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arXiv:cond-mat/0009054 · Journal of Physics A Mathematical and General

Exact finite-size-scaling corrections to the critical two-dimensional Ising model on a torus

Jesús Salas

Published Feb 9, 2001• 49 citations
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Abstract

We analyse the finite-size corrections to the energy and specific heat of the critical two-dimensional spin-½ Ising model on a torus. We extend the analysis of Ferdinand and Fisher to compute the correction of order L -3 to the energy and the corrections of order L -2 and L -3 to the specific heat. We also obtain general results on the form of the finite-size corrections to these quantities: only integer powers of L -1 occur, unmodified by logarithms (except of course for the leading log L term in the specific heat); and the energy expansion contains only odd powers of L -1 . In the specific-heat expansion any power of L -1 can appear, but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion.

DOI
10.1088/0305-4470/34/7/307

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We analyse the finite-size corrections to the energy and specific heat of the critical two-dimensional spin-½ Ising model on a torus. We extend the analysis of Ferdinand and Fisher to compute the correction of order L -3 to the energy and the corrections of order L -2 and L -3 to the specific heat. We also obtain general results on the form of the finite-size corrections to these quantities: only integer powers of L -1 occur, unmodified by logarithms (except of course for the leading log L term in the specific heat); and the energy expansion contains only odd powers of L -1 . In the specific-heat expansion any power of L -1 can appear, but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion.
We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model. Key Words: Ising model; finite-size scaling; corrections to scaling; renormalization group; irrelevant operators; scaling functions. It is well-known that phase transitions in statistical-mechanical systems can occur only in the infinite-volume limit. In any finite system, all thermodynamic quantities (such as the magnetic susceptibility and the specific heat) are analytic functions of all parameters (such as the temperature and the magnetic field); but near a critical
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​.
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 analyze the exact partition function of the Ising model on a square lattice under helical boundary conditions obtained by Liaw [Phys. Rev. E 73, 055101(R) (2006)]. Based on such an expression, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive an exact asymptotic expansion of the logarithm of the partition function and its first to fourth derivatives at the critical point. From such results, we find that the shift exponent for the specific heat is lambda=1 for all values of the helicity factor d . We also find that finite-size corrections for the free energy, the internal energy, and the specific heat of the model in a crucial way depend on the helicity factor of the lattice.
Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.