two possible configurations that respects inversion symmetry correspond to either having electrons wave functions localized at every site, or alternatively having them localized at the center of each bond. Those two situations have respectively electric polarization 0 and P_q /2 modulo the quantum of polarization P_q . Of course it seems very simple to go continuously from one to the other situation by simply shifting all electrons by half the lattice constant. But this way breaks inversion symmetry at every steps between the two configurations. Alternatively one may keep inversion symmetry at all stage if one first spreads the wave functions in a symmetric way before reconcentrating them around the new localization centers. Nevertheless this latter protocol goes through a fully delocalized state that is gapless. In conclusion, it seems impossible to go continuously from the configuration $P_\textrm{tot} = 0$ to the configuration $P_\textrm{tot} = P_q/2$ while simultaneously keeping the whole system gapped and inversion symmetric at each intermediate steps. Such a topological transition can be implemented in a 1D model with coupled s and p bands [21, 141]. The trivial phase is the atomic insulator (a filled s band) with vanishing polarization and the topological phase is a covalent insulator (a filled sp hybridized bonding band) with quantized polarization.

Actually, these two 1D insulating phases have exponentially-localized and inversion-symmetric Wannier functions. They are therefore both eligible for being atomic limits [188]. One is a naive atomic limit (the atomic insulator with Wannier center on the ions), while the other has been called an obstructed atomic limit (the covalent insulator with Wannier center on the bonds) [140]. What distinguish the two is whether the Wannier center is located on the ions or mid-bonds. The conclusion is that in this example, there is no obstruction in finding symmetric and localized Wannier functions. Still, in a finite chain, as shown by Shockley, the covalent insulator hosts edge states, but not the atomic insulator [141].

A general theory based on the obstruction in finding symmetric localized Wannier functions has recently been developed [140, 188] and is reviewed in [183] (it is sometimes called topological quantum chemistry). It is based on the key idea of a mismatch or a compatibility between the symmetry of atomic orbitals in real-space and the topology of bands in the BZ. First, starting from well-localized atomic orbitals, the notion of elementary band representations is used to build all possible atomic limits. These are essentially the cases where symmetric localized Wannier functions exist. Second, starting from the opposite view of delocalized electrons in Bloch states, one considers all possible band structures, which defines quasi-band representations. Third, one analyses the compatibility of those two views of band theory. Topological semimetals correspond to the case where compatibility relations cannot be satisfied. When they can be satisfied, as there are more quasi-band representations than elementary band representations, one defines atomic limits (i.e. trivial insulators) as being the quasi-band representations that are elementary band representations. There exists several such limits and hence the notion of obstructed atomic limits. Quasi-band representations that are not elementary band representations are called topological insulators. In order to detect such topological insulators it is possible to use symmetry-based indicators [188] following the initial idea of Fu and Kane for inversion-symmetric topological insulators [32]. Another interesting notion that emerges from this framework is that of fragile topological bands [193]. Using these ideas, several catalogues of topological materials based on the 230 space groups appeared recently [194–196].

An atomic limit corresponds to exponentially-localized and symmetric Wannier functions. A trivial insulator is such that it can be smoothly continued to an atomic limit. Due to the different possible protecting symmetries, there may be several distinct atomic limits and therefore several types of trivial insulators (see the notion of an obstructed atomic limit). A topological insulator is such that it can not be continued to an atomic limit without closing a gap or breaking a protecting symmetry. In a SPT insulator there is an obstruction to finding symmetric and exponentially-localized Wannier functions.

This second definition of a topological insulator is actually more stringent than the first one. For example, according to the first definition, a 1D inversion-symmetric band insulator with quantized electric polarization (e.g. the CDW chain realizing an ionic insulator, or the Shockley model of coupled s and p bands in the covalent insulator phase) is a topological insulator. But according to the second definition, a 1D inversion-symmetric band insulator is always trivial: it is either a naive atomic limit if the Wannier centers are on the ions (e.g. the CDW chain, or the Shockley model in the atomic insulator limit) or an obstructed atomic limit if the Wannier center are on the bonds (e.g. the SSH chain realizing a molecular insulator, or the Shockley model in the covalent insulator limit).