We have studied numerically the remanent magnetization in the six- and eight-dimensional Ising spin glass and we have compared it with the behaviour observed in the SK model, that we have also computed analytically. We also report the value of the dynamical critical exponent z in six dimensions measured in three different ways: from the behaviour of the energy and the susceptibility as a function of the Monte Carlo time and by studying the overlap - overlap correlation function as a function of space and time. These three results are in very good agreement with the mean field prediction z = 4. Finally we have checked numerically the analytical prediction, obtained by assuming spontaneously broken replica symmetry, for the most singular part of the propagator in the spin-glass phase. This last result supports the existence of spontaneously broken replica symmetry in finite-dimensional spin glasses.
We investigate the balanced
M=4,
p=4 spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range
p-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for
σ=0,0.25, and 0.55. For these values of the long-range exponent
σ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the
λ-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of
λ≡ω2/ω1<1 suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of
λ>1 is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent
σ=0.85, which roughly corresponds to a three dimensional system, we find that the renormalized value of
λ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature
TK in three dimensions might be zero and the complete absence of phase transitions in structural glasses.
By using numerical simulations we show that the four-dimensional Edwards Anderson spin glass in magnetic field undergoes a mean-field-like phase transition. We use a dynamical approach: we simulate large lattices (of volume V ) and work out the behaviour of the system in a limit where both t and V go to infinity, but where the limit is taken first. By showing that the dynamic overlap q converges to a value smaller than the static one we exhibit replica symmetry breaking. The critical exponents are compatible with those obtained by mean-field computations.
Spin glasses are the paradigm of complex systems. These materials present really slow dynamics. However, the nature of the spin glass phase in finite dimensional systems is still controversial. Different theories describing the low temperature phase have been proposed: droplet, replica symmetry breaking and chaotic pairs. We present analytical studies of critical properties of spin glasses, in particular, critical exponents at and below the phase transition, existence of a phase transition in a magnetic field, computation of the lower critical dimension (in presence/absence of a magnetic field). We also introduce some rigorous results based on the concept of metastate. Finally, we report some numerical results regarding the construction of the Aizenman-Wehr metastate, scaling of the correlation functions in the spin glass phase and existence of a phase transition in a field, confronting these results with the predictions of different theories.
Which is the field-theory for the spin-glass phase transition in a magnetic field? This is an open question in less than six dimensions. So far, perturbative computations have not found a stable fixed-point for the renormalization group flow. We tackle this problem through a numerical analysis of the Ising spin glass in four spatial dimensions (data obtained from the Janus collaboration) and in the Bethe lattice. We find strong numerical evidence supporting that the phase transition of the four dimensional Ising spin glass in a field is described by a replica-symmetric Hamiltonian.