From hard spheres to hard-core spins

Abstract

A system of hard spheres exhibits physics that is controlled only by their density. This comes about because the interaction energy is either infinite or zero, so all allowed configurations have exactly the same energy. The low density phase is liquid, while the high density phase is crystalline, an example of "order by disorder" as it is driven purely by entropic considerations. Here we study a family of hard spin models, which we call hardcore spin models, where we replace the translational degrees of freedom of hard spheres with the orientational degrees of freedom of lattice spins. Their hardcore interaction serves analogously to divide configurations of the many spin system into allowed and disallowed sectors. We present detailed results on the square lattice in $d=2$ for a set of models with ${\mathbb{Z}}_n$ symmetry, which generalize Potts models, and their $U(1)$ limits, for ferromagnetic and antiferromagnetic senses of the interaction, which we refer to as exclusion and inclusion models. As the exclusion/inclusion angles are varied, we find a Kosterlitz-Thouless phase transition between a disordered phase and an ordered phase with quasi-long-ranged order, which is the form order by disorder takes in these systems. These results follow from a set of height representations, an ergodic cluster algorithm, and transfer matrix calculations.

arXiv comment20 pages, 12 figures

References in corpus (37)

• Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods
• Mermin-Wagner fluctuations in 2D amorphous solids
• The two dimensional XY model at the transition temperature: A high precision Monte Carlo study
• On the Hohenberg-Mermin-Wagner theorem and its limitations
• Critical properties of the two-dimensional $q$-state clock model
• Large-scale Monte Carlo simulation of two-dimensional classical XY model using multiple GPUs
• Zero-Temperature Phase Transitions of Antiferromagnetic Ising Model of General Spin on a Triangular Lattice
• Percolation with long-range correlated disorder
• Pocket Monte Carlo algorithm for classical doped dimer models
• The 3-State Square-Lattice Potts Antiferromagnet at Zero Temperature
• A constrained Potts antiferromagnet model with an interface representation
• Numerical study of the phase transitions in the two-dimensional Z(5) vector model
• Ground State Entropy of Potts Antiferromagnets and the Approach to the 2D Thermodynamic Limit
• The Binder Cumulant at the Kosterlitz-Thouless Transition
• O(N) and RP^{N-1} Models in Two Dimensions
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models III. Triangular-Lattice Chromatic Polynomial
• Spin and chiral orderings of the antiferromagnetic XY model on the triangular lattice and their critical properties
• Delocalization of two-dimensional random surfaces with hard-core constraints
• Ground State Entropy of the Potts Antiferromagnet on Triangular Lattice Strips
• Critical Behaviour of the Two Dimensional Step Model
• Geometric properties of the Fortuin-Kasteleyn representation of the Ising model
• Unusual corrections to scaling in the 3-state Potts antiferromagnet on a square lattice
• Disorder-free spin glass transitions and jamming in exactly solvable mean-field models
• Two-dimensional Potts antiferromagnets with a phase transition at arbitrarily large q
• Dynamic critical exponents of Swendsen-Wang and Wolff algorithms by nonequilibrium relaxation
• Berezinskii-Kosterlitz-Thouless Transition with a Constraint Lattice Action
• Partition function zeros of the p-state clock model in the complex temperature plane
• Phase diagram of the triangular-lattice Potts antiferromagnet
• Topological Lattice Actions for the 2d XY Model
• Entropy-driven phase transition in low-temperature antiferromagnetic Potts models
• Novel phase transitions in XY Antiferromagnets on Plane Triangulations
• A condition for long-range order in discrete spin systems with application to the antiferromagnetic Potts model
• Proper 3-colorings of ${\mathbb{Z}}^2$ are Bernoulli
• Logarithmic variance for the height function of square-ice
• Long-range order in discrete spin systems
• Rigidity of proper colorings of ${\mathbb{Z}}^d$
• Delocalization of uniform graph homomorphisms from ${\mathbb{Z}}^2$ to $\mathbb{Z}$

Cited by in corpus (3)

• Sequence of phase transitions in a model of interacting rods
• Cusp singularities in the distribution of orientations of asymmetrically-pivoted hard discs on a lattice
• Arresting dynamics in hardcore spin models