An introduction to lattice gauge theory and spin systems
- 1 October 1979
- journal article
- research article
- Published by American Physical Society (APS) in Reviews of Modern Physics
- Vol. 51 (4), 659-713
- https://doi.org/10.1103/revmodphys.51.659
Abstract
This article is an interdisciplinary review of lattice gauge theory and spin systems. It discusses the fundamentals, both physics and formalism, of these related subjects. Spin systems are models of magnetism and phase transitions. Lattice gauge theories are cutoff formulations of gauge theories of strongly interacting particles. Statistical mechanics and field theory are closely related subjects, and the connections between them are developed here by using the transfer matrix. Phase diagrams and critical points of continuous transitions are stressed as the keys to understanding the character and continuum limits of lattice theories. Concepts such as duality, kink condensation, and the existence of a local, relativistic field theory at a critical point of a lattice theory are illustrated in a thorough discussion of the two-dimensional Ising model. Theories with exact local (gauge) symmetries are introduced following Wegner's Ising lattice gauge theory. Its gauge-invariant "loop" correlation function is discussed in detail. Three—dimensional Ising gauge theory is studied thoroughly. The renormalization group of the two dimensional planar model is presented as an illustration of a phase transition driven by the condensation of topological excitations. Parallels are drawn to Abelian lattice gauge theory in four dimensions. Non-Abelian gauge theories are introduced and the possibility of quark confinement is discussed. Asymptotic freedom of O(n) Heisenberg spin systems in two dimensions is verified for and is explained in simple terms. The direction of present-day research is briefly reviewed.
Keywords
This publication has 43 references indexed in Scilit:
- Experiments with a Gauge-Invariant Ising SystemPhysical Review Letters, 1979
- Topological excitations in the Abelian Higgs modelPhysical Review D, 1978
- Phase transitions in Abelian lattice gauge theoriesNuclear Physics B, 1977
- Gauge fixing, the transfer matrix, and confinement on a latticePhysical Review D, 1977
- Phase transition in the two-dimensional Coulomb gas, and the interfacial roughening transitionPhysical Review B, 1976
- Impossibility of spontaneously breaking local symmetriesPhysical Review D, 1975
- Gauge fields on a lattice. III. Strong-coupling expansions and transition pointsPhysical Review D, 1975
- Broken Scale Invariance in Scalar Field TheoryPhysical Review D, 1970
- Exact Results in the Kondo Problem. II. Scaling Theory, Qualitatively Correct Solution, and Some New Results on One-Dimensional Classical Statistical ModelsPhysical Review B, 1970
- All Possible Symmetries of theMatrixPhysical Review B, 1967