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

摘要

We consider a classical random-bond Ising model (RBIM) with binarydistribution of $\pm K$ bonds on the square lattice at finite temperature. Inthe phase diagram of this model there is the so-called Nishimori line whichintersects the phase boundary at a multicritical point. It is known that thecorrelation functions obey many exact identities on this line. We use asupersymmetry method to treat the disorder. In this approach the transfermatrices of the model on the Nishimori line have an enhanced supersymmetryosp($2n+1|2n$), in contrast to the rest of the phase diagram, where thesymmetry is osp($2n|2n$) (where $n$ is an arbitrary positive integer). Ananisotropic limit of the model leads to a one-dimensional quantum Hamiltoniandescribing a chain of interacting superspins, which are irreduciblerepresentations of the osp($2n+1|2n$) superalgebra. By generalizing thissuperspin chain, we embed it into a wider class of models. These include othermodels that have been studied previously in one and two dimensions. We suggestthat the multicritical behavior in two dimensions of a class of thesegeneralized models (possibly not including the multicritical point in the RBIMitself) may be governed by a single fixed point, at which the supersymmetry isenhanced still further to osp($2n+2|2n$). This suggestion is supported by acalculation of the renormalization-group flows for the corresponding nonlinearsigma models at weak coupling.