We study the realization of anyon-permuting symmetries of topological phases on the lattice using tensor networks. Working on the virtual level of a projected entangled pair state, we find matrix product operators (MPOs) that realize all unitary topological symmetries for the toric and color codes. These operators act as domain walls that enact the symmetry transformation on anyons as they cross. By considering open boundary conditions for these domain wall MPOs, we show how to introduce symmetry twists and defect lines into the state.
We provide a superselection theory of symmetry defects in 2+1D symmetry enriched topological (SET) order in the infinite volume setting. For a finite symmetry group
G with a unitary on-site action, our formalism produces a
G-crossed braided tensor category
GSec. This superselection theory is a direct generalization of the usual superselection theory of anyons, and thus is consistent with this standard analysis in the trivially graded component
GSec1. This framework also gives us a completely rigorous understanding of symmetry fractionalization. To demonstrate the utility of our formalism, we compute
GSec explicitly in both short-range and long-range entangled spin systems with symmetry and recover the relevant skeletal data.
We investigate the algebraic theory of symmetry-enriched topological (SET) order in (2+1)D bosonic topological phases of matter and its applications to topological quantum computing. Our goal is twofold: first, to demonstrate how an abstract categorical approach can be applied to understand phenomena in (2+1)D topological phases of matter, and second, to show how ideas from physics can be useful for categorification.After reviewing modular tensor categories (MTCs) and their role as algebraic theories of anyons in topological phases of matter, we recall their associated quantum representations and their interpretation as quantum gates for a topological quantum computer. Next we recall the characterization of SET order in terms of
G-crossed braided extensions of MTCs and the mathematical formalism of topological quantum computing (TQC) with anyons and symmetry defects.We then apply modular tensor category theory to construct algebraic models of symmetry defects in multi-layer (2+1)D topological order with layer-exchange permutation symmetry. Our main result frames a correspondence between bilayer symmetry enriched topological order and monolayer topological order on surfaces with genus, illuminating a connection between quantum symmetry and topological order that first appeared in the work of condensed matter theorists Barkeshi, Ji, and Qian.
This thesis contributes to the understanding of symmetry-enriched topological phases focusing on their descriptions in terms of tensor network states. The Projected Entangled Pair State (PEPS) formalism allows us to locally encode the main properties of the models (like topological order, symmetries and their interplay) in the tensors. We have used that encoding to classify, construct and detect some classes of symmetry-enriched topological phases in 2D PEPS. For that purpose, we have studied what the allowed freedom in the tensors generating the same tensor network state is. These results are the so-called 'fundamental theorems' and they give the necessary knowledge to properly study symmetries (actions that leave the states invariant). We focus our study on the family of PEPS describing quantum double models of
G, the so-called
G-injective PEPS, and on global on-site symmetries coming from a finite group
Q: as a result of the classification we have a finite number of phases closely related to the theory of group extensions. We also provide the maps that appear in group extension theory with physical meaning and we characterize their actions on the ground subspace and on the anyons. We construct a representative of each class of our classification. Moreover, we connect our construction with an interesting physical phenomenon, the so-called anyon condensation, which describes topological phase transitions. We have also proposed a family of gauge invariant quantities and their corresponding order parameters (in the form of expectation values of a local operator in 2D) to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns. We conclude this thesis by touching on some mathematical open problems in PEPS.
Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a
G-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension
E of
G by
N: an
E-graded fusion category induces a
G-SET input by passing to the quotient symmetry
G, and the resulting input naturally carries a compatible
N-grading that implements the further condensation inside the SET phase while preserving
G. We illustrate the construction using three quantum-double examples: the trivial extension
Z2×Z2, the non-Abelian semidirect product
S3, and the nontrivial central extension
Z4. The
Z4 example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.
Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations—anyons—and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the global symmetry group G , which we take to be discrete, does not change topological superselection sectors—i.e. does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group H 2 ( G , A a b e l i a n ) being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G , one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic ‘twist’ defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418 ), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.