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Register a new userEntropic Topological Invariants in Three Dimensions
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We evaluate the entanglement entropy of exactly solvable Hamiltonians corresponding to general families of three-dimensional topological models. We show that the modification to the entropic area law due to three-dimensional topological properties is richer than the two-dimensional case. In addition to the reduction of the entropy caused by non-zero vacuum expectation value of contractible loop operators a new topological invariant appears that increases the entropy if the model consists of non-trivially braiding anyons. As a result the three-dimensional topological entanglement entropy provides only partial information about the two entropic topological invariants.
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We present a fully many-body formulation of topological invariants for various topological phases of fermions protected by antiunitary symmetry, which does not refer to single particle wave functions. For example, we construct the many-body $mathbb{Z}_2$ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele $mathbb{Z}_2$ invariant written in terms of single-particle Bloch wave functions. We show that an important ingredient for the construction of the many-body topological invariants is a fermionic partial transpose which is basically the standard partial transpose equipped with a sign structure to account for anti-commuting property of fermion operators. We also report some basic results on various kinds of pin structures -- a key concept behind our strategy for constructing many-body topological invariants -- such as the obstructions, isomorphism classes, and Dirac quantization conditions.
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.
The computation of certain obstruction functions is a central task in classifying interacting fermionic symmetry-protected topological (SPT) phases. Using techniques in group-cohomology theory, we develop an algorithm to accelerate this computation. Mathematically, cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear basis, known as the resolutions of the group. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinite to finite. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups.
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.
Topological qauntum field theory(TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In $2+1$D, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasi-particle braiding statistics, chiral central charge and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in $3+1$D and it has been shown that loop like excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and loop like excitation in $3+1$D. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.
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