Topological Protection by Local Support Symmetry and Destructive Interference

TL;DR

This work introduces local support symmetry (LSS), a framework in which a symmetry acts only on a subregion of a system, yet protected topological features persist in the full system when the inter-part coupling satisfies a specific compatibility condition. Central to the mechanism is destructive interference, which enforces compactly supported Bloch states confined to the symmetry-compatible region and preserves topology despite symmetry breaking elsewhere. The authors develop general insulating and semimetal theories under LSS, then demonstrate through several tight-binding models (Model-I: TRS-protected insulator; Model-II: $\text{C}_2$-protected Dirac fermions; Model-III: nonsymmorphic Dirac fermions) and a realistic fluorinated biphenylene network that local protection can yield robust band crossings and nontrivial topology. They show that even when global symmetry is broken (e.g., by fluorination), local symmetry on a subregion can maintain Dirac nodes with vanishingly small gaps, highlighting the practical relevance for material realizations and extensions to three-dimensional nodal features. The work connects LSS protection to sub-symmetry concepts and provides a quantitative basis for assessing robustness against perturbations, with implications for designing materials where partial symmetries stabilize desired topological properties.

Abstract

Conventionally, symmetry-protected topological phases and band crossings are protected by global symmetries acting on the entire system. Here, we show that symmetries preserved only on a partial region of a system, termed local support symmetries, can protect topological features of the full system, even in the presence of symmetry-breaking couplings. We establish a unified framework by deriving explicit conditions for such protection in both insulating and metallic phases and show that destructive interference of Bloch wave functions plays a key role. Using representative tight-binding models, we demonstrate band crossings and topological bands protected by local support crystalline and time-reversal symmetries, and further present a realistic material realization in a fluorinated biphenylene network, where a band crossing is protected by a local support C$_2$ symmetry.

Figures (5)

• Figure 1: Concept of local support symmetry and symmetry-protected band features. Conventionally, SPTs and band-crossings are protected by a global symmetry that uniformly acts on the entire system. For example, the orange object preserves a C$_2$ symmetry about the dashed line throughout the whole region. In the case of the local support symmetry (LSS), the symmetry operation is applied to the part $\mathcal{S}_1$ (blue region) of the system while the other part $\mathcal{S}_2$ (white region) is unaffected. With the aid of destructive interference, which can prevent the propagation of a Bloch wave function into $\mathcal{S}_2$ from $\mathcal{S}_1$, the LSS in $\mathcal{S}_1$ can still protect band-crossings or band topology.
• Figure 2: Model-I and local support time-reversal symmetry. (a) The lattice and hopping structure of the modified Lieb lattice model. Four sublattices in a unit cell(gray box) are labeled by numbers in the sites. Electrons can hop along the solid and dashed lines, and the corresponding hopping parameters are given on the right-hand side. For the imaginary hopping processes, the direction of the hopping is represented by the arrows. $\alpha$ is the strength of the spin-orbit coupling, and spin-up and -down are denoted by $\sigma=1$ and $-1$, respectively. (b) The band structure for $\alpha=0.5$, $e_{\uparrow}=3$, $e_{\downarrow}=2$, and $t=0.3$. Two highly dispersive bands indicated by the arrows are doubly degenerate. (c) The band structure of the ribbon geometry with 100 unit cells along the $x$-direction. The band-crossing enforced by S$_z$ symmetry, slightly away from $k=0$, is indicated by the red arrow. The edge bands in the two yellow dashed boxes are highlighted in (d) and (e).
• Figure 3: Model-II and local support $C_2$ symmetry. (a) The $\mathcal{S}_1$ part of Model-II. The hopping amplitudes between A and C sites are one, while those from B sites to A or C ones are $t_1$. We assume that the onsite energies at A, B, and C sites are identically zero. Therefore, the system preserves C$_2$ symmetry about the axis indicated by a grey dashed line. The yellow box is the unit cell. (b) The entire lattice and hopping structures of Model-II. D sites consist of the $\mathcal{S}_2$ part. The onsite energy at D sites is also zero. The hopping processes from D to B sites are denoted by $t_2$ and $t_3$. (c) The band dispersion for $\{t_1,t_2,t_3\} = \{1.2,2,0.5\}$. A Dirac point along $\Gamma\bar{\mathrm{S}}$, where $\bar{\mathrm{S}}$ represents $k=(-\pi,\pi)$, is denoted by DP. Three Bloch states ($\psi_1$ and $\psi_2$) corresponding to red dots are plotted in (d) and (e). Red and blue circles indicate the positive and negative signs of the wave function, respectively, with their sizes representing its magnitude. $\psi_1$ spans only the sites belonging to $\mathcal{S}_1$ due to the destructive interference indicated by red and blue curved arrows in (d). (f) The fragility of the Dirac band-crossing against the additional hopping processes described by $t_4$ and $t_4-\delta t$.
• Figure 4: Model-III and local support nonsymmorphic symmetry. (a) The lattice and hopping structures of the herringbone lattice model. In the unit cell, indicated by a yellow box, there are six sublattice sites denoted by A$_1$, B$_1$, C$_1$, and D$_1$ belonging to the $\mathcal{S}_1$ part and A$_2$ and B$_2$ belonging to the $\mathcal{S}_2$ part. Hopping processes with amplitudes $t_1$, $t_2$, and $t_3$ are expressed by black and red lines. The blue line represents the screw axis considered in our model. (b) The band structure along $k_y$-axis ($k_x=0$) for the band parameters $\{t_1,t_2,t_3\}=\{1,1,-1\}$. There are doubly degenerate flat bands at zero energy. The eigenvalues of the screw operation $\{\mathrm{C}_{2y}|\frac{1}{2}\hat{y}\}$ for each band are represented by the complex numbers with the same color as the corresponding band. (c) The fragility of the band-crossing at $k_y=\pi$ in (b), as a function of the additional hopping processes represented by $t_4$.
• Figure 5: Fluorinated biphenylene network. (a) A unit cell (green box) of the biphenylene network passivated by two fluorine atoms. Grey(sky blue) spheres represent carbon(fluorine) atoms. Hopping processes are illustrated by red arrows. The unit cell is divided into two parts, $\mathcal{S}_1$ and $\mathcal{S}_2$. The carbon sites in $\mathcal{S}_1$ and $\mathcal{S}_2$ are indicated by A$_i$ and U$_i$, respectively. (b) The DFT band structure. Along $\Gamma$Y, two massive type-II Dirac fermions appear at distinct energy levels, labeled as MDP$_i$. The MDP$_1$ feature in the DFT band structure is highlighted in the right-hand panel. $k_1$ is an arbitrary momentum near the MDP$_1$. (c) The tight-binding(TB) band dispersion calculated using the hopping parameters $\{t_1, t_2, t_3, t_4, t_5, t_6, t_7, t_8\} = \{-3.4, -3.1, -3.1, -3.2, -0.8, -0.06, 0.1725, 0.17\}$. The MDP$_1$ of the tight-binding result is enlarged in the left-hand panel, where $k_2$ is an arbitrary momentum near the MDP$_2$. (d) The tight-binding band spectrum computed by suppressing the perturbative hopping terms associated with $t_6$, $t_7$, and $t_8$. Several Dirac nodes (DP$_i$'s) are marked with blue dots. (b) to (d) share the same $y$-axis. From (e) to (g), we plot Bloch wave functions $\Psi_i$'s indicated by red dots in (d). Red and blue circles denote the positive and negative signs of the wave function, respectively, while the circle size reflects the magnitude of the wave function. The grey dashed vertical lines are the local support mirror symmetry axes. Destructive interferences are described by curved arrows.