Doubled lattice Chern-Simons-Yang-Mills theories with discrete gauge group

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• 作者：S. Caspar ; D. Mesterhá ; zy ; ^{mesterh@itp.unibe.ch" class="auth_mail" title="E-mail the corresponding author} ; T.Z. Olesen ; N.D. Vlasii ; U.-J. Wiese
• 关键词：Chern&ndash ; Simons theory ; Lattice gauge theory ; Topological phases ; Confinement ; Quantum information ; Toric code
• 刊名：Annals of Physics
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：374
• 期：Complete
• 页码：255-290
• 全文大小：832 K

文摘

We construct doubled lattice Chern–Simons–Yang–Mills theories with discrete gauge group G$G$ in the Hamiltonian formulation. Here, these theories are considered on a square spatial lattice and the fundamental degrees of freedom are defined on pairs of links from the direct lattice and its dual, respectively. This provides a natural lattice construction for topologically-massive gauge theories, which are invariant under parity and time-reversal symmetry. After defining the building blocks of the doubled theories, paying special attention to the realization of gauge transformations on quantum states, we examine the dynamics in the group space of a single cross, which is spanned by a single link and its dual. The dynamics is governed by the single-cross electric Hamiltonian and admits a simple quantum mechanical analogy to the problem of a charged particle moving on a discrete space affected by an abstract electromagnetic potential. Such a particle might accumulate a phase shift equivalent to an Aharonov–Bohm phase, which is manifested in the doubled theory in terms of a nontrivial ground-state degeneracy on a single cross. We discuss several examples of these doubled theories with different gauge groups including the cyclic group Z(k)⊂U(1)$\mathbb{Z}\left(k\right)\subset U\left(1\right)$, the symmetric group ae2e61883fa070b11da1803fdf6" title="Click to view the MathML source">S₃⊂O(2)$S_3\subset O\left(2\right)$, the binary dihedral (or quaternion) group aec24fad9e8ef71d7d">${\bar{D}}_2\subset SU\left(2\right)$, and the finite group Δ(27)⊂SU(3)$\Delta \left(27\right)\subset SU\left(3\right)$. In each case the spectrum of the single-cross electric Hamiltonian is determined exactly. We examine the nature of the low-lying excited states in the full Hilbert space, and emphasize the role of the center symmetry for the confinement of charges. Whether the investigated doubled models admit a non-Abelian topological state which allows for fault-tolerant quantum computation will be addressed in a future publication.