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Register a new userSymmetry fractionalization and anomaly detection in three-dimensional topological phases
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In a phase with fractional excitations, topological properties are enriched in the presence of global symmetry. In particular, fractional excitations can transform under symmetry in a fractionalized manner, resulting in different Symmetry Enriched Topological (SET) phases. While a good deal is now understood in $2D$ regarding what symmetry fractionalization patterns are possible, the situation in $3D$ is much more open. A new feature in $3D$ is the existence of loop excitations, so to study $3D$ SET phases, first we need to understand how to properly describe the fractionalized action of symmetry on loops. Using a dimensional reduction procedure, we show that these loop excitations exist as the boundary between two $2D$ SET phases, and the symmetry action is characterized by the corresponding difference in SET orders. Moreover, similar to the $2D$ case, we find that some seemingly possible symmetry fractionalization patterns are actually anomalous and cannot be realized strictly in $3D$. We detect such anomalies using the flux fusion method we introduced previously in $2D$. To illustrate these ideas, we use the $3D$ $Z_2$ gauge theory with $Z_2$ global symmetry as an example, and enumerate and describe the corresponding SET phases. In particular, we find four non-anomalous SET phases and one anomalous SET phase, which we show can be realized as the surface of a $4D$ system with symmetry protected topological order.
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Symmetry fractionalization describes the fascinating phenomena that excitations in a 2D topological system can transform under symmetry in a fractional way. For example in fractional quantum Hall systems, excitations can carry fractional charges while the electrons making up the system have charge one. An important question is to understand what symmetry fractionalization (SF) patterns are possible given different types of topological order and different symmetries. A lot of progress has been made recently in classifying the SF patterns, providing deep insight into the strongly correlated experimental signatures of systems like spin liquids and topological insulators. We review recent developments on this topic. First, it was shown that the SF patterns need to satisfy some simple consistency conditions. More interesting, it was realized that some seemingly consistent SF patterns are actually `anomalous, i.e. they cannot be realized in strictly 2D systems. We review various methods that have been developed to detect such anomalies. Applying such an understanding to 2D spin liquid allows one to enumerate all potentially realizable SF patterns and propose numerical and experimental probing methods to distinguish them. On the other hand, the anomalous SF patterns were shown to exist on the surface of 3D systems and reflect the nontrivial order in the 3D bulk. We review examples of this kind where the bulk states are topological insulators, topological superconductors, or have other symmetry protected topological orders.
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 an SET of a discrete, onsite, 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 are captured by the fourth cohomology group $H^4( G, ,U(1))$, which also precisely labels the set of 3D SPT phases, with symmetry group $G$. 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 ($U(1)_2$) topological order with a reduced symmetry $mathbb{Z}_2 times mathbb{Z}_2 subset SO(3)$, 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 non-trivial by computing three loop braiding statistics. Possible extensions to anti-unitary symmetries are also discussed.
We review the dimensional reduction procedure in the group cohomology classification of bosonic SPT phases with finite abelian unitary symmetry group. We then extend this to include general reductions of arbitrary dimensions and also extend the procedure to fermionic SPT phases described by the Gu-Wen super-cohomology model. We then show that we can define topological invariants as partition functions on certain closed orientable/spin manifolds equipped with a flat connection. The invariants are able to distinguish all phases described within the respective models. Finally, we establish a connection to invariants obtained from braiding statistics of the corresponding gauged theories.
We study the Z2 topologically ordered surface state of three-dimensional bosonic SPT phases with the discrete symmetries G1 x G2. It has been argued that the topologically ordered surface state cannot be realized on a purely two-dimensional lattice model. We carefully examine the statement and show that the surface state should break G2 if the symmetry G1 is gauged. This manifests the conflict of the symmetry G1 and G2 on the surface of the three-dimensional SPT phase. Given that there is no such phenomena in the purely two-dimensional model, it signals that the symmetries are encoded anomalously on the surface of the three-dimensional SPT phases and that the surface state can never be realized on the purely two-dimensional models.
We study Abelian braiding statistics of loop excitations in three-dimensional (3D) gauge theories with fermionic particles and the closely related problem of classifying 3D fermionic symmetry-protected topological (FSPT) phases with unitary symmetries. It is known that the two problems are related by turning FSPT phases into gauge theories through gauging the global symmetry of the former. We show that there exist certain types of Abelian loop braiding statistics that are allowed only in the the presence of fermionic particles, which correspond to 3D intrinsic FSPT phases, i.e., those that do not stem from bosonic SPT phases. While such intrinsic FSPT phases are ubiquitous in 2D systems and in 3D systems with anti-unitary symmetries, their existence in 3D systems with unitary symmetries was not confirmed previously due to the fact that strong interaction is necessary to realize them. We show that the simplest unitary symmetry to support 3D intrinsic FSPT phases is $mathbb{Z}_2timesmathbb{Z}_4$. To establish the results, we first derive a complete set of physical constraints on Abelian loop braiding statistics. Solving the constraints, we obtain all possible Abelian loop braiding statistics in 3D gauge theories, including those that correspond to intrinsic FSPT phases. Then, we construct exactly soluble state-sum models to realize the loop braiding statistics. These state-sum models generalize the well-known Crane-Yetter and Dijkgraaf-Witten models.
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