The authors propose that the quantized Berry phase serves as a many-body topological invariant that characterizes the higher-order symmetry-protected topological phases in two- and three-dimensions, and provides a clear insight of bulk-corner correspondence. The quantized Berry phase has wide applicability ranging from fermionic models with and without interactions to spin models.

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Physical review. B./Physical review. B

Many-body higher-order topological invariant for Cn -symmetric insulators

Ammar Jahin, Yuan-Ming Lu, Yuxuan Wang

Published May 8, 2024• 6 citations

Publisher

Abstract

{C}_{n}

-symmetric higher-order topological insulators that is valid for both noninteracting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of

{C}_{n}

rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as

e/n

with

{C}_{n}

symmetry, leading to a

{\mathbb{Z}}_{n}

classification for higher-order topological insulators in two dimensions.

DOI: 10.1103/physrevb.109.205123

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{C}_{n}

{C}_{n}

rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as

e/n

with

{C}_{n}

symmetry, leading to a

{\mathbb{Z}}_{n}

classification for higher-order topological insulators in two dimensions.

Many-body higher-order topological invariant for

C_n

-symmetric insulators

C_n

symmetric higher-order topological insulators, which is valid for both non-interacting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of

C_n

rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as

e/n

with

C_n

symmetry, leading to a

\mathbb{Z}_n

classification for higher-order topological insulators in two dimensions.

We study the entanglement spectrum (ES) of two-dimensional

{C}_{n}

-symmetric second-order topological insulators (TIs). We show that some characteristic higher-order topological observables, e.g., the filling anomaly and its associated fractional corner charge, can be determined from the ES of atomic and fragile TIs. By constructing the relationship between the configuration of Wannier orbitals and the number of protected in-gap states in the ES for different symmetric cuts in real space, we express the fractional corner charge in terms of the number of protected in-gap states of the ES. We show that our formula is robust in the presence of electron-electron interactions as long as the interactions preserve

{C}_{n}

rotation symmetry and charge-conservation symmetry. Moreover, we discuss the possible signatures of higher-order topology in the many-body ES. Our methods allow the identification of some classes of higher-order topology without requiring the usage of nested Wilson loops or nested entanglement spectra.

{C}_{4}

rotation symmetry, and present a topological field theory description of the mixed geometry-charge responses. Our theory provides a unified description of the corner and disclination charges in terms of a physical geometry (which encodes disclinations), and an effective geometry (which encodes corners). We extend this analysis to interacting systems, and predict the response of fractional quadrupole insulators, which exhibit charge

e/2(2k+1)

bound to corners and disclinations.

Recently, it has been shown how topological phases of matter with crystalline symmetry and

U(1)

charge conservation can be partially characterized by a set of many-body invariants, the discrete shift

{\mathcal{S}}_{\mathrm{o}}

and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where

\mathrm{o}

labels a high-symmetry point. Crucially, these can be defined even with nonzero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of

{\mathcal{S}}_{\mathrm{o}}

and ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.