Gauge models for spin-glasses

Abstract

A Landau-Ginzburg description of a spin-glass which incorporates naturally the concept of "frustration," or the incompatibility of different local stable spin configurations in neighboring regions, is presented. For a planar spin, the effective Hamiltonian has a form analogous to that of the Landau-Ginzburg functional for a superconductor in a magnetic field, except that the role of the vector potential is taken by a quenched random variable $Q\left(x\right)\mathrm{}$ which represents the wave vector of the spin-density wave of minimum local free energy. The model is thus a simple transcription to a Landau-Ginzburg picture of the basic notion of a spin-glass as a material whose properties are determined by competition between ferromagnetic and antiferromagnetic interactions. The probability distribution of $Q\left(x\right)\mathrm{}$ is chosen not to depend on $Q\left(x\right)\mathrm{}$ directly (in order not to favor any particular value of $Q$), but to be Gaussian in the curl of $Q\left(x\right)\mathrm{}$. The variance $f$ of this distribution, the mean-square vorticity in $Q\left(x\right)\mathrm{}$, is a measure of the degree of frustration. [Any longitudinal part of $Q\left(x\right)\mathrm{}$ is gauged away by rotating the local spin axes appropriately.] For a classical vector (Heisenberg) spin system, the analogous description is a Hamiltonian of $O\left(3\right)\mathrm{}$ Yang-Mills form, again with the gauge random variable. Two calculations are presented. The first tests the stability of the $f=0\mathrm{}$ theory (thermodynamically identical to an ordinary ferromagnet) against the introduction of a small amount of frustration. The result is that the $f=0\mathrm{}$ fixed point is unstable, and no new fixed point (of order $4-d$) appears. Thus the spin-glass transition does not appear to be related to any normal sort of critical point with a particular local-spin-density configuration as a "hidden" order parameter. The second is a mean-field analysis of a transition to a state characterized by an Edwards-Anderson order parameter; its qualitative features are similar to those of mean-field theories for other models for spin-glasses. The conditions for the thermodynamic stability of such a state remain unknown.