Droplet Phenomenology and Mean Field in a Frustrated Disordered System

Abstract

The low lying excited states of the three-dimensional minimum matching problem are studied numerically. The excitations' energies grow with their size and confirm the droplet picture. However, some low energy, infinite size excitations create multiple valleys in the energy landscape. These states violate the droplet scaling ansatz, and are consistent with mean field predictions. A similar picture may apply to spin glasses whereby the droplet picture describes the physics at small length scales, while mean field describes that at large length scales. 75.10.Nr, 64.60.Cn, 02.60.Pn A most useful approach in the study of disordered systems is the replica method. It has been successfully applied [1] to the Sherrington and Kirkpatrick (SK) model [2] of spin glasses, yielding exact results and revealing remarkable properties such as multiplicity of nearly degenerate ground states, lack of self-averaging, and ultrametricity. However it is not clear whether these "mean field" properties hold for more realistic spin glass models like the one of Edwards and Anderson [3] where finite dimensional effects may be dominant. To tackle systems in finite dimensions, a number of approaches based on scaling and the renormalization group have been proposed [4][5][6]. In these phenomenological pictures, it is assumed that there is a unique ground state (up to a global symmetry) and that excitation energies satisfy a scaling ansatz. For our purposes, the essential ingredient of this ansatz is that a "droplet", defined as the lowest energy excitation of characteristic size L containing a given spin, is assumed to have an energy which scales as L θ , with θ > 0. Hereafter we refer to such approaches as the "droplet picture".Although the "mean field" and droplet pictures are very different, they both agree that there are numerous local minima in the energy landscape separated by significant energy barriers. The corner-stone of disagreement between the two approaches concerns the energy of excitations whose size is comparable to that of the whole system. In the mean field picture, there are such system-size excitations whose excitation energies are finite, i.e., do not grow with the system size. Thus there are many nearly degenerate ground states and the energy landscape consists of numerous similar low energy valleys. On the contrary, in the droplet picture, the characteristic energy for such system-size excitations grows as a positive power of the size of the system. As a consequence, the probability of having such an excitation with an energy below a fixed value goes to zero as the system size grows. Thus the ground state is almost never nearly degenerate with another significantly different local minimum. Furthermore, from the point of view of the droplet scaling ansatz, the existence of many nearly degenerate ground states would lead to θ ≤ 0, and yet, for the spin glass phase to exist at non-zero temperatures, droplet excitations must be suppressed, leading to θ > 0.These points show that an unambiguous determination of the lowest ...

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