Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators

TL;DR

This work introduces topological crystalline markers (TCMs), a position-space formalism built from projected rotation operators and the ground-state projector, to diagnose bulk crystalline topology in $C_n$-symmetric insulators and superconductors. By expressing bulk invariants such as the Chern number modulo $n$, bulk polarization, and sector charge in terms of fully traced and partially traced TCMs, the authors provide a basis-independent, local diagnostic that remains valid even when translation symmetry is broken or when finite-size lattices miss high-symmetry points. The paper develops explicit mappings between TCMs and momentum-space symmetry data, including rotation invariants and $\,\Gamma$-point irrep multiplicities, and extends the framework to twisted boundary conditions to handle arbitrary lattice sizes. It further demonstrates the practical use of TCM densities and meshes to spatially resolve bulk invariants and to analyze domain-wall configurations in inhomogeneous systems, with concrete 1D and 2D model examples. The approach holds potential for extensions to interacting, amorphous, and quasicrystalline systems, and provides a versatile tool for diagnosing crystalline topology beyond conventional momentum-space indicators.

Abstract

Crystalline symmetry can be used to predict bulk and surface properties of topological phases. For non-interacting cases, symmetry-eigenvalue analysis of Bloch states at high symmetry points in the Brillouin zone simplifies the calculation of topological quantities. However, when open boundaries are present, and only the point group part of the symmetry group remains, it is unclear how to utilize crystalline symmetries to diagnose band topology. In this work, we introduce topological crystalline markers to characterize bulk topology in $C_n$-symmetric ($n=2,3,4,6$) crystalline insulators and superconductors with and without translation symmetry. These markers are expressed using a crystalline symmetry operator and the ground state projector, and are defined locally in position space. First, we provide a general method to calculate topological markers in periodic systems with an arbitrary number of unit cells. This includes cases where momentum quantization does not span all necessary high-symmetry points for computing the topological quantities, which we address using twisted boundary conditions. Second, we map these markers to the Chern number, bulk polarization, and sector charge for two-dimensional $C_n$-symmetric insulators in symmetry classes A, AI, AII, and superconductors in class D. Finally, we show how to numerically calculate the markers in finite-size systems with translation-symmetry (and even rotation-symmetry) breaking defects, and how to diagnose the bulk topology from the marker. Our results demonstrate how to compute bulk topological crystalline invariants locally in position space, thereby providing broader scope to diagnosing bulk crystalline topology that works even in inhomogeneous systems where there is no global rotation symmetry.

Figures (9)

• Figure 1: Illustration of (a) $C_2$-symmetric, (b) $C_4$-symmetric, (c) $C_3$-symmetric, and (d) $C_6$-symmetric lattices and the corresponding sets of invariant positions $\mathcal{X}[c_n(\mathbf{r}_0)]$ given by Eq. \ref{['eq:invariant_positions']} for lattices with dimensions $N_1 \times N_2$ specified by Eq. \ref{['eq:perfect_constraint']}. The set of invariant positions $\mathcal{X}[c_n(\mathbf{r}_0)]$ for each $C_n$-symmetric lattice is determined with respect to the unit cell at $\mathbf{R} = \mathbf{0}$ located at the lower left corner for the $C_2$, $C_3$, and $C_4$ lattices shown in (a)-(c) respectively, while the unit cell at $\mathbf{R} = \mathbf{0}$ is located at the center of the $C_6$-symmetric lattice in (d). For each lattice depicted in (a)-(d), the invariant positions correspond to WPs of multiplicity $1$, and when the dimensions $N_1 \times N_2$ satisfy Eq. \ref{['eq:perfect_constraint']}, the set $\mathcal{X}[c_n(\mathbf{r}_0)]$ is formed from Wyckoff positions of the same symbol (i.e., same shape and color) as per Eq. \ref{['eq:invariant_positions']}. The different choices of $\mathbf{r}_0$ correspond to different symbols.
• Figure 2: Illustration of a unit cell for each $C_n$-symmetric lattice and the corresponding Wyckoff positions (WPs). (a) $C_2$ symmetry. The WPs $1a,1b,1c,1d$ are located at $\mathbf{x}_{1a} = \mathbf{0}$, $\mathbf{x}_{1b} = \frac{1}{2} \mathbf{a}_1$, $\mathbf{x}_{1c} = \frac{1}{2} \mathbf{a}_2$, and $\mathbf{x}_{1d} = \frac{1}{2} (\mathbf{a}_1 + \mathbf{a}_2)$, respectively. (b) $C_4$ symmetry. The WPs $1a$ and $1b$ are located at $\mathbf{x}_{1a} = \mathbf{0}$ and $\mathbf{x}_{1b} = \frac{1}{2}(\mathbf{a}_1 + \mathbf{a}_2)$, respectively. The WP $2c$ is composed of two $C_4$-related positions, $\mathbf{x}_{2c,1} = \frac{1}{2} \mathbf{a}_1$ and $\mathbf{x}_{2c,2} = \frac{1}{2} \mathbf{a}_2$. (c) $C_3$ symmetry. The WPs $1a,1b,1c$ are located at $\mathbf{x}_{1a} = \mathbf{0}$, $\mathbf{x}_{1b} = \frac{1}{3} (\mathbf{a}_1 + \mathbf{a}_2)$, and $\mathbf{x}_{1c} = \frac{1}{3} (-\mathbf{a}_1 + 2 \mathbf{a}_2)$, respectively. (d) $C_6$ symmetry. The WP $1a$ is located $\mathbf{x}_{1a} = \mathbf{0}$. The WP $2b$ is composed of $\mathbf{x}_{2b,1} = \frac{1}{3} (\mathbf{a}_1 + \mathbf{a}_2)$, $\mathbf{x}_{2b,2} = \frac{1}{3} (-\mathbf{a}_1 + 2 \mathbf{a}_2)$. The WP $3c$ is composed of $\mathbf{x}_{3c,1} = \frac{1}{2} \mathbf{a}_1$, $\mathbf{x}_{3c,2} = \frac{1}{2} \mathbf{a}_2$, and $\mathbf{x}_{3c,3} = \frac{1}{2} (- \mathbf{a}_1 + \mathbf{a}_2)$. The primitive lattice vectors $\mathbf{a}_{1,2}$ are set as $\mathbf{a}_1 = (1,0)$ and $\mathbf{a}_2 = (0,1)$ for $C_{2,4}$, and $\mathbf{a}_1 = (1,0)$ and $\mathbf{a}_2 = (\tfrac{1}{2}, \tfrac{\sqrt{3}}{2})$ for $C_{3,6}$.
• Figure 3: Brillouin zones and their high symmetry momenta for the corresponding $C_n$-symmetric lattices shown in Fig. \ref{['fig:invariant_positions']} with dimensions $N_1 \times N_2$ given by Eq. \ref{['eq:perfect_constraint']}. (a) $C_2$-symmetric BZ (b) $C_4$-symmetric BZ (c) $C_3$-symmetric BZ, and (d) $C_6$-symmetric BZ. The reciprocal lattice vectors $\mathbf{b}_{1,2}$ that span each Brillouin zone are set as $\mathbf{b}_1 = (2\pi,0)$ and $\mathbf{b}_2= (0,2\pi)$ for $C_{2,4}$, and $\mathbf{b}_1 = \frac{2\pi}{\sqrt{3}} (\sqrt{3}, -1)$ and $\mathbf{b}_2=\frac{4\pi}{\sqrt{3}} (0,1)$ for $C_{3,6}$. Shaded regions indicate the domains that generate the entire Brillouin zone upon rotation around the fixed point at the center $\Gamma$ of the Brillouin zones.
• Figure 4: Illustration of the filling anomaly in a 1D obstructed atomic insulator (e.g., SSH model) with (a) an even number of unit cells ($N=6$), (b) an odd number of unit cells ($N=7$), and a 2D $C_2$ obstructed atomic insulator with (c) an even number of unit cells ($N_1 = N_2 = 2$), and (d) an odd number of unit cells ($N_1 = N_2 = 3$). The yellow dashed lines in all the figures indicate a fundamental unit $C_2$ sector. In (a) and (b), the 1D obstructed atomic insulator has a filling anomaly of $\eta = 1$ since an additional electron must be filled to restore the $C_2$ symmetry, at the expense of losing charge neutrality. Similar reasoning holds for (c) and (d), in which the 2D $C_2$ obstructed atomic insulator has a filling anomaly of $\eta = 1 \pmod 2$. In open boundary conditions, the neutral configuration of localized electrons cannot satisfy the $C_2$ symmetry even if the boundary electrons are rearranged. This can also be seen from the fact that the number of electrons on the boundary is an odd integer, however, $C_2$ symmetry requires an even number of boundary electrons. To respect $C_2$ symmetry, we must deviate from neutrality, i.e., a filling anomaly must occur.
• Figure 5: Simplified illustration of a finite-size, 1D atomic insulator such as the SSH chain with $C_2$ symmetry on a periodic lattice shown using a circular geometry to emphasize the periodicity. Details of the inner structure of the unit cell, such as sublattice sites, have been omitted. The blue circles denote the $1a$ WPs, and the red stars denote the $1b$ WPs. The green circles indicate electronic Wannier orbital centers. The parameter $\theta$ indicates the boundary condition twist/flux at the $C_2$ rotation center ($\theta=0$ periodic, $\theta=\pi$ anti-periodic). (a) An even number of unit cells and corresponding $C_2$-invariant momenta in BZ. (b) An odd number of unit cells and the corresponding $C_2$-invariant momenta for each boundary condition. In both (a) and (b), $x=0$ denotes the coordinates of the unit cell serving as the origin of the lattice. Positions of electronic Wannier orbital centers in (c) even and (d) odd unit cell cases for both the trivial (left) and obstructed (right) atomic insulator phases. In order to distinguish the phases of the SSH chain in (d), $\theta=0$ and $\theta=\pi$ must both be considered (i.e., periodic and anti-periodic boundary conditions).