Polynomial-time approximation algorithms for the antiferromagnetic Ising model on line graphs

We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.

Lee-Yang zeros of the antiferromagnetic Ising model

We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.

Anisotropic scaling of the two-dimensional Ising model II: surfaces and boundary fields

Based on the results published recently [SciPost Phys. 7, 026 (2019)], the influence of surfaces and boundary fields are calculated for the ferromagnetic anisotropic square lattice Ising model on finite lattices as well as in the finite-size scaling limit. Starting with the open cylinder, we independently apply boundary fields on both sides which can be either homogeneous or staggered, representing different combinations of boundary conditions. We confirm several predictions from scaling theory, conformal field theory and renormalisation group theory: we explicitly show that anisotropic couplings enter the scaling functions through a generalised aspect ratio, and demonstrate that open and staggered boundary conditions are asymptotically equal in the scaling regime. Furthermore, we examine the emergence of the surface tension due to one antiperiodic boundary in the system in the presence of symmetry breaking boundary fields, again for finite systems as well as in the scaling limit. Finally, we extend our results to the antiferromagnetic Ising model.

Phase transitions of antiferromagnetic Ising spins on the zigzag surface of an asymmetrical Husimi lattice

An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting a regular square lattice, has been developed to investigate the thermodynamics and phase transitions on surface by the methodology of recursive lattice, which we have previously applied to study polymers near a surface. The model retains the advantages of simple formulation and exact calculation of the conventional Bethe-like lattices. An antiferromagnetic Ising model is solved on the surface of this lattice to evaluate thermal properties such as free energy, energy density and entropy, from which we have successfully identified a first-order order–disorder transition other than the spontaneous magnetization, and a secondary transition on the supercooled state indicated by the Kauzmann paradox.

Критические свойства в модели Изинга на треугольной решетке с переменным межслойным обменным взаимодействием

The Monte Carlo replica algorithm studies of phase transitions and critical phenomenaofthe layered triangular antiferromagnetic Ising model withvariable interlayer exchange interaction. Investigations were carried out for the ratios of the value of the intralayerJ1 and interlayer J2 exchange interactions in the range of values r = J2 / J1 = 0.01 ÷ 1.0. It was found that in the above range r is observed second-order phase transition.Static critical exponents for the heat capacityα, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent ηare computed by means of the finite-size scaling theory. It is shown that the universality class of the critical behavior of the model is maintained in the range of 0.05 <r≤1.0.The results obtained suggest that3D frustrated Ising models on stacked triangularlattice with interlayer-to-intralayer exchange ratio 0.05<r≤1.0. preserves the universality class of criticalbehavior. At lower r, a crossover from 3D to 2D criticalbehavior is observed.