Properties of the multicritical point of ±J Ising spin glasses on the square lattice

Abstract

We use numerical transfer-matrix methods to investigate properties of the multicritical point of binary Ising spin glasses on a square lattice, whose location we assume to be given exactly by a conjecture advanced by Nishimori and Nemoto. We calculate the two largest Lyapunov exponents, as well as linear and nonlinear zero-field uniform susceptibilities, on strip of widths $4\ensuremath{\leqslant}L\ensuremath{\leqslant}16$ sites, from which we estimate the conformal anomaly $c$, the decay-of-correlations exponent $\ensuremath{\eta}$, and the linear and nonlinear susceptibility exponents $\ensuremath{\gamma}∕\ensuremath{\nu}$ and ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}$, with the help of finite-size scaling and conformal invariance concepts. Our results are $c=0.46(1)$; $0.187\ensuremath{\lesssim}\ensuremath{\eta}\ensuremath{\lesssim}0.196$; $\ensuremath{\gamma}∕\ensuremath{\nu}=1.797(5)$; ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}=5.59(2)$. A direct evaluation of correlation functions on the strip geometry, and of the statistics of the zeroth moment of the associated probability distribution, gives $\ensuremath{\eta}=0.194(1)$, consistent with the calculation via Lyapunov exponents. Overall, these values tend to be inconsistent with the universality class of percolation, though by small amounts. The scaling relation ${\ensuremath{\gamma}}^{\mathrm{nl}}∕\ensuremath{\nu}=2\ensuremath{\gamma}∕\ensuremath{\nu}+d$ (with space dimensionality $d=2$) is obeyed to rather good accuracy, thus showing no evidence of multiscaling behavior of the susceptibilities.