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arXiv:2209.00026 · Physical review. B./Physical review. B

Topological field theories of three-dimensional rotation symmetric insulators: Coupling curvature and electromagnetism

Julian May-Mann, Mark R. Hirsbrunner, Xuchen Cao, Taylor L. Hughes

Published May 25, 2023• 5 citations
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Abstract

Quantized responses are important tools for understanding and characterizing the universal features of topological phases of matter. In this work, we show that three-dimensional ($3\mathrm{D}$) insulators with ${C}_{n}$ rotation symmetry along a fixed axis can possess a mixed geometry-charge response, in which disclination lines of the $3\mathrm{D}$ lattice carry electric polarization. These disclinations bind a fractional charge at the gapped surfaces of the insulator because of the surface-charge-polarization theorem. This response is accompanied by a dual response that binds fractional angular momentum to magnetic monopoles in the bulk (analogously to the Witten effect) and to magnetic fluxes on gapped surfaces. We show that these responses are described by a $3\mathrm{D}$ topological response term that couples the lattice curvature to the electromagnetic field strength. Additional mirror or particle-hole symmetry quantizes the mixed geometry-charge responses and defines a new class of rotation-invariant topological crystalline insulators (rTCIs). Notably, the surface charge bound to disclinations of the rTCIs is half the minimal amount that can occur in purely two-dimensional insulators. We construct lattice models of these rTCIs and numerically verify that they exhibit mixed geometry-charge responses. We also demonstrate that the particle-hole symmetric rTCI supports anomalous surface topological order and that the mirror symmetric rTCI can be smoothly deformed into a higher order octopole insulator with quantized corner charges. Additionally, we construct symmetry indicators for diagnosing the mirror symmetric rTCIs.

DOI
10.1103/physrevb.107.205149

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Two-dimensional higher-order topological insulators can display a number of exotic phenomena, such as half-integer charges localized at both corners and disclination defects. In this paper, we analyze these phenomena, focusing on the paradigmatic example of the quadrupole insulator with C4{C}_{4}C4​ 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)e/2(2k+1)e/2(2k+1) bound to corners and disclinations.
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Robust fractional charge localized at disclination defects has recently been found as a topological response in C6{C}_{6}C6​-symmetric 2D topological crystalline insulators (TCIs). In this article, we thoroughly investigate the fractional charge on disclinations in Cn{C}_{n}Cn​-symmetric TCIs, with or without time-reversal symmetry, and including spinless and spin-12\frac{1}{2}21​ cases. We compute the fractional disclination charges from the Wannier representations in real space and use band representation theory to construct topological indices of the fractional disclination charge for all 2D TCIs that admit a (generalized) Wannier representation. We find the disclination charge is fractionalized in units of en\frac{e}{n}ne​ for Cn{C}_{n}Cn​-symmetric TCIs, and for spin-12\frac{1}{2}21​ TCIs, with additional time-reversal symmetry, the disclination charge is fractionalized in units of 2en\frac{2e}{n}n2e​. We furthermore prove that with electron-electron interactions that preserve the Cn{C}_{n}Cn​ symmetry and many-body bulk gap, though we can deform a TCI into another which is topologically distinct in the free fermion case, the fractional disclination charge determined by our topological indices will not change in this process. Moreover, we use an algebraic technique to generalize the indices for TCIs with nonzero Chern numbers, where a Wannier representation is not applicable. With the inclusion of the Chern number, our generalized fractional disclination indices apply for all Cn{C}_{n}Cn​-symmetric TCIs. Finally, we briefly discuss the connection between the Chern number dependence of our generalized indices and the Wen-Zee term.
Topological order provides a new paradigm to view phases of matter. Unlike conventional symmetry breaking order, these states are not distinguished by different patterns of symmetry breaking, instead by their intricate mathematical structure, topology. By the bulk-boundary correspondence, the nontrivial topology of the bulk results in robust gapless excitations on symmetry preserving surfaces. We utilize both of these views to study topological phases together with the analysis of their quantized physical responses to perturbations.First we study the edge excitations of strongly interacting abelian fractional quantum Hall liquids on an infinite strip geometry. We use the infinite density matrix renormalization group method to numerically measure edge exponents in model systems, including subleading orders. Using analytic methods we derive a generalized Luttinger's theorem that relates momenta of edge excitations.Next we consider topological crystalline insulators protected by space group symmetry. After reviewing the general formalism, we present results about the quantization of the magnetoelectric response protected by orientation-reversing space group symmetries. We construct and analyze insulating and superconducting tight-binding models with glide symmetry in three dimensions to illustrate the general result. Following this, we derive constraints on weak indices of three dimensional topological insulators imposed by space group symmetries. We focus on spin-orbit coupled insulators with and without time reversal invariance and consider both symmorphic and nonsymmorphic symmetries.Finally, we calculate the response of metals and generalize the notion of the magnetoelectric effect to noninteracting gapless systems. We use semiclassical dynamics to study the magnetopiezoelectric effect, the current response to elastic strain in static external magnetic fields.
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.
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.