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” σ |

G=Z acts on sites | |

μ | σ |

μ | Kink (or string) operator |

Large field (Low T) Broken symmetry μ | Large J/K (High T) Strong coupling Kinks condensed Gauge charges confined Area law for Wilson loops |

Small field (High T) Disordered μ | 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 |

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