Abstract
We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic finite-size part, and – if present – the surface tension, which emerges due to at least one antiperiodic boundary in the system. For the scaling limit we extend the proper finite-size scaling theory to the anisotropic case and show how this anisotropy can be absorbed into suitable scaling variables.
Highlights
The two-dimensional Ising model on the square lattice is by far the most examined and best understood non-trivial system in statistical physics
Albeit there are still open questions, there is a whole plethora of properties known exactly, starting with the exact partition function on the torus in the thermodynamic limit calculated by Onsager [1,2], over the universal finite-size scaling at its continuous phase transition from a ferromagnetic low-temperature to a paramagnetic high-temperature phase [3, 4], to the exact solutions at criticality due to conformal field theory for arbitrary geometries and boundary conditions (BCs) [5,6,7,8]
The dimer mapping will be our starting point, and we will be putting a lot of effort into its analysis relating the topology of the underlying graph and the boundary conditions of the spin system
Summary
The two-dimensional Ising model on the square lattice is by far the most examined and best understood non-trivial system in statistical physics. Albeit there are still open questions, there is a whole plethora of properties known exactly, starting with the exact partition function on the torus in the thermodynamic limit calculated by Onsager [1,2], over the universal finite-size scaling at its continuous phase transition from a ferromagnetic low-temperature to a paramagnetic high-temperature phase [3, 4], to the exact solutions at criticality due to conformal field theory for arbitrary geometries and boundary conditions (BCs) [5,6,7,8]. Other experiments were made on thin films of 3He-4He mixtures near the tricritical point [39], again in the X Y -universality class, and binary liquids, whose demixing transition is in the Ising-universality class [40] The latter experimental system was expanded to the direct measurement of interactions between spherical particles and a potentially chemically striped surface as well as the observation of aggregation processes in the above-mentioned colloidal suspensions [41]. The cylindric geometry and boundary field effects will be discussed in [28]
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