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Acta Physica Sinica (Overseas Edition)

Exact results for a symmetric 27-vertex model

Xiang‐Dong Mi, Yang Zhan-Ru

Published Sep 1, 1996• 0 citations
Publisher

Abstract

Wu study a symmetric 3-state 27-vertex model and summarize three ways contributing to obtaining information of a higher-spin (more-vertex) model in terms of lower-spin (less-vertex) models. We find that this model is equivalent to a symmetric 2-state 8-vertex model or an arbitrary 3-state spin models on Kagomé lattice in different cases. And 21 exact critical lines and a multicritical point (add something that I cannot recognize) are obtained in ten-dimensional weight space.

DOI
10.1088/1004-423x/5/9/002

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Wu study a symmetric 3-state 27-vertex model and summarize three ways contributing to obtaining information of a higher-spin (more-vertex) model in terms of lower-spin (less-vertex) models. We find that this model is equivalent to a symmetric 2-state 8-vertex model or an arbitrary 3-state spin models on Kagomé lattice in different cases. And 21 exact critical lines and a multicritical point (add something that I cannot recognize) are obtained in ten-dimensional weight space.
A method is presented by which critical and multicritical points of spin-1 (three-state) vertex models and classical O(n) models on two-dimensional lattices are determined. It is a straightforward generalization of the ideas that earlier led to the determination of critical points and critical exponents of a honeycomb O(n) model. On the square lattice the methods leads to tricritical as well as critical loci. For n=2 a larger critical manifold is found than for other values of n. At the critical and multicritical points thus produced the models turn out to be soluble. The method is applicable to O(n) models and spin-1 vertex models on any planar lattice.
We study the multicritical behavior of a class of two-dimensional ice-type vertex models on different lattices using renormalization-group theory. The models are classified by an integer m, with m=2 corresponding to the known square lattice case. For m>2, the specific-heat exponent is a=(m-2)/(m-1) with an upper critical dimensional confluent (lnt)1/2{)}^{1/2})1/2 divergence for m=3. The nature of the transition is similar to the mth-order multicritical point, yet the exponents are not those known from c1 conformal invariance. The models are anisotropic with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$=1 and ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=12. A few special features of the models are discussed.
A two-dimensional Ising model on a particular topological structure, a ``triangles-in-triangles'' Kagom\'e structure, which is called a triangular-Kagom\'e lattice, is investigated. In this model, we consider two interaction parameters Jaa{J}_{aa}Jaa​ and Jab{J}_{ab}Jab​, corresponding to the interactions among spins on the inner sites and between spins on the inner and vertex sites of the triangular in the Kagom\'e lattice, respectively. By summing over all spins at inner sites in the partition function, we arrive at the partition function of the Kagom\'e lattice. The effective interaction of the corresponding Kagom\'e lattice is always ferromagnetic, even for antiferromagnetic Jaa{J}_{aa}Jaa​. The critical properties of the system depend only on ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$. When ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}>\ensuremath{-}1$, the system has long range order at low temperature. However, when ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$, the partition function of the triangular-Kagom\'e lattice can be related to that of the Kagom\'e lattice with effective interaction at a temperature higher than its critical temperature. Therefore, the system will be in paramagnetic phase at all temperatures for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$. The phase diagram for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$ is given exactly.
The spin-3/2 Ising model described by the most general Hamiltonian with up-down symmetry, −βH=Σ〈ij〉{JS i S j +KS 2 i S 2 j +LS 3 i S 3 j +M/2(S i S 3 j +S j S 3 i )}−ΔΣi S 2 i , is investigated on a square lattice. It is shown that this model is reducible to an eight-vertex model on a surface in the parameter space spanned by the coupling constants J, K, L, and M. It is shown that this model is equivalent to an exactly solvable free fermion model along two lines in the parameter space. Consequently, the critical behavior and, in particular, the critical temperature for the second-order phase transitions of the model is found exactly.
We solved the Yang-Baxter equation for the RRR-matrices of three-state vertex models with ice condition, and obtained a complete list of solvable nineteen-vertex models and associated quantum spin Hamiltonians of spin one.
A critical dilute O(n) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O(n) spin model on the kagome lattice to the exactly solvable critical q-state Potts model on the honeycomb lattice with q=(n+1)2. The intermediate steps involve the random-cluster model on the honeycomb lattice and a fully packed loop model with loop weight n'=sqrt(q) and a dilute loop model with loop weight n , both on the kagome lattice. This mapping enables the determination of a branch of critical points of the dilute O(n) model, as well as some of its critical properties. These properties differ from those of the generic O(n) critical points. For n=0, our model reproduces the known universal properties of the theta point describing the collapse of a polymer. For n not equal 0 it displays a line of multicritical points, with the same universal behavior as a branch of critical points that was found earlier in a dilute O(n) model on the square lattice. These findings are supported by a finite-size-scaling analysis in combination with transfer-matrix calculations.