We construct in the
K matrix formalism concrete examples of symmetry-enriched topological phases, namely intrinsically topological phases with global symmetries. We focus on the Abelian and nonchiral topological phases and demonstrate by our examples how the interplay between the global symmetry and the fusion algebra of the anyons of a topologically ordered system determines the existence of gapless edge modes protected by the symmetry and that a (quasi)group structure can be defined among these phases. Our examples include phases that display charge fractionalization and more exotic nonlocal anyon exchange under global symmetry that correspond to general group extensions of the global symmetry group.
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.
The hallmark of a two-dimensional (2d) topologically ordered phase is the existence of deconfined ``anyon'' excitations that have exotic braiding and exchange statistics, different from those of ordinary bosons or fermions. As opposed to conventional Landau-Ginzburg-Wilson phases, which are classified on the basis of the spontaneous breaking of an underlying symmetry, topologically ordered phases, such as those occurring in the fractional quantum Hall effect, are absolutely stable, not requiring any such symmetry. Recently, though, it has been realized that symmetries, which may still be present in such systems, can give rise to a host of new, distinct, many-body phases, all of which share the same underlying topological order. These ``symmetry enriched'' topological (SET) phases are distinguished not on the basis of anyon braiding statistics alone, but also by the symmetry properties of the anyons, such as their fractional charges, or the way that different anyons are permuted by the symmetry. Thus a useful approach to classifying SETs is to determine all possible such symmetry actions on the anyons that are algebraically consistent with the anyon statistics. Remarkably, however, there exist symmetry actions that, despite being algebraically consistent, cannot be realized in any physical system, and hence do not lead to valid 2d SETs. One class of such ``anomalous'' SETs, characterized by certain disallowed symmetry fractionalization patterns, finds a physical interpretation as an allowed surface state of certain three-dimensional (3d) short-range entangled phases, but another, characterized by some seemingly valid but anomalous permutation actions of the symmetry on the anyons and encoded in an
H3(G,A) group cohomology class, has so far eluded a physical interpretation. In this work, we find a way to physically realize these anomalously permuting SETs at the surfaces of certain 3d long-range entangled phases, expanding our understanding of general anomalous SETs in two dimensions.
Symmetry-enriched topological phases (SET) exhibit both long-range entanglement and intricate interplay with global symmetries, such as anyon permutation symmetry or fractionalization of quantum numbers, going beyond the Landau paradigm of symmetry breaking. Here, the authors study two-dimensional SET phases based on the novel concept of symmetric local unitary transformations. Using the fixed-point structure of wave functions, they construct systematically exactly solvable models that can describe generic nonchiral SET phases. The technique applies to on-site as well as spatial symmetries and, furthermore, to anomalous symmetries that can only be realized on the surface of three-dimensional symmetry-protected phases.
Topologically ordered phases of matter, in particular so-called symmetry-enriched topological phases, can exhibit quantum number fractionalization in the presence of global symmetry. In
Z2 topologically ordered states in two dimensions, fundamental translations
Tx and
Ty acting on anyons can either commute or anticommute. This property, crystal momentum fractionalization, can be seen in a periodicity of the excited-state spectrum in the Brillouin zone. We present a numerical method to detect the presence of this form of symmetry enrichment given a projected entangled pair state; we study the minima of the spectrum of correlation lengths of the transfer matrix for a cylinder. As a benchmark, we demonstrate our method using a modified toric code model with perturbation. An enhanced periodicity in momentum clearly reveals the nontrivial anticommutation relation
Tx,Ty=0 for the corresponding quasiparticles in the system.