Metastable states of Ising spin glasses and random ferromagnets.

Abstract

Using a recursive method, we have calculated and compared the number of single-spin-flip metastable states for one-dimensional (1D) (chain) and 2D (strip) Ising spin glasses and random ferromagnets at zero temperature. For the 1D case, an extensive study was made of the distribution of the metastable states ${N}_{s}$(\ensuremath{\varepsilon},m) with regard to their magnetization m per spin and energy \ensuremath{\varepsilon} per spin, and the effect of an applied magnetic field on this distribution was investigated. The distributions of metastable states in energy and in magnetization for the 2D systems (strips of width up to 6 and length ${10}^{4}$ spins) are qualitatively similar to those of the chains. Our results suggest that the ground state of the spin glass is extremely sensitive to changes in the magnetic field. The number of low-energy metastable states depends on the energy above the ground-state energy ${\ensuremath{\varepsilon}}_{0}$ as (1/N)ln${N}_{s}$ (\ensuremath{\varepsilon})\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}(\ensuremath{\varepsilon}-${\ensuremath{\varepsilon}}_{0}$${)}^{1/\ensuremath{\lambda}}$. For d=1, 1/\ensuremath{\lambda} is known exactly. For d=2 random ferromagnets, the low-energy metastable states have different character for wide and narrow strips: In wide strips, they arise primarily from flipping two-spin clusters relative to the ground state, while for narrow strips (open at both ends) they are due to flipping all spins to the left (or right) of a line. The value of the exponent 1/\ensuremath{\lambda} predicted from this picture agrees with the numerical value at each width. For d=2 spin glasses, our results suggest that 1/\ensuremath{\lambda} is greater than 1/2.