Session F7: Focus Session: Novel Topological Phases: Theory

Quantum criticality of topological phase transitions in 3D interacting electronic systems

Topological phase transitions in condensed matters accompany emerging singularities of the electronic wave function, often manifested by gap-closing points in the momentum space. In conventional topological insulators in three dimensions (3D), the low energy theory near the gap-closing point can be described by relativistic Dirac fermions coupled to the long range Coulomb interaction, hence the quantum critical point of topological phase transitions provides a promising platform to test the novel predictions of quantum electrodynamics. Here we show that a new class of quantum critical phenomena emanates in topological materials breaking either the inversion symmetry or the time-reversal symmetry. At the quantum critical point, the theory is described by the emerging low energy fermions, dubbed the anisotropic Weyl fermions, which show both the relativistic and Newtonian dynamics simultaneously. The interplay between the anisotropic dispersion and the Coulomb interaction brings about a new screening phenomenon distinct from the conventional Thomas-Fermi screening in metals and logarithmic screening in Dirac fermions.   

Adiabatic Pumping of Chern-Simons Axion Coupling

The Chern-Simons axion (CSA) coupling $\theta$ makes a contribution of topological origin to the magnetoelectric response of insulating materials. Here we study the adiabatic pumping of the CSA coupling along a parametric loop characterized by a non-zero second Chern number $C^{(2)}$ from the viewpoint of the hybrid Wannier representation. The hybrid Wannier charge centers (WCCs), when plotted over the 2D projected Brillouin zone, were previously shown to give an insightful visualization of the topological character of a 3D insulator. By defining Berry connections and curvatures on these WCC sheets, we derive a new formula for $\theta$, emphasizing that it is naturally decomposed into a topological Berry-curvature dipole term and a nontopological correction term. By explicit calculations on a model tight-binding Hamiltonian, we show how the Berry curvature on the WCC sheets is transported by a lattice vector via a series of Dirac sheet-touching events, resulting in the pumping of $e^2/h$ units of CSA coupling during one closed cycle. The new formulation may provide a particularly efficient means of computing the CSA coupling $\theta$ in practice, since there is no need to establish a smooth gauge in the 3D Brillouin zone.   

Anyon and Loop Braiding Statistics in Field Theories with a Topological $\Theta$-term

For gapped quantum many-body systems, the topological properties of the state are usually encoded by the exotic statistics between its excitations. In 2d, braiding statistics of quasi-particle excitations can be anyonic and uniquely determine the topological phase. This method is successfully applied in 2d to distinguish different Symmetry Protected Topological Phases. Recently, a generalized idea about braiding statistics of loop excitations in 3d gapped system was proposed. We demonstrate that the anyon statistics and three-loop statistics of various 2d and 3d topological phases can be derived using semiclassical Nonlinear Sigma Model field theories with a Topological $\Theta$-term. In our formalism, the braiding statistics has a natural geometric meaning: The braiding process of anyons or loops leads to a nontrivial field configuration in the space-time, which will contribute a braiding phase factor due to the $\Theta$-term. We also provide several physical pictures to understand the cyclic relation of the loop statistics.   

Interplay between geometry and topology in topological crystalline phases

In this talk I will discuss new developments that illustrate the interplay between topology, geometry, and symmetry in topological phases of matter. I will discuss the classification of some topological insulator/superconductor phases via their spatial symmetries and the consequences for topological defects such as disclinations and dislocations.~ Additionally, I will show how spatial symmetries can protect quantized topological responses in topological insulator phases.~If time permits,~I will discuss how interactions can generate a spatial protected topological phase in a symmetry class which only has trivial phases in the non-interacting limit.   

Microscopic Realization of 2-Dimensional Bosonic Topological Insulators

It is well known that a Bosonic Mott insulator can be realized by condensing vortices of a boson condensate. Usually, a vortex becomes an anti-vortex (and vice-versa) under time reversal symmetry, and the condensation of vortices results in a trivial Mott insulator. However, if each vortex/anti-vortex interacts with a spin trapped at its core, the time reversal transformation of the composite vortex operator will contain an extra minus sign. It turns out that such a composite vortex condensed state is a bosonic topological insulator (BTI) with gapless boundary excitations protected by $U(1) Z_2^T$ symmetry. We point out that in BTI, an external $\pi$ flux monodromy defect carries a Kramers doublet. We propose lattice model Hamiltonians to realize the BTI phase, which might be implemented in cold atom systems or spin-$1$ solid state systems.   

Ab Initio Studies of the Tunability of Topological Phases of Complex Materials

Recently, there have been intensive studies of new existing and hypothetical, as-yet-unsynthesized materials with topological phases. Using density functional theory-based approaches, we perform detailed calculations on several promising candidate compounds, including Bi- and Cd- based Dirac and Weyl semimetals and oxide topological insultaors, that are predicted to exhibit topological or near-topological states. We compute surface states of candidate materials such as Dirac semimetals Na3Bi and Cd3As2 along various surfaces. For systems with strong correlations, we examine the efficacy of DFT-based hybrid functionals and the GW approximation for accurate prediction of band inversion. For a selection of systems, we also explore quantitatively the tunability of topological phases via hydrostatic pressure, biaxial strain, broken crystal symmetry, or doping.   

Fermionic Symmetry Protected Topological Phase Induced by Interaction

It is known that interaction can reduce the classification of topological phases in free femion systems, for instance, the Z classes of 1D Kitaev Majorana chains with time reversal symmetry reduce to $Z_8$ under interaction. However, strong interactions can give rise to new SPT phases which have no counterparts in free fermion systems. In this talk, I illustrate this result through an concrete example. The symmetry group we consider is $U(1) Z_2^T$. There are no topological phases for non-interacting fermions with this symmetry. When interactions are turned on, a nontrivial topological phase appears owning to the existence of nontrivial projective representation. We illustrate this result by studying a three-legged ladder of spineless fermions with strong interactions. We show that there are two gapped SPT phases, the trivial one is adiabatically connected to the band insulator, while the states in the nontrivial phase cannot be adiabatically evolved into the trivial phase without breaking symmetry.   

Generic Symmetry Breaking Instability of Topological Insulators due to a Novel van Hove Singularity

We point out that in the deep band-inverted state, topological insulators are generically vulnerable against symmetry breaking instability, due to a divergently large density of states of 1D-like exponent near the chemical potential. This feature at the band edge is associated with a novel van Hove singularity resulting from the development of a Mexican-hat band dispersion. We demonstrate this generic behavior via prototypical 2D and 3D models. This realization not only explains the existing experimental observations of additional phases, but also suggests a route to activate additional functionalities to topological insulators via ordering, particularly for the long-sought topological superconductivities.   

Symmetry Protected Topological States of Interacting Fermions and Bosons

We study the classification for a large class of interacting fermionic and bosonic symmetry protected topological (SPT) states. We define a SPT state as whether or not it is separated from the trivial state through a bulk phase transition, which is a general definition applicable to SPT states with or without spatial symmetries. We show that in all dimensions short range interactions can reduce the classification of free fermion SPT states, and we demonstrate these results by making connection between fermionic and bosonic SPT states. We first demonstrate that our formalism gives the correct classification for several known SPT states, with or without interaction, then we will generalize our method to SPT states that involve the spatial inversion symmetry.   

Symmetry, Defects, and Gauging of Topological Phases

We examine the interplay of symmetry and topological order in 2+1D topological phases of matter. We define the topological symmetry group, characterizing symmetry of the emergent topological quantum numbers, and describe its relation with the microscopic symmetry of the physical system. We derive a general framework to classify symmetry fractionalization in topological phases, including phases that are non-Abelian and symmetries that permute the quasiparticle types and/or are anti-unitary. We develop a theory of extrinsic defects (fluxes) associated with elements of the symmetry group G, which provides a general classification of symmetry-enriched topological phases derived from a topological phase of matter with symmetry. The algebraic theory of the defects (G-crossed braided tensor category), allows one to compute many properties, such as the topologically distinct types of defects, their fusion rules, quantum dimensions, zero modes, braiding transformations, a generalized Verlinde formula, and modular transformations of the G-crossed extensions of topological phases. We also examine the promotion of the global symmetry to a local gauge invariance, wherein the extrinsic defects are turned into deconfined quasiparticle excitations, which results in a different topological phase.   

Kondo Breakdown in Topological Kondo Insulators

Motivated by the observation of light surface states of SmB$_6$, we examine the effects of surface Kondo breakdown in topological Kondo insulators. We present both numerical and analytic results which show that the decoupling of the localized moments at the surface disturbs the compensation between light and heavy electrons and dopes the Dirac cone. Dispersion of these uncompensated surface states are dominated by inter-site hopping, which leads to a much lighter quasiparticles. These surface states are also highly durable against effects of magnetism and decreasing the thickness of the sample.   

Dirac Fermions without bulk backscattering in rhombohedral topological insulators

The realization of a spintronic device using topological insulators is not trivial, because there are inherent difficulties in achieving the surface transport regime. The majority of 3D topological insulators materials (3DTI) despite of support helical metallic surface states on an insulating bulk, forming topological Dirac fermions protected by the time-reversal symmetry, exhibit electronic scattering channels due to the presence of residual continuous bulk states near the Dirac-point. From ab initio calculations, we studied the microscopic origin of the continuous bulk states in rhombohedral topological insulators materials with the space group $D^{5}_{3d}(R\bar{3}m)$, showing that it is possible to understand the emergence of residual continuous bulk states near the Dirac-point into a six bands effective model, where the breaking of the $R_{3}$ symmetry beyond the $\Gamma$ point has an important role in the hybridization of the $p_x$, $p_y$ and $p_z$ atomic orbitals. Within these model, the mechanisms known to eliminate the bulk scattering, for instance: the stacking faults (SF), electric field and alloy, generated the similar effect in the effective states of the 3DTI. Finally, we show how the surface electronic transport is modified by perturbations of bulk with SF.   

Berry curvature induced nonlinear Hall effect in time-reversal invariant materials

It is well-known that a non-vanishing Hall conductivity requires time-reversal symmetry breaking. However, in this work, we demonstrate that a Hall-like transverse current can occur in second-order response to an external electric field in a wide class of time-reversal invariant and inversion breaking materials. This nonlinear Hall effect arises from the dipole moment of the Berry curvature in momentum space, which generates a net anomalous velocity when the system is in a current-carrying state. We show that the nonlinear Hall coefficient is a rank-two pseudo-tensor, whose form is determined by point group symmetry. We will describe the optimal conditions and candidate materials to observe this effect.