Folding the Square-Diagonal Lattice

The topological theory of phase transitions was proposed on the basis of different arguments, the most important of which are: a direct evidence of the relation between topology and phase transitions for some exactly solvable models; an explicit relation between entropy and topological invariants of certain submanifolds of configuration space, and, finally, two theorems stating that, for a wide class of physical systems, phase transitions should necessarily stem from topologi...

We study the residual entropy of a two-dimensional Ising model with crossing and four-spin interactions, both for the case that in zero magnetic field and that in an imaginary magnetic field i({\pi}/2)kT. The spin configurations of this Ising model can be mapped into the hydrogen configurations of square ice with the defined standard direction of the hydrogen bonds. Making use of the equivalence of this Ising system with the exactly solved eight-vertex model and taking the lo...

A Hamiltonian cycle of a graph is a closed path that visits every vertex once and only once. It serves as a model of a compact polymer on a lattice. I study the number of Hamiltonian cycles, or equivalently the entropy of a compact polymer, on various lattices that are not homogeneous but with a sublattice structure. Estimates for the number are obtained by two methods. One is the saddle point approximation for a field theoretic representation. The other is the numerical diag...

We decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons. We end up with a 24 vertex model which is known in the literature as the fully packed double loop model. In the particular case in which the fugacities of the polygons are the same, the model admits an exact solution. The solution is obtained using coordinate Bethe ansatz and provides a c...

We study an effective Hamiltonian generating time evolution of states on intermediate time scales in the strong-coupling limit of the spin-1/2 XXZ model. To leading order, it describes an integrable model with local interactions. We solve it completely by means of a coordinate Bethe Ansatz that manifestly breaks the translational symmetry. We demonstrate the existence of exponentially many jammed states and estimate their stability under the leading correction to the effectiv...

In this text, we provide a fully rigorous, complete and self-contained proof of E.H.Lieb's statement that (topological) entropy of square ice (or six vertex model, XXZ spin chain for anisotropy parameter $\Delta=1/2$) is equal to $\frac{3}{2}\log_2 (4/3)$. For this purpose, we gather and expose in full detail various arguments dispersed in the literature on the subject, and complete several of them that were left partial.

In this article we present analytical results on the exact tensor network representation, spin order parameter and color correlation function of the first example of 2D ground states with entanglement phase transitions from area law to extensive entanglement entropy. The tensor network constructed is one dimension higher than the lattice of the physical system, allowing entangled physical degrees of freedoms to be paired with one another arbitrarily far away. Contraction rule...

The on lattice $\phi^4$ model is a paradigmatic example of continuous real variables model undergoing a continuous symmetry braking phase transition (SBPT). In this paper we study the $\mathbb{Z}_2$-symmetric mean-field version without the quadratic term of the local potential. Obviously, the simplification is directly extensible to the other symmetry groups for which the model undergoes a SBPT. We show that the $\mathbb{Z}_2$-SBPT is not affected by the quadratic term, and t...

This paper deals with themes such as approximate counting/evaluation of the total number of flat-foldings for random origami diagrams, evaluation of the values averaged over various instances, obtaining forcing sets for general origami diagrams, and evaluation of average computational complexity. An approach to the above problems using a physical model and an efficient size reduction method for them is proposed. Using a statistical mechanics model and a numerical method of ap...

We solve a 4-(bond)-vertex model on an ensemble of 3-regular Phi3 planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent -- a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers ...