Session Z10: Topological Insulators - General Theory

Topological insulators in staggered flux systems

Topological insulators are generally characterized by the $Z_{2}$ index, which requires time-reversal symmetry. On the other hand, the staggered flux states, known as orbital antiferromagnetic or charge flux phases, break both time-reversal and translational symmetry. In this work, we investigate the behavior of topological insulators within staggered flux. Interestingly, gapless edge states consisting of counter-propagating states with opposite spins survive, and in some regions, a phase with two such pairs of edge states emerges. We examine the robustness of these phases in the presence of disorder and study the topological phase transitions by varying the disorder strength. These systems demonstrate topological properties similar to but different from the ones predicted by the well-known $Z_{2}$ topological theory.   

General symmetry fractionalizations of topologically ordered systems in two dimensions

A framework is presented to describe symmetry fractionalizations for a generic topological order in two dimensions, via studying the operator algebra of quantum systems. We give a precise definition of symmetry fractionalizations, including those relevant to space group symmetry and time reversal symmetry. Examples are given to apply this framework to exactly solvable local bosonic models with abelian or non-abelian topological order. In addition, the general relations among fractional quantum numbers carried by different anyon species are derived. This framework is applicable in particular to gapped quantum spin liquids, fractional Chern insulators, and fractional topological insulators.   

Dissipative Floquet Topological Systems

Motivated by recent pump-probe spectroscopies, we study the effect of phonon dissipation and potential cooling on the nonequilibrium distribution function in a Floquet topological state. To this end, we apply a Floquet kinetic equation approach to study two dimensional Dirac fermions irradiated by a circularly polarized laser, a system which is predicted to be in a laser induced quantum Hall state. We find that the initial electron distribution shows an anisotropy with momentum dependent spin textures whose properties are controlled by the switching-on protocol of the laser. The phonons then smoothen this out leading to a non-trivial isotropic nonequilibrium distribution which has no memory of the initial state and initial switch-on protocol, and yet is distinct from a thermal state. An analytical expression for the distribution at the Dirac point is obtained that is relevant for observing quantized transport.   

Anomalous Symmetry Fractionalization and Surface Topological Order

In addition to fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in unusual ways such as carrying fractional quantum numbers, leading to a variety of symmetry enriched topological (SET) phases. While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all consistent symmetry fractionalizations can be realized in 2D systems. Instead, certain SETs are anomalous in that they can only occur on the surface of a 3D symmetry protected topological (SPT) phase. In this paper we describe a procedure for identifying an anomalous SET which has a discrete unitary symmetry group $G$. The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to defining a consistent topological theory involving both the original anyons and the gauge fluxes. We point out that a class of obstructions are captured by the fourth cohomology group $H^4( G, \,U(1))$, which also labels the set of 3D SPT phases, providing an explicit link to surface topological orders. We illustrate this using the simplest possible example - the projective semion model - where a $Z_2 \times Z_2$ symmetry acts on a chiral semion in a way which is only possible on the surface of a 3D SPT phase.   

Modular Anomalies in the Topological Classification of 2$+$1D and 3$+$1D Edge Theories

Classification of topological phases of matter in the presence of interactions is an area of intense interest. While much progress has been made on classification of interacting bosonic systems, the classification of fermionic systems is less established. One possible means of classification is via studying the partition function under modular transforms, as the presence of an anomalous phase arising in the edge theory of a D-dimensional system under modular transforms, or modular anomaly, signals the presence of a (D$+$1)-dimensional nontrivial bulk. In this work, we discuss the modular transforms of conformal field theories along a (2$+$1)-D and (3$+$1)-D edge. By both analytical and numerical methods, free chiral complex fermions in (2$+$1)-D and (3$+$1)-D are shown to be modular invariant, however, we show in (3$+$1)-D that a background U(1) gauge field results in the presence of a modular anomaly that is the manifestation of a quantum Hall effect in a (4$+$1)-D bulk.   

Z2 topological invariants and gauge transformation of time reversal polarization

The $Z_{2}$ topological indices for 2D and 3D (strong and weak) topological materials can be calculated from the time reversal polarization as shown by Fu and Kane. These polarizations are defined up to a sign, which represents a gauge choice, since these signs do not influence the topological indices. We discuss the origin of this gauge invariance and its physical interpretation for materials with inversion center.   

An exactly solvable model for twisted symmetry-enriched phases

Topological phases in 2D have a long history of exotic behaviour, producing anyons and protected edge states. This trend continues when we impose an extra symmetry $G$, producing a symmetry-enriched topological (SET) phase. While the ground state will remain invariant under $G$, the set of anyons $A$ may transform non-trivially. The different ways of implementing the symmetry are classified by the elements of the group cohomology $H^2_{\rho}(G,A)$, where $\rho$ describes the action of $G$ on the set of anyons. Previously constructed models fix $\rho$ to be the identity, meaning that $G$ can only modify anyons by a phase, whereas we could easily envision a case where $G$ permutes anyon types, which we call twisted SETs. In this talk, we will propose a modified string-net model which allows $G$ to act on the anyons in exactly that manner, for any choice of $\rho$. We will also introduce a constructive method of gauging the global symmetry, which allows us to verify that the obtained twisted SETs are distinct by showing that discrete gauge theories produced by gauging $G$ are distinct.   

``Gauging'' Non-on-site Symmetries and Symmetry Protected Topological Phases

We gauge non-on-site symmetries, such as parity symmetries, for a general (1+1)D conformal field theory (CFT) which is the boundary of (2+1)D symmetry protected topological (SPT) phases. This provides an efficient method to diagnose stability of SPT phases with the discrete non-on-site symmetries. To gauge the non-on- site symmetries, we are naturally led to consider field theories defined on a non-orientied manifold, such as Klein bottle. The partner states of the ``vortices'' (or twist operators) of the gauged non-on-site symmetries, the so-called crosscap states, provide information about the classification of the corresponding SPT phases. Our method also provide a way to gauging time-reversal symmetry, which is ``topologically'' related to parity symmetry by CPT theorem.   

Twist liquids and gauging anyonic symmetries

Topological phases of matter in $(2+1)$D are frequently equipped with global symmetries that relabel anyons without changing the fusion and braiding structures. Twist defects are static symmetry fluxes that permute the labels of orbiting anyons. {\em Gauging} or {\em melting} these symmetries by quantizing defects into dynamical excitations leads to a wide class of more exotic topological phases known as {\em twist liquids}. We formulate a general gauging framework, characterize the anyon structure of twist liquids and provide solvable lattice models that capture the gauging phase transitions. Generalizing a discrete gauge theory, we represent the anyons in a twist liquid by compositions of not only fluxes and charges but also quasiparticle supersectors. We show the gauging transition amplifies the total quantum dimension by $|G|$, the order of the symmetry group, and thus modifies the topological entanglement entropy.   

Fluctuating Domain Wall Wavefunctions for Symmetry Protected Topological Phases

Symmetry protected topological (SPT) phases have been argued to be classified by the group cohomology of the symmetry group. In general, it has been challenging to connect this classification directly and intuitively to physical properties. In this talk, we provide a simple picture of SPT ground state wave functions in terms of fluctuating domain walls, for SPT phases in one and two dimensions with a finite internal symmetry group. The structure of group cohomology has a simple physical manifestation in the wave functions we construct. We also employ the fluctuating domain wall picture to analyze physical properties of SPT phases, and relate these directly to the group cohomology structure of the wave function.   

Holographic entanglement renormalization of topological insulators

In this work we study the real-space entanglement renormalization group (RG) flows and associated emergent holographic geometry of topological band insulators in (2+1) dimensions with continuum multi-scale entanglement renormalization ansatz (cMERA). Given a ground state of a topological insulator at the UV layer, we study how the Berry curvature as well as the quantum metric evolve in the bulk of cMERA. Besides the nontrivial topological properties in the bulk of cMERA, it is found that the UV state flows to a nontrivial IR state which carries a nonzero Berry flux. Our result is in parallel with the picture in lattice MERA that a nontrivial UV state corresponds to a nontrivial IR state. On the other hand, if we try to construct the UV state with a trivial IR state, we find there is a ``phase transition'' feature in the bulk of cMERA.   

Classification of topological phases with reflection symmetry

In $Z_{2}$ topological band insulators, the time-reversal symmetry protects their topological structure. In these years such a notion is extended to correlated systems including bosonic systems, and these nontrivial phases are referred to as symmetry protected topological (SPT) phases. Parallel to this progress, a topological crystalline insulator, protected by spatial symmetry, is found for SnTe. Thus, SPT phases protected by this type of symmetry are naturally expected, and classifications of such phases are desired. In this article, we address this issue by focusing on a reflection symmetry. Our analysis based on the Chern-Simons approach proposes periodic tables for bosonic and fermionic SPT phases in two dimensions. Besides that, we show an SPT phase with the reflection symmetry is stabilized in a spin model of honeycomb lattice.   

Gauging and Orbifolding Topological Phases

Topological phases of matter in $(2+1)$D are commonly equipped with global symmetries, such as electric-magnetic duality in gauge theories and bilayer symmetry in fractional quantum Hall states. $\textit{Gauging}$ these symmetries into local dynamical ones is one way of obtaining exotic phases from conventional systems. We study this using the bulk-boundary correspondence and $\textit{orbifolding}$ the $(1+1)$D edge described by a conformal field theory (CFT). Our procedure puts twisted boundary conditions into the partition function, and predicts the fusion, spin and braiding behavior of anyonic excitations after gauging. We demonstrate this for the twofold-symmetric $Z_N$ gauge theory and the $S_3$-symmetric $so(8)_1$ state.   

Sensing Coulomb impurities with 1/f noise in 3D Topological Insulator

Electrical transport in the non-trivial surface states of bulk Topological Insulator (TI) reveal several intriguing properties ranging from bipolar field effect transistor action, weak antilocalization in quantum transport, to the recently discovered quantum anomalous Hall effect. Many of these phenomena depend crucially on the nature of disorder and its screening by the Dirac Fermions at the TI surface. We have carried out a systematic study of low-frequency 1/f noise in Bi$_{\mathrm{1.6}}$Sb$_{\mathrm{0.4}}$Te$_{\mathrm{2}}$Se$_{\mathrm{1}}$ single crystals, to explore the dominant source of scattering of surface electrons and monitor relative contributions of the surface and bulk channels. Our results reveal that while trapped coulomb impurities at the substrate-TI interface are dominating source of scattering for thin (10 nm) TI, charged crystal disorder contribute strongly in thick TI (110 nm) channels. An unexpected maximum at 25K in noise from thick TI devices indicate scattering of the surface states by a cooperative charge dynamics in the bulk of the TI, possibly associated with the Selenium vacancies. Our experiment demonstrates, for the first time, impact of the bulk charge distribution on the surface state transport in TIs that could be crucial to the implementation of these materials in electronic applications.