Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials

We first describe our main concept, the formation and spectral isolation of stable defect states against a background of flat bands, in the general context of two-dimensional bipartite lattices [29, 30]. Such systems consist of two sublattices (A sites and B sites) that are coupled together to result in the following off-diagonal Bloch Hamiltonian (see the appendix for further details),

$$\begin{eqnarray}h\left(\mathbf{k}\right)=\left(\begin{array}{c} 0 & {{t}_{AB}}\left(\mathbf{k}\right) \\ {{t}_{BA}}\left(\mathbf{k}\right) & 0 \end{array}\right),\quad E\varphi =h\varphi,\quad \varphi =\left(\begin{array}{c} {{\varphi}_{A}} \\ {{\varphi}_{B}} \end{array}\right).\end{eqnarray} \tag{ 1 }$$

Here ${{t}_{AB}}\left(\mathbf{k}\right)=t_{BA}^{\dagger}\left(\mathbf{k}\right)$ describes the coupling of the A sites to the B sites, whose amplitudes are collected into vectors ${{\varphi}_{A}}$ and ${{\varphi}_{B}}$. The coupling term ${{t}_{AB}}\left(\mathbf{k}\right)$ is thus an ${{n}_{A}}\times {{n}_{B}}$ matrix whose dimensions are given by the count of sublattice sites in the unit cell. We set ${{n}_{A}}\geqslant {{n}_{B}}$ and identify the A and B sites with the majority and minority sublattice, respectively. Generically, there are then ${{n}_{A}}-{{n}_{B}}$ zero modes for any wavevector $\mathbf{k}$, given by the solutions of ${{t}_{AB}}\left(\mathbf{k}\right){{\varphi}_{A}}=0$ while ${{\varphi}_{B}}=0$. These dispersionless states constitute the sublattice-polarized flat bands. They are complemented by 2n_B dispersive bands of extended states that occur in pairs of opposite energies, while the Bloch wavefunctions carry equal weight on both sublattices. All of these features are intimately linked to the chiral symmetry, ${{\tau}_{z}}h\left(\mathbf{k}\right){{\tau}_{z}}=-h\left(\mathbf{k}\right)$, where the Pauli matrix ${{\tau}_{z}}$ acts in sublattice space; in particular, only the zero modes display a finite sublattice polarization $\mathcal{P}=\,|{{\varphi}_{A}}{{|}^{2}}-\,|{{\varphi}_{B}}{{|}^{2}}$.

As $\mathbf{k}$ is varied over the Brillouin zone, the dispersive bands can touch at E  =  0, meaning that they cross the flat band. This corresponds to situations where the columns of ${{t}_{AB}}\left(\mathbf{k}\right)$ become linearly dependent of each other. We focus on scenarios where this happens at discrete Dirac points $\mathbf{k}={{\mathbf{k}}_{p}}$ (p integer). If a Dirac point occurs at real $\mathbf{k}$, one finds two additional zero modes, obtained by the degeneracy of the finite-energy states as $\mathbf{k}\to {{\mathbf{k}}_{p}}$. Exploiting that these states are related by ${{\tau}_{z}}$, they can be combined into a mode ${{\varphi}_{+}}$ on the majority sublattice, as well as a mode ${{\varphi}_{-}}$ on the minority sublattice—which then obeys ${{\varphi}_{-,A}}=0$, ${{t}_{AB}}\left({{\mathbf{k}}_{p}}\right){{\varphi}_{-,B}}=0$. We now aim to spatially confine and spectrally isolate the state ${{\varphi}_{-}}$.

In order to create, as a first step, spatially localized variants of this mode, we have to consider situations where the Dirac point is lifted, so that it lies in the complex $\mathbf{k}$-plane and describes evanescent zero modes. We identify three important features of these modes: (i) they appear in pairs of distinct wave vectors ${{\mathbf{k}}_{\pm }}$, (ii) they are supported by opposite sublattices, and (iii) in a finite geometry, only one of them can be compatible with the boundary conditions. In detail, the mode ${{\varphi}_{+}}$ localized on the majority sublattice corresponds to the case of linearly dependent rows of ${{t}_{BA}}\left({{\mathbf{k}}_{+}}\right)$, while the mode ${{\varphi}_{-}}$ localized on the minority sublattice corresponds to linearly dependent rows of ${{t}_{AB}}\left({{\mathbf{k}}_{-}}\right)$. As ${{t}_{BA}}\left(\mathbf{k}\right)={{\left[{{t}_{AB}}\left({{\mathbf{k}}^{\ast}}\right)\right]}^{\dagger}}$, the complex wave vectors ${{\mathbf{k}}_{+}}$ and ${{\mathbf{k}}_{-}}=\mathbf{k}_{+}^{\ast}$ are distinct and describe states that decay into opposite directions. In a finite system, the mode ${{\varphi}_{+}}$ is automatically compatible with the boundary conditions if the system is terminated on the majority sublattice, while for ${{\varphi}_{-}}$ this is the case for termination on the minority sublattice—the sites just beyond the boundary then lie on the sublattice where the amplitude of the mode vanishes. This zero mode is then exponentially localized around a corner of the system, and retains its finite sublattice polarization $\mathcal{P}=\pm 1$.

To move this localized zero mode to an arbitrary position within the system, we can create a crossing of line defects that join four regions, in which the mode decays as one moves away from the resulting point defect, as illustrated in figure 1(a). The matching conditions are then automatically met if the interface is formed by the appropriate sublattice.

Recall that this zero mode can still hybridize with the flat band. The flat band can now be gapped out by introducing symmetry-breaking terms t_AA (but not t_BB) into equation (1), leading to a Bloch Hamiltonian of the form

$$\begin{eqnarray}h\left(\mathbf{k}\right)=\left(\begin{array}{c} {{t}_{AA}}\left(\mathbf{k}\right) & {{t}_{AB}}\left(\mathbf{k}\right) \\ {{t}_{BA}}\left(\mathbf{k}\right) & 0 \end{array}\right).\end{eqnarray} \tag{ 2 }$$

Note that this Bloch Hamiltonian fulfills

$$\begin{eqnarray}&&{{\left[{{\tau}_{z}}h\left(\mathbf{k}\right){{\tau}_{z}}\right]}_{BB}}={{\left[-h\left(\mathbf{k}\right)\right]}_{BB}},\end{eqnarray} \tag{ 3 }$$

which we take as the formal definition of a partial chiral symmetry. While the final configuration does not exhibit a global chiral symmetry, the symmetry is thus still operational on the minority sublattice, and if the point-defect mode was localized on this sublattice neither its wavefunction nor its energy are affected. This then leaves behind a spectrally isolated and spatially localized zero mode supported by the minority sublattice.

The regular Lieb lattice [30] consists of a square sublattice B, with additional sites A₁ and A₂ (forming the A sublattice) placed into the unit cell (gray rectangle in figure 1(a)) so that they subdivide each horizontal or vertical edge. This regular lattice has recently been realized in photonic lattices, allowing to directly observe the compacton-like states [39, 40]. Figure 1(a) shows a dimerized version of the Lieb lattice (with a central defect obtained as discussed below), and figure 1(b) its experimental implementation. This consists of an array of 65 microwave cylindrical resonators (5 mm height, 8 mm diameter) made of ZrSnTiO (refractive index 6), placed between two metallic plates (top plate not shown in figure 1(b)).

The resonators possess a bare resonance at ${{\nu}_{0}}=$ $6.65$ GHz, which is well isolated within a large frequency interval, and are coupled via the evanescent field, with a distance-dependence that has been characterized in detail in [47, 48]. If we denote by ${{ \Psi }_{0}}$ the wavefunction associated to the resonance, the z-component of the magnetic field reads (see [48], equation (4), for the complete expression of the field) ${{B}_{z}}(r,z)={{B}_{0}}\sin \left(\frac{\pi}{h}z\right){{ \Psi }_{0}}(r)$ with

$$\begin{eqnarray}{{ \Psi }_{0}}(r)=\left\{\begin{array}{c} {{J}_{0}}\left({{\gamma}_{j}}r\right) & \text{inside}, \\ \alpha {{K}_{0}}\left({{\gamma}_{k}}r\right) & \text{outside}, \end{array}\right. \end{eqnarray} \tag{ 4 }$$

where ${{ \Psi }_{0}}(0)=1$. Here J₀ and K₀ are Bessel functions, r is the distance from the center of the resonator, ${{\gamma}_{j}}=\sqrt{{{\left(\frac{2\pi {{\nu}_{0}}n}{c}\right)}^{2}}-{{\left(\frac{\pi}{h}\right)}^{2}}}$, and ${{\gamma}_{k}}=\sqrt{{{\left(\frac{\pi}{h}\right)}^{2}}-{{\left(\frac{2\pi {{\nu}_{0}}}{c}\right)}^{2}}}$ (n denoting the refractive index). Recall that far from the origin, K₀(x) is essentially approximated by $\exp \left(-x\right)/\sqrt{x}$. The coupling energy between two adjacent discs depends on the disc separation d and can be described, accordingly to equation (4), by a modified Bessel function $|{{K}_{0}}\left({{\gamma}_{k}}d/2\right){{|}^{2}}$ [49]. As one infers from in figures 1(a) and (b), in the Lieb lattice the dominant coupling is between nearest neighbors (u, ${{u}^{\prime}}$, v, ${{v}^{\prime}}$), but next-nearest neighbor couplings of sites on the A sublattice (represented by w, ${{w}^{\prime}}$, ${{w}^{\prime \prime}}$ and ${{w}^{\prime \prime \prime}}$ in figure 1(a)) are appreciable, while direct couplings on the B sublattice are negligible.

Using a vectorial network analyzer, the reflected signal S₁₁ at position ${{\mathbf{r}}_{1}}$ is measured by adequately positioning a movable loop-antenna. From the amplitude and phase of S₁₁ measured over a given frequency range, we have direct access to the local density of states (LDOS)

$$\begin{eqnarray}&&\rho \left({{\mathbf{r}}_{1}},E\right)\propto \frac{|{{S}_{11}}(E){{|}^{2}}}{{{\langle |{{S}_{11}}{{|}^{2}}\rangle}_{E}}}\varphi _{11}^{\prime}(E),\end{eqnarray} \tag{ 5 }$$

where ${{\langle \ldots \rangle}_{E}}$ indicates an average over the whole accessible frequency spectrum, ${{\varphi}_{11}}=\text{Arg}\left({{S}_{11}}\right)$ is the phase of the reflected signal, and $\varphi _{11}^{\prime}$ denotes its derivative with respect to the frequency. The density of states (DOS) is obtained by averaging over all positions ${{\mathbf{r}}_{1}}$. Moreover, our set-up allows us to visualize the wavefunction associated with each eigenfrequency. According to the definition of the local density of states $\rho \left({{\mathbf{r}}_{1}},E\right)={\sum}_{n}|{{ \Psi }_{n}}\left({{\mathbf{r}}_{1}}\right){{|}^{2}}\delta \left(E-{{E}_{n}}\right)$, the resonance curve exhibits a maximum whose value is related to the intensity of the wavefunction at the specific position ${{\mathbf{r}}_{1}}$. The intensity distribution for the wavefunction associated with each individual eigenfrequency thus becomes directly accessible.

Figures 1(c)–(e) show the predicted evolution of the band structure with the couplings, obtained by diagonalizing the corresponding Bloch Hamiltonian (see the appendix for details). Figure 1(c) corresponds to the situation of a regular Lieb lattice with uniform couplings $u=v={{u}^{\prime}}={{v}^{\prime}}=1$ and w⁽ⁱ⁾  =  0. The band structure consists of a flat band of states supported by the A sublattice and two dispersive bands of extended states touching at a conical Dirac point, positioned at the M point ${{\mathbf{K}}_{0}}=\left(\pi,\pi \right)$. As shown on figure 1(d), by introducing dimerized couplings u  =  v  =  4/3, ${{u}^{\prime}}={{v}^{\prime}}=2/3$, a gap $2 \Delta$, with ${{ \Delta }^{2}}={{\left(u-{{u}^{\prime}}\right)}^{2}}+{{\left(v-{{v}^{\prime}}\right)}^{2}}$, opens up between the dispersive bands, while the flat band and the point-defect state remain fixed at zero energy. This corresponds to a virtual Dirac point at ${{\mathbf{k}}_{\pm }}={{\mathbf{K}}_{0}}\pm \text{i}\mathcal{A}$, with $\mathcal{A}=\left[\ln \left(u/{{u}^{\prime}}\right),\ln \left(v/{{v}^{\prime}}\right)\right]$. Additional next-nearest-neighbour couplings w  =  0.4, ${{w}^{\prime}}={{w}^{\prime \prime \prime}}=0.2$, ${{w}^{\prime \prime}}=0.1$ make the flat band dispersive and move it to finite energies ($|E|\geqslant |w+{{w}^{\prime \prime}}-{{w}^{\prime}}-{{w}^{\prime \prime \prime}}|\equiv {{ \Delta }_{0}}$), while also breaking the symmetry of the dispersive bands, but do not affect the defect state, as shown in figure 1(e). Without further modification, this defect would be localized around the top right corner of the system; as illustrated in figure 1(a), the defect state can then be moved along the edges and into the bulk by creating dimerization line defects separating regions where the role of u and ${{u}^{\prime}}$ or v and ${{v}^{\prime}}$ are interchanged.

Figure 2 shows the experimental results for configurations where the point defect sits at the corner (figures 2(a)–(c)) or at the center (figures 2(d)–(f)) of the system. Figures 2(a) and (d) show the corresponding density of states (DOS). In agreement with the scenario proposed in this paper, the next-nearest neighbor couplings break the symmetry of the extended bands and isolate the zero mode (light gray zone) from the original flat band (dark gray zone), which now spreads over the whole system but remains mainly confined to the A sublattice (figures 2(c) and (f)). As expected, the state associated with the spectrally isolated peak displays a spatially localized profile with intensity confined to the B sublattice (figures 2(b) and (e)).

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By construction, the created defect state is insensitive to any disorder in the couplings $w,{{w}^{\prime}},{{w}^{\prime \prime}},{{w}^{\prime \prime \prime}}$. Disorder in the couplings $u,{{u}^{\prime}},v,{{v}^{\prime}}$ modifies the wavefunction, but does not affect its energy or sublattice polarization as long as each plaquette fulfills the constraint

$$\begin{eqnarray}&&{{u}_{nm}}v_{nm}^{\prime}u_{nm+1}^{\prime}{{v}_{n+1m}}={{v}_{nm}}u_{nm}^{\prime}{{u}_{nm+1}}v_{n+1m}^{\prime},\end{eqnarray} \tag{ 6 }$$

with couplings enumerated by the unit-cell index of the corresponding A sites (see figure 1(a)). In the continuum limit, where the displacement $\mathcal{A}$ of the virtual Dirac point can be interpreted as an imaginary vectorpotential, this condition amounts to a vanishing pseudomagnetic field induced by the deformation, $\nabla \times \mathcal{A}=0$.

Figure 3(a) shows the experimental DOS of a dimerized Lieb lattice with a central defect and disorder that respects the constraint (6). The disordered positions were generated on the level of the tight-binding model, where the couplings $u_{nm}^{\prime}\in \left[(1-W){{u}^{\prime}},(1+W){{u}^{\prime}}\right],{{v}_{nm}}\in$ $\left[(1-W)v,(1+W)v\right]$, $v_{nm}^{\prime}\in \left[(1-W){{v}^{\prime}},\right.$$\left. (1+W){{v}^{\prime}}\right]$ were chosen randomly from box distributions with W  =  0.175, while the couplings u_nm were determined from the constraint (6). These couplings were then translated into a compatible configuration of random displacements, which also introduces disorder into the couplings w⁽ⁱ⁾. The disks were then placed accordingly. In comparison with figure 2(d), we see that the disorder is sufficiently strong to mix the extended states, but does not affect the energy of the zero mode, which remains spectrally isolated. Furthermore, the spatial distribution of this mode shown in figure 3(b) exhibits the same localization as observed for the nondisordered lattice in figure 2(e).

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Encouragingly for practical applications, the defect state remains well isolated and localized also for generic disorder. In figures 3(c) and (d), this is confirmed experimentally for disorder of the same strength W  =  0.175, obtained by generating displaced positions of the disks as before but ignoring the constraint (6). As such disorder is still correlated (displacing a disk modifies several couplings systematically), we also carried out numerical calculations in which all couplings were randomly perturbed. As shown in figure 3(e), the spectral isolation of the defect state remains intact even at larger values of disorder; furthermore, the defect state remains predominantly localized on the B sublattice, as shown in figure 3(f).

This robustness can be understood as an added benefit from the spectral isolation and sublattice polarization of the defect state, as this automatically suppresses its sensitivity to generic coupling disorder in a perturbative treatment. Due to the sublattice polarization, the disorder only contributes from the second order, where energy levels further repel. Condition (6) leaves only one effective disorder freedom per plaquette, so that the relevant matrix elements only become comparable to the spectral isolation energy ${{ \Delta }_{0}}$ when the disorder is sizeable ($W\gtrsim 0.5$ under the conditions investigated here). An equivalent amount of onsite disorder, which breaks the original chiral symmetry, has a stronger effect on the spectral isolation, but only weakly affects the sublattice polarization as the matrix elements relevant for the leading order again vanish. The latter type of disorder is small in all experiments.

The partial chiral symmetry remains intact if one includes a possibly inhomogeneous onsite potential on the A sublattice. This includes the choice of an imaginary onsite potential that corresponds to absorption, in generalization of non-hermitian PT-symmetric optics [50–56]. In absence of the next-nearest neighbour couplings, the extended states remain sublattice-balanced for $\gamma <2 \Delta$ and decay according to $\text{Im}\,E=-\gamma /2$; the states from the flat band decay twice as fast ($\text{Im}\,E=-\gamma$), while the point-defect state stays pinned at E  =  0 and thus remains unaffected. In the presence of additional amplification, this corresponds to a topological mechanism of mode selection for lasing [23, 24]. In our experiments, we implement the absorption by placing elastomer patches on the top of each A site [26], thus inducing a uniform absorption on the A sublattice. The selection mechanism can be clearly seen in figure 4 where two situations are investigated. Figures 4(a) and (b) shows the effect of the absorption on a dimerized Lieb lattice with central defect (same configuration as in figure 2(d)), while figure 4(c) corresponds to the disordered system shown in figures 3(a) and (b). In both cases, the original flat band and the two dispersive bands are considerably suppressed, while the zero mode is not affected both spectrally as well as in its spatial distribution. Therefore, absorption on the A sublattice can be used to further enhance the spectral isolation of the defect state.

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The dimerized Lieb lattice with nearest-neighbour couplings is reproduced in figure B1. In the tight-binding description, each site X  =  A₁, A₂ or B in a given unit cell provides a basis state $|{{X}_{n,m}}\rangle$, with the cells enumerated by a pair of integers n,m. The tight-binding Hamiltonian is then given by

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} H=\underset{nm}{\sum}\,\left[\left({{u}_{nm}}|{{B}_{n,m}}\rangle +u_{nm}^{\prime}|{{B}_{n+1,m}}\rangle \right)\langle {{A}_{1;n,m}}| \right. \\ \quad \ \;+\,\left({{v}_{nm}}|{{B}_{n,m}}\rangle +v_{nm}^{\prime}|{{B}_{n,m+1}}\rangle \right)\langle {{A}_{2;n,m}}| \\ \quad \ \;+\,\left({{w}_{nm}}|{{A}_{2;n,m}}\rangle +w_{nm}^{\prime}|{{A}_{2;n+1,m}}\rangle \right. \\ \left. \quad \ \;+\,w_{nm}^{\prime \prime}|{{A}_{2;n+1,m-1}}\rangle +w_{nm}^{\prime \prime \prime}|{{A}_{2;n,m-1}}\rangle \right) \\ \left. \quad \ \;\times \,\langle {{A}_{1;n,m}}|\right]+\text{h}.\text{c}. \end{array}\end{eqnarray} \tag{ B.1 }$$

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For the infinitely periodic system with u_nm  =  u etc, we can seek solutions in the form of a Bloch wave

$$\begin{eqnarray}&&|\psi \left(\mathbf{k}\right)\rangle =\underset{X={{A}_{1}},{{A}_{2}},B}{\sum}\,\underset{nm}{\sum}\,{{\text{e}}^{\text{i}\mathbf{k}\centerdot (n,m)}}{{\varphi}_{X}}\left(\mathbf{k}\right)|{{X}_{n,m}}\rangle,\end{eqnarray} \tag{ B.2 }$$

which results in the Bloch eigenvalue problem $E\varphi =h\varphi$,

$$\begin{eqnarray}\begin{array}{*{35}{l}} h\left(\mathbf{k}\right) & = & U\left({{k}_{x}}\right)|{{A}_{1}}\rangle \langle B|+V\left({{k}_{y}}\right)|{{A}_{2}}\rangle \langle B|+W\left(\mathbf{k}\right)|{{A}_{1}}\rangle \langle {{A}_{2}}|+\text{h}.\text{c}., \\ |\varphi \left(\mathbf{k}\right)\rangle & = & {{\varphi}_{{{A}_{1}}}}\left(\mathbf{k}\right)|{{A}_{1}}\rangle +{{\varphi}_{{{A}_{2}}}}\left(\mathbf{k}\right)|{{A}_{2}}\rangle +{{\varphi}_{B}}\left(\mathbf{k}\right)|B\rangle, \\ U\left({{k}_{x}}\right) & = & u+{{u}^{\prime}}{{\text{e}}^{\text{i}{{k}_{x}}}},\quad V\left({{k}_{y}}\right)=v+{{v}^{\prime}}{{\text{e}}^{\text{i}{{k}_{y}}}}, \\ W\left(\mathbf{k}\right) & = & w+{{w}^{\prime}}{{\text{e}}^{\text{i}{{k}_{x}}}}+{{w}^{\prime \prime}}{{\text{e}}^{\text{i}{{k}_{x}}-i{{k}_{y}}}}+{{w}^{\prime \prime \prime}}{{\text{e}}^{-\text{i}{{k}_{y}}}}, \end{array}\end{eqnarray} \tag{ B.3 }$$

where we set the lattice constant $a\equiv 1$.

For the well-studied case of identical nearest-neighbour couplings $u={{u}^{\prime}}=v={{v}^{\prime}}$ and w⁽ⁱ⁾  =  0, the corresponding band structure consists of a flat band of states $|{{\varphi}_{0}}\left(\mathbf{k}\right)\rangle \propto V\left(-{{k}_{y}}\right)|{{A}_{1}}\rangle -U\left(-{{k}_{x}}\right)|{{A}_{2}}\rangle$ localized on the A sublattice, and a conical dispersion relation

$$\begin{eqnarray}&&{{E}^{2}}=\,|U\left({{k}_{x}}\right){{|}^{2}}\,+\,|V\left({{k}_{y}}\right){{|}^{2}}\end{eqnarray} \tag{ B.4 }$$

of extended states, with a Dirac point at the M point ${{\mathbf{K}}_{0}}=\left(\pi,\pi \right)$ in the corner of the Brillouin zone. The two degenerate extended states at the Dirac point can be combined into a sublattice-polarized states $|{{\varphi}_{+}}\rangle$, localized on the A sublattice, and $|{{\varphi}_{-}}\rangle =\,|B\rangle$, where the latter corresponds to a real-space wavefunction with $|{{\psi}_{-}}\rangle ={\sum}_{nm}{{\left(-1\right)}^{nm}}|{{B}_{n,m}}\rangle$. This state is compatible with the boundary conditions in finite systems terminated on the B sublattice, as used in the main text and also shown in figure B1.

To see how this state becomes confined by a dimerization pattern (alternating couplings u, ${{u}^{\prime}}$ along the x direction and v, ${{v}^{\prime}}$ along the y direction), we first confirm from equation (B.4) that this opens a gap $2 \Delta$ for the extended states, with ${{ \Delta }^{2}}={{E}^{2}}\left({{\mathbf{K}}_{0}}\right)={{\left(u-{{u}^{\prime}}\right)}^{2}}+{{\left(v-{{v}^{\prime}}\right)}^{2}}$. The state $|{{\varphi}_{-}}\rangle$ still survives as a state with a complex wave number ${{\mathbf{k}}_{\pm }}={{\mathbf{K}}_{0}}\pm i\left(\ln u/{{u}^{\prime}},\ln v/{{v}^{\prime}}\right)$, according to the analytical continuation of the Dirac point into the complex plane. The corresponding real-space wavefunction

$$\begin{eqnarray}&&|{{\psi}_{-}}\rangle =\underset{nm}{\sum}\,{{\left(-u/{{u}^{\prime}}\right)}^{n}}{{\left(-v/{{v}^{\prime}}\right)}^{m}}|{{B}_{n,m}}\rangle \end{eqnarray} \tag{ B.5 }$$

still vanishes on the A sublattice, while on the B sublattice it now has an exponentially varying envelope. In a rectangular finite system, this state remains compatible with the boundary conditions if the system terminates on the B sublattice, and then localizes around a corner of the system. We choose an orientation so that this is the top right corner (this corresponds to $u>{{u}^{\prime}}>0$, $v>{{v}^{\prime}}>0$ and can always be achieved by a rotation of the system and redefinition of the unit cell), as indicated in figure B1(a).

To confirm that the state $|{{\psi}_{-}}\rangle$ can be moved along the edges and into the bulk, we consider line defects separating regions where the role of u and ${{u}^{\prime}}$ or v and ${{v}^{\prime}}$ are interchanged, see figures B1(b) and (c). Along a vertical line defect, we encounter A sites that are coupled by ${{u}^{\prime}}$ both to the right and to the left, while along a horizontal line defect they are coupled by ${{v}^{\prime}}$ both to the bottom and to the top. We denote by n  =  m  =  0 the crossing point of two such line defects, or the point where a single line defect meets the upper or right boundary. This results in a point-defect state with real-space wavefunction

$$\begin{eqnarray}&&|{{\psi}_{-}}\rangle =\underset{nm}{\sum}\,{{\left(-u/{{u}^{\prime}}\right)}^{|n|}}{{\left(-v/{{v}^{\prime}}\right)}^{|m|}}|{{B}_{n,m}}\rangle,\end{eqnarray} \tag{ B.6 }$$

which now is exponentially localized into all directions. Arbitrary next-to-nearest neighbour couplings w⁽ⁱ⁾ between A₁ and A₂ sites break the symmetry of the extended bands and introduce a dispersion to the flat band, which moves away to finite energies, as illustrated in figure 1 of the main text. However, as a consequence of the partial chiral symmetry, the energy and wavefunction of the point-defect state (B.6) remains unchanged.

These statements are verified by the numerical results in the middle and bottom row of figure B1. In each case, we find a spectrally isolated zero mode that is localized on the B sublattice, and exhibits the expected spatial profile (B.6).

By construction, the point-defect state (B.6) is insensitive to any disorder in the couplings $w_{nm}^{(i)}$. From the condition ${{T}_{AB}}{{\psi}_{B}}=0$, we can further verify that disorder in the couplings ${{u}_{nm}},u_{nm}^{\prime},{{v}_{nm}},v_{nm}^{\prime}$ does not affect the energy or sublattice polarization of the state as long as each plaquette fulfills the constraint (6). Compatible types of disorder include quasi-one-dimensional disorder in which the hoppings ${{u}_{nm}},u_{nm}^{\prime}$ only depend on the unit-cell index n while the hoppings ${{v}_{nm}},v_{nm}^{\prime}$ only depend on the unit-cell index m. As described in the main text, the constraint also allows for much richer disorder configurations, two of which are illustrated in figures B2(a) and (b), while figure B2(c) shows a case of generic disorder.

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While we focussed on the implementation of the point-defect state in a photonic setting, we remark that is also interesting to consider this system in an electronic context. In the ground state the single-particle states are then occupied according to the Pauli principle up to a Fermi energy E_F. From a simple counting argument [34], we then find that a uniform charge density at finite filling requires a half-population of the point-defect state. At Fermi energy ${{E}_{\text{F}}}={{0}^{\pm }}$, the state thus serves as a fractional charge  ±1/2 against a uniform background density on the minority sublattice. This resembles the situation for a Z₄ vortex in a dimerized square lattice (which does not display a flat band) [34, 35].

An important paradigm of a dimerized system with topologically protected point-defect zero modes is the Su–Schrieffer Heeger (SSH) model [32], which consists of a one-dimensional tight-binding chain with alternating couplings

$$\begin{eqnarray}&&H=\underset{n}{\sum}\,\left(u|{{A}_{n}}\rangle +{{u}^{\prime}}|{{A}_{n-1}}\rangle \right)\langle {{B}_{n}}|+\text{h}.\text{c}.\end{eqnarray} \tag{ B.7 }$$

Here we give an interpretation of the dimerized Lieb lattice as a two-dimensional generalization of the SSH model. This connection is most usefully established in the continuum limit, which sheds further light on the topological features of the system.

For the SSH model, the continuum limit is obtained by a gauge transformation $|A\rangle \to {{\text{e}}^{\text{i}{{k}_{x}}/2}}|A\rangle$, followed by a gradient expansion in ${{k}_{x}}\to \pi -\text{i}{{\partial}_{x}}$. This results in a massive Dirac Hamiltonian, the Jackiw–Rebbi model [31] with

$$\begin{eqnarray}&&E\psi =H\psi,\quad H=\text{i}{{v}_{0}}{{\sigma}_{x}}{{\partial}_{x}}-m{{\sigma}_{y}},\quad \psi =\left(\begin{array}{c} {{\psi}_{A}}(x) \\ {{\psi}_{B}}(x) \end{array}\right),\end{eqnarray} \tag{ B.8 }$$

where $m={{u}^{\prime}}-u$ and ${{v}_{0}}=\left(u+{{u}^{\prime}}\right)/2$. For constant mass m, the dispersion relation is given by ${{E}^{2}}={{m}^{2}}+v_{0}^{2}k_{x}^{2}$. When two regions of opposite mass are joined together, $m(x)=M\,\text{sgn}\,x$ with M  >  0, one finds an exponentially localized defect mode of the form ${{\psi}_{A}}(x)=0$, ${{\psi}_{B}}(x)\propto \exp \left(-|x|M/{{v}_{0}}\right)$.

For the dimerized Lieb lattice, we obtain the continuum limit from the Bloch Hamiltonian (B.3) by performing the gauge transformation $|{{A}_{1}}\rangle \to {{\text{e}}^{\text{i}{{k}_{x}}/2}}|{{A}_{1}}\rangle$, $|{{A}_{2}}\rangle \to {{\text{e}}^{\text{i}{{k}_{y}}/2}}|{{A}_{2}}\rangle$ and expanding around $\mathbf{k}\to {{\mathbf{K}}_{0}}-\text{i}\nabla$. We then obtain a massive Dirac Hamiltonian

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \mathcal{H}={{\mathcal{H}}_{X}}+{{\mathcal{H}}_{Y}}+{{\mathcal{H}}_{XY}}, \end{array}\end{eqnarray} \tag{ B.9 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\mathcal{H}}_{X}}=\text{i}{{v}_{X}}{{\lambda}_{4}}{{\partial}_{x}}-{{m}_{X}}{{\lambda}_{5}}, \end{array}\end{eqnarray} \tag{ B.10 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\mathcal{H}}_{Y}}=\text{i}{{v}_{Y}}{{\lambda}_{6}}{{\partial}_{y}}-{{m}_{Y}}{{\lambda}_{7}}, \end{array}\end{eqnarray} \tag{ B.11 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\mathcal{H}}_{XY}}={{m}_{XY}}{{\lambda}_{1}}+\text{i}{{\lambda}_{2}}\left({{\tilde{{v}}\,}_{X}}{{\partial}_{x}}+{{\tilde{{v}}\,}_{Y}}{{\partial}_{y}}\right), \end{array}\end{eqnarray} \tag{ B.12 }$$

where ${{\lambda}_{i}}$ are the Gell–Mann matrices. The effective velocities are ${{v}_{X}}=\left(u+{{u}^{\prime}}\right)/2$ and ${{v}_{Y}}=\left(v+{{v}^{\prime}}\right)/2$, as well as ${{\tilde{v}}_{X}}=\left(-w-{{w}^{\prime}}+{{w}^{\prime \prime}}+{{w}^{\prime \prime \prime}}\right)/2$ and ${{\tilde{v}}_{Y}}=\left(w-{{w}^{\prime}}-{{w}^{\prime \prime}}+{{w}^{\prime \prime \prime}}\right)/2$, while the masses are given by ${{m}_{X}}={{u}^{\prime}}-u$, ${{m}_{Y}}={{v}^{\prime}}-v$, ${{m}_{XY}}=$ $w-{{w}^{\prime}}+{{w}^{\prime \prime}}-{{w}^{\prime \prime \prime}}$.

This model recovers the low-energy part of the band structure. In absence of the couplings w⁽ⁱ⁾, we obtain a flat band of zero modes on the majority sublattice, and dispersive bands with ${{E}^{2}}=m_{X}^{2}+m_{Y}^{2}+v_{X}^{2}k_{x}^{2}+v_{Y}^{2}k_{Y}^{2}$. For the system with a vertical dimerization line-defect as in figure B1(b), ${{m}_{X}}(x)={{M}_{X}}\,\text{sgn}\,x$ with ${{M}_{X}}=u-{{u}^{\prime}}>0$, so that the sign of the mass changes at the interface. This creates a conduction channel within the gap of extended states, described by exponentially confined modes

$$\begin{eqnarray}\begin{array}{*{35}{l}} |\psi \rangle & \propto {{\text{e}}^{-\frac{{{M}_{X}}}{{{v}_{X}}}|x|}}{{\text{e}}^{-\text{i}{{k}_{y}}y}} \\ {} & \times \left[|B(x,y)\rangle +\frac{\text{i}{{m}_{Y}}-{{v}_{Y}}{{k}_{y}}}{E}|{{A}_{2}}(x,y)\rangle \right], \end{array}\end{eqnarray} \tag{ B.13 }$$

which form two symmetric bands with ${{E}^{2}}=m_{Y}^{2}+v_{Y}^{2}k_{y}^{2}$. The channel is thus described by a Jackiw–Rebbi model for the transport along y.

Within this channel, an additional line defect parallel to the x axis should therefore produce a point-defect zero-mode that decays exponentially in all directions. We can verify this directly. For a central dimerization defect as in figure B1(c), the effective masses are given by ${{m}_{X}}(x)={{M}_{X}}\,\text{sgn}\,x$ and ${{m}_{Y}}\left(\,y\right)={{M}_{Y}}\,\text{sgn}\,y$, with ${{M}_{X}}=u-{{u}^{\prime}}>0$ and ${{M}_{Y}}=v-{{v}^{\prime}}>0$. While $\left[{{\mathcal{H}}_{X}},{{\mathcal{H}}_{Y}}\right]\ne 0$, we have ${{\left(\left[{{\mathcal{H}}_{X}},{{\mathcal{H}}_{Y}}\right]\right)}_{BB}}=0$, so that the point-defect zero mode on the minority sublattice can be obtained by separation of variables,

$$\begin{eqnarray}&&|{{\psi}_{-}}(x,y)\rangle \propto {{\text{e}}^{-\frac{{{M}_{X}}}{{{v}_{X}}}|x|}}{{\text{e}}^{-\frac{{{M}_{Y}}}{{{v}_{Y}}}|\,y|}}|B(x,y)\rangle.\end{eqnarray} \tag{ B.14 }$$

This indeed corresponds to the continuum limit of the tight-binding solution (B.6). We can again verify directly that this state is not affected by the symmetry-breaking terms H_XY.

Thus, within the continuum limit the point-defect state can be associated with a nontrivial background mass pattern in the system. Note that when this mass pattern is smoothed out, the mass-gap order parameter ${{m}_{X}}\left(\mathbf{r}\right)+i{{m}_{Y}}\left(\mathbf{r}\right)$ has a non-trivial topology, with a finite winding number as one encircles the origin along a closed loop. This observation establishes an additional connection to systems with charge fractionalization [33].

We conclude by identifying the continuum interpretation of the constraint (6). In the tight-binding model, the position of the virtual Dirac point can be written as ${{\mathbf{k}}_{\pm }}={{\mathbf{K}}_{0}}\pm \text{i}\mathcal{A}$, where the displacement $\mathcal{A}=\left[\ln \left(u/{{u}^{\prime}}\right),\ln \left(v/{{v}^{\prime}}\right)\right]$ can be interpreted as an imaginary pseudo vector potential (in analogy to the description of strained graphene). We then can rewrite equation (6) as

$$\begin{eqnarray}&&{{\mathcal{A}}_{y;n+1,m}}-{{\mathcal{A}}_{y;n,m}}={{\mathcal{A}}_{x;n,m+1}}-{{\mathcal{A}}_{x;n,m}}\to {{\partial}_{x}}{{\mathcal{A}}_{y}}={{\partial}_{y}}{{\mathcal{A}}_{x}},\end{eqnarray} \tag{ B.15 }$$

therefore $\nabla \times \mathcal{A}=0$. Thus, the constraint on the disorder can be naturally interpreted as the condition of a vanishing deformation-induced pseudomagnetic field.