Trivial Ground State Structure in the Two-Dimensional Ising Spin Glass

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Abstract

We study how the ground state of the two-dimensional Ising spin glass with Gaussian interactions in zero magnetic field changes on altering the boundary conditions. The probability that relative spin orientations change in a region far from the boundary goes to zero with the (linear) size of the system L like L^{-lambda}, where lambda = -0.70 +/- 0.08. We argue that lambda is equal to d-d_f where d (=2) is the dimension of the system and d_f is the fractal dimension of a domain wall induced by changes in the boundary conditions. Our value for d_f is consistent with earlier estimates. These results show that, at zero temperature, there is only a single pure state (plus the state with all spins flipped) in agreement with the predictions of the droplet model.

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Chaotic Nature of the Spin-Glass Phase

A. Bray, M. Moore (1987)

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Multiple states and thermodynamic limits in short-ranged Ising spin-glass models

D. L. Stein, C. Newman (1992)

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Simplicity of State and Overlap Structure in Finite-Volume Realistic Spin Glasses

D. L. Stein, C. Newman (1997)

We present a combination of heuristic and rigorous arguments indicating that both the pure state structure and the overlap structure of realistic spin glasses should be relatively simple: in a large finite volume with coupling-independent boundary conditions, such as periodic, at most a pair of flip-related (or the appropriate number of symmetry-related in the non-Ising case) states appear, and the Parisi overlap distribution correspondingly exhibits at most a pair of delta-functions at plus/minus the self-overlap. This rules out the nonstandard SK picture introduced by us earlier, and when combined with our previous elimination of more standard versions of the mean field picture, argues against the possibility of even limited versions of mean field ordering in realistic spin glasses. If broken spin flip symmetry should occur, this leaves open two main possibilities for ordering in the spin glass phase: the droplet/scaling two-state picture, and the chaotic pairs many-state picture introduced by us earlier. We present scaling arguments which provide a possible physical basis for the latter picture, and discuss possible reasons behind numerical observations of more complicated overlap structures in finite volumes.