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Physical Review

Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition

Lars Onsager

Published Feb 1, 1944• 6,515 citations
Publisher

Abstract

The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width ($n$ atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum. The choice of different interaction energies ($\ifmmode\pm\else\textpm\fi{}J,\ifmmode\pm\else\textpm\fi{}{J}^{\ensuremath{'}}$) in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature $T={T}_{c}$ given by the condition $sinh(\frac{2J}{k{T}_{c}}) sinh(\frac{2{J}^{\ensuremath{'}}}{k{T}_{c}})=1.$ The energy is a continuous function of $T$; but the specific heat becomes infinite as $\ensuremath{-}log |T\ensuremath{-}{T}_{c}|$. For strips of finite width, the maximum of the specific heat increases linearly with $log n$. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is ${(\mathrm{exp} (\frac{2J}{\mathrm{kT}}) tanh(\frac{2{J}^{\ensuremath{'}}}{\mathrm{kT}}))}^{n}.$

DOI
10.1103/physrev.65.117

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The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width (nnn atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case n=\ensuremath∞n=\ensuremath{\infty}n=\ensuremath∞ an integral replaces a sum. The choice of different interaction energies (\ifmmode±\else\textpm\fiJ,\ifmmode±\else\textpm\fiJ\ensuremath′\ifmmode\pm\else\textpm\fi{}J,\ifmmode\pm\else\textpm\fi{}{J}^{\ensuremath{'}}\ifmmode±\else\textpm\fiJ,\ifmmode±\else\textpm\fiJ\ensuremath′) in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature T=TcT={T}_{c}T=Tc​ given by the condition sinh(2JkTc)sinh(2J\ensuremath′kTc)=1.sinh(\frac{2J}{k{T}_{c}}) sinh(\frac{2{J}^{\ensuremath{'}}}{k{T}_{c}})=1.sinh(kTc​2J​)sinh(kTc​2J\ensuremath′​)=1. The energy is a continuous function of TTT; but the specific heat becomes infinite as \ensuremath−log∣T\ensuremath−Tc∣\ensuremath{-}log |T\ensuremath{-}{T}_{c}|\ensuremath−log∣T\ensuremath−Tc​∣. For strips of finite width, the maximum of the specific heat increases linearly with lognlog nlogn. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is (exp(2JkT)tanh(2J\ensuremath′kT))n.{(\mathrm{exp} (\frac{2J}{\mathrm{kT}}) tanh(\frac{2{J}^{\ensuremath{'}}}{\mathrm{kT}}))}^{n}.(exp(kT2J​)tanh(kT2J\ensuremath′​))n.
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The specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of nnn parallel layers with free boundary conditions and a surface magnetic field H1=kBTh1{H}_{1}={k}_{B}{\mathrm{Th}}_{1}H1​=kB​Th1​ imposed on the last layer, is analyzed for large nnn in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form $\ensuremath{\Delta}f(n,t,{h}_{1})\ensuremath{\approx}{n}^{\ensuremath{-}2}(\mathrm{ln}n){W}_{1}(nt,{n}^{\frac{1}{2}}{h}_{1})+{n}^{\ensuremath{-}2}{W}_{2}(nt,{n}^{\frac{1}{2}}{h}_{1})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The scaling functions Wi(x,y){W}_{i}(x,y)Wi​(x,y) are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed $t\ensuremath{\ne}0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order ${e}^{\frac{\mathrm{pna}}{\ensuremath{\xi}(T)}}$, where aaa is the lattice spacing and $\ensuremath{\xi}(T)$ is the bulk correlation length; the value of the small rational constant ppp is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").
The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.
We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field H\mathfrak{H}H. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of H\mathfrak{H}H and proportional to the total number of lattice sites, while the other depends on H\mathfrak{H}H and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature Tc{T}_{c}Tc​. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For T=TcT={T}_{c}T=Tc​, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small H\mathfrak{H}H. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at T=TcT={T}_{c}T=Tc​, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for T<TcT<{T}_{c}T<Tc​, may be analytically continued beyond the point H=0\mathfrak{H}=0H=0. We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.
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