Symmetric Topological Orders Research Articles

Anomaly of $(2+1)$-dimensional symmetry-enriched topological order from $(3+1)$-dimensional topological quantum field theory

Symmetry acting on a (2+1)DD topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1)DD on-site symmetry. In this paper, we develop a (3+1)DD topological quantum field theory to calculate the anomaly indicators of a (2+1)DD topological order with a general symmetry group GG, which may be discrete or continuous, Abelian or non-Abelian, contain anti-unitary elements or not, and permute anyons or not. These anomaly indicators are partition functions of the (3+1)DD topological quantum field theory on a specific manifold equipped with some GG-bundle, and they are expressed using the data characterizing the topological order and the symmetry actions. Our framework is applied to derive the anomaly indicators for various symmetry groups, including \mathbb{Z}_2\times\mathbb{Z}_2ℤ2×ℤ2, \mathbb{Z}_2^T\times\mathbb{Z}_2^Tℤ2T×ℤ2T, SO(N)SO(N), O(N)^TO(N)T, SO(N)\times \mathbb{Z}_2^TSO(N)×ℤ2T, etc, where \mathbb{Z}_2ℤ2 and \mathbb{Z}_2^Tℤ2T denote a unitary and anti-unitary order-2 group, respectively, and O(N)^TO(N)T denotes a symmetry group O(N)O(N) such that elements in O(N)O(N) with determinant -1−1 are anti-unitary. In particular, we demonstrate that some anomaly of O(N)^TO(N)T and SO(N)\times \mathbb{Z}_2^TSO(N)×ℤ2T exhibit symmetry-enforced gaplessness, i.e., they cannot be realized by any symmetry-enriched topological order. As a byproduct, for SO(N)SO(N) symmetric topological orders, we derive their SO(N)SO(N) Hall conductance.

Fractional Hall conductivity and spin-c structure in solvable lattice Hamiltonians

The Kapustin-Fidkowski no-go theorem forbids U(1) symmetric topological orders with non-trivial Hall conductivity in (2+1)d from admitting commuting projector Hamiltonians, where the latter is the paradigmatic method to construct exactly solvable lattice models for topological orders. Even if a topological order would intrinsically have admitted commuting projector Hamiltonians, the theorem forbids so once its interplay with U(1) global symmetry which generates Hall conductivity is taken into consideration. Nonetheless, in this work, we show that for all (2+1)d U(1) symmetric abelian topological orders of such kind, we can construct a lattice Hamiltonian that is controllably solvable at low energies, even though not “exactly” solvable; hence, this no-go theorem does not lead to significant difficulty in the lattice study of these topological orders. Moreover, for the fermionic topological orders in our construction, we introduce the lattice notion of spin-c structure — a concept important in the continuum that has previously not been adequately introduced in the lattice context.

Anomaly indicators for topological orders with U(1) and time-reversal symmetry

We study anomalies in time-reversal ($\mathbb{Z}_2^T$) and $U(1)$ symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly $(2+1)$-D system but can be realized on the surface of a $(3+1)$-D symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators --- functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e. $U(1)\rtimes\mathbb{Z}_2^T$ and $U(1)\times\mathbb{Z}_2^T$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of $(3+1)$-D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk $(3+1)$-D SPT phase on a particular manifold and in the presence of a particular background gauge field for the $U(1)$ symmetry.

Topological order from disorder and the quantized Hall thermal metal: Possible applications to the ν=5/2 state

Although numerical studies modeling the quantum hall effect at filling fraction $5/2$ predict either the Pfaffian (Pf) or its particle hole conjugate, the anti-Pfaffian (aPf) state, recent experiments appear to favor a quantized thermal hall conductivity with quantum number $K=5/2$ , rather than the value $K=7/2$ or $K=3/2$ expected for the Pf or aPF state, respectively. While a particle hole symmetric topological order (the PH-Pfaffian) would be consistent with the experiments, this state is believed to be energetically unfavorable in a homogenous system. Here we study the effects of disorder that are assumed to locally nucleate domains of Pf and aPf. When the disorder is relatively weak and the size of domains is relatively large, we find that when the electrical Hall conductance is on the quantized plateau with $\sigma_{xy} = (5/2)(e^2/h)$, the value of $K$ can be only 7/2 or 3/2, with a possible first-order-like transition between them as the magnetic field is varied. However, for sufficiently strong disorder an intermediate state might appear, which we analyze within a network model of the domain walls. Predominantly, we find a thermal metal phase, where $K$ varies continuously and the longitudinal thermal conductivity is non-zero, while the electrical Hall conductivity remains quantized at $(5/2)e^2/h$. However, in a restricted parameter range we find a thermal insulator with $K=5/2$, a disorder stabilized phase which is adiabatically connected to the PH-Pfaffian. We discuss a possible scenario to rationalize these special values of parameters.

Anomaly Indicators for Time-Reversal Symmetric Topological Orders.

Some time-reversal symmetric topological orders are anomalous in that they cannot be realized in strictly two-dimensional systems; instead, they can only be realized on the surface of three-dimensional symmetry-protected topological phases. We propose two quantities, which we call anomaly indicators, that can detect if a time-reversal symmetric topological order is anomalous in this sense. Both anomaly indicators are expressed in terms of the quantum dimensions, topological spins, and time-reversal properties of the anyons in the given topological order. The first indicator, η_{2}, applies to bosonic systems while the second indicator, η_{f}, applies to fermionic systems in the DIII class. We conjecture that η_{2}, together with a previously known indicator η_{1}, can detect the two known Z_{2} anomalies in the bosonic case, while η_{f} can detect the Z_{16} anomaly in the fermionic case.

Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1 dimensions

We propose a generic construction of exactly soluble \emph{local bosonic models} that realize various topological orders with gappable boundaries. In particular, we construct an exactly soluble bosonic model that realizes a 3+1D $Z_2$ gauge theory with emergent fermionic Kramer doublet. We show that the emergence of such a fermion will cause the nucleation of certain topological excitations in space-time without pin$^+$ structure. The exactly soluble model also leads to a statistical transmutation in 3+1D. In addition, we construct exactly soluble bosonic models that realize 2 types of time-reversal symmetry enriched $Z_2$-topological orders in 2+1D, and 20 types of simplest time-reversal symmetry enriched topological (SET) orders which have only one non-trivial point-like and string-like topological excitations. Many physical properties of those topological states are calculated using the exactly soluble models. We find that some time-reversal SET orders have point-like excitations that carry Kramer doublet -- a fractionalized time-reversal symmetry. We also find that some $Z_2$ SET orders have string-like excitations that carry anomalous (non-on-site) $Z_2$ symmetry, which can be viewed as a fractionalization of $Z_2$ symmetry on strings. Our construction is based on cochains and cocycles in algebraic topology, which is very versatile. In principle, it can also realize emergent topological field theory beyond the twisted gauge theory.

Dual Dirac Liquid on the Surface of the Electron Topological Insulator

We discuss a non-fermi liquid gapless metallic surface state of the topological band insulator. It has an odd number of gapless Dirac fermions coupled to a non-compact U(1) gauge field. This can be viewed as a vortex dual to the conventional Dirac fermion surface state. This surface duality is a reflection of a bulk dual description discussed recently for the gauged topological insulator. All the other known surface states can be conveniently accessed from the dual Dirac liquid, including the surface quantum hall state, the Fu-Kane superconductor, the gapped symmetric topological order and the "composite Dirac liquid". We also discuss the physical properties of the dual Dirac liquid, and its connection to the half-filled Landau level.

Translation-symmetry-protected topological orders in quantum spin systems

In this paper we systematically study a simple class of translation-symmetry protected topological orders in quantum spin systems using slave-particle approach. The spin systems on square lattice are translation invariant, but may break any other symmetries. We consider topologically ordered ground states that do not spontaneously break any symmetry. Those states can be described by Z2A or Z2B projective symmetry group. We find that the Z2A translation symmetric topological orders can still be divided into 16 sub-classes corresponding to 16 new translation-symmetry protected topological orders. We introduced four $Z_2$ topological indices $\zeta_{\v{k}}=0,1$ at $\v {k}=(0,0)$, $(0,\pi)$, $(\pi, 0)$, $(\pi ,\pi)$ to characterize those 16 new topological orders. We calculated the topological degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry protected Z2A topological orders for a system on 3D lattice.