Partition function zeros at first-order phase transitions: A general analysis

Abstract

We present a general, rigorous theory of partition function zeros for lattice spin models depending on one complex parameter. First, we formulate a set of natural assumptions which are verified for a large class of spin models in a companion paper [BBCKK2, math-ph/0304007]. Under these assumptions, we derive equations whose solutions give the location of the zeros of the partition function with periodic boundary conditions, up to an error which we prove is (generically) exponentially small in the linear size of the system. For asymptotically large systems, the zeros concentrate on phase boundaries which are simple curves ending in multiple points. For models with an Ising-like plus-minus symmetry, we also establish a local version of the Lee-Yang Circle Theorem. This result allows us to control situations when in one region of the complex plane the zeros lie precisely on the unit circle, while in the complement of this region the zeros concentrate on less symmetric curves.

52 pages, 3 eps figs, to appear in Commun. Math. Phys.; see also math-ph/0304007 and math-ph/0004003

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Cited by in corpus (16)

• Statistical Mechanics of Equilibrium and Nonequilibrium Phase Transitions: The Yang-Lee Formalism
• The Lee-Yang and Pólya-Schur Programs. I. Linear Operators Preserving Stability
• The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications
• Mean-field driven first-order phase transitions in systems with long-range interactions
• Lee-Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model
• Determination of universal critical exponents using Lee-Yang theory
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial
• Convergence of Bergman geodesics on CP^1
• Limiting Measure of Lee--Yang Zeros for the Cayley Tree
• Yang-Lee zeros and the critical behavior of the infinite-range two- and three-state Potts models
• Signatures of partition functions and their complexity reduction through the KP II equation
• Motion of Lee-Yang zeros
• From Complexification to Self-Similarity: New Aspects of Quantum Criticality
• Monotonicity of Ursell functions in the Ising model
• Disordered Monomer-Dimer model on Cylinder graphs
• Partition Function Zeros of Paths and Normalization Zeros of ASEPS