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arXiv:2307.00486

Quadrupole Insulator without Corner States in the Energy Spectrum

Yu-Liang Tao, Jiong-Hao Wang, Xu, Yong

Published Jul 2, 2023• 2 citations
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Abstract

The quadrupole insulator is a well-known instance of higher-order topological insulators in two dimensions, which possesses midgap corner states in both the energy spectrum and entanglement spectrum. Here, by constructing and exploring a model Hamiltonian under a staggered $\mathbb{Z}_2$ gauge field that respects momentum-glide reflection symmetries, we surprisingly find a quadrupole insulator that lacks zero-energy corner modes in its energy spectrum, despite possessing a nonzero quadrupole moment. Remarkably, the existence of midgap corner modes is found in the entanglement spectrum. Since these midgap states cannot be continuously eliminated, the quadrupole insulator cannot be continuously transformed into a trivial topological insulator, thereby confirming its topological nature. We show that the breakdown of the correspondence between the energy spectrum and entanglement spectrum occurs due to the closure of the edge energy gap when the Hamiltonian is flattened. Finally, we present a model that demonstrates an insulator with corner modes in the energy spectrum even in the absence of the quadrupole moment. In this phase, the entanglement spectrum does not display any midgap states. The results suggest that the bulk-edge correspondence of quadrupole insulators generally manifests in the entanglement spectrum rather than the energy spectrum.

DOI
10.48550/arxiv.2307.00486

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The quadrupole insulator is a well-known instance of higher-order topological insulators in two dimensions, which possesses midgap corner states in both the energy spectrum and entanglement spectrum. Here, by constructing and exploring a model Hamiltonian under a staggered Z2\mathbb{Z}_2Z2​ gauge field that respects momentum-glide reflection symmetries, we surprisingly find a quadrupole insulator that lacks zero-energy corner modes in its energy spectrum, despite possessing a nonzero quadrupole moment. Remarkably, the existence of midgap corner modes is found in the entanglement spectrum. Since these midgap states cannot be continuously eliminated, the quadrupole insulator cannot be continuously transformed into a trivial topological insulator, thereby confirming its topological nature. We show that the breakdown of the correspondence between the energy spectrum and entanglement spectrum occurs due to the closure of the edge energy gap when the Hamiltonian is flattened. Finally, we present a model that demonstrates an insulator with corner modes in the energy spectrum even in the absence of the quadrupole moment. In this phase, the entanglement spectrum does not display any midgap states. The results suggest that the bulk-edge correspondence of quadrupole insulators generally manifests in the entanglement spectrum rather than the energy spectrum.
Higher-order topological insulators are materials with a bulk energy gap featuring gapless modes at (d−n)(d-n)(d−n)-dimensional edges, where n>1n > 1n>1. Despite this modified bulk-boundary correspondence, their nontrivial topology can nevertheless be observed through the bipartite entanglement spectrum. Inspired by works suggesting fluctuations in conserved quantities could be used to study topology in one-dimensional systems, which likewise feature zero-dimensional edge modes, we extend this treatment to a two-dimensional second-order topological insulator where corner states play an analogous role. We show that the standard bipartite entanglement entropy and particle number fluctuation lack striking signals of topological phase transitions, both in the static case and in the time evolution after a quench, obscured by an area-law term from the subsystem edges. By introducing a quadripartite construction of the subsystem, we show that it is possible to isolate the topological contributions, producing sharp transition peaks in both quantities, observable by studying subsystems much smaller than the full system. As particle number fluctuations are accessible in existing experiments, these results establish a concrete and scalable path to experimental detection of the entanglement in higher-order topological insulators.
Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa [arXiv:2302.00800], we propose a well-defined, many-body bulk invariant for Cn{C}_{n}Cn​-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 Cn{C}_{n}Cn​ 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/ne/ne/n with Cn{C}_{n}Cn​ symmetry, leading to a Zn{\mathbb{Z}}_{n}Zn​ classification for higher-order topological insulators in two dimensions.
We propose the entanglement dipole polarization to describe the topological quadrupole phase. The quadrupole moment can be regarded as a pair of the dipole moment, in which the total dipole moment is canceled. The entanglement polarization, we propose, is useful to detect such a constituent dipole polarization. We first introduce partitions of sites in the unit cell and divide the system into two subsystems. Then, introducing an entanglement Hamiltonian by tracing out one of the subsystems partly, we compute the dipole polarization of the occupied states associated with the entanglement Hamiltonian, which is referred to as the entanglement polarization. Although the total dipole polarization is vanishing, those of the subsystems can be finite. The entanglement dipole polarization is quantized by reflection symmetries. We also introduce the entanglement polarization of the edge states, which reveals that the edge states themselves are gapped and topologically nontrivial. Therefore, such edge states yield the zero energy edge states if the system has boundaries. This is the origin of the corner states.
Many-body higher-order topological invariant for CnC_nCn​-symmetric insulators
Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa, we propose a well-defined many-body bulk invariant for CnC_nCn​ 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 CnC_nCn​ 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/ne/ne/n with CnC_nCn​ symmetry, leading to a Zn\mathbb{Z}_nZn​ classification for higher-order topological insulators in two dimensions.
The zero-mode corner states in the gap of two-dimensional non-Hermitian Su-Schrieffer-Heeger model are robust to infinitesimal perturbations that preserve chiral symmetry. However, we demonstrate that this general belief is no longer valid in large-sized systems. To reveal the higher-order topology of non-Hermitian systems, we establish a correspondence between the stable zero-mode singular states and the topologically protected corner states of energy spectrum in the thermodynamic limit. Within this framework, the number of zero-mode singular values is directly linked to the number of mid-gap corner states. The winding numbers in real space can be defined to count the number of stable zero-mode singular states. Our results formulate a bulk-boundary correspondence for both static and Floquet non-Hermitian systems, where topology arises intrinsically from the non-Hermiticity, even without symmetries.
In a d -dimensional topological insulator of order d , there are zero-energy states on its corners which have close relationship with its entanglement behaviors. We studied the bipartite entanglement spectra for different subsystem shapes and found that only when the entanglement boundary has corners matching the lattice, exact zero modes exist in the entanglement spectrum corresponding to the zero-energy states caused by the same physical corners. We then considered finite-size systems in which cases these corner states are coupled together by long-range hybridizations to form multipartite entangled states. We proposed a scheme to calculate the quadripartite entanglement entropy on the square lattice, which is well described by a four-sites toy model and thus provides another way to identify the higher order topological insulators from the multipartite entanglement point of view.