(3+1)D topological phases of matter can host a broad class of nontrivial topological defects of codimension1, 2, and 3, of which the wellknown point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible faulttolerant logical operations in topological quantum errorcorrecting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of \mathbb{Z}_2 ℤ 2 gauge theory with fermionic charges, in \mathbb{Z}_2 \times \mathbb{Z}_2 ℤ 2 × ℤ 2 gauge theory with bosonic charges, and also in nonAbelian discrete gauge theories based on dihedral ( D_n D n ) and alternating ( A_6 A 6 ) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an H^4 H 4 cohomology class that characterizes part of an underlying 3group symmetry of the topological order. The equations involving background gauge fields for the 3group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with nonAbelian flux loops (defining part of a noninvertible higher symmetry), examples of noninvertible codimension2 defects, and examples of the interplay of codimension2 defects with codimension1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D A_6 A 6 gauge theory.
more »
« less
Relative anomalies in (2+1)D symmetry enriched topological states
Certain patterns of symmetry fractionalization in topologicallyordered phases of matter are anomalous, in the sense that they can onlyoccur at the surface of a higher dimensional symmetryprotectedtopological (SPT) state. An important question is to determine how tocompute this anomaly, which means determining which SPT hosts a givensymmetryenriched topological order at its surface. While special casesare known, a general method to compute the anomaly has so far beenlacking. In this paper we propose a general method to compute relativeanomalies between different symmetry fractionalization classes of agiven (2+1)D topological order. This method applies to all types ofsymmetry actions, including anyonpermuting symmetries and generalspacetime reflection symmetries. We demonstrate compatibility of therelative anomaly formula with previous results for diagnosing anomaliesfor \mathbb{Z}_2^{T} ℤ 2 T spacetime reflection symmetry (e.g. where timereversal squares to theidentity) and mixed anomalies for U(1) \times \mathbb{Z}_2^{T} U ( 1 ) × ℤ 2 T and U(1) \rtimes \mathbb{Z}_2^{T} U ( 1 ) ⋊ ℤ 2 T symmetries. We also study a number of additional examples, includingcases where spacetime reflection symmetries are intertwined innontrivial ways with unitary symmetries, such as \mathbb{Z}_4^{T} ℤ 4 T and mixed anomalies for \mathbb{Z}_2 \times \mathbb{Z}_2^{T} ℤ 2 × ℤ 2 T symmetry, and unitary \mathbb{Z}_2 \times \mathbb{Z}_2 ℤ 2 × ℤ 2 symmetry with nontrivial anyon permutations.
more »
« less
 NSFPAR ID:
 10142858
 Date Published:
 Journal Name:
 SciPost Physics
 Volume:
 8
 Issue:
 2
 ISSN:
 25424653
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


null (Ed.)Strong interactions between electrons occupying bands of opposite (orlike) topological quantum numbers (Chern =\pm1 = ± 1 ),and with flat dispersion, are studied by using lowest Landau level (LLL)wavefunctions. More precisely, we determine the ground states for twoscenarios at halffilling: (i) LLL’s with opposite sign of magneticfield, and therefore opposite Chern number; and (ii) LLL’s with the samemagnetic field. In the first scenario – which we argue to be a toy modelinspired by the chirally symmetric continuum model for twisted bilayergraphene – the opposite Chern LLL’s are Kramer pairs, and thus thereexists timereversal symmetry ( \mathbb{Z}_2 ℤ 2 ).Turning on repulsive interactions drives the system to spontaneouslybreak timereversal symmetry – a quantum anomalous Hall state describedby one particle per LLL orbital, either all positive Chern {++\cdots+}\rangle  + + ⋯ + ⟩ or all negative {\cdots}\rangle  − − ⋯ − ⟩ .If instead, interactions are taken between electrons of likeChernnumber, the ground state is an SU(2) S U ( 2 ) ferromagnet, with total spin pointing along an arbitrary direction, aswith the \nu=1 ν = 1 spin \frac{1}{2} 1 2 quantum Hall ferromagnet. The ground states and some of theirexcitations for both of these scenarios are argued analytically, andfurther complimented by density matrix renormalization group (DMRG) andexact diagonalization.more » « less

We study the gauging of a global U(1) symmetry in a gapped system in(2+1)d. The gauging procedure has been wellunderstood for a finiteglobal symmetry group, which leads to a new gapped phase with emergentgauge structure and can be described algebraically using themathematical framework of modular tensor category (MTC). We develop acategorical description of U(1) gauging in a MTC, taking into accountthe dynamics of U(1) gauge field absent in the finite group case. Whenthe ungauged system has a nonzero Hall conductance, the gauged theoryremains gapped and we determine the complete set of anyon data for thegauged theory. On the other hand, when the Hall conductance vanishes, weargue that gauging has the same effect of condensing a special Abeliananyon nucleated by inserting 2\pi 2 π U(1) flux. We apply our procedure to theSU(2) _k k MTCs and derive the full MTC data for the \mathbb{Z}_k ℤ k parafermion MTCs. We also discuss a dual U(1) symmetry that emergesafter the original U(1) symmetry of an MTC is gauged.more » « less

We propose and prove a family of generalized LiebSchultzMattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions.The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin1/2 particles per unit cell, forbids a symmetric shortrangeentangled ground state in such a system.Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a nontrivial SPT phase with anomalous boundary excitations.Depending on models, they can be either strong or ``higherorder'' crystalline SPT phases, characterized by nontrivial surface/hinge/corner states.Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory.We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.more » « less

A bstract We analyze topological mass terms of BF type arising in supersymmetric Mtheory compactifications to AdS 5 . These describe spontaneously broken higherform gauge symmetries in the bulk. Different choices of boundary conditions for the BF terms yield dual field theories with distinct global discrete symmetries. We discuss in detail these symmetries and their ’t Hooft anomalies for 4d $$ \mathcal{N} $$ N = 1 SCFTs arising from M5branes wrapped on a Riemann surface without punctures, including theories from M5branes at a ℤ 2 orbifold singularity. The anomaly polynomial is computed via inflow and contains background fields for discrete global 0, 1, and 2form symmetries and continuous 0form symmetries, as well as axionic background fields. The latter are properly interpreted in the context of anomalies in the space of coupling constants.more » « less