One-dimensional extended Su–Schrieffer–Heeger models as descendants of a two-dimensional topological model

The topological phase diagrams and finite-size energy spectra of one-dimensional extended Su–Schrieffer–Heeger (SSH) models with long-range hoppings on the trimer lattice are investigated in detail. Due to the long-range hoppings, the band structure of the original SSH model becomes more complicated and new phases with the large Zak phase can emerge. Furthermore, a seeming violation of bulk-edge correspondence occurs in the one-dimensional topological system whose band topology stems from the inversion symmetry. The one-dimensional models are mapped onto a two-dimensional topological model when a parameter of the one-dimensional models is regarded as an additional degree of freedom. As Fourier components of the derived two-dimensional model, phase boudaries and the finite-size spectra of one-dimensional models can be recovered from the model in the higher spatial dimensions. Then the origin of edge modes of one-dimensional models can be understood from two dimensions and we give a reasonable explanation of the violation of bulk-edge correspondence in one spatial dimension. In fact, we may give a general perspective that the topological properties of one-dimensional (lower-dimensional) systems can be found their origin from two-dimensional (higher-dimensional) systems.


Introduction
In the family of topological insulator models, the one-dimensional Su-Schrieffer-Heeger (SSH) model may be the simplest one, which was originally supposed to describe the soliton in polyacetylene [1], rather than the topological structure of energy bands.The original SSH model has the bipartite chain structure with alternating electron hoppings between nearest neighboring sites.Along with the rapid development of researches on topological insulators, it was found that the model can possess the topologically non-trivial band structure characterized by Zak phase and there is a zero-energy edge mode under the open boundary condition, if the inter-cell hopping amplitude is larger than the intra-cell hopping amplitude [2,3].
The extension of the original SSH model includes some important aspects.Firstly, more terms can be introduced into the model, e.g. the long-range hoppings and the spin-orbit coupling (SOC).It was found that the so-called 'odd hoppings' [4] which preserve the chiral symmetry can induce the topological phase transitions and make the phase diagram much richer [4][5][6][7][8].A systematic study of engineering topological phases with any topological invariant by adding further neighbor hoppings into the SSH model has been done [9].In fact, Chen and Chiou [10] have already proven that arbitrary odd long-range hoppings lead to arbitrary winding numbers and then the arbitrarily large Zak phase.For the spinfull SSH model, the SOC which preserves the chiral symmetry was also introduced into the SSH model [7,[11][12][13][14].The investigation on the nearest neighbor SOC given by Ahmadi et al [7] shows that the SOC can also lead to topologically non-trivial phases like the long-range hoppings.It was shown that the nearest neighbor SOC lifts the spin degeneracy of Bloch bands and leads to more topological phases due to new behaviors of the crossing between bands [11,12].Another term that can be added in the original SSH model is the disorder.There are two types, i.e. the diagonal and off-diagonal disorders [4,8,15].It is generally concluded that the disorder

The Hamiltonian and the symmetry
The original SSH model on the one-dimensional dimerized chain includes two alternating neareast neighbor bonds and then the lattice is composed of two sublattices [1,42].The original SSH model can be extended to models on the one-dimensional chain, which possess more sites in an unit cell, e.g. the SSH3 [29,[32][33][34][35][36]43] and the SSH4 model [31].These one-dimensional SSH models can generally be described by off-diagonal Aubry-André [39] or Harper model [40] as where ĉ † i (ĉ i ) is the creation (annihilation) operator of the spinless electron at site i. t is the real hopping amplitude between nearest neighbor electrons and λ od is the strength of the modulation of the electron hopping.The modulation is represented by the cosine function and parametrized by the frequency b and phase ϕ (−π < ϕ ⩽ π).From the periodicity of cosine function, the original SSH model is obtained when b = 1/2, while the SSH3 model and the SSH4 model are obtained if b = 1/3 and 1/4 respectively.In the case of b = 1/3, there are three types of nearest neighbor bonds on the chain and then the lattice is composed of three sublattices denoted by A, B, C respectively, as shown in figure 1. t AB (t BA ), t BC (t CB ) and t CA (t AC ) denote amplitudes of the three nearest neighbor hoppings respectively.We set t AB = t BA , t BC = t CB and t CA = t AC and write amplitudes of the nearest neighbor hoppings as t AB , t BC or t CA only below.Corresponding to the equation (1), t AB = t + λ od cos(2π/3 + ϕ), t BC = t + λ od cos(4π/3 + ϕ), and t CA = t + λ od cos(ϕ).The generalized Aubry-André or Harper model with b = 1/3 in photonic waveguide arrays and the direct measurement of the bulk Chern number have been proposed [44].
The SSH3 model can be extended by introducing the long-range hoppings (We focus on the SSH3 model only).The Hamiltonian becomes Here, where The symbol '• • • ' denotes the terms induced by the longer-range hoppings and we ignore these terms in this work.For the concrete calculation, we set the lattice constant a = 1 below.There is an important symmetry which is critical for the topology of bands and energy spectra.That is the inversion symmetry.For lattice structure of our extended model, the inversion center can be chosen to be a lattice site of sublattice A, B or C, and the discussions are equivalent for the arbitrary choice of the inversion center.If the site on sublattice B is chosen as the inversion center, the inversion operator in the matrix representation is and the inversion symmetry is represented by For the original SSH3 model, the inversion symmetry holds if t AB = t BC .For the model with NN neighbor hoppings, t AB = t BC and t ′ AB = t ′ BC are both needed to preserve the inversion symmetry.Besides the inversion symmetry, the one-dimensional extended SSH3 model preserves the time-reversal symmetry.It is easy to derive that the Hamiltonian (3) satisfies TH(k)T −1 = H(k) * = H(−k).Here, T is the time-reversal operator, which is actually a complex-conjugation operator and satisfies T 2 = 1 for spinless electrons.However, the particle-hole symmetry is generally broken by the NN neighbor hoppings.In the Hamiltonian (2), the terms of NN neighbor hoppings That makes the band structures of paticles and holes different, i.e. the particle-hole symmetry is broken.Combining the time-reversal T and the particle-hole transformation C, we conclude that the chiral symmetry (operator: S = T • C) is also broken.In the Altland-Zirnbauer (AZ) symmetry classes [45,46], the extended SSH3 model we studied here belongs to the class AI, since T = +1, C = 0 and S = 0.The system does not possess topologically non-trivial phase in one dimension.However, the inversion symmetry modifies the original AZ classification and leads to new topologically non-trivial phases protected by the inversion symmetry [47].So, the topologcially non-trivial band structrue of our extended model stems from the inversion symmetry.

The extended SSH3 model with NN neighbor hoppings-the case of inversion symmetry(ϕ = 0 or π) 2.2.1. The topological phase diagrams
In the case of inversion symmetry, the topological invariant is well-defined for the one-dimensional system.The Zak phase is chosen as the topological invariant in this work.The standard definition of Zak phase [48] for an energy band is given by where u n (k) is the eigenvector of the Bloch Hamiltonian (3) with the eigenvalue E n (k) and the index n labels the energy bands with Due to the numerical calculation on the Bloch Hamiltonian, we shall adopt the discrete formulation of the Zak phase as [49] Z where u nl is the nth eigenvector of the Hamiltonian (3) at the successively discretized point k l in the Brillouin zone.Firstly, we investigate the phases of the extended model, which characterized by the topological invariant (Z  2(a)), the topologically non-trivial phase (π, 2π, π) connects adiabatically to the one of the original SSH3 model, whose Zak phase of the lower band was computed as π [34].In the case of t AB > t CA (figure 2(b)), the trivial phase (0, 0, 0) in the bottom-left region of the phase diagram also connects adiabatically to the trivial phase of the original SSH3 model [34].
The long-range hoppings can drive topological phase transitions and induce new topological phases, since they make the band structure more complex and lead to more band crossings in the Brillouin zone.The topological phase boundaries are obtained from the crossings between two of the three bands of the extended SSH3 model.In the case of t AB < t CA (figure 2(a)), six phase boundaries are confirmed.The boundary I(IV) is given by the simple expression 2t + λ od /2 = t ′ AB + t ′ CA , on which the middle and lower bands (upper and middle bands) touch at k = ±π/3.The boundary II(V) is given by 3λ od /2 = −t ′ AB + t ′ CA , on which the upper and middle bands (middle and lower bands) touch at k = ±π/3.On the boundary III, 3λ od /2 = t ′ AB − t ′ CA , the middle and lower bands touch at k = 0.The another phase boundary (boundary VI in figure 2(a)) has no the analytic expression.On this boundary, the upper and middle bands touch at some k (π/6 < |k| < π/3).For the case of t AB > t CA (figure 2(b)), the topological phase boundaries are listed below.
I: 2t − λ od /2 = t ′ AB + t ′ CA , middle and lower bands touch at k = ±π/3.II: 3λ od /2 = −t ′ AB + t ′ CA , middle and lower bands touch at k = 0. III: 3λ od /2 = t ′ AB − t ′ CA , middle and lower bands touch at k = ±π/3.VI: t ′ AB = t + λ od /2, middle and lower bands touch at |k| < π/3 slightly.VII: non-analytic expression, upper and middle bands touch at π/6 < |k| < π/3.Furthermore, the calculation of the topological invariant shows that the Zak phase of the single band in the one-dimensional system is well-defined only if the band is separated from others.For example, consider the topological phase transition from phase (π, 2π, π) to phase (π, π, 2π) (figure 2(a)).During the transition, the lower band maintains the separation from other bands and the middle and upper bands touch on the boundary IV.In the process, the Zak phase of the lower band maintains Z 1 = π both in the two phases and on the boundary IV.The Zak phases of the upper and middle bands are also well-defined in the two phases, while they become ill-defined on the boundary IV due to the crossing of the two bands at k = ±π/3.

The finite-size energy spectra and edge modes
For a finite system, the edge modes can occur due to the non-zero topological invariant of bulk bands.We now impose the open boundary condition on the extended SSH3 model to obtain the finite-size energy spectra.The results for some topological phases in the case of t AB < t CA are shown in figure 3. On the one hand, the finite-size energy spectrum and edge modes at those chosen points reflect the bulk-edge correspondence of topological systems, that is, the different bulk bands (characterized by Zak phase here) correspond to different patterns of the edge modes.For example, the points α 1 and α 4 in the phase diagram (figure 2(a)) belong to the topologically non-trivial phases (π, 2π, π) and (2π, 2π, 2π) respectively.And naturally, the edge modes of the two phases reside in different energy bands, although there are both four edge modes for the two phases (see figures 3(a) and (d)).On the other hand, we find the seeming violation of the bulk-edge correspondence.For example, the models at points α 1 , α 2 and α 3 possessing the same topological phase (π, 2π, π) have different edge modes, i.e. there are four, two and none edge modes respectively.Figures 3(a)-(c) illustrate the details of these edge modes.In fact, the violation of the bulk-edge correspondence can be found in almost topological phases in the phase diagram in the case t AB < t CA .
For the case of t AB > t CA , the finite-size spectra of the models at some points in the phase diagram(figure 2(b)) are shown in figure 4. For the topologically trivial phase (0, 0, 0), as expected, the finite-size spectrum does not possess in-gap states and there are no edge modes.The spectra of finite-size systems at points β 1 and β 2 in figure 2(b) illustrate the situations (figures 4(a) and (b)).However, we checked that there are also no edge modes in the topologically non-trivial phase (π, π, 0).As an illustration, the finite-size spectrum at point β 3 in the phase diagram (figure 2(b)) is shown in figure 4(c).Naturally, there are in-gap states and edge modes for topologically non-trivial phase (2π, 2π, 2π) and (π, π, 2π).For example, at point β 4 (β 5 for (π, π, 2π)) there are two edge modes as shown in figures 4(d) and (e).However, like topologically non-trivial phases (π, π, 0), there may also be no edge modes for the two topologically non-trivial phases, e.g.spectra at point β 6 (see figure 4(f)).

The extended SSH3 model with NN neighbor hoppings-the case of broken inversion symmetry (0 < |ϕ| < π)
In the case of the sublattice B as the inversion center, the following three cases can break the inversion symmetry.Case I: ) is chosen to break the inversion symmetry here.Corresponding to the general description by off-diagonal Aubry-André or Harper model (equation ( 1)), ϕ ̸ = 0 and ϕ ̸ = π are required.Firstly, let us investigate the energy spectra of corresponding extended SSH3 models without the inversion symmetry, when ϕ in the off-diagonal Aubry-André or Harper model is given.It is found that there are only two gap-closing points in the t ′ AB − t ′ CA parameter space for these inversion-symmetry-broken SSH3 models (corresponding to ϕ ̸ = 0 and π).For example, the upper and middle bands cross at k = ±π/3 at point (t ′ AB , t ′ CA ) = (0.65, 2.04), and the middle and lower bands cross at k = ±π/3 at point (t ′ AB , t ′ CA ) = (0.65, 1.35) when |ϕ| = π/6, as shown in figure 5(a).The gap-closing points of all of the extended SSH3 models without the inversion symmetry are shown in figure 5(b).The set of these points constitutes a closed curve in t ′ AB − t ′ CA parameter space.Now, we investigate the finite-size energy spectra of the extended SSH3 models without the inversion symmetry.The case of |ϕ| = π/6 is studied in detail.The finite-size energy spectra of models at the six chosen points in figure 5(a) are obtained, as shown in figure 6.The results show that models with different long-range hoppings t ′ AB and t ′ CA can possess in-gap states or edge modes, although the inversion symmetry is broken and the topological invariant becomes ill-defined.However, unlike the case of the inversion symmetry, the in-gap states are not degenerate in these inversion-symmetry-broken models.Of course, models at different points in the t ′ AB − t ′ CA plane may have different patterns of edge modes, which can easily be observed from figure 6.For other values of ϕ, we have also investigated the finite-size spectra of corresponding models (not shown here).The most interesting property is that there also exist in-gap states or edge modes in these extended SSH3 models without the inversion symmetry.Furthermore, it can be checked that there are also in-gap states or edge modes in the models without inversion symmetry if the inversion symmetry is broken by other means, i.e. case II:

A brief summary
From the investigations on specific examples above, some important properties of the extended SSH3 models (actually can be generalized to other extended SSH models) can be summarized as follows.(i) The different topological phases have different patterns of edge modes.(ii) In the same topological phases, the models can possess different patterns of edge modes.(iii) For some topologically non-trivial phases, there may be no edge modes.(iv) When the inversion symmetry is broken, in which case the topological invariant is ill-defined, the model may possesses in-gap states or edge modes.(v) In the case of inversion symmetry, the in-gap states or the edge modes are degenerate, while they are not degenerate for the case of broken inversion symmetry.
In the list of properties, (ii)-(iv) may seem to be the violation of the bulk-edge correspondence.In fact, these properties can be understood from a general framework as discussed below.Especially, we will give the answer why a topologically non-trivial phase can possess different patterns of edge modes (even no edge modes) and how in-gap states or edge modes emerge in the topologically trivial phase when the topological invariant is ill-defined.

The mapping from one dimension to two dimensions
If the variable ϕ of the equation ( 1) is considered as an additional momentum, the general description of the SSH model can be viewed as a Fourier component of some two-dimensional Hamiltonians.If the one-dimensional SSH chain is placed along the x-direction and then the momentum ϕ corresponds to the y-direction, the electron operator can be extended as ).The one-dimensional extended SSH model (equation ( 2)) can be written as Summing over momenta ϕ, there is

Introducing the Fourier transform c
Finally, we actually get a mapping from one-dimensional models (H) to a two-dimensional model (H 2D ).
Corresponding to the one-dimensional extended SSH3 model discussed in detail here, b = 1/3 is required, and the i and i ′ are the NN neighboring sites in the x-direction.It is natural to find a key point about the mapping.That is, when the periodic boundary condition in the y-direction is imposed on the derived two-dimensional model, the one-dimensional extended SSH model will be obtained if a ϕ is given.

The 1D extended models as the descendants of the 2D model from the viewpoints of topological phase diagrams
In fact, the derived two-dimensional model describes the electrons with long-range hoppings on the two-dimensional lattice in an applied magnetic field.Due to the magnetic field, the translation symmetry enlarges the original unit cell in the x-direction to form a magnetic unit cell.The magnetic unit cell is 1/b times larger than the original one and carries one magnetic flux quanta.Accordingly, the Brillouin zone should be reduced to [−bπ, bπ] in k x -direction.In the case of b = 1/3, the lattice structure and Brillouin zone are shown in figure 7.In fact, there may be some interesting properties and phenomena about the energy spectrum of this derived two-dimensional model, e.g. the famous Hofstadter butterfly for the varying b [50,51].However, we will not discuss these interesting objects in the work.We will focus on the topological aspect of this model induced by the magnetic field, i.e. the Chern or TKNN number [52].
The Chern number of a band is defined as where the Berry vector potential , and u n (k) is the nth eigenvectors of the Bloch Hamiltonian.As the case of one-dimensional models, eigenenergies E 1 , E 2 and E 3 are set as The Bloch Hamiltonian of the derived two-dimensional model with NN neighbor hoppings in the case of b = 1/3 can be obtained from the Hamiltonian ( 12) via the Fourier transform as It is less necessary to carry out analytically eigenvectors of the Bloch Hamiltonian to calculate the Chern number of an energy band from equation ( 13).We adopt the numerical method by Fukui et al [53] to handle the equation (13).From the Chern number and the crossing between the three bands, we obtain the topological phase diagram of two dimensional model as shown in figure 8.
It is interesting that the topological phase diagram of the derived two-dimensional model with NN neighbor hoppings may contain all of the phase boundaries or band crossing points of our one-dimensional extended SSH3 models for any ϕ.In fact, it can be confirmed that every phase boundary or band crossing point of the one-dimensional extended SSH3 model can be recovered from the topological phase diagram of the derived two-dimensional model.
Firstly, let us investigate one of the boundaries, i.e. the boundary I in the phase diagram figure 8 (the phase diagram is redrawn in figure 9(a)) to illustrate how this works.On the boundary I (t ′ AB > t − λ od /2) in the phase diagram (see figure 9(a)), the middle and lower bands touch at X(±π/3, 0) and Y(0, ±π) in the Brillouin zone, as shown in figure 9(b).For crossing point X(±π/3, 0), the crossing of bands at k x = ±π/3 and k y = 0 corresponds to the situation that middle and lower bands of the one-dimensional extended SSH3 model (ϕ = 0) touch at k = ±π/3.That is, the boundary V in the phase diagram (figure 2(a)) is recovered.For crossing point Y(0, ±π), it corresponds to the case that middle and lower bands of the one-dimensional extended SSH3 model (ϕ = π) touch at k = 0.Then, the boundary II (t ′ AB > t − λ od /2) in the phase diagram (figure 2(b)) is recovered.On the boundary I (0 < t ′ AB < t − λ od /2) in the figure 9(a), the middle band touches the upper and lower bands at X(±π/3, 0) and Y(0, ±π) respectively (see figure 9(c)).For crossing point X(±π/3, 0), it corresponds to the case that middle and upper bands of the one-dimensional extended SSH3 model (ϕ = 0) touch at k = ±π/3 and then the boundary II of phase diagram (figure 2(a)) is recovered.For crossing point Y(0, ±π), the situation is actually the same as before, and then the boundary II (0 Let us continue to make the illustration.On the boundary II in the topological phase diagram of the two-dimensional model, the situation is different from the boundary I and others.For different points on the Finally, from the boring checks on every boundary, general conclusions are given below.For the two-dimensional model, if the bands cross at k y = 0 in the Brillouin zone, the boundaries (including boundary III in figure 9(a)) are those of the one-dimensional model with inversion symmetry (in the case of ϕ = 0).If the bands cross at k y = ±π, the boundaries of the one-dimensional model with inversion symmetry (in the case of ϕ = π) are recovered from the boundaries (including boundary IV in figure 9(a)) of the two-dimensional model.If the bands cross at k x = ±π/3 and 0 < |k y | < π, the boundary in the t ′ AB − t ′ CA space is actually the set of crossing points of all of the one-dimensional models without inversion symmetry (0 < |ϕ| < π).
It is easy to understand the features.If the periodic boundary condition is imposed on the twodimensional model only in the y-direction (the corresponding momentum is k y ) and then k y is replaced by ϕ, the one-dimensional model is recovered for a given ϕ.The derivation is actually the reverse process of the mapping from one dimension to two dimensions.In this sense, the one-dimensional extended SSH3 models are regarded as the descendants of the two-dimensional topological model.

An explanation of spectra and edge modes of the 1D extended SSH3 model
In the section 3, we summarized that there is a seeming violation of the bulk-edge correspondence in the one-dimensional extended SSH3 model.Now, we can give the explanation of the absence of edge modes in a topologically non-trivial phase and the origin of the edge modes in a topologically trivial phase, from the role of one-dimensional models as the descendants of the two-dimensional topological model.The groups phase diagrams are chosen to illustrate the explanation.
Firstly, let us investigate a topological phase of the two-dimensional model to gain the insight into the relationship of spectra between the one-and two-dimensional models.In the topologically non-trivial phase (−1, 2, −1), the finite-size spectrum of the two-dimensional model at point δ 1 is shown in figure 10(a).Due to the bulk-edge correspondence, there are overall in-gap states and edge modes in the two energy gaps.It can be seen that two degenerate edge modes exist in both gaps if k y = 0. Including the continuous part of the spectrum, we can see that, in the particular case k y = 0 of the two-dimensional model, the spectrum is just that of the one-dimensional extended SSH3 model (ϕ = 0) at α 1 (see figure 3(a)).Similarly, the spectrum of the one-dimensional extended SSH3 model (ϕ = π) at β 1 (see figure 4(a)) can be obtained from the spectrum of the two-dimensional model if k y = π (see figure 10 4 and 6).The general conclusion is that at any point in the phase diagram of the two-dimensional model the spectrum for a given k y recovers the spectrum of the corresponding one-dimensional extended SSH3 model with ϕ = k y .This is actually the manifestation of the role of the one-dimensional model as the descendant of the derived two-dimensional model.The result leads us to regard the existence of edge modes of one-dimensional models as the outcome of the bulk-edge correspondence of the two-dimensional model, no matter whether there is the inversion symmetry or the topological invariant can be well-defined for one-dimensional models.
Now we can give a reasonable explanation of the violation of bulk-edge correspondence in the one-dimensional models.First, we investigate the phenomenon that there are different edge modes in the same topologically non-trivial phases.An example is that the one-dimensional model (ϕ = 0) at α 1 has edge modes different from that of the model at α 2 , although the two models are both in the topologically non-trivial phase (π, 2π, π) (see figures 2(a) and 3(a), (b)).Points α 1 and α 2 respectively correspond to δ 1 and δ 2 in the phase diagram of two-dimensional model (see figure 8).From the spectra of the two-dimensional model at δ 1 and δ 2 (figures 10(a) and (b)) when k y = 0, we can find the difference between edge modes of the one-dimensional models (ϕ = 0) at α 1 and α 2 .Another example is the case of the one-dimensional model (ϕ = 0) at α 3 .At α 3 the one-dimensional model (ϕ = 0) is in the topologically non-trivial phase as the model at α 1 (or α 2 ).However, unlike the situation of α 1 , there is no edge modes (see figure 3(c)).In this case, the absence of lower edge modes is due to the closing of the indirect gap between the lower and middle bands, while the upper edge modes can be understood from the two-dimensional model.At δ 3 which corresponds to α 3 , the two-dimensional model is in topologically non-trivial phase (2, −1, −1).There exist edge modes in the upper gap.That satisfies the bulk-edge correspondence of the two-dimensional  10(c)) (note that the absence of lower edge modes in the two-dimensional model is also due to the closing of the indirect gap between the middle and lower bands).However, there is no edge modes if k y = 0 (see figure 10(c)).That is why the edge modes of the one-dimensional model (ϕ = 0) at α 3 are absent.
Secondly, we give an answer to the puzzle that there may be no edge modes for the one-dimensional models even in a topologically non-trivial phase.In fact, the situation has been encountered above.We give other examples here.The one-dimensional model (ϕ = 0) at α 6 has no edge modes, although the model is in topologically non-trivial phase (π, 2π, π) (see figures 2(a) and 3(f)).This can also be understood from the spectrum of the two-dimensional topological model.At corresponding point δ 6 , the two-dimensional model is in topologically non-tivial phase (2, −1, −1) (see figure 8), whose finite-size spectrum is shown in figure 10(f).When k y = 0, the spectrum of the one-dimensional model (ϕ = 0) is recovered (the reader can compare figure 3(f) with figure 6(f)).And it is obvious that there is no edge modes for one-dimenional model, while there are overall edge modes for the two-dimensional model due to the bulk-edge correspondence.Another example is the one-dimensional model (ϕ = π) at β 6 .The model is in topologically non-trivial phase (π, π, 2π) (see figure 2(b)), but its spectrum (figure 4(f)) shows that there is no any edge mode.This can also be understood from the spectrum of the two-dimensional model at δ 6 as above, but k y = π should be chosen (see figure 10(f)).
Lastly, let us focus on the origin of edge modes in a phase of a one-dimensional topologically trivial model.When ϕ ̸ = 0 and π, e.g.|ϕ| = π/6, the topological invariant is ill-defined due to the breaking of the inversion symmetry.However, there exist edge modes (see figure 6 for the case of |ϕ| = π/6).Let us come back to the spectra of the two-dimensional model again.At points in the phase diagram (figure 8) (e.g. from δ 1 to δ 6 ), the two-dimensional models are in different topologically non-trivial phases and there exist overall edge modes in the finite-size spectra due to the bulk-edge correspondence (see figures 10(a)-(f)).We can see that for most of k y (k y ̸ = 0 and π), edge modes are present.Additionally, they are generally not degenerate.We have known that a k y corresponds to a one-dimensional extended SSH3 model (ϕ = k y ), and then we can say that the edge mode in the one-dimensional topologically trivial model has its origin from the two-dimensional topologically non-trivial model.

Conclusions and outlook
In this work, we studied the one-dimensional extended SSH models with long-range hoppings (the NN neighbor hoppings) on the trimer lattice (the SSH3 model) and established a connection between them and a two-dimensional topological model.Using the connection, we explained some confusing properties of spectra.We gave a general expression of the one-dimensional extended SSH models in the form of off-diagonal Aubry-André or Harper model, where a parameter ϕ is the modulated parameter to produce various types of the SSH model.For one-dimensional extended SSH3 models with the inversion symmetry (ϕ = 0 or π), the topological phase diagrams were established and the band crossings in the parameter space spanned by NN neighbor hoppings were also obtained.We found the violation of the bulk-edge correspondence in the one-dimensional model.To understand the phenomena, the one-dimensional models were mapped onto a two-dimensional model which actually describes lattice electrons with long-range hoppings in an applied magnetic field (in the mapping, the parameter ϕ is regarded as an additional degree of freedom corresponding to the spatial y-direction).From the band structure and the well-defined topological invariant--Chern number, the topological phase diagram of the derived two-dimensional model was obtained.From the viewpoint that these one-dimensional models can be regarded as the descendants of the derived two-dimensional topological model, we recovered phase boundaries and crossing points of bands of all of the one-dimensional extended SSH3 models.Especially, the finite-size spectra of all of the one-dimensional extended SSH3 models (regardless of the inversion symmetry) can be recovered by the spectrum of the two-dimensional topological model if the degree of freedom k y is selected.The relationship between the one-and two-dimensional models gives a reasonable explanation of the origin of the presence, change or absence of the edge modes in the one-dimensional extended SSH3 models, ranther than from the bulk-edge correspondence.
There are many discussions on the band topology of other one-dimensional SSH models and their extended models, e.g. the classical SSH model with long-range hoppings, the SSH4 model and the SSH ladder.In essence, these models can be mapped onto a two-dimensional model as we did in the work.Then it is interesting to investigate generally these one-dimensional topological models (even the SSH ladder) from the higher spatial dimensions regardless of symmetries, e.g. the chiral or sublattice symmetry.The key point is that one shall view the one-dimensional models from the topological properties of two-dimensional models.There may be an interesting re-examination on the bulk-edge correspondence of one-dimensional topological systems.

Figure 1 .
Figure 1.The lattice structure of the extended SSH3 model.The model includes the next-nearest neighbor hoppings only.The sublattices are denoted by A, B, C respectively.

Figure 2 .
Figure 2. Topological phase diagrams of the extended SSH3 model with NN neighbor hoppings and the inversion symmetry, (a) tAB < tCA(ϕ = 0) and (b) tAB > tCA(ϕ = π).Here, we set t = 1, λ od = 0.8 in the concrete calculation.The topological phases are characterized by the Zak phases of each bands as (Z1, Z2, Z3).The middle and upper bands touch at k = ±π/3 and |k| < π/3 slightly on red solid and red dotted lines respectively.The middle and lower bands touch at k = ±π/3, k = 0 and |k| < π/3 slightly on black solid, blue solid and black dotted lines respectively.Inset of (b) shows the details of the lower part of the phase diagram marked by the gray rectangle.Some points α1, α2, • • • and β1, β2, • • • are selected for further discussions below.

Figure 3 .
Figure 3.The finite-size spectra of models at points α1 (a), α2 (b), α3 (c), α4 (d), α5 (e) and α6 (f) in the phase diagram (figure 2(a)).In the inset figures, the amplitude of the wave function of the in-gap state at the lattice site is shown.These in-gap states are actually the edge modes.The red, orange and green lines denote the amplitudes of electron distribution of edge modes on the sublattices A, B and C respectively.

Figure 7 .
Figure 7. (a)The lattice structure of the derived two-dimensional model with NN neighbor hoppings in the x-direction for the case b = 1/3.The primitive vectors a1 = (3a, 0) and a2 = (0, a).In the lattice structure, an elementary plaquette contains b (1/3 here) magnetic flux quanta and a magnetic unit cell contains one magnetic flux quanta.(b)The two elementary plaquette of the lattice.The magnetic field is perpendicular to the lattice plane.(c)The reduced Brillouin zone of the two-dimensional model.
phase diagram (figure 2(b)) can be recovered.We conclude that, on the boundary I in the phase diagram of the derived two-dimensional model, the boundaries II and V in the phase diagram (figure 2(a)) of the one-dimensional extended SSH3 model (ϕ = 0) are recovered if k y = 0 and the boundary II in the phase diagram (figure 2(b)) of the one-dimensional model (ϕ = π) is recovered if k y = ±π.

Figure 9 .
Figure 9.The phase boundaries of the derived two-dimensional model and the energy spectra on the boundaries I and II.Here, points b, c, d and e on the boundaries are chosen for concrete calcultions.The high-symmetry points Γ, X, S and Y in the Brillouin zone are shown in figure 7(c).