Hucht, Alfred The square lattice Ising model on the rectangle. I: Finite systems. (English) Zbl 1357.81154 J. Phys. A, Math. Theor. 50, No. 6, Article ID 065201, 23 p. (2017). Summary: 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 \(L\times M\) and temperature. We start with the dimer method of Kasteleyn, McCoy and 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 \(M/2\times M/2\) 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. Cited in 3 ReviewsCited in 7 Documents MathOverflow Questions: Determinant of a certain Vandermonde matrix Determinant of structurally symmetric \(n\)-banded matrix? How to calculate one Cauchy type determinant MSC: 81T55 Casimir effect in quantum field theory 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics Keywords:exact solution; boundary conditions; critical Casimir effect; two-dimensional Ising model; transfer matrix Software:MathOverflow PDFBibTeX XMLCite \textit{A. Hucht}, J. Phys. A, Math. Theor. 50, No. 6, Article ID 065201, 23 p. (2017; Zbl 1357.81154) Full Text: DOI arXiv References: [1] Ising, E., Beitrag zur Theorie des Ferromagnetismus, Z. Phys., 31, 253, (1925) · Zbl 1439.82056 · doi:10.1007/BF02980577 [2] Onsager, L., Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev., 65, 117, (1944) · Zbl 0060.46001 · doi:10.1103/PhysRev.65.117 [3] McCoy, B. M.; Wu, T. T., The Two-Dimensional Ising Model, (1973), Cambridge, MA: Harvard University Press, Cambridge, MA [4] Baxter, R. J., Exactly Solved Models in Statistical Mechanics, (1982), London: Academic, London · Zbl 0538.60093 [5] Fisher, M. E.; de Gennes, P-G, Phénomènes aux parois dans un mélange binaire critique, C. R. Acad. Sci. Paris B, 287, 207, (1978) [6] Casimir, H. B G., On the attraction between two perfectly conducting plates, Proc. K. Ned. Akad. Wet., 51, 793, (1948) · Zbl 0031.19005 [7] Evans, R.; Stecki, J., Solvation force in two-dimensional Ising strips, Phys. Rev. B, 49, 8842-8851, (1994) · doi:10.1103/PhysRevB.49.8842 [8] Au-Yang, H.; Fisher, M. E., Wall effects in critical systems: scaling in Ising model strips, Phys. Rev. B, 21, 3956, (1980) · doi:10.1103/PhysRevB.21.3956 [9] Brankov, J. G.; Dantchev, D. M.; Tonchev, N. S., Theory of Critical Phenomena in Finite-Size Systems—Scaling and Quantum Effects, (2000), Singapore: World Scientific, Singapore · Zbl 0967.82002 [10] Ferdinand, A. E.; Fisher, M. E., Bounded and inhomogeneous Ising models. I. Specific-heat anomaly of a finite lattice, Phys. Rev., 185, 832, (1969) · doi:10.1103/PhysRev.185.832 [11] Lu, W. T.; Wu, F. Y., Ising model on nonorientable surfaces: exact solution for the Möbius strip and the Klein bottle, Phys. Rev. E, 63, (2001) · doi:10.1103/PhysRevE.63.026107 [12] Kleban, P.; Vassileva, I., Free energy of rectangular domains at criticality, J. Phys. A: Math. Gen., 24, 3407, (1991) · doi:10.1088/0305-4470/24/14/027 [13] Hucht, A.; Grüneberg, D.; Schmidt, F. M., Aspect-ratio dependence of thermodynamic Casimir forces, Phys. Rev. E, 83, (2011) · doi:10.1103/PhysRevE.83.051101 [14] Hobrecht, H.; Hucht, A., Critical Casimir force scaling functions of the two-dimensional Ising model at finite aspect ratios, J. Stat. Mech., (2016) · Zbl 1457.82098 [15] Hucht, A., The square lattice Ising model on the rectangle II: finite-size scaling limit · Zbl 1369.82005 [16] Vernier, E.; Jacobsen, J. L., Corner free energies and boundary effects for Ising, Potts and fully-packed loop models on the square and triangular lattices, J. Phys. A: Math. Theor., 45, (2012) · Zbl 1235.82015 · doi:10.1088/1751-8113/45/4/045003 [17] Baxter, R. J., The bulk, surface and corner free energies of the square lattice Ising model, J. Phys. A: Math. Theor., 50, (2017) · Zbl 1357.82008 · doi:10.1088/1751-8113/50/1/014001 [18] Kasteleyn, P. W., Dimer statistics and phase transitions, J. Math. Phys., 4, 287, (1963) · doi:10.1063/1.1703953 [19] Molinari, L. G., Determinants of block tridiagonal matrices, Linear Algebr. Appl., 429, 2221, (2008) · Zbl 1154.15009 · doi:10.1016/j.laa.2008.06.015 [20] Fisher, M. E., Statistical mechanics of dimers on a plane lattice, Phys. Rev., 124, 1664, (1961) · Zbl 0105.22403 · doi:10.1103/PhysRev.124.1664 [21] Baxter, R. J., Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant {\it D}_{PQ}, J. Phys. A: Math. Theor., 43, (2010) · Zbl 1194.82012 · doi:10.1088/1751-8113/43/14/145002 [22] Hucht, F., Determinant of a certain Vandermonde matrix, (2016) [23] Hobrecht, H.; Hucht, A., Direct simulation of critical Casimir forces, Europhys. Lett., 106, 56005, (2014) · doi:10.1209/0295-5075/106/56005 [24] Hobrecht, H.; Hucht, A., Many-body critical Casimir interactions in colloidal suspensions, Phys. Rev. E, 92, (2015) · doi:10.1103/PhysRevE.92.042315 [25] Enting, I. G., Generalised Möbius functions for rectangles on the square lattice, J. Phys. A: Math. Gen., 11, 563, (1978) · Zbl 0368.05008 · doi:10.1088/0305-4470/11/3/016 [26] Baxter, R. J.; Sykes, M. F.; Watts, M. G., Magnetization of the three-spin triangular Ising model, J. Phys. A: Math. Gen., 8, 245, (1975) · doi:10.1088/0305-4470/8/2/015 [27] Weisstein, E. W., Euler transform. From MathWorld—a Wolfram web resource [28] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers, (1979), Oxford: Open University Press, Oxford · Zbl 0423.10001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.