Anomalous Symmetry Fractionalization and Surface Topological Order

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 $H^4(G,\mathrm{U}(1))$, 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 [$\mathrm{U}(1{)}_2$] topological order with a reduced symmetry ${\mathbb{Z}}_2\times {\mathbb{Z}}_2\subset \mathrm{SO}(3)$, 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.

Popular Summary

Two-dimensional quantum many-body systems may support exotic excitations known as anyons, which are realized in, for example, fractional quantum Hall states. Anyons possess two key properties: They exhibit exotic statistics, distinct from those of ordinary bosons or fermions, and they can carry fractional amounts of charge. These two properties are not independent: The anyons’ statistics impose constraints on the fractional charges that they can carry. However, one can ask whether these constraints are sufficient (i.e., are there states of matter with anyons that are impossible to realize despite exhibiting ostensibly admissible patterns of charge fractionalization). Here, we show that indeed such anomalous theories exist that are impossible to realize in two dimensions.

Furthermore, we show that anomalous theories of this variety, although impossible to realize in a purely two-dimensional setting, can be realized as surface states of a three-dimensional system. We construct an exactly solved lattice model for such a three-dimensional system that is gapped with no interesting bulk excitations, a so-called “symmetry protected topological” phase. Our model yields a concrete connection to a class of rigorous mathematical results, which, given a particular pattern of symmetry fractionalization, can (i) determine if it is anomalous, and, if so, (ii) identify the three-dimensional symmetry-protected topological phase that “cures” the anomaly. When this anomaly vanishes, we show that the system can indeed be realized purely in two dimensions, which constrains the set of possible gapped spin-liquid phases in frustrated magnets that typically have unbroken symmetries in addition to supporting anyons. We illustrate this general theory for a specific example, a variant of the Kalmeyer-Laughlin chiral spin liquid.

We expect that our result can be generalized to apply to anti-unitary time-reversal symmetry in topological phases as well.

Figure 1

In this paper, we study 2D anomalous SETs as surface state of nontrivial 3D SPT phase. On the one hand, we detect anomaly in surface SET from gauging obstruction; on the other hand, we construct the bulk model and detect the nontrivial SPT order with three-loop braiding statistics.

Figure 2

The fusion rule ${\mathrm{\Omega }}_x\times {\mathrm{\Omega }}_x=s$ derived from the symmetry action on the semion in theory $X$ [defined in Eq. (3)]. (a) Bringing a gauge flux ${\mathrm{\Omega }}_x$ around the center semion is equivalent to acting locally on it with the corresponding symmetry $g_x$. (b) Bringing two ${\mathrm{\Omega }}_x$ gauge fluxes around the center semion gives rise to a $-1$ phase factor, because in theory $X$ a semion carries half of the corresponding $g_x$ charge. This $-1$ can be reproduced by bringing another semion around the center one, giving rise to the $s$ on the right-hand side of the fusion rule.

Figure 3

Braiding and fusion statistics of the semion theory. (a) The exchange of two semions leads to a phase factor of $i$. (b) The two ways of fusing three semions differ by a sign.

Figure 4

Graphical representation of the pentagon equation involving four fluxes. According to Theorem 8.10 of Ref. [23], this pentagon equation can fail by an overall phase $\nu (f,g,h,k)$, and the phase can be computed as a product of $F$ and $R$ matrices of the original theory, applied to anyons determined by the fractionalization class $\omega$. These $F$ and $R$ matrices can intuitively be thought of as contributions coming from five additional steps in the pentagon equation, in which the various $\omega$ anyons are reassociated or exchanged as required. We stress that this is just a mnemonic for the precise formula given in Theorem 8.10 of Ref. [23].

Figure 5

Some configurations in the semion ground state. The vertex condition ensures that only configurations with an even number of edges on which $n_i=1$ (shown in blue) can meet at a vertex, such that the ground state is a superposition of loops. The relative amplitudes of these loop configurations are given by the phase factor $\mathrm{\Theta }(P){\mathrm{\Phi }}_{O,O^{\prime }}{\mathrm{\Phi }}_{U,U^{\prime }}$ in Eq. (15).

Figure 6

(a) We define our model on the point-split cubic lattice, because the Hamiltonian is simplest for a lattice with only trivalent vertices. (b) The plaquette term is defined with respect to a fixed angle of projection. In this angle, there are two edges ($O$ and $U$) that are projected into each plaquette from above and below, respectively, shown in red. These (and their partner edges $O^{\prime }$ and $U^{\prime }$, shown here in blue) are used to define the phase factors ${\mathrm{\Phi }}_{O,O^{\prime }},{\mathrm{\Phi }}_{U,U^{\prime }}$ in the plaquette operator.

Figure 7

The semion string operator on the surface. The curve $C$ connecting the surface vertices $V_1$ and $V_2$ is shown in red. A second copy of $C$ (shown in dashed blue), displaced from $C$ along the $(-1,-1,1)$ direction, determines the set of $R$ vertices. These are vertices where the dashed blue curve crosses over an edge $e_i$ (shown in green) of the surface. The edge $i+1$ is the edge in $C$ that shares a vertex with $e_i$ and is crossed second when following $C$ such that $C$ turns to the right at these crossings.

Figure 8

Beginning with the configuration shown in (a), we exchange two open string ends at the surface. Returning the strings to their original configuration (b) multiplies the eigenstate of the Walker-Wang model by $\pm i$, demonstrating that these defects have semionic statistics.

Figure 9

(a) Low-energy and (b) high-energy configurations at each vertex. Dotted lines represent links without strings, while solid lines represent links with strings. Empty circles represent spinless vertex particles, while solid circles represent spin-$1/2$’s. Two connected circles form a singlet.

Figure 10

Possible configuration where no isolated spin-$1/2$’s appear near the end of strings, showing that such excitations are confined. The solid lines have a semion label, and dashed lines have a trivial label. The violated vertices are indicated with green arrows, and red arrows indicate edges that are not in a ground state of $H_0$.

Figure 11

Symmetry action and surface semion strings. (a) Acting with the symmetry operator $R_x$ on all spins inside the gray region creates a bulk domain surface (the boundaries of the gray box) that ends on a surface domain wall (shown in red). This operation twists each singlet-chain link piercing the domain surface by $U_x$ (or $U_x^{†}$). Performing this transformation twice multiplies the wave function by a phase factor $(-1{)}^{n_s}$, where $n_s$ is the number of edges piercing the domain surface that carry semion labels. (b) Diagrammatically, this can be represented by encircling each edge with a small semion loop. (c) Acting with a product of plaquette operators on the shaded plaquettes reproduces this operator in the bulk but leaves a large loop running below the surface along the domain wall. (d) To obtain the correct phase factor, an additional semion string (shown here in blue) must be added at the surface along the domain wall. Note that, for simplicity, we suppress the point splitting of the cubic lattice in this figure.

Figure 12

Intersecting pairs of domain walls. (a) Here, we first act with $g$ on all spins inside the blue-shaded region, twisting all singlet chains along the set of red edges by $g$. We next act with $h$ on all spins inside the mauve-shaded region. This will twist the singlet chains on a second set of bonds (shown in green) by $h$, and also generate a phase factor of $(-1{)}^{n_s}$ for each red edge inside the mauve region. (b) When acting on states where $B_P\equiv 1$, this is equivalent to running a semion string (thick blue line) around the boundary.

Figure 13

The operator ${\mathbf{Z}}_{xy}$ acts on the gauge field by changing by $Z\in G$ the values of the gauge field variables on the indicated links; in other words, it inserts a $Z$ flux in the $z$ direction. Similarly, operators that insert flux in the other directions act in the other planes.

Figure 14

The Walker-Wang part of the membrane operator that inserts a flux of $X$ on a state that already has a flux of $Z$ in a perpendicular direction. Naive successive application of the twist operators defined in the text leave a line of violated plaquettes at the intersection of the two fluxes. These are satisfied by running a semion string operator (blue line) through them. This blue semion string is glued into the lattice skeleton of our model using the local Walker-Wang graphical rules.

Figure 15

Graphical illustration of the two orderings, Eqs. (44) and (45). They result in two states that differ by $\pm 1$ in the nonanomalous model, but $\pm i$ in the anomalous projective semion model $Y$ theory, as explained in the text.