Thursday, April 15, 2010

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




``spins μ defined on sites

``spins” σ3 = ±1 defined on links

G=Z2 gauge transformation

acts on sites

μ1 = ±1

σ3 σ3 σ3 σ3


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)


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.

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