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arXiv:2001.11938 · Physical review. B./Physical review. B

Topological quantum field theory, symmetry breaking, and finite gauge theory in 3+1D

Ryan Thorngren

Published Jun 25, 2020• 29 citations
arXivHTMLPDF

Abstract

We derive a canonical form for 2-group gauge theory in $3+\text{1D}$ which shows that it is equivalent either to Dijkgraaf-Witten theory or to the so-called ``EF1'' topological order of Lan and Wen. According to that classification, recently argued from a different point of view by Johnson-Freyd, this amounts to a very large class of all $3+\text{1D}$ topological quantum field theories (TQFTs). We use this canonical form to compute all possible anomalies of 2-group gauge theory which may occur without spontaneous symmetry breaking, providing a converse of the recent symmetry-enforced-gaplessness constraints of C\'ordova and Ohmori and also uncovering some possible new examples. On the other hand, in cases where the anomaly is matched by a TQFT, we try to provide the simplest possible such TQFT. For example, with anomalies involving time reversal, ${\mathbb{Z}}_{2}$ gauge theory almost always works.

DOI
10.1103/physrevb.101.245160

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We derive a canonical form for 2-group gauge theory in 3+1D3+\text{1D}3+1D which shows that it is equivalent either to Dijkgraaf-Witten theory or to the so-called ``EF1'' topological order of Lan and Wen. According to that classification, recently argued from a different point of view by Johnson-Freyd, this amounts to a very large class of all 3+1D3+\text{1D}3+1D topological quantum field theories (TQFTs). We use this canonical form to compute all possible anomalies of 2-group gauge theory which may occur without spontaneous symmetry breaking, providing a converse of the recent symmetry-enforced-gaplessness constraints of C'ordova and Ohmori and also uncovering some possible new examples. On the other hand, in cases where the anomaly is matched by a TQFT, we try to provide the simplest possible such TQFT. For example, with anomalies involving time reversal, Z2{\mathbb{Z}}_{2}Z2​ gauge theory almost always works.
We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.
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 higher-group global symmetry, which we study in detail. We derive the d d -group global symmetry and its ’t Hooft anomaly for topological finite group gauge theories in (d+1) ( d + 1 ) space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group 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)) [ O 5 ] ∈ H 5 ( B G , U ( 1 ) ) obstruction that has appeared in prior work. We also show how the d d -group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the d d -group symmetry, such as a controlled Z gate in the (3+1) D \mathbb{Z}_2 ℤ 2 toric code.
A bstract In this work, we employ the Tannaka-Krein reconstruction to compute the quantum double 𝒟(𝒢) of a finite 2-group 𝒢 as a Hopf monoidal category. We also construct a 3+1D lattice model from the Dijkgraaf-Witten TQFT functor for the 2-group 𝒢, generalizing Kitaev’s 2+1D quantum double model. Notably, the string-like local operators in this lattice model are shown to form 𝒟(𝒢). Specializing to 𝒢 = ℤ 2 , we demonstrate that the topological defects in the 3+1D toric code model are modules over 𝒟(ℤ 2 ).
Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of non-invertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging the 0-form symmetry gives the 4+1d “inflow” symmetry topological field theory for the non-invertible symmetry. We find a two levels of anomalies: (1) the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and (2) the “Frobenius-Schur indicator” of the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for non-invertible symmetries in 3+1d to be compatible with symmetric gapped phases, and invertible gapped phases. Along the way, we see that the defects characterizing \mathbb{Z}_{4} ℤ 4 ordinary symmetry host worldvolume theories with time-reversal symmetry \mathsf{T} 𝖳 obeying the algebra \mathsf{T}^{2}=C 𝖳 2 = C or \mathsf{T}^{2}=(-1)^{F}C, 𝖳 2 = ( − 1 ) F C , with C C a unitary charge conjugation symmetry. We classify the anomalies of this symmetry algebra in 2+1d and further use these ideas to construct 2+1d topological orders with non-invertible time-reversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain their dynamics.
We use the Symmetry Topological Field Theory (SymTFT) to study and classify gapped phases in (2+1)d for a class of categorical symmetries, referred to as being of bosonic type. The SymTFTs for these symmetries are given by twisted and untwisted (3+1)d Dijkgraaf-Witten (DW) theories for finite groups G G . A finite set of boundary conditions (BCs) of these DW theories is well-known: these simply involve imposing Dirichlet and Neumann conditions on the (3+1)d gauge fields. We refer to these as minimal BCs. The key new observation here is that for each DW theory, there exists an infinite number of other BCs, that we call non-minimal BCs. These non-minimal BCs are all obtained by a ‘theta construction’, which involves stacking the Dirichlet BC with 3d TFTs having G G 0-form symmetry, and gauging the diagonal G G symmetry. On the one hand, using the non-minimal BCs as symmetry BCs gives rise to an infinite number of non-invertible symmetries having the same SymTFT, while on the other hand, using the non-minimal BCs as physical BCs in the sandwich construction gives rise to an infinite number of (2+1)d gapped phases for each such non-invertible symmetry. Our analysis is thoroughly exemplified for G=\mathbb{Z}_2 G = ℤ 2 and more generally any finite abelian group, for which the resulting non-invertible symmetries and their gapped phases already reveal an immensely rich structure.
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group GGG and its group 4-cocycle ω4∈H4[G;U(1)]ω_4 \in \mathcal H^4[G;U(1)]ω4​∈H4[G;U(1)] up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.