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Physical review. B, Condensed matter

Tricritical behavior in two dimensions. I. Model-dependent behavior from finite size effects

A. L. Lewis

Published Nov 1, 1978• 2 citations
Publisher

Abstract

The onset of two-dimensional tricritical behavior in $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ strips of spins in an Ising-like model is investigated. This is done through numerical computation of the largest eigenvalue of the single-spin advance transfer matrix. Primary attention is paid to the specific heat at constant nonordering field. The usual phase diagram, in zero ordering field, locates the tricritical point at the intersection of a first-order line and a $\ensuremath{\lambda}$ line; it is found here that a curve of large-field, rounded and bounded specific-heat peaks also seems to intersect this point. This curve, when it exists, may be helpful in experimentally locating tricritical points in any dimension. Evidence is also presented, at zero nonordering field, for the validity of the Onsager specific-heat temperature shift ${N}^{\ensuremath{-}2}\mathrm{ln}N$ in the model.

DOI
10.1103/physrevb.18.5091

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The onset of two-dimensional tricritical behavior in N\ifmmode×\else\texttimes\fi\ensuremath∞N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}N\ifmmode×\else\texttimes\fi\ensuremath∞ strips of spins in an Ising-like model is investigated. This is done through numerical computation of the largest eigenvalue of the single-spin advance transfer matrix. Primary attention is paid to the specific heat at constant nonordering field. The usual phase diagram, in zero ordering field, locates the tricritical point at the intersection of a first-order line and a \ensuremathλ\ensuremath{\lambda}\ensuremathλ line; it is found here that a curve of large-field, rounded and bounded specific-heat peaks also seems to intersect this point. This curve, when it exists, may be helpful in experimentally locating tricritical points in any dimension. Evidence is also presented, at zero nonordering field, for the validity of the Onsager specific-heat temperature shift N\ensuremath−2lnN{N}^{\ensuremath{-}2}\mathrm{ln}NN\ensuremath−2lnN in the model.
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Ferdinand and Fisher have discovered some unusual lattice-shape effects on the finite-size-induced specific-heat peak shift in the two-dimensional Ising-model critical region. It is shown that they are well approximated by keeping only the two largest eigenvalues of the transfer matrix, and may be expressed entirely in terms of a one-dimensional Ising array of domain walls. Approximate expressions for the effective coupling and number of spins are given. It is also pointed out that in general domain walls should be thermodynamically important for such finite-size effects.
We present a finite-size-scaling study of a two-dimensional square-lattice-gas model with nearest-neighbor repulsion and next-nearest-neighbor attraction. This system may be used as a simplified model of adsorbates on transition-metal (100) surfaces. An accurate determination of the tricritical temperature is obtained by the utilization of the degeneracy of the three largest eigenvalues of the transfer matrix at a tricritical point. We obtain the phase diagram and critical and tricritical exponents, all quite consistent with previously known results. We also find that a recent conjecture relating the critical exponent $\ensuremath{\eta}$ to the correlation-length amplitude seems to work well even at the tricritical point.
Ferdinand and Fisher have analyzed the finite-size behavior of the two-dimensional Ising model with periodic boundary conditions for TTT near Tc{T}_{c}Tc​. In the thermodynamic limit, the leading correction term to the specific heat has an unusual and unexpected behavior as the shape of the $m\ifmmode\times\else\texttimes\fi{}n$ lattice is changed. For shape parameter s=mn=1s=\frac{m}{n}=1s=nm​=1, the peak in the specific-heat correction term occurs at a reduced temperature ${\ensuremath{\tau}}_{max}=2{K}_{c}n\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}>0$. With increasing sss the peak moves to negative $\ensuremath{\tau}$ values and then increases to 0. We show that this effect may be understood by keeping only the two largest eigenvalues of the transfer matrix. To this approximation, which is good for s=1s=1s=1 and improves exponentially with increasing sss, all the shape dependence of the free energy is due to a factor that can be expressed entirely in terms of a domain-wall energy. Furthermore, the functional form of this factor is the same as that of the finite-length correction to the one-dimensional Ising model. Thus the dependence of ${\ensuremath{\tau}}_{max}$ on sss is qualitatively similar to the dependence of Tmax1{T}_{max}^{1}Tmax1​ the location of the one-dimensional Ising-model specific-heat peak on NNN, the number of spins. We also argue that the two-dimensional partition function is expressible at least for s>1s>1s>1 as a factor due to domains, and another due to a one-dimensional Ising array of domain walls. Approximate expressions are obtained for the effective number of spins and the effective coupling of the array. We also point out that there is reason to believe the one-dimensional Ising array of domain walls at fixed $\ensuremath{\tau}$ connects smoothly with a similar array for T<TcT<{T}_{c}T<Tc​.
INTRODUCTION Finite-size scaling (FSS) is a well established technique for the extraction of critical exponents from finite volume analyses [1]. Such exponents characterise critical phenomena at a second-order phase transition. The simplest model exhibiting such a transition is the Ising model in two dimensions, which, despite a long history and extensive study, still o#ers new results and insights. Here, we study the model under the special boundary conditions of Brascamp and Kunz [2] to extract new information and to help resolve some hitherto puzzling features of FSS. Let CL (#) be the specific heat at inverse temperature # for a system of linear extent L. FSS of the specific heat is characterized by the location of its peak, #L , its height CL (# L ) and its value at the infinite-volume critical point CL (# c ). The peak position, #L , is a pseudocritical point which typically approaches # c as L ## as |# # c L -# , (1) where # is the shift exponent. In two dimensions,
We have performed multicanonical simulations to study the critical behavior of the two-dimensional Ising model with dipole interactions. This study concerns the thermodynamic phase transitions in the range of the interaction δ where the phase characterized by striped configurations of width h = 1 is observed. Controversial results obtained from local update algorithms have been reported for this region, including the claimed existence of a second-order phase transition line that becomes first order above a tricritical point located somewhere between δ = 0.85 and 1. Our analysis relies on the complex partition function zeros obtained with high statistics from multicanonical simulations. Finite size scaling relations for the leading partition function zeros yield critical exponents ν that are clearly consistent with a single second-order phase transition line, thus excluding such a tricritical point in that region of the phase diagram. This conclusion is further supported by analysis of the specific heat and susceptibility of the orientational order parameter.