Random-bond Ising model in two dimensions, the Nishimori line, and supersymmetry

Details Random-bond Ising model in two dimensions, the Nishimori line, and supersymmetry Random-bond Ising model in two dimensions, the Nishimori line, and supersymmetry

2000-07-14

arxiv.org/abs/cond-mat/0007254v2

Phys. Rev. B 63, 104422 (2001).

Physics

Condensed Matter

Disordered Systems and Neural Networks

28 pages; largely independent of cond-mat/0007255. V. 2: small additions, references updated; as published

Scientific paper

10.1103/PhysRevB.63.104422

We consider a classical random-bond Ising model (RBIM) with binary distribution of $\pm K$ bonds on the square lattice at finite temperature. In the phase diagram of this model there is the so-called Nishimori line which intersects the phase boundary at a multicritical point. It is known that the correlation functions obey many exact identities on this line. We use a supersymmetry method to treat the disorder. In this approach the transfer matrices of the model on the Nishimori line have an enhanced supersymmetry osp($2n+1|2n$), in contrast to the rest of the phase diagram, where the symmetry is osp($2n|2n$) (where $n$ is an arbitrary positive integer). An anisotropic limit of the model leads to a one-dimensional quantum Hamiltonian describing a chain of interacting superspins, which are irreducible representations of the osp($2n+1|2n$) superalgebra. By generalizing this superspin chain, we embed it into a wider class of models. These include other models that have been studied previously in one and two dimensions. We suggest that the multicritical behavior in two dimensions of a class of these generalized models (possibly not including the multicritical point in the RBIM itself) may be governed by a single fixed point, at which the supersymmetry is enhanced still further to osp($2n+2|2n$). This suggestion is supported by a calculation of the renormalization-group flows for the corresponding nonlinear sigma models at weak coupling.

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