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Exact Universal Characterization of Chiral-Symmetric Higher-Order Topological Phases

Jia-Zheng Li, Xun-Jiang Luo, Fengcheng Wu, Meng Xiao

TL;DR

The paper tackles the lack of a universal, rigorous framework for classifying higher-order topological phases in chiral-symmetric systems across arbitrary geometries. It introduces a Bott index vector constructed from polynomials in real-space position operators under open boundary conditions and proves an exact correspondence with zero-energy corner-state patterns, via a corner-state configuration vector. A key result is a concrete relation between the Bott index and corner-state patterns, plus a sum-rule that distributes topological information among bulk, boundary, and corner states under different boundary connectivities. The authors illustrate the theory with lattice models of square, hexagonal, and other polygonal shapes, demonstrate that Bott index vectors capture spatial topology beyond prior multipole invariants, and provide an algorithm to generate the necessary polynomials and matrices; they anticipate wide applicability to HOTPs in condensed matter, photonics, acoustics, and superconducting platforms.

Abstract

Utilizing Bott index vectors formulated through a series of polynomials of position operators under open boundary conditions, we establish a universal, rigorous, and complete correspondence between the Bott index vector and topological zero-energy corner states in systems with chiral symmetry. Our framework covers systems of arbitrary shapes, including topological phases that are beyond the characterization by previously proposed invariants such as multipole moments or multipole chiral numbers. A key feature of our approach is its ability to capture the real-space patterns of zero-energy corner states, providing a deeper understanding of higher-order topological phases. We provide a rigorous analytical proof of its higher-order correspondence and sum rules for Bott index vectors under different boundary conditions. To demonstrate the effectiveness of our theory, we examine several model systems with representative patterns of zero-energy corner states that lie outside the scope of previous classification frameworks.

Exact Universal Characterization of Chiral-Symmetric Higher-Order Topological Phases

TL;DR

The paper tackles the lack of a universal, rigorous framework for classifying higher-order topological phases in chiral-symmetric systems across arbitrary geometries. It introduces a Bott index vector constructed from polynomials in real-space position operators under open boundary conditions and proves an exact correspondence with zero-energy corner-state patterns, via a corner-state configuration vector. A key result is a concrete relation between the Bott index and corner-state patterns, plus a sum-rule that distributes topological information among bulk, boundary, and corner states under different boundary connectivities. The authors illustrate the theory with lattice models of square, hexagonal, and other polygonal shapes, demonstrate that Bott index vectors capture spatial topology beyond prior multipole invariants, and provide an algorithm to generate the necessary polynomials and matrices; they anticipate wide applicability to HOTPs in condensed matter, photonics, acoustics, and superconducting platforms.

Abstract

Utilizing Bott index vectors formulated through a series of polynomials of position operators under open boundary conditions, we establish a universal, rigorous, and complete correspondence between the Bott index vector and topological zero-energy corner states in systems with chiral symmetry. Our framework covers systems of arbitrary shapes, including topological phases that are beyond the characterization by previously proposed invariants such as multipole moments or multipole chiral numbers. A key feature of our approach is its ability to capture the real-space patterns of zero-energy corner states, providing a deeper understanding of higher-order topological phases. We provide a rigorous analytical proof of its higher-order correspondence and sum rules for Bott index vectors under different boundary conditions. To demonstrate the effectiveness of our theory, we examine several model systems with representative patterns of zero-energy corner states that lie outside the scope of previous classification frameworks.
Paper Structure (3 sections, 6 theorems, 102 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 3 sections, 6 theorems, 102 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Definitions
  2. The relationship between $q_{xy}$ and $N_{xy}$
  3. Discussions

Key Result

Theorem 1

If no zero-energy corner states exist in a system with chiral symmetry, a finite coupling range, and an energy gap, then the following conclusion holds true in the limit of large system size $L\to \infty$. where $f$ and $g$ in $\hat{M}$ can be arbitrary polynomials with the same degree, i.e., $\operatorname{deg}(f)=\operatorname{deg}(g)$. Here, $\sigma$ denotes the set of eigenvalues of a matrix.

Figures (3)

  • Figure 1: (a) Illustrations of three distinct corner state patterns in a square system, each characterized by a linearly independent configuration vector. The four-component Bott index vector, $\boldsymbol{\nu}_4$, as defined in Eq. \ref{['eqs:correspondence']}, uniquely classifies all possible corner state configurations, with $\mathcal{M}$ given by Eq. \ref{['eqs:M_square']}. (b) The top row shows that under open boundary conditions, the system's states consist of both bulk and boundary states. As shown in the middle row, these states can be represented as linear combinations of states from various mixed boundary conditions, where boundaries connected by black lines with double arrows are glued together. Consequently, the Bott index vector $\boldsymbol{\nu}_6$ for the fully open system can be decomposed in a corresponding manner, as depicted in the bottom row. The bulk and boundary states are schematically represented by orange checker-board and cyan lines, respectively.
  • Figure 2: (a) Left panel: Schematic of the model in Eq. \ref{['eqs:hami_without']}. For clarity, we do not show all non-nearest-neighbor hoppings. The dashed lines represent an additional phase factor of $-1$ such that the system possesses the staggered $\pi$ flux. Right panel: Density of states for this system and corresponding $\nu_{2xy}$, $q_{xy}$ and $N_{xy}$ as functions of $t_x$ with $t_y=0.1$, $t_x^{\prime}=t_y^{\prime}$, $w_d=0.8$. $\nu_x$ and $\nu_y$ always equal $0$. The system length $L=50$. (b) Left panel: Phase diagram of the model governed by Eq. \ref{['eqs:hami_mirr']} as a function of $\delta$ and $m_2$ with $m_1=1$. The topological invariant $\boldsymbol{\nu}_4$ is used to identify three distinct phases, colored green, yellow, and blue. Right panel: Illustration of how the green and yellow phases correspond to different patterns of zero-energy corner states (ZECSs) and their associated configuration vectors, $\boldsymbol{\chi}_4$. (c) Left panel: Schematic of the model in Eq. \ref{['eqs:hami_c6']}. Right panel: The upper part shows the density of states for this system and the configuration vectors $\boldsymbol{\chi}_{6}$ for three higher-order topological phases. The lower part displays the corresponding Bott indices $\nu^{(1,\cdots,5)}_6$ as functions of $t_2^{\prime}$. We fix $t_1=1/10$, $t_2=1/3$, $t_3=1/4$, $t_4=3/10$, $t_5=1/4$, $t_6=1/2$, $t_1^{\prime}=t_3^{\prime}=1$ and the system length $L=30$.
  • Figure S1: The two systems are shaped as a regular pentagon and a cube, respectively. The Bott index vector $\boldsymbol{\nu}$ and configuration vectors $\boldsymbol{\chi}$ of these systems adhere to the analytical relationship outlined in the main text. For $H_{penta}$, the system size $L$ is $30$, and for $H_{3D}$, it is $15$.

Theorems & Definitions (13)

  • Theorem 1
  • Definition M1
  • Remark M1
  • Theorem M1
  • proof
  • Corollary M1
  • proof
  • Theorem M2
  • proof
  • Lemma M1
  • ...and 3 more