Anyonic order parameters for discrete gauge theories on the lattice

Bais, F.A.; Romers, Jesper

doi:10.1016/j.aop.2009.01.002

Cited by 5 publications

(10 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In ordinary lattice gauge theories [34][35][36][37] the field algebra is link-based. This framework has also been used in studies of Chern-Simons gauge theories on the lattice [14,[38][39][40][41][42]. In our lattice formulation of a doubled Chern-Simons gauge theories, on the other hand, the field algebra is cross-based.…”

Section: Introductionmentioning

confidence: 99%

From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

2015

View full text Add to dashboard Cite

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 R 2 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 2π k . 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.

show abstract

Section: Introductionmentioning

confidence: 99%

From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

2015

View full text Add to dashboard Cite

show abstract

“…[3] requires one to have access to an infinite set of ground-states labeled by a continuous parameter, and is difficult to implement. Recently, Bais et al [20] also discussed extracting S matrix in numerical simulations, by explicit braiding of excitations. In contrast, here we will just use the set of ground states on a torus, to determine the braiding and fusing of gapped excitations.…”

mentioning

confidence: 99%

Quasiparticle statistics and braiding from ground-state entanglement

et al. 2012

View full text Add to dashboard Cite

Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the Topological Entanglement Entropy (TEE) on partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasi-particle excitations of the topological phase. The transformation of these states allows for a determination of the modular S and U matrices which encode quasi-particle properties. We demonstrate our method by extracting the modular S matrix of a chiral spin liquid phase using a Monte Carlo scheme to calculate TEE, and prove that the quasi-particles obey semionic statistics. This method offers a route to a nearly complete determination of the topological order in certain cases.

show abstract

“…Once the condensate is identified, we have shown how to unambiguously reconstruct the S-matrix of the low-energy theory in a broken or unbroken phase by measurements of the complete set of braided loop operators, using the anyonic loop operators we proposed in the earlier work [16]. Due to an auxiliary gauge symmetry these operators are particularly well suited for detecting the nontrivial splittings of fields that correspond to fixed points under fusion with the condensate.…”

Section: Discussionmentioning

confidence: 99%

“…We return to D(H ) and its representations in section 2.2. An important consequence of this underlying symmetry is that it suggested that in the pure DGT a corresponding set of gauge invariant loop operators should exist, thereby generalizing the Wilson/'t Hooft operators to all anyonic species, and indeed such operators have been constructed [16]. With these operators in hand, one can now probe the full phase structure of discrete gauge theories in a lattice formulation and that is what will be done in this paper.…”

Section: Introductionmentioning

confidence: 91%

The modularS-matrix as order parameter for topological phase transitions

Bais

Romers

2012

New J. Phys.

View full text Add to dashboard Cite

We study topological phase transitions in discrete gauge theories in two spatial dimensions induced by the formation of a Bose condensate. We analyse a general class of euclidean lattice actions for these theories which contain one coupling constant for each conjugacy class of the gauge group. To probe the phase structure we use a complete set of open and closed anyonic string operators. The open strings allow one to determine the particle content of the condensate, whereas the closed strings enable us to determine the matrix elements of the modular S-matrix, in both the unbroken and broken phases. From the measured broken S-matrix we may read off the sectors that split or get identified in the broken phase, as well as the sectors that are confined. In this sense the modular S-matrix can be employed as a matrix valued nonlocal order parameter from which the low-energy effective theories that occur in different regions of parameter space can be fully determined. To verify our predictions, we studied a non-abelian anyon model based on the quaternion group H =D 2 of the order of eight by Monte Carlo simulation. We probe part of the phase diagram for the pure gauge theory and find a variety of phases with magnetic condensates leading to various forms of (partial) confinement in complete agreement with the algebraic breaking analysis. Also the order of various transitions is established.

show abstract

Anyonic order parameters for discrete gauge theories on the lattice

Cited by 5 publications

References 23 publications

From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

Quasiparticle statistics and braiding from ground-state entanglement

The modularS-matrix as order parameter for topological phase transitions

Contact Info

Product

Resources

About