Recent Progress in Spin Glasses

Abstract

We review recent findings on spin glass models. Both the equilibrium properties and the dynamic properties are covered. We focus on progress in theoretical, in particular numerical, studies, while its relationship to real magnetic materials is also mentioned.The motivation that pulls researchers toward spin glasses is possibly not the potential future use of spin glass materials in "practical" applications. It is rather the expectation that there must be something very fundamental in systems with randomness and frustration. It seems that this expectation has been acquiring a firmer ground as the difficulty of the problem is appreciated more clearly. For example, a close relationship has been established between spin glass problems and a class of optimization problems known to be N P -hard. It is now widely accepted that a good (i.e. polynomial) computational algorithm for solving any one of N P -hard problems most probably does not exist. Therefore, it is quite natural to expect that the Ising spin glass problem, as one of the problem in the class, would be very hard to solve, which indeed turns out to be the case in a vast number of numerical studies. None the less, it is often the case that only numerical studies can decide whether a certain hypothetical picture applies to a given instance. Consequently, the major part of what we present below is inevitably a de-

Analytical estimates of the locations of phase transition points in the ground state for the bimodal Ising spin glass model in two dimensions

“…The obtained location is consistent with a previous estimate such as 0.894(2), 9) and the obtained location is larger than but marginally consistent with 0.85 15) and 0.870, 21) while the obtained location is inconsistent with 0.896(2), 18) 0.86(2) 7) and 0.854(2). 20) As described above, the obtained locations of p (1) c and p (2) c are reasonably close to the previously estimated locations, although the approximate approach by regarding the present bonds as random bonds is applied. Thus, it is considered that this approximate approach is valid in this study.…”

“…We derive the lower bounds of the locations with no approximation by using the present bonds. The present bonds are slightly more complicated than the random bonds, so the exact values of P c with no approximation as 0.895 3995 · · · < p (1) c (no approximation) and 0.893 0756 · · · < p (2) c (no approximation). From one of these inequalities, it is shown that the present percolation and the plaquette percolation 7) yield different results, since the plaquette percolation gives p (2) c = 0.86(2).…”

“…According to Kawashima and Rieger,9) the location of p (1) c is estimated as 0.896 (1), and the location of p (2) c is estimated as 0.894 (2). The difference is estimated as 0.002 (2). According to Amoruso and Hartmann 18) by using an exact numerical method, the location of p (1) c is estimated as 0.897 (1), and the location of p (2) c is estimated as 0.896 (2).…”

“…The difference is estimated as 0.002 (2). According to Amoruso and Hartmann 18) by using an exact numerical method, the location of p (1) c is estimated as 0.897 (1), and the location of p (2) c is estimated as 0.896 (2). The difference is estimated as 0.001 (2).…”

“…In this article, we do not mention the problem 6) of the presence or absence of the spin glass phase at a finite temperature for the square lattice. In this article, we mention the locations of the phase transition points p (1) c and p (2) c at zero temperature for the square lattice and mention the intermediate phase between p (1) c and p (2) c . The exact locations of p (1) c and p (2) c have not been analytically obtained.…”