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Physical review. B, Condensed matter

New model Hamiltonian with four-spin interactions: Phase diagram and formation of domains and walls

Santiago D’Elía, Horacio Ceva

Published Oct 1, 1983• 3 citations
Publisher

Abstract

A model Hamiltonian related to the Ashkin-Teller and eight-vertex models is introduced and studied in a two-dimensional square lattice of $N\ifmmode\times\else\texttimes\fi{}N$ sites. It is formulated in terms of a vector variable $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ with four states, and has two-body (${K}_{2}$) and four-body ($\ensuremath{\lambda}$) interactions. If ${K}_{2}=0$, the ground state of this system is highly degenerate and almost all of these states have domains and walls. We also study the region $x\ensuremath{\gg}1$ of the phase diagram ${K}_{2}={K}_{2}(x)$, where $x=\frac{\ensuremath{\lambda}}{{K}_{2}}$, obtaining conditions for the existence of these walls. We find that for $\frac{\ensuremath{\lambda}\ensuremath{\sim}x}{N}\ensuremath{\gg}1$ there is a crossover toward an Ising-type ordering, without walls.

DOI
10.1103/physrevb.28.4001

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A model Hamiltonian related to the Ashkin-Teller and eight-vertex models is introduced and studied in a two-dimensional square lattice of N\ifmmode×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN sites. It is formulated in terms of a vector variable S\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}S\ensuremath→​ with four states, and has two-body (K2{K}_{2}K2​) and four-body (\ensuremathλ\ensuremath{\lambda}\ensuremathλ) interactions. If K2=0{K}_{2}=0K2​=0, the ground state of this system is highly degenerate and almost all of these states have domains and walls. We also study the region x\ensuremath≫1x\ensuremath{\gg}1x\ensuremath≫1 of the phase diagram K2=K2(x){K}_{2}={K}_{2}(x)K2​=K2​(x), where x=\ensuremathλK2x=\frac{\ensuremath{\lambda}}{{K}_{2}}x=K2​\ensuremathλ​, obtaining conditions for the existence of these walls. We find that for \ensuremathλ\ensuremath∼xN\ensuremath≫1\frac{\ensuremath{\lambda}\ensuremath{\sim}x}{N}\ensuremath{\gg}1N\ensuremathλ\ensuremath∼x​\ensuremath≫1 there is a crossover toward an Ising-type ordering, without walls.
We study a two-dimensional XXZ -Ising model on a square-hexagon (denoted for simplicity by 4-6) lattice with spin 1/2. The phase diagram at zero temperature is discussed, where five states are found, two types of ferrimagnetic states, two types of antiferromagnetic states, and one ferromagnetic state. To solve this model, we have mapped onto the eight-vertex model with union Jack interaction term, and it was verified that the model cannot be completely mapped onto eight-vertex model. However, by imposing an exact solution condition, we have found the region where the XXZ -Ising model on 4-6 lattice is exactly soluble with one free parameter, particularly for the case of symmetric eight-vertex model condition. In this manner we have explored the properties of the system and have analyzed the interacting competition parameters which preserve the region where there is an exact solution. Unfortunately the present model does not satisfy the free fermion condition of the eight-vertex model, unless for a trivial solution. Even so, we are able to discuss the critical point region, beyond the region of exact resolvability.
The gonihedric Ising Hamiltonians defined in three and higher dimensions by Savvidy and Wegner provide an extensive, and little explored, catalogue of spin models on (hyper)cubic lattices with many interesting features. In three dimensions the kappa=0 gonihedric Ising model on a cubic lattice has been shown to possess a degenerate low-temperature phase and a first order phase transition, as well as interesting dynamical properties. The dual Hamiltonian to this may be written as an anisotropic Ashkin-Teller model and also has a degenerate low-temperature phase as a result of similar symmetries to the original plaquette action. It is possible to write an alternative dual formulation which utilizes three flavours of spins, rather than the two of the Ashkin-Teller model. This still possesses anisotropic couplings, but all the interaction terms are now four spin couplings and it acquires an additional, local gauge symmetry. We investigate this alternative dual Hamiltonian using zero temperature and mean-field methods together with Monte-Carlo simulations and discuss its properties and the relation to the Ashkin-Teller variant.
In this work we study a two-dimensional XXZ-Ising spin-1/2 model with quartic interactions. The model is composed of a two-dimensional lattice of edge-sharing unitary cells, where each cell consists of two triangular prisms, converging in a basal plane with four Ising spin-1/2 (open circles); the apical positions are also occupied by four Heisenberg spin-1/2 (solid circles). Interaction of the base plane containing the multispin Ising interaction has the parameter J_{4}, and the other pairwise interactions have parameter J. For the proposed model we construct the phase diagram at zero temperature and give all possible spin configurations. In addition, we investigate two regions where the model can be solved exactly, the free fermion condition (FFC) and the symmetrical eight-vertex condition (SEVC). For this purpose we perform a straightforward mapping for a zero-field eight-vertex model. The necessary conditions for the equivalence are analyzed for all ranges of the interaction parameters. Unfortunately, the present model does not satisfy the FFC unless the trivial case; however, it was possible to give a region where the model can be solved approximately. We study the SEVC and verify that this condition is always satisfied. We also explore and discuss the critical conditions giving the region where these critical points are relevant.
The study of the Ashkin-Teller model (ATM) of spin-3/2 on a hypercubic lattice is undertaken via Monte Carlo simulation.The phase diagrams are displayed and discussed in the physical parameter space.Rich physical properties are recovered, namely the second order transition and multicritical points.The phase diagrams have been obtained by varying the strength describing the four spin interaction and the single ion potential.This model shows a new high temperature partially ordered phase, called 〈S〉 and a new Baxter 3/2 ground state which do not exist either in the spin-1/2 ATM or in the spin-1 ATM.
We study a model Hamiltonian consisting of NNN Ising models coupled pairwise through a four-spin interaction K4{K}_{4}K4​. When N=2N=2N=2 this model is the well-known Ashkin-Teller model which shows nonuniversal critical behavior in two dimensions in the neighborhood of K4=0{K}_{4}=0K4​=0. To see if this behavior persists for $N\ensuremath{\ne}2$ we perform a first-order-perturbation expansion around the decoupling point in two dimensions. As an aid in interpreting the results of this perturbation expansion we have determined the phase diagram of the system through mean-field theory and Monte Carlo studies in both two and three dimensions for N=3N=3N=3. The results show that N=2N=2N=2 is special because the coupling between Ising models is marginal over a range of values of K4{K}_{4}K4​. We discuss the effect of the coupling K4{K}_{4}K4​ for $N\ensuremath{\ne}2$.
We study a one-dimensional quantum Hamiltonian problem which is equivalent to a highly anisotropic version of the two-dimensional Ashkin-Teller model. This problem is studied by using its duality and other symmetry properties, by a consideration of its limiting cases, and by mapping it into an XXZ\mathrm{XXZ}XXZ linear chain which is equivalent to a highly anisotropic six-vertex model. In addition eleventh-order strong-coupling series are used to derive numerical data about the Hamiltonian problem. By these means a phase diagram is obtained. The essentially new feature of this diagram is a "critical fan," i.e., a region where a line of continuously varying criticality "fans out" and becomes an area of critical behavior.