Berlin 2018 – scientific programme

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DY: Fachverband Dynamik und Statistische Physik

DY 18: Critical Phenomena and Phase Transitions

DY 18.4: Talk

Monday, March 12, 2018, 16:15–16:30, BH-N 333

Analytic finite-size scaling functions in the anisotropic Ising rectangle — •Fred Hucht — Fakultät für Physik, Universität Duisburg-Essen, 47048 Duisburg

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 × 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_strip(L,M)+F_strip^res(L,M), where the residual part F_strip^res(L,M) contains the nontrivial finite-L contributions for fixed M. While F_strip^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.

In the finite-size scaling limit L,M → ∞, T → T_c, with fixed temperature scaling variable x∝(T/T_c−1)M and fixed aspect ratio ρ∝ L/M, we derive exponentially fast converging series for the related universal Casimir potential and Casimir force scaling functions. At the critical point T=T_c we confirm predictions from conformal field theory. 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.

A. Hucht, J. Phys. A: Math. Theor. 50, 065201 (2017), arXiv:1609.01963; 50, 265205 (2017), arXiv:1701.08722