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An introduction to lattice gauge theory and spin systems
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 d…
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“The lattice Hamiltonian of Eq. ( 1) can be written in the reciprocal space as”
From Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions (2025) · cites this paper as [6]
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“As of now, non-Abelian lattice gauge fields have been realized experimentally using cold atoms [4,5] and electric circuits [6].”
From Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions (2025) · cites this paper as [6]
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Metadata
- Authors
- Kogut, J. B.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- Corresponding author
- Kogut, John B.
- ORCID
- Kogut, John B.
- Venue
- Reviews of Modern Physics
- Publisher
- American Physical Society (APS)
- Published
- 1979-10-01
- Volume
- 51
- Issue
- 4
- Pages
- 659-713
- ISSN
- 0034-6861 (ISSN-L)
- OpenAlex
- W2125986857
- Cited by
- 2,029
- References
- 49
- Citations by year
- 2026: 65 · 2025: 116 · 2024: 102 · 2023: 92 · 2022: 95 · 2021: 76 · 2020: 79 · 2019: 94 · 2018: 55 · 2017: 59 · 2016: 50 · 2015: 42 · 2014: 36 · 2013: 55 · 2012: 36
- FWCI
- 39.29
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Reviews of Modern Physics (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Quantum Chromodynamics and Particle Interactions
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Particle physics theoretical and experimental studies
- Keywords
- lattice gauge theory, spin systems, Ising model, transfer matrix, phase transitions, critical phenomena, duality, planar model, renormalization group, topological excitations, Abelian lattice gauge theory, Yang-Mills theory, quark confinement, asymptotic freedom
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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