Ising spin glass by the transfer matrix approach

Abstract

The short-range Ising spin glass on the diamond hierarchical lattice is investigated via a transfer-matrix-based method. An exact set of discrete maps leads to all thermodynamical functions for any finite lattice, from which the thermodynamic limit can be obtained. The method, which encompasses the random choice of the coupling constants between neighboring sites, has been applied to both Gaussian and bimodal probability distributions. Results for the thermodynamic potentials of the model defined on lattices with fractal dimension ${d}_{f}=2,2.58\dots{}$ and $3,$ below and above the estimated lower critical dimension ${(d}_{l}\ensuremath{\sim}2.5),$ are discussed and fully analyzed. Typical Schottky-like profiles are observed in the temperature behavior of the specific heat for both distributions in all lattices. Finite residual entropy is found to persist for the bimodal distribution case. When ${d}_{f}>{d}_{l}$ and for large number of iterations the correlation length \ensuremath{\xi} increases exponentially in a wide temperature interval. The divergence of \ensuremath{\xi} at a finite temperature ${T}_{c}$ associated with the spin-glass phase transition is investigated within an approximate scheme. The numerical values for ${T}_{c}$ and \ensuremath{\nu} are brought in comparison with those previously obtained by other methods.