Higher-Order Topological Systems and Their Sub-Symmetry-Protected Topology

TL;DR

The paper addresses how sub-symmetry-protected topology extends to higher-order topological phases and gapless semimetals. It introduces a sub-symmetry-protecting perturbation acting on a single sublattice and analyzes the BBH second-order topological insulator and a stacked BBH-based second-order topological semimetal, using Jackiw–Rebbi boundary theory and state-resolved quadrupole moments as diagnostic tools. Key findings include three zero-energy corner states remaining protected on sublattices $2$, $3$, and $4$ with quantized $q_{xy}^i$, while the corner on sublattice $1$ loses zero energy and quantization; in the SOTSM, one second-order Fermi arc becomes dispersive and non-quantized while the others stay protected with quantized quadrupole moments. The work establishes sub-symmetry as a design principle for symmetry-resilient higher-order boundary states and suggests experimental platforms such as topolectrical circuits and photonic lattices for realization and testing.

Abstract

Symmetry and topology are essential principles in topological physics. Recently, the idea of sub-symmetry-protected topology -- where some of the original symmetries are broken while a remaining subset, called sub-symmetries, continues to protect specific boundary states -- has been developed. Here, we extend sub-symmetry-protected topology to higher-order topological systems from second-order topological insulators to semimetals. By introducing a sub-symmetry-protecting perturbation that acts on a single sublattice and selectively preserves specific topological boundary states, we track the evolution of these states and their topological features using numerical and analytical methods, and we show that state-resolved quadrupole moments diagnose which corner or hinge modes remain topological. As a representative example of a second-order topological insulator, we begin with the Benalcazar-Bernevig-Hughes model. We demonstrate that, under a sub-symmetry-protecting perturbation, sub-symmetry-protected corner states remain pinned at zero energy and maintain quantized state-resolved quadrupole moments. In contrast, corner states on sub-symmetry-broken boundaries shift away from zero energy and lose their quantized character. We further extend this framework to a three-dimensional second-order topological semimetal, constructed by stacking second-order topological insulator layers, and analyze how second-order Fermi arc states -- hinge-localized modes that link the projections of bulk Dirac points, in contrast to conventional surface Fermi arcs -- evolve under a sub-symmetry-protecting perturbation. While one second-order Fermi arc becomes dispersive and loses its quadrupolar character under a sub-symmetry-breaking perturbation, the remaining second-order Fermi arcs retain chiral symmetry and preserve quantized quadrupolar characters.

Figures (5)

• Figure 1: Schematics of the topological systems and their boundary states under sub-symmetry-protecting perturbation. Left (right) panels show the cases without (with) sub-symmetry-protecting perturbations. (a) One-dimensional topological insulator. Edge states with (without) topological charge are shown as filled (open) red circles. (b) Two-dimensional SOTI with corner states. Corner states protected (unprotected) by sub-symmetry exhibit quantized (non-quantized) quadrupole moments, shown by the filled (open) red circles. (c) Three-dimensional SOTSM and the corresponding second-order Fermi arc states. Green circles represent the surface projections of bulk Dirac points, connected by second-order Fermi arc states (red lines). The topological character of each second-order Fermi arc is distinguished by the quadrupole moment along $k_z$, as shown in the inset. The second-order Fermi arc with a non-quantized quadrupole moment is represented by a dotted red line, with its non-quantized quadrupole value also indicated in the inset.
• Figure 2: Sub-symmetry-protected corner states in the BBH model.(a) Schematics of the BBH model. Each square unit cell contains four sublattice sites labeled 1 through 4. Black (intercell, $\lambda$) and red (intracell, $\gamma$) lines indicate hopping amplitudes; dotted lines denote negative amplitude. (b) Energy spectrum of finite square geometry. Three zero-energy sub-symmetry-protected states appear (blue), along with one finite-energy trivial state (red). The inset illustrates the system's geometry, which is a square. The parameters are $\gamma=-0.5, \lambda=1$, and $m_1=0.2$. The system is finite in the $xy$ plane ($20 \times 20$ unit cells). The total number of states is $2n=1600$. (c,d) Spatial distributions of the wavefunctions corresponding to the sub-symmetry-protected (c, blue) and the trivial (d, red) corner states. (e) Energy spectrum for a modified system with corner-cut geometry. The inset shows the geometry, with dashed lines indicating the regions that have been removed. Each removed region is comprised of a total of six unit cells. The sub-symmetry-protected states (blue) remain at zero energy, while the trivial state (red) is lifted. The parameters are identical to panel (b), and the total number of states is $2n=1504$. (f,g) Wavefunction distributions of the sub-symmetry-protected (f, blue) and the trivial (g, red) in-gap states in the corner-cut geometry. The sub-symmetry-protected states retain their localization, demonstrating robustness against geometric deformation and confirming their geometry-independent topological origin.
• Figure 3: Charge density distribution of the occupied states and quadrupole moment with respect to the sub-symmetry-protecting perturbation.(a,b) Charge density distribution ($\rho$) for the occupied states (a) without and (b) with the sub-symmetry-protecting perturbation ($m_1=0.2$). In (a), the bulk has a uniform charge, and the corner states have an identical absolute charge distribution. For (b), the bulk charge distribution is not uniform and will result in a non-zero bulk quadrupole moment. (c) State-resolved quadrupole moments of the bulk and corner states as a function of $m_1$. The black line indicates the bulk quadrupole moment $q_{xy}^\text{bulk}$. The red and blue lines correspond to the absolute value of $q_{xy}^i$ for the trivial and sub-symmetry-protected corner states, respectively. The parameters excluding $m_1$ and geometry are the same as Fig. \ref{['fig1:2DQI']}(b).
• Figure 4: Modification of the second-order Fermi arcs in a SOTSM under sub-symmetry-protecting perturbation.(a) Three-dimensional Brillouin zone with labeled time-reversal-invariant momenta (TRIM) and the bulk band structure for parameters $\gamma = -0.5$, $\lambda = 1$, and $\lambda_z = 1$. Dirac cones appear along the $\Gamma Z$ high-symmetry line. (b) (Left) Band structure in rod geometry, with a system finite in the $xy$ plane ($20 \times 20$ unit cells) and periodic along $z$, without sub-symmetry-protecting perturbation ($m_1 = 0$). Band dispersion shows fourfold-degenerate flat-band second-order Fermi arcs at zero energy. (Right) Energy spectrum at $k_z = 0$, highlighting four symmetry-protected hinge-localized zero-energy second-order Fermi arc states (green). The inset shows their corresponding spatially localized wavefunctions. The total number of states at $k_z = 0$ is $2n = 1600$. (c) State-resolved quadrupole moments $q_{xy}$ of the hinge states as a function of $k_z$. All second-order Fermi arc states exhibit quantized values corresponding to a fractional charge of $e/4$. (d) Same as panel (b), but with a finite sub-symmetry-protecting perturbation ($m_1 = 0.2$). One hinge-localized second-order Fermi arc state (red) is shifted away from zero energy due to the perturbation, while the other three (green) remain protected. Insets show the spatial wavefunction distributions of the perturbed and unperturbed second-order Fermi arc states. (e) State-resolved quadrupole moments $q_{xy}$ of the hinge states as a function of $k_z$ under perturbation. The sub-symmetry-protected second-order Fermi arcs (blue) retain quantized quadrupole values ($e/4$), while the perturbed second-order Fermi arc state (red) becomes non-quantized, indicating topological degradation under sub-symmetry-breaking perturbation.
• Figure 5: Edge symmetry classification, sublattice basis, and corner Dirac mass structure.(a) Without sub-symmetry-protecting perturbation, each edge belongs to the BDI class, with its Hamiltonian defined in the two-component sublattice basis. All four corners host zero Dirac mass (filled circles). (b) Under the sub-symmetry-protecting perturbation, the edges involving sublattice-1 change their classification from BDI to AI. Consequently, only the corresponding corners acquire a finite Dirac mass (open circle), while the other corners remain massless and topologically protected (filled circles).