Intertwined multiple Chern–Simons gauge fields induce matrix statistics among particles. We analyze this theory on a torus, focusing on the vacuum structure and the Hilbert space. The theory can be mimicked, although not completely, by an effective theory with one Chern–Simons gauge field. The correspondence between the Wilson line integrals, vacuum degeneracy and wave functions for these two theories are discussed. Further, it is found in both of these cases that the two total momenta and Hamiltonian commute only in the physical Hilbert space.
We regularize compact and non-compact Abelian Chern-Simons-Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group
R, each local Hilbert space is analogous to the one of a charged "particle" moving in the link-pair group space
R2 in a constant "magnetic" background field. In the compact case, the link-pair group space is a torus
U(1)2 threaded by
k units of quantized "magnetic" flux, with
k being the level of the Chern-Simons theory. The holonomies of the torus
U(1)2 give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from
U(1) to
Z(k). The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern-Simons limit of a large "photon" mass, this results in a
Z(k)-symmetric variant of Kitaev's toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle
k2π. Non-Abelian
U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.
Abstract We study the multi-boundary entanglement structure of the states prepared in (1+1) and (2+1) dimensional Chern-Simons theory with finite discrete gauge group G. The states in (1+1)-d are associated with Riemann surfaces of genus g with multiple S 1 boundaries and we use replica trick to compute the entanglement entropy for such states. In (2+1)-d, we focus on the states associated with torus link complements which live in the tensor product of Hilbert spaces associated with multiple T 2. We present a quantitative analysis of the entanglement structure for both abelian and non-abelian groups. For all the states considered in this work, we find that the entanglement entropy for direct product of groups is the sum of entropy for individual groups, i.e. EE(G 1 × G 2) = EE(G 1) + EE(G 2). Moreover, the reduced density matrix obtained by tracing out a subset of the total Hilbert space has a positive semidefinite partial transpose on any bi-partition of the remaining Hilbert space.
Abstract Topological entanglement structure amongst disjoint torus boundaries of three manifolds have already been studied within the context of Chern-Simons theory. In this work, we study the topological entanglement due to interaction between the quasiparticles inside three-manifolds with one or more disjoint S 2 boundaries in SU(N) Chern-Simons theory. We focus on the world-lines of quasiparticles (Wilson lines), carrying SU(N) representations, creating four punctures on every S 2. We compute the entanglement entropy by partial tracing some of the boundaries. In fact, the entanglement entropy depends on the SU(N) representations on these four-punctured S 2 boundaries. Further, we observe interesting features on the GHZ-like and W-like entanglement structures. Such a distinction crucially depends on the multiplicity of the irreducible representations in the tensor product of SU(N) representations.
Lattice Hamiltonians (e.g. Levin and Wen 2005 Phys. Rev. B 71 045110) can be constructed that have a low energy description, that is a doubled Chern–Simons (CS) theory—two independent opposite chirality topological sectors. We show that the partition function of these theories is an expectation of Wilson loops that form a link in 2+1 dimensional spacetime known in the mathematical literature as chain–mail. This geometric construction establishes a concrete connection between the lattice models and continuum Chern–Simons theories, allowing us to use well-established results on the latter to obtain a physical interpretation of the lattice model Hilbert space and Hamiltonian, its topological invariance, exactness under coarse-graining and how two opposite chirality sectors of the doubled theory arise. These features of the lattice models can thus be situated in the broader context of topological invariants obtained from CS theories.
We consider Chern-Simons theory for gauge group G at level k on 3-manifolds M n with boundary consisting of n topologically linked tori. The Euclidean path integral on M n defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where M n is the link-complement of some n-component link inside the three-sphere S 3. The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k (G = U(1) k ) we give a general formula for the entanglement entropy associated to an arbitrary (m|n − m) partition of a generic n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod k) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k). For G = SU(2) k , we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a “W-like” entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have “GHZ-like” entanglement (i.e., tracing out one torus does lead to a separable state).
We compute various averages over bulk geometries of quantum states prepared by the Chern-Simons path integral, for any level k and compact simple gauge group G . We do so by carefully summing over all topologically distinct bulk geometries which have n disjoint boundary tori and a decomposition into space × time of fixed spatial topology. We find that to leading order in the complexity of the state, the typical state contains many types of multiparty entanglement, proving a conjecture of Balasubramanian Combined with the fact that periodic link states are Greenberger–Horne–Zeilinger (GHZ)-like, this result implies that a generic fibered link state contains non-GHZ entanglement if and only if its link complement is hyperbolic. Additionally, we compute an averaged wave function which captures the leading order statistics of boundary observables in the n torus Chern-Simons Hilbert space.