Basic aspects of lattice gauge theories

Gauge theories can describe phase transitions which do not involve symmetry breaking.

A key aspect is described by Elitzur’s theorem: in a lattice gauge theory any operator which is not locally gauge invariant must have an expectation value of zero.

The Z2 lattice gauge theory in 2+1 dimension can be mapped onto the two dimensional transverse field Ising model at zero temperature. The Table below describes the relationship between key quantities in the two theories. An important aspect of the Z2 lattice gauge theory is that it exhibits a confinement-deconfinement transition.

2dim transverse field Ising at T=0 (3d classical Ising)	2+1-dim Z2 lattice gauge (3d Z2 lattice gauge theory)
Dual lattice	Lattice
site	Plaquette
``spins” μ defined on sites	``spins” σ3 = ±1 defined on links
	G=Z2 gauge transformation acts on sites
μ1 = ±1	σ3 σ3 σ3 σ3
μ3	Kink (or string) operator
Large field (Low T) Broken symmetry μ3 ≠ 0	Large J/K (High T) Strong coupling Kinks condensed Gauge charges confined Area law for Wilson loops
Small field (High T) Disordered μ3 =0	Small J/K (Low T) Weak coupling Gauge charges deconfined Topological order Perimeter law
Transverse field, K	Vison (Z2 vortex) energy
Spin-spin interaction, J	Vison hopping energy

A useful pedagogical review which describes the relationship between lattice gauge theories and lattice spin models is by John Kogut.