Study of the de Almeida-Thouless line in the one-dimensional diluted power-law XY spin glass.

We study the de Almeida-Thouless (AT) line in the one-dimensional power-law diluted XY spin-glass model, in which the probability that two spins separated by a distance r interact with each other, decays as 1/r^{2σ}. Tuning the exponent σ is equivalent to changing the space dimension of a short-range model. We develop a heat bath algorithm to equilibrate XY spins; using this in conjunction with the standard parallel tempering and overrelaxation sweeps, we carry out large-scale Monte Carlo simulations. For σ=0.6, which is in the mean-field regime above six dimensions-it is similar to being in 10 dimensions-we find clear evidence for an AT line. For σ=0.75 and σ=0.85, which are in the non-mean-field regime and similar to four and three dimensions, respectively, our data is like that found in previous studies of the Ising and Heisenberg spin glasses when reducing the temperature at fixed field. For σ=0.75, there is evidence from finite-size-scaling studies for an AT transition but for σ=0.85, the evidence for a transition is nonexistent. We have also studied these systems at fixed temperature varying the field and discovered that at both σ=0.75 and at σ=0.85 there is evidence of an AT transition! Confusingly, the correlation length and spin-glass susceptibility as a function of the field are both entirely consistent with the predictions of the droplet picture and hence the nonexistence of an AT line. In the usual finite-size critical point scaling studies used to provide evidence for an AT transition, there is seemingly good evidence for an AT line at σ=0.75 for small values of the system size N, which is strengthening as N is increased, but for N>2048 the trend changes and the evidence then weakens as N is further increased. We have also studied with fewer bond realizations the system at σ=0.70, which is the analog of a system with short-range interactions just below six dimensions, and found that it is similar in its behavior to the system at σ=0.75 but with larger finite-size corrections. The evidence from our simulations points to the complete absence of the AT line in dimensions outside the mean-field region and to the correctness of the droplet picture. Previous simulations which suggested there was an AT line can be attributed to the consequences of studying systems which are just too small. The collapse of our data to the droplet scaling form is poor for σ=0.75 and to some extent also for σ=0.85, when the correlation length becomes of the order of the length of the system, due to the existence of excitations which only cost a free energy of O(1), just as envisaged in the TNT picture of the ordered state of spin glasses. However, for the case of σ=0.85 we can provide evidence that for larger system sizes, droplet scaling will prevail even when the correlation length is comparable to the system size.