Anomalous Symmetry Fractionalization and Surface Topological Order

Abstract

In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways, leading to a variety of symmetry-enriched topological (SET) phases. While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems. Instead, certain "anomalous" SETs can only occur on the surface of a 3D symmetry-protected topological (SPT) phase. In this paper, we describe a procedure for determining whether a SET of a discrete, on-site, unitary symmetry group G is anomalous or not. The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes. Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions is captured by the fourth cohomology group H 4 ðG; Uð1ÞÞ, which also precisely labels the set of 3D SPT phases, with symmetry group G. An explicit procedure for calculating the cohomology data from a SET is given, with the corresponding physical intuition explained. We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3D bulk SPT whose surface termination realizes it. We illustrate this idea using the chiral spin liquid [Uð1Þ 2 ] topological order with a reduced symmetry Z 2 × Z 2 ⊂ SOð3Þ, which can act on the semion quasiparticle in an anomalous way. We construct exactly solved 3D SPT models realizing the anomalous surface terminations and demonstrate that they are nontrivial by computing three-loop braiding statistics. Possible extensions to antiunitary symmetries are also discussed.

Symmetry fractionalization and anomaly detection in three-dimensional topological phases

“…References [10,11,22,44] provide a detailed explanation for this formula and its relation to symmetry fractionalization in 2D. We only want to mention that the projective phase factors e iα a (g 1 ,g 2 ) in Eq.…”

“…It was realized that some of the SF patterns are anomalous, i.e., they cannot be realized in strictly 2D systems [14][15][16][17][18][19][20][21][22][23][24][25][26]. Various anomaly detection methods have been proposed to identify such SF patterns.…”

“…Various anomaly detection methods have been proposed to identify such SF patterns. The central idea of most methods is based on introducing symmetry fluxes into the system and trying to gauge the global symmetry [10,[22][23][24][25][26]. If the SF pattern can be realized in a strictly 2D model, then the global symmetry can be consistently gauged, resulting in a larger topological theory including the original quasiparticles and the symmetry fluxes and charges.…”

Dual Dirac Liquid on the Surface of the Electron Topological Insulator

“…The price we pay by symmetrically gapping out the surface is to introduce intrinsic topological order, i.e., with fractionalized anyon excitations on the surface. This was first demonstrated in bosonic analogues of the topological insulators [4][5][6][7], which are also known as symmetry-protected topological phases [8][9][10], and subsequently in fermionic systems [11] including the FKM topological insulator [12][13][14][15]. More recently, another interesting interacting surface state called "composite Dirac liquid" was proposed [16].…”

Topological orders with global gauge anomalies

“…For example, the "anomalous" boundary topological order of 3d topological insulator and topological superconductor 3 He-B phase has already been studied [10][11][12][13][14]. And one natural boundary state of a 3d bosonic SPT state [15,16] is a Z 2 topological orders whose e and m excitations carry fractional quantum numbers of the symmetry [17,18] (some systems can also have a different boundary topological order with semions [19]), and this particular kind of fractionalization cannot exist in 2d, even though it is consistent with all the fusion rules of e and m excitations.…”