SciPost Phys. 16, 122 (2024) ·
published 7 May 2024

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It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a highergroup. Here we study the highergroup structure in (3+1)D $\mathbb{Z}_2$ gauge theory with an emergent fermion, and point out that pumping chiral $p+ip$ topological states gives rise to a $\mathbb{Z}_{8}$ 0form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3group 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 of $T^3$ (3torus) and $T^2 \rtimes_{C_2} S^1$ (2torus bundle over the circle) respectively, and pumping $p+ip$ states. Our considerations also imply the possibility of a logical $T$ gate by placing the code on $\mathbb{RP}^3$ and pumping a $p+ip$ topological state.
SciPost Phys. 16, 089 (2024) ·
published 3 April 2024

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A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we show how such gauge theories possess a highergroup global symmetry, which we study in detail. We derive the $d$group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in $(d+1)$ spacetime dimensions, including nonAbelian gauge groups and DijkgraafWitten twists. We focus on the 1form symmetry generated by invertible (Abelian) magnetic defects and the higherform symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetryprotected topological (SPT) phases. We show that due to a generalization of the Witten effect and chargeflux attachment, the 1form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such highergroup symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the $[O_5] ∈ H^5(BG, U(1))$ obstruction that has appeared in prior work. We also show how the $d$group symmetry is related to faulttolerant nonPauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the $d$group symmetry, such as a controlled Z gate in the (3+1) D $\mathbb{Z}_2$ toric code.
SciPost Phys. 14, 065 (2023) ·
published 11 April 2023

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(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 $\mathbb{Z}_2$ gauge theory with fermionic charges, in $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge theory with bosonic charges, and also in nonAbelian discrete gauge theories based on dihedral ($D_n$) and alternating ($A_6$) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an $H^4$ 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 $A_6$ gauge theory.
SciPost Phys. 8, 028 (2020) ·
published 18 February 2020

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Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetryprotected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetryenriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionalization classes of a given (2+1)D topological order. This method applies to all types of symmetry actions, including anyonpermuting symmetries and general spacetime reflection symmetries. We demonstrate compatibility of the relative anomaly formula with previous results for diagnosing anomalies for $\mathbb{Z}_2^{\bf T}$ spacetime reflection symmetry (e.g. where timereversal squares to the identity) and mixed anomalies for $U(1) \times \mathbb{Z}_2^{\bf T}$ and $U(1) \rtimes \mathbb{Z}_2^{\bf T}$ symmetries. We also study a number of additional examples, including cases where spacetime reflection symmetries are intertwined in nontrivial ways with unitary symmetries, such as $\mathbb{Z}_4^{\bf T}$ and mixed anomalies for $\mathbb{Z}_2 \times \mathbb{Z}_2^{\bf T}$ symmetry, and unitary $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry with nontrivial anyon permutations.