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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 %'egner's Ising lattice gauge theory. Its gauge-invariant "loop" correlation function is discussed in detail. Threedimensional 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 n) and is explained in. simple terms. The direction of present-day research is briefly reviewed. CONTENTS V. VI. Introduction-An Overview of this Article Phenomenology and Physics of Phase Transitions A. Facts about critical behavior B. Correlation length scaling and the droplet picture The Transfer Matrix-Field Theory and Statistical Mechanics A. General remarks B. The path integral and transfer matrix of the s imple harmonic oscillator C. The transfer matrix for field theories The Two-Dimensional Ising Model A. Transfer matrix and 7-continuum formulation B. Self-duality of the Ising model C. Strong coupling expansions for the mass gap, weak coupling expansions for the magnetization D. Kink condensation and disorder E. Self-duality of the isotropic Ising model F. Exact solution of the Ising model in two dimensions Wegner's Ising Lattice Gauge Theory A. Global symmetries, local symmetries, and the energetics of spontaneous symmetry breaking B. Constructing an Ising model with a local symmetry C. Elitzur 's theoremthe impossibility of spontaneously breaking a local symmetry D. Gauge-invar iant correlation functions E. Quantum Hamiltonian and phases of the threedimensional Ising gauge theory Abelian Lattice Gauge Theory A. General formulation B. Gauge-invariant correlation functions, physical interpretations, and phase diagrams C. The quantum Hamiltonian formulation and quark confinement The Planar Heisenberg Model in Two Dimensions A. Introductory comm. ents and motivation B. The physical picture of Kosterlitz and Thoul...

We study QCD-like theories with pseudoreal fermions at finite baryon density. Such theories include two-color QCD with quarks in the fundamental representation of the color group as well as any-color QCD with quarks in the adjoint color representation. In all such theories the lightest baryons are diquarks. At zero chemical potential µ they are, together with the pseudoscalar mesons, the Goldstone modes of a spontaneously broken enlarged chiral symmetry group. Using symmetry principles, we derive the low-energy effective Lagrangian for these particles. We find that a second order phase transition occurs at a value of µ equal to half the mass of the Goldstone modes. For values of µ beyond this point the scalar diquarks Bose condense and the diquark condensate is nonzero. We calculate the dependence of the chiral condensate, the diquark condensate, the baryon charge density, and the masses of the diquark and pseudoscalar excitations on µ at finite bare quark mass and scalar diquark source. The relevance of our results to lattice QCD calculations and to real three-color QCD at finite baryon density is discussed.

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