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arXiv:1008.4961 · Physics of Particles and Nuclei

Fermions and disorder in ising and related models in two dimensions

V. N. Plechko

Published Nov 25, 2010• 4 citations
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Abstract

The aspects of phase transitions in the two-dimensional Ising models modified by quenched and annealed site disorder are discussed in the framework of fermionic approach based on the reformulation of the problem in terms of integrals with anticommuting Grassmann variables.
DOI
10.1134/s1063779610070166

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The notion of the integral over the anticommuting Grassmann variables is applied to analyze the fermionic structure of the 2D Ising model with quenched site dilution. In the NNN-replica scheme, the model is explicitly reformulated as a theory of interacting fermions on a lattice. For weak dilution, the continuum-limit approximation implies the log-log singularity in the specific heat near TcT_cTc​.
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 eective eld{theoretical (continuum{limit) fermionic formulations for the 2D Ising models near the critical point.