Dimensional Topological Phases Research Articles

(3+1)-dimensional path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)-dimensional topological phases

(3+1)-dimensional path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)-dimensional topological phases

Universal Platform of Point-Gap Topological Phases from Topological Materials.

Whereas point-gap topological phases are responsible for exceptional phenomena intrinsic to non-Hermitian systems, their realization in quantum materials is still elusive. Here, we propose a simple and universal platform of point-gap topological phases constructed from Hermitian topological insulators and superconductors. We show that (d-1)-dimensional point-gap topological phases are realized by making a boundary in d-dimensional topological insulators and superconductors dissipative. A crucial observation of the proposal is that adding a decay constant to boundary modes in d-dimensional topological insulators and superconductors is topologically equivalent to attaching a (d-1)-dimensional point-gap topological phase to the boundary. We furthermore establish the proposal from the extended version of the Nielsen-Ninomiya theorem, relating dissipative gapless modes to point-gap topological numbers. From the bulk-boundary correspondence of the point-gap topological phases, the resultant point-gap topological phases exhibit exceptional boundary states or in-gap higher-order non-Hermitian skin effects.

Fractional quantum Hall states on CP2 space

We study four-dimensional fractional quantum Hall states on CP2 geometry from microscopic approaches. While in 2d the standard Laughlin wave function, given by a power of Vandermonde determinant, admits a product representation in terms of the Jastrow factor, this is no longer true in higher dimensions. In 4d we can define two different types of Laughlin wavefunctions, the Determinant-Laughlin (Det-Laughlin) and Jastrow-Laughlin (Jas-Laughlin) states. We find that they are exactly annihilated by, respectively, two-particle and three-particle short ranged interacting Hamiltonians. We then mainly focus on the ground state, low energy excitations and the quasi-hole degeneracy of Det-Laughlin state. The quasi-hole degeneracy exhibits an anomalous counting, indicating the existence of multiple forms of quasi-hole wavefunctions. We argue that these are captured by the mathematical framework of the commutative algebra of N-points in the plane. We also generalize the pseudopotential formalism to dimensions higher than two, by considering coherent state wavefunction of bound states. The microscopic wavefunctions and Hamiltonians studied in this work pave the way for systematic study of high dimensional topological phase of matter that is potentially realizable in cold atom and optical experiments.

Generalized Gibbs ensemble description of real-space entanglement spectra of (2+1) -dimensional chiral topological systems with SU(2) symmetry

We provide a quantitative analysis of the splittings in low-lying numerical entanglement spectra (ES), at given momentum, of a number of quantum states that can be identified, based on "Li-Haldane state-counting", as ground states of (2+1)-dimensional chiral topological phases with global SU(2) symmetry. The ability to account for numerical ES splittings solely within the context of conformal field theory (CFT) is an additional diagnostic of the underlying topological theory, of finer sensitivity than "state-counting". We use the conformal boundary state description of the ES, which can be viewed as a quantum quench. In this language, the ES splittings arise from local conservation laws in the chiral CFT besides the energy, which we view as a Generalized Gibbs Ensemble (GGE). Global SU(2) symmetry imposes strong constraints on the number of such conservation laws, so that only a small number of parameters can be responsible for the splittings. We work out these conservation laws for chiral SU(2) Wess-Zumino-Witten CFTs at levels one and two, and for the latter we notably find that some of the conservation laws take the form of local integrals of operators of fractional dimension, as proposed by Cardy for quantum quenches. We analyze numerical ES from systems with SU(2) symmetry including chiral spin-liquid ground states of local 2D Hamiltonians and two chiral Projected Entangled Pair States (PEPS) tensor networks, which exhibit the "state-counting" of the SU(2)-level-one and -level-two theories. We find that the low-lying ES splittings can be well understood by the lowest of our conservation laws, and we demonstrate the importance of accounting for the fractional conservation laws at level two. Thus the states we consider, including the PEPS, appear chiral also under our more sensitive diagnostic.

Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras

Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.

Torsion, energy magnetization, and thermal Hall effect

We study the effective action of hydrostatic response to torsion in the absence of spin connections in gapped $(2+1)$-dimensional topological phases at low temperatures. In previous studies, a torsional Chern-Simons term with a temperature-squared (${T}^{2}$) coefficient was proposed as an alternative action to describe the thermal Hall effect with the idea of balancing the diffusion of heat by a torsional field. However, the question remains whether this action leads to a local bulk thermal response that is not suppressed by the gap. In our hydrostatic effective action, we show that the ${T}^{2}$ bulk term is invariant under variations up to boundary terms considering the geometry dependence of local temperature, which precisely describes the edge thermal current. Furthermore, there are no local boundary diffeomorphism anomalies or bulk inflow thermal currents at equilibrium, and also there is no edge-to-edge adiabatic thermal current pumping upon changing the gravitational background. These results are consistent with exponentially suppressed thermal current for gapped phases.

Chiral Majorana edge modes and vortex Majorana zero modes in superconducting antiferromagnetic topological insulator

The antiferromagnetic topological insulator (AFTI) is topologically protected by the combined time-reversal and translational symmetry . In this paper we investigate the effects of the s-wave superconducting pairings on the multilayers of AFTI, which breaks symmetry and can realize quantum anomalous Hall insulator with unit Chern number. For the weakly coupled pairings, the system corresponds to the topological superconductor (TSC) with the Chern number C = ±2. We answer the following questions whether the local Chern numbers and chiral Majorana edge modes of such a TSC distribute around the surface layers. By the numerical calculations based on a theoretic model of AFTI, we find that when the local Chern numbers are always dominated by the surface layers, the wavefunctions of chiral Majorana edge modes must not localize on the surface layers and show a smooth crossover from spatially occupying all layers to only distributing near the surface layers, similar to the hinge states in a three dimensional second-order topological phases. The latter phase, denoted by the hinged TSC, can be distinguished from the former phase by the measurements of the local density of state. In addition we also study the superconducting vortex phase transition in this system and find that the exchange field in the AFTI not only enlarges the phase space of topological vortex phase but also enhances its topological stability. These conclusions will stimulate the investigations on superconducting effects of AFTI and drive the studies on chiral Majorana edge modes and vortex Majorana zero modes into a new era.

Disentangling (2+1)D topological states of matter with entanglement negativity

We use the entanglement negativity, a bipartite measure of entanglement in mixed quantum states, to study how multipartite entanglement constrains the real-space structure of the ground state wavefunctions of $(2+1)$-dimensional topological phases. We focus on the (Abelian) Laughlin and (non-Abelian) Moore-Read states at filling fraction $\nu=1/m$. We show that a combination of entanglement negativities, calculated with respect to specific cylinder and torus geometries, determines a necessary condition for when a topological state can be disentangled, i.e., factorized into a tensor product of states defined on cylinder subregions. This condition, which requires the ground state to lie in a definite topological sector, is sufficient for the Laughlin state. On the other hand, we find that a general Moore-Read ground state cannot be disentangled even when the disentangling condition holds.

Topological gaps by twisting

Twisted bilayered systems such as bilayered graphene exhibit remarkable properties such as superconductivity at magic angles and topological insulating phases. For generic twist angles, the bilayers are truly quasiperiodic, a fact that is often overlooked and that has consequences which are largely unexplored. Herein, we uncover that twisted n-layers host intrinsic higher dimensional topological phases, and that those characterized by second Chern numbers can be found in twisted bi-layers. We employ phononic lattices with interactions modulated by a second twisted lattice and reveal Hofstadter-like spectral butterflies in terms of the twist angle, which acts as a pseudo magnetic field. The phason provided by the sliding of the layers lives on 2n-tori and can be used to access and manipulate the edge states. Our work demonstrates how multi-layered systems are virtual laboratories for studying the physics of higher dimensional quantum Hall effect, and can be employed to engineer topological pumps via simple twisting and sliding.

Dynamically characterizing topological phases by high-order topological charges

We propose a new theory to characterize equilibrium topological phase with non-equilibrium quantum dynamics by introducing the concept of high-order topological charges, with novel phenomena being predicted. Through a dimension reduction approach, we can characterize a $d$-dimensional ($d$D) integer-invariant topological phase with lower-dimensional topological number quantified by high-order topological charges, of which the $s$th-order topological charges denote the monopoles confined on the $(s-1)$th-order band inversion surfaces (BISs) that are $(d-s+1)$D momentum subspaces. The bulk topology is determined by the $s$th order topological charges enclosed by the $s$th-order BISs. By quenching the system from trivial phase to topological regime, we show that the bulk topology of post-quench Hamiltonian can be detected through a high-order dynamical bulk-surface correspondence, in which both the high-order topological charges and high-order BISs are identified from quench dynamics. This characterization theory has essential advantages in two aspects. First, the highest ($d$th) order topological charges are characterized by only discrete signs of spin-polarization in zero dimension (i.e. the $0$th Chern numbers), whose measurement is much easier than the $1$st-order topological charges that are characterized by the continuous charge-related spin texture in higher dimensional space. Secondly, a more striking result is that a first-order high integer-valued topological charge always reduces to multiple highest-order topological charges with unit charge value, and the latter can be readily detected in experiment. The two fundamental features greatly simplify the characterization and detection of the topological charges and also topological phases, which shall advance the experimental studies in the near future.

Interfaces and the extended Hilbert space of Chern-Simons theory

The low energy effective field theories of (2 + 1) dimensional topological phases of matter provide powerful avenues for investigating entanglement in their ground states. In [1] the entanglement between distinct Abelian topological phases was investigated through Abelian Chern-Simons theories equipped with a set of topological boundary conditions (TBCs). In the present paper we extend the notion of a TBC to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases. These boundary conditions furnish a defining relation for the extended Hilbert space of the quantum theory and allow the calculation of entanglement directly in the gauge theory. Because we allow for trivial interfaces, this includes a generic construction of the extended Hilbert space in any (compact) Chern-Simons theory quantized on a Riemann surface. Additionally, this provides a constructive and principled definition for the Hilbert space of effective ground states of gapped phases of matter glued along gapped interfaces. Lastly, we describe a generalized notion of surgery, adding a powerful tool from topological field theory to the gapped interface toolbox.

Representations of the loop braid groups from braided tensor categories

The loop braid group is the motion group of unknotted oriented circles in R3. In this paper, we study their representations through the approach inspired by two dimensional topological phases of matter. In principle, the motion of loops in R3 reduces to the motions of points in a two dimensional sliced plane. We realize this physical picture in terms of braided tensor categories and their braid group representations.

Characterizing topological phases by quantum quenches: A general theory

We investigate a generic dynamical theory to characterize topological quantum phases by quantum quenches, and study the emergent topology of quantum dynamics when the quenches start from a deep or shallow trivial phase to topological regimes. Two dynamical schemes are examined: One is to characterize topological phases via quantum dynamics induced by a single quench along an arbitrary axis, and the other applies a sequence of quenches with respect to all (pseudo)spin axes. These two schemes are both built on the so-called dynamical bulk-surface correspondence, which shows that the $d$-dimensional ($d$D) topological phases with integer invariants can be characterized by the dynamical topological pattern emerging on $(d-1)$D band inversion surfaces (BISs). We show that the first dynamical scheme works for both deep and shallow quenches, the latter of which is initialized in an incompletely polarized trivial phase. For the second scheme, however, when the initial phase for the quench study varies from the deep trivial (fully polarized) to shallow trivial (incompletely polarized) regime, a new dynamical topological transition, associated with topological charges crossing BISs, is predicted in quench dynamics. A generic criterion of the dynamical topological transition is precisely obtained. Above the criterion, quantum dynamics on BISs well characterizes the topology of the post-quench Hamiltonian. Below the criterion, the quench dynamics may depict a new dynamical topology; the post-quench topology can be characterized by the emergent topological invariant plus the total charges moving outside the region enclosed by BISs. We illustrate our results by numerically calculating the 2D quantum anomalous Hall model. This work broadens the way to classify topological phases by non-equilibrium quantum dynamics, and has feasibility for experimental realization.

Chiral edge states in (2+1)-dimensional topological phases

Chiral edge states of [Formula: see text]-dimensional Abelian and non-Abelian topological phases can be represented by chiral conformal field theories with integer and noninteger values of central charge, respectively. In this work we describe certain edge states in terms of constrained fermionic fields that realize chiral coset CFT structures. This construction arises naturally in the so-called quantum wires approach for topological phases and allows for representing fractionalized edge states directly in terms of fermionic degrees of freedom. At the same time, the constrained fermions description introduces some subtleties concerning gauge anomalies since it involves the coupling of chiral fermions to gauge fields. We describe in this paper how to handle these issues.

Multifractality at the Weyl-semimetal–diffusive-metal transition for generic disorder

A Weyl semimetal is a three dimensional topological gapless phase. In the presence of strong enough disorder it undergoes a quantum transition towards a diffusive metal phase whose universality class depends on the range of disorder correlations. Similar to other quantum transitions driven by disorder, the critical wave functions at the semimetal-diffusive metal transition exhibit multifractality. Using renormalization group methods we study the corresponding multifractal spectrum as a function of the range of disorder correlations for generic disorder including random scalar and vector potentials. We also discuss the relation between the geometric fluctuations of critical wave functions and the broad distribution of the local density of states (DOS) at the transition. We derive a new scaling relation for the typical local DOS and argue that it holds for other disorder-driven transitions in which both the average and typical local DOS vanish on one side of the transition. As an illustration we apply it to the recently discussed unconventional quantum transition in disordered semiconductors with power-law dispersion relation near the band edge.

Symmetry fractionalization, defects, and gauging of topological phases

We examine the interplay of symmetry and topological order in $2+1$ dimensional topological phases of matter. We present a definition of the \it topological symmetry \rm group, which characterizes the symmetry of the emergent topological quantum numbers of a topological phase, and we describe its relation with the microscopic symmetry of the underlying physical system. We derive a general framework to characterize and 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, which provides a general classification of symmetry-enriched topological phases derived from a topological phase of matter $\mathcal{C}$ with symmetry group $G$. The algebraic theory of the defects, known as a $G$-crossed braided tensor category $\mathcal{C}_{G}^{\times}$, allows one to compute many properties, such as the number of topologically distinct types of defects associated with each group element, their fusion rules, quantum dimensions, zero modes, braiding exchange transformations, a generalized Verlinde formula for the defects, 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 $G$-defects are turned into deconfined quasiparticle excitations, which results in a different topological phase $(\mathcal{C}_{G}^{\times})^{G}$. A number of instructive and/or physically relevant examples are studied in detail.

Compactifying fracton stabilizer models

We investigate two dimensional compactifications of three dimensional fractonic stabilizer models. We find the two dimensional topological phases produced as a function of compactification radius for the X-cube model and Haah's cubic code. Furthermore, we uncover translation symmetry-enrichment in the compactified cubic code that leads to twisted boundary conditions.

Chiral anomaly induced oscillations in the Josephson current in Weyl semimetals

Weyl semimetals are a three dimensional topological phase of matter with linearly dispersed Weyl points which appear in pairs and carry opposite chirality. The separation of paired Weyl points allows charge transfer between them in the presence of parallel electric and magnetic fields, which is known as the chiral anomaly. In this paper, we theoretically study the influence of the chiral anomaly induced chiral charge imbalance on the Josephson current in a Weyl superconductor-Weyl semimetal-Weyl superconductor junction. In Weyl superconductors, two types of pairings are considered, namely, zero momentum BCS-like pairing and finite momentum FFLO-like pairing. For BCS-like pairing, we find that the Josephson current exhibits $0$-$\pi$ transitions and oscillates as a function of $\lambda_{0} L $, where $\lambda_{0}$ is the chirality imbalance induced by the parallel electric and magnetic fields and $L$ is the length of the Weyl semimetal. The amplitude of the Josephson current also depends on the angle $\beta$ between the line connecting two paired Weyl points and the transport direction along the junction. For FFLO-like pairing, the chirality imbalance induced periodic oscillations are absent and the Josephson current is also independent of the angle $\beta$. These findings are useful in detecting the chiral anomaly and distinguishing the superconducting pairing mechanism of Weyl semimetals.

Higher-order topological phases: A general principle of construction

We propose a general principle for constructing higher-order topological (HOT) phases. We argue that if a $D$-dimensional first-order or regular topological phase involves $m$ Hermitian matrices that anti-commute with additional $p-1$ mutually anti-commuting matrices, it is conceivable to realize an $n$th-order HOT phase, where $n=1, \cdots, p$, with appropriate combinations of discrete symmetry-breaking Wilsonian masses. An $n$th-order HOT phase accommodates zero modes on a surface with codimension $n$. We exemplify these scenarios for prototypical three-dimensional gapless systems, such as a nodal-loop semimetal possessing SU(2) spin-rotational symmetry, and Dirac semimetals, transforming under (pseudo-)spin-$\frac{1}{2}$ or 1 representations. The former system permits an unprecedented realization of a fourth-order phase, without any surface zero modes. Our construction can be generalized to HOT insulators and superconductors in any dimension and symmetry class.

Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter

We study a class of anomalies associated with time-reversal and spatial reflection symmetry in (2+1)D topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a $\mathbb{Z}_2$ symmetry which can be associated with either time-reversal (denoted $\mathbb{Z}_2^{\bf T})$ or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a notion of symmetry fractionalization. We present simple sufficient conditions which determine when $\mathbb{Z}_2^{\bf T}$ symmetry localization anomalies exist. We present an infinite series of TQFTs with such anomalies, some examples of which include USp$(4)_2$ and SO$(4)_4$ Chern-Simons (CS) theory. The theories that we find with these $\mathbb{Z}_2^{\bf T}$ anomalies can be obtained by gauging the unitary $\mathbb{Z}_2$ subgroup of a different TQFT with a $\mathbb{Z}_4^{\bf T}$ symmetry. We show that the anomaly can be resolved in several ways: (1) the true symmetry of the theory is $\mathbb{Z}_4^{\bf T}$, or (2) the theory can be considered to be a theory of fermions, with ${\bf T}^2 = (-1)^{N_f}$ corresponding to fermion parity. Finally, we demonstrate that theories with the $\mathbb{Z}_2^{\bf T}$ localization anomaly can be compatible with $\mathbb{Z}_2^{\bf T}$ if they are "pseudo-realized" at the surface of a (3+1)D symmetry-enriched topological phase. The "pseudo-realization" refers to the fact that the bulk (3+1)D system is described by a dynamical $\mathbb{Z}_2$ gauge theory and thus only a subset of the quasiparticles are confined to the surface.