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Register a new userThe square lattice Ising model on the rectangle II: Finite-size scaling limit
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Authors
Alfred Hucht
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Based on the results published recently [J. Phys. A: Math. Theor. 50, 065201 (2017)], the universal finite-size contributions to the free energy of the square lattice Ising model on the $Ltimes M$ rectangle, with open boundary conditions in both directions, are calculated exactly in the finite-size scaling limit $L,Mtoinfty$, $Tto T_mathrm{c}$, with fixed temperature scaling variable $xpropto(T/T_mathrm{c}-1)M$ and fixed aspect ratio $rhopropto L/M$. We derive exponentially fast converging series for the related Casimir potential and Casimir force scaling functions. At the critical point $T=T_mathrm{c}$ we confirm predictions from conformal field theory by Cardy & Peschel [Nucl. Phys. B 300, 377 (1988)] and by Kleban & Vassileva [J. Phys. A: Math. Gen. 24, 3407 (1991)]. The presence of corners and the related corner free energy has dramatic impact on the Casimir scaling functions and leads to a logarithmic divergence of the Casimir potential scaling function at criticality.
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The partition function of the square lattice Ising model on the rectangle with open boundary conditions in both directions is calculated exactly for arbitrary system size $Ltimes M$ and temperature. We start with the dimer method of Kasteleyn, McCoy & Wu, construct a highly symmetric block transfer matrix and derive a factorization of the involved determinant, effectively decomposing the free energy of the system into two parts, $F(L,M)=F_mathrm{strip}(L,M)+F_mathrm{strip}^mathrm{res}(L,M)$, where the residual part $F_mathrm{strip}^mathrm{res}(L,M)$ contains the nontrivial finite-$L$ contributions for fixed $M$. It is given by the determinant of a $frac{M}{2}times frac{M}{2}$ matrix and can be mapped onto an effective spin model with $M$ Ising spins and long-range interactions. While $F_mathrm{strip}^mathrm{res}(L,M)$ becomes exponentially small for large $L/M$ or off-critical temperatures, it leads to important finite-size effects such as the critical Casimir force near criticality. The relations to the Casimir potential and the Casimir force are discussed.
Based on the results obtained in [Hucht, J. Phys. A: Math. Theor. 50, 065201 (2017)], we show that the partition function of the anisotropic square lattice Ising model on the $L times M$ rectangle, with open boundary conditions in both directions, is given by the determinant of a $M/2 times M/2$ Hankel matrix, that equivalently can be written as the Pfaffian of a skew-symmetric $M times M$ Toeplitz matrix. The $M-1$ independent matrix elements of both matrices are Fourier coefficients of a certain symbol function, which is given by the ratio of two characteristic polynomials. These polynomials are associated to the different directions of the system, encode the respective boundary conditions, and are directly related through the symmetry of the considered Ising model under exchange of the two directions. The results can be generalized to other boundary conditions and are well suited for the analysis of finite-size scaling functions in the critical scaling limit using SzegH{o}s theorem.
We calculate very long low- and high-temperature series for the susceptibility $chi$ of the square lattice Ising model as well as very long series for the five-particle contribution $chi^{(5)}$ and six-particle contribution $chi^{(6)}$. These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150000 CPU hours on computer clusters. For $chi^{(5)}$ 10000 terms of the series are calculated {it modulo} a single prime, and have been used to find the linear ODE satisfied by $chi^{(5)}$ {it modulo} a prime. A diff-Pade analysis of 2000 terms series for $chi^{(5)}$ and $chi^{(6)}$ confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the $n$-particle components of the susceptibility, up to a small set of ``additional singularities. We find the presence of singularities at $w=1/2$ for the linear ODE of $chi^{(5)}$, and $w^2= 1/8$ for the ODE of $chi^{(6)}$, which are {it not} singularities of the ``physical $chi^{(5)}$ and $chi^{(6)},$ that is to say the series-solutions of the ODEs which are analytic at $w =0$. Furthermore, analysis of the long series for $chi^{(5)}$ (and $chi^{(6)}$) combined with the corresponding long series for the full susceptibility $chi$ yields previously conjectured singularities in some $chi^{(n)}$, $n ge 7$. We also present a mechanism of resummation of the logarithmic singularities of the $chi^{(n)}$ leading to the known power-law critical behaviour occurring in the full $chi$, and perform a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility $chi$.
Energy eigenvalues and order parameters are calculated by exact diagonalization for the transverse Ising model on square lattices of up to 6x6 sites. Finite-size scaling is used to estimate the critical parameters of the model, confirming universality with the three-dimensional classical Ising model. Critical amplitudes are also estimated for both the energy gap and the ground-state energy.
This paper deals with $tilde{chi}^{(6)}$, the six-particle contribution to the magnetic susceptibility of the square lattice Ising model. We have generated, modulo a prime, series coefficients for $tilde{chi}^{(6)}$. The length of the series is sufficient to produce the corresponding Fuchsian linear differential equation (modulo a prime). We obtain the Fuchsian linear differential equation that annihilates the depleted series $Phi^{(6)}=tilde{chi}^{(6)} - {2 over 3} tilde{chi}^{(4)} + {2 over 45} tilde{chi}^{(2)}$. The factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper. The depleted differential operator is shown to have a structure similar to the corresponding operator for $tilde{chi}^{(5)}$. It splits into factors of smaller orders, with the left-most factor of order six being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral $E$. The right-most factor has a direct sum structure, and using series calculated modulo several primes, all the factors in the direct sum have been reconstructed in exact arithmetics.
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