An official website of the United States government
We explore exact generalized symmetries in the standard 2+1d lattice\mathbb{Z}_2 gauge theory coupled to the Ising model, and compare them with their continuum field theory counterparts. One model has a (non-anomalous) non-invertible symmetry, and we identify two distinct non-invertible symmetry protected topological phases. The non-invertible algebra involves a lattice condensation operator, which creates a toric code ground state from a product state. Another model has a mixed anomaly between a 1-form symmetry and an ordinary symmetry. This anomaly enforces a nontrivial transition in the phase diagram, consistent with the “Higgs=SPT” proposal. Finally, we discuss how the symmetries and anomalies in these two models are related by gauging, which is a 2+1d version of the Kennedy-Tasaki transformation.
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 symmetry-protectedtopological (SPT) state. An important question is to determine how tocompute this anomaly, which means determining which SPT hosts a givensymmetry-enriched 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 anyon-permuting symmetries and generalspace-time reflection symmetries. We demonstrate compatibility of therelative anomaly formula with previous results for diagnosing anomaliesfor \mathbb{Z}_2^{T} ℤ 2 T space-time reflection symmetry (e.g. where time-reversal 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 space-time reflection symmetries are intertwined innon-trivial 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 non-trivial anyon permutations.
more »
« less
- PAR ID:
- 10142858
- Date Published:
- Journal Name:
- SciPost Physics
- Volume:
- 8
- Issue:
- 2
- ISSN:
- 2542-4653
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
(3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known 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 fault-tolerant logical operations in topological quantum error-correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 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 non-Abelian 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 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of the interplay of codimension-2 defects with codimension-1 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
-
We investigate the interplay of generalized global symmetries in 2+1 dimensions by introducing a lattice model that couples a ZN clock model to a ZN gauge theory via a topological interaction. This coupling binds the charges of one symmetry to the disorder operators of the other, and when these composite objects condense, they give rise to emergent generalized symmetries with mixed ’t Hooft anomalies. These anomalies result in phases with ordinary symmetry breaking, topological order, and symmetry-protected topological (SPT) order, where the different types of order are not independent but intimately related. We further explore the gapped boundary states of these exotic phases and develop theories for phase transitions between them. Additionally, we extend our lattice model to incorporate a non-invertible global symmetry, which can be spontaneously broken, leading to domain walls with non-trivial fusion rules.more » « less
-
It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D\mathbb{Z}_2 gauge theory with an emergent fermion, and point out that pumping chiralp+ip topological states gives rise to a\mathbb{Z}_{8} 0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization ofT^3 (3-torus) andT^2 \rtimes_{C_2} S^1 (2-torus bundle over the circle) respectively, and pumpingp+ip states. Our considerations also imply the possibility of a logicalT gate by placing the code on\mathbb{RP}^3 and pumping ap+ip topological state.more » « less
-
We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible0 -form symmetries. These symmetries can arise as global symmetries in quantum spin liquids, given by the quotient of the projective symmetry group by a non-normal subgroup as invariant gauge group. We point out that such coset non-invertible symmetries in topological orders can exhibit symmetry fractionalization: each anyon can carry a “fractional charge” under the coset non-invertible symmetry given by a gauge invariant superposition of fractional quantum numbers. We present various examples using field theories and quantum double lattice models, such as fractional quantum Hall systems with charge conjugation symmetry gauged and finite group gauge theory from gauging a non-normal subgroup. They include symmetry enrichedS_3 andO(2) gauge theories. We show that such systems have a fractionalized continuous non-invertible coset symmetry and a well-defined electric Hall conductance. The coset symmetry enforces a gapless edge state if the boundary preserves the continuous non-invertible symmetry. We propose a general approach for constructing coset symmetry defects using a “sandwich” construction: non-invertible symmetry defects can generally be constructed from an invertible defect sandwiched by condensation defects. The anomaly free condition for finite coset symmetry is also identified.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.21468/SciPostPhys.8.2.028
