Abstract The Sherrington-Kirkpatrick (SK) spin-glass model exhibits well-studied phase transitions that are mostly established using replica-based methods. Regardless of the method used for detection, the intrinsic phase of a system exists whether or not replicas are considered. Therefore, in this study, we propose a novel method for phase detection based on the variation of the local field experienced by each spin in a configuration of a single replica. The mean and the variance of these local fields are powerful indicators that effectively distinguish different phases, including ferromagnetic, paramagnetic, and spin-glass phases. By analyzing the mean and variance of these local fields, we develop a machine learning algorithm to generate the phase diagram, which shows strong agreement with the theoretical solutions for the Sherrington-Kirkpatrick model. Also, this algorithm offers a more computationally efficient approach for phase detection in spin-glass systems.
The Sherrington-Kirkpatrick (SK) spin-glass model exhibits well-studied phase transitions that are mostly established using replica-based methods. Regardless of the method used for detection, the intrinsic phase of a system exists whether or not replicas are considered. Therefore, in this study, we propose a novel method for phase detection based on the variation of the local field experienced by each spin in a configuration of a single replica. The mean and the variance of these local fields are powerful indicators that effectively distinguish different phases, including ferromagnetic, paramagnetic, and spin-glass phases. By analyzing the mean and variance of these local fields, we develop a machine learning algorithm to generate the phase diagram, which shows strong agreement with the theoretical solutions for the SK model. This algorithm offers a more computationally efficient approach for phase detection in spin-glass systems.
We construct a complete numerically exact solution of a mean-field quantum spin glass model---the transverse field Sherrington-Kirkpatrick model---by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in $\ensuremath{\omega}$, scaling for small frequencies $\ensuremath{\omega}$, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet---the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound---in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.
We consider a system of linear chains of Ising spins with short-range nearest-neighbor ferromagnetic interactions of strength K along the chains, and with infinite-range random interchain interactions of a Sherrington-Kirkpatrick type. Also included in the model is a local Gaussian random field with variance \ensuremath{\Delta}. In the replica symmetric case, the system is mapped exactly on the one-dimensional Ising model in a random field and a recursion method for calculating the spin-correlation functions is developed. For \ensuremath{\Delta}=0, an analytic relation for freezing temperature
Tf(K) is obtained. The replica-symmetric spin-glass phase is shown to be stable above the freezing temperature
Tf(K,\ensuremath{\Delta}), which is determined numerically.
We discuss a phase transition in spin glass models that have been rarely considered in the past, namely, the phase transition that may take place when two real replicas are forced to be at a larger distance (i.e., at a smaller overlap) than the typical one. In the first part of the work, by solving analytically the Sherrington-Kirkpatrick model in a field close to its critical point, we show that, even in a paramagnetic phase, the forcing of two real replicas to an overlap small enough leads the model to a phase transition where the symmetry between replicas is spontaneously broken. More importantly, this phase transition is related to the de Almeida-Thouless (dAT) critical line. In the second part of the work, we exploit the phase transition in the overlap between two real replicas to identify the critical line in a field in finite dimensional spin glasses. This is a notoriously difficult computational problem, because of considerable finite size corrections. We introduce a new method of analysis of Monte Carlo data for disordered systems, where the overlap between two real replicas is used as a conditioning variate. We apply this analysis to equilibrium measurements collected in the paramagnetic phase in a field, h > 0 and T c ( h ) < T < T c ( h = 0 ) , of the d = 1 spin glass model with long range interactions decaying fast enough to be outside the regime of validity of the mean field theory. We thus provide very reliable estimates for the thermodynamic critical temperature in a field.
Abstract This chapter discusses the problem of spin glasses. If the interactions between spins are not uniform in space, the analysis of the previous chapter does not apply. In particular, when the interactions are ferromagnetic for some bonds and antiferromagnetic for others, the spin orientation cannot be uniform in space, unlike the ferromagnetic system. Under such a circumstance it sometimes happens that spins become randomly frozen — random in space but frozen in time. This is the intuitive picture of the spin glass phase. The chapter investigates the condition for the existence of the spin glass phase as an extension of the mean-field theory. In particular, the properties of the so-called replica-symmetric solution are explained in detail for the Sherrington–Kirkpatrick (SK) model.
The behavior of the nonlinear susceptibility ${\ensuremath{\chi}}_{3}$ and its relation to the spin-glass transition temperature
Tf in the presence of random fields are investigated. To accomplish this task, the Sherrington-Kirkpatrick model is studied through the replica formalism, within a one-step replica-symmetry-breaking procedure. In addition, the dependence of the Almeida-Thouless eigenvalue ${\ensuremath{\lambda}}_{\mathrm{AT}}$ (replicon) on the random fields is analyzed. Particularly, in the absence of random fields, the temperature
Tf can be traced by a divergence in the spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{SG}}$, which presents a term inversely proportional to the replicon ${\ensuremath{\lambda}}_{\mathrm{AT}}$. As a result of a relation between ${\ensuremath{\chi}}_{\mathrm{SG}}$ and ${\ensuremath{\chi}}_{3}$, the latter also presents a divergence at
Tf, which comes as a direct consequence of ${\ensuremath{\lambda}}_{\mathrm{AT}}=0$ at
Tf. However, our results show that, in the presence of random fields, ${\ensuremath{\chi}}_{3}$ presents a rounded maximum at a temperature
T∗ which does not coincide with the spin-glass transition temperature
Tf (i.e.,
T∗>Tf for a given applied random field). Thus, the maximum value of ${\ensuremath{\chi}}_{3}$ at
T∗ reflects the effects of the random fields in the paramagnetic phase instead of the nontrivial ergodicity breaking associated with the spin-glass phase transition. It is also shown that ${\ensuremath{\chi}}_{3}$ still maintains a dependence on the replicon ${\ensuremath{\lambda}}_{\mathrm{AT}}$, although in a more complicated way as compared with the case without random fields. These results are discussed in view of recent observations in the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound.