Dresden 2017 – scientific programme

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DY: Fachverband Dynamik und Statistische Physik

DY 11: Critical phenomena

DY 11.3: Talk

Monday, March 20, 2017, 16:00–16:15, HÜL 186

The square lattice Ising model on the rectangle — •Fred Hucht — Fakultät für Physik, Universität Duisburg-Essen, 47048 Duisburg

The partition function of the square lattice Ising model on the rectangle 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. It is given by the determinant of a M/2 × M/2 matrix and can be mapped onto an effective spin model with M Ising spins and long-range interactions. 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, the involved expressions simplify and lead to the scaling functions of the Casimir potential and of the Casimir force. At criticality, a prediction from conformal field theory is confirmed.

Alfred Hucht, "The square lattice Ising model on the rectangle I: Finite systems", J Phys A: Math. Theo., 2016, arXiv:1609.01963, accepted

Alfred Hucht, "The square lattice Ising model on the rectangle II: Finite-size scaling limit", in preparation