Journal of Physical Studies 1(4), 554–563 (1997) DOI: https://doi.org/10.30970/jps.01.554 FERMIONIC INTEGRALS AND ANALYTIC SOLUTIONS FOR TWO-DIMENSIONAL ISING MODELS

V. N. Plechko
Joint Institute for Nuclear Research, JINR-Dubna, Bogoliubov Laboratory of Theoretical Physics
141980, Dubna, Moscow Region, Russia

We review some aspects of the fermionic interpretation of the two-dimensional Ising model. The use is made of the notion of the integral over the anticommuting Grassmann variables. For simple and more complicated 2D Ising lattices, the partition function can be expressed as a fermionic Gaussian integral. Equivalently, the 2D Ising model can be reformulated as a free-fermion theory on a lattice. For regular lattices the analytic solution then readily follows by passing to the momentum space for fermions. We also comment on the effective field–theoretical (continuum–limit) fermionic formulations for the 2D Ising models near the critical point.

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