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arXiv:1909.08637 · Physical Review Research

Electric polarization as a nonquantized topological response and boundary Luttinger theorem

Xue-Yang Song, Yin-Chen He, Ashvin Vishwanath, Chong Wang

Published Apr 2, 2021• 58 citations
arXivHTMLPDF

Abstract

We develop a nonperturbative approach to the bulk polarization of crystalline electric insulators in d 1 dimensions.Formally, we define polarization via the response to background fluxes of both charge and lattice translation symmetries.In this approach, the bulk polarization is related to properties of magnetic monopoles under translation symmetries.Specifically, in 2D, the monopole is a source of 2π flux, and the polarization is determined by the crystal momentum of the 2π flux.In 3D, the polarization is determined by the projective representation of translation symmetries on Dirac monopoles.Our approach also leads to a concrete scheme to calculate polarization in 2D, which in principle can be applied even to strongly interacting systems.For open boundary conditions, the bulk polarization leads to an altered boundary Luttinger theorem (constraining the Fermi surface of surface states) and also to modified Lieb-Schultz-Mattis theorems on the boundary, which we derive.

DOI
10.1103/physrevresearch.3.023011

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We develop a nonperturbative approach to the bulk polarization of crystalline electric insulators in d 1 dimensions.Formally, we define polarization via the response to background fluxes of both charge and lattice translation symmetries.In this approach, the bulk polarization is related to properties of magnetic monopoles under translation symmetries.Specifically, in 2D, the monopole is a source of 2π flux, and the polarization is determined by the crystal momentum of the 2π flux.In 3D, the polarization is determined by the projective representation of translation symmetries on Dirac monopoles.Our approach also leads to a concrete scheme to calculate polarization in 2D, which in principle can be applied even to strongly interacting systems.For open boundary conditions, the bulk polarization leads to an altered boundary Luttinger theorem (constraining the Fermi surface of surface states) and also to modified Lieb-Schultz-Mattis theorems on the boundary, which we derive.
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 (3D3\mathrm{D}3D) insulators with Cn{C}_{n}Cn​ rotation symmetry along a fixed axis can possess a mixed geometry-charge response, in which disclination lines of the 3D3\mathrm{D}3D 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 3D3\mathrm{D}3D 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.
Nonperturbative constraints on many-body physics, such as the famous Lieb-Schultz-Mattis theorem, are valuable tools for studying strongly correlated systems. To this end, we present a number of nonperturbative results that constrain the low-energy physics of systems having conserved dipole moments. We find that for these systems, a unique translationally invariant gapped ground state is only possible if the polarization of the system is integer. Furthermore, if a lattice system also has U(1)\text{U}(1)U(1) subsystem charge conservation symmetry, a unique gapped ground state is only possible if the particle filling along these subsystems is integer. We also apply these methods to spin systems, and determine criteria for the existence of a different type of magnetic response plateau in the presence of a nonuniform magnetic field. Finally, we formulate a version of Luttinger's theorem for one-dimensional systems consisting of dipoles.
Recently, it has been shown how topological phases of matter with crystalline symmetry and U(1)U(1)U(1) charge conservation can be partially characterized by a set of many-body invariants, the discrete shift So{\mathcal{S}}_{\mathrm{o}}So​ and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where o\mathrm{o}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 So{\mathcal{S}}_{\mathrm{o}}So​ 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.
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.
In this paper, we investigate in general how thermodynamic quantities such as the polarization, magnetization, and the magnetoelectric tensor are affected by the boundaries. We show that when the calculation with periodic boundary conditions does not involve a Berry's phase, the quantity in question is determined unambiguously by the bulk, even in the presence of gapless surface states. When the calculation involves a Berry's phase, the bulk can only determine the quantity up to some quantized value, given that (i) there are no gapless surface states, (ii) the surfaces do not break the symmetries preserved by the bulk, and (iii) the system is kept at charge neutrality. If any of the above conditions is violated, the quantity is then determined entirely by the details at the boundary. Due to the strong dependence on the boundary, this kind of thermodynamic quantity, such as the isotropic magnetoelectric coefficient, cannot be measured in the bulk without careful control at the boundary. One thus cannot distinguish between a topological insulator and a trivial insulator in three dimensions by any local measurement in the bulk.
The notion of symmetry protected distinctions among quantum phases has been well appreciated in understanding many-body systems. As a prominent example, recent studies on topological crystalline insulators (TCIs) indicate that these insulators are nontrivial if and only if the relevant crystalline (and some other) symmetries are preserved. By bulk-boundary correspondence, the authors study the properties of the bulk magnetic monopoles of some TCIs that are conceptually interesting and/or experimentally feasible, and argue that this bulk characterization is a more direct and unified physical description of a TCI. In doing so, the authors prove the stability of these TCIs with respect to interactions.