Metastable minima of the Heisenberg spin glass in a random magnetic field

Abstract

We have studied zero-temperature metastable minima in classical m-vector component spin glasses in the presence of m-component random fields for two models, the Sherrington-Kirkpatrick (SK) model and the Viana-Bray (VB) model. For the SK model we have calculated analytically its complexity (the log of the number of minima) for both the annealed case where one averages the number of minima before taking the log and the quenched case where one averages the complexity itself, both for fields above and below the de Almeida-Thouless (AT) field, which is finite for m>2. We have done numerical quenches starting from a random initial state (infinite temperature state) by putting spins parallel to their local fields until there is no further decrease of the energy and found that in zero field it always produces minima that have zero overlap with each other. For the m=2 and m=3 cases in the SK model the final energy reached in the quench is very close to the energy E_{c} at which the overlap of the states would acquire replica symmetry-breaking features. These minima have marginal stability and will have long-range correlations between them. In the SK limit we have analytically studied the density of states ρ(λ) of the Hessian matrix in the annealed approximation. Despite the fact that in the presence of a random field there are no continuous symmetries, the spectrum extends down to zero with the usual sqrt[λ] form for the density of states for fields below the AT field. However, when the random field is larger than the AT field, there is a gap in the spectrum, which closes up as the AT field is approached. The VB model behaves differently and seems rather similar to studies of the three-dimensional Heisenberg spin glass in a random vector field.

Edwards statistical mechanics for jammed granular matter

“…4b). The test consists to validate the Edwards measure in the metastable states as done in (Sharma et al, 2016), following the use of the Edwards assumption to calculate the ground state of the hard sphere model in (Charbonneau et al, 2014b) and using the exact results of (Newman and Stein, 1999). This test can be done for the 1-SF metastable state in the exactly solvable Sherrington-Kirkpatrick (SK) spin-glass model (Sherrington and Kirkpatrick, 1975), which is the canonical mean field model of spin glasses.…”

“…3. The analytical study in (Sharma et al, 2016) of zero-temperature metastable minima in classical Heisenberg spin glass in a random magnetic field. Such a study confirms that the energy reached dynamically is in agreement with a computation of metastable states using Edwards equiprobability, see Eq.…”

“…Such a study confirms that the energy reached dynamically is in agreement with a computation of metastable states using Edwards equiprobability, see Eq. (12) in (Sharma et al, 2016). Newman and Stein (Newman and Stein, 1999) probing a weaker formulation of Edwards uniform measure: the final states that a zero-temperature dynamics in spin-glass model arrive at a given energy are solely determined by the dynamical protocol and are accessed with equal probability for a given energy.…”

Self-organized critical behavior and marginality in Ising spin glasses

“…Our interest in the static aspects of SOC behavior was triggered by our previous studies [23] of vector spin glasses in the Sherrington-Kirkpatrick (SK) model. We found that the quench in those models reached metastable minima whose energy per spin E c was very close to that calculated for the energy which separates minima with zero overlap with each other from those which have a full replica symmetry overlap with each other [24].…”

Realizable solutions of the Thouless-Anderson-Palmer equations

“…What led us to carry out this investigation was the work of Sharma et al [15,16]. It was found in these papers that a quench from an initial random (i.e.…”