Lattice Gauge Theories and Spin Models

Abstract

The Wegner $Z_2$ gauge theory-$Z_2$ Ising spin model duality in $(2+1)$ dimensions is revisited and derived through a series of canonical transformations. The Kramers-Wannier duality is similarly obtained. The Wegner $Z_2$ gauge-spin duality is directly generalized to SU(N) lattice gauge theory in $(2+1)$ dimensions to obtain the SU(N) spin model in terms of the SU(N) magnetic fields and their conjugate SU(N) electric scalar potentials. The exact and complete solutions of the $Z_2,U(1),SU(N)$ Gauss law constraints in terms of the corresponding spin or dual potential operators are given. The gauge-spin duality naturally leads to a new gauge invariant magnetic disorder operator for SU(N) lattice gauge theory which produces a magnetic vortex on the plaquette. A variational ground state of the SU(2) spin model with nearest neighbor interactions is constructed to analyze SU(2) gauge theory.

24 pages, 15 figures, 1 table. Enlarged and completely reorganized version with many clarifications and extended discussions (one new appendix, 3 new figures and many new references). The version published in Phys. Rev. D

References in corpus (4)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• A cold-atom quantum simulator for SU(2) Yang-Mills lattice gauge theory
• Optical Abelian Lattice Gauge Theories
• Loop Approach to Lattice Gauge Theories

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