I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).
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.
The infinite-range m-vector spin glass in the presence of a Gaussian random field (characterized by a mean
h0 and width \ensuremath{\Delta}) is investigated through the replica method. The phase diagram of the model, for
h0=0 and \ensuremath{\Delta} finite, is obtained within replica-symmetry approximation. It is shown that the paramagnetic phase is enhanced, contrary to what happens to the spin-glass, mixed-ferromagnetic, and ferromagnetic phases, for growing values of \ensuremath{\Delta}. For
h0\ensuremath{\ne}0, the changes introduced in the Gabay-Toulouse line by finite values of \ensuremath{\Delta} are investigated. In particular, it is shown that the critical exponent characteristic of the Gabay-Toulouse line is nonuniversal, changing continuously with the distribution width \ensuremath{\Delta}. \textcopyright{} 1996 The American Physical Society.
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.
The infinite-range quantum Ising spin glass in transverse ($\ensuremath{\Gamma}$) and longitudinal (
h) fields is studied by means of thermo-field-dynamics as a substitute for the
n-replica trick. With the use of a one-loop approximation for the dynamic self-interaction the stability of the mean-field-type solution against the action of fluctuations is investigated in the parameter space (${k}_{B}T, \ensuremath{\Gamma}, h$).
Results of Monte Carlo simulations of the one-dimensional long-range Ising spin glass with power-law interactions in the presence of a (random) field are presented. By tuning the exponent of the power-law interactions, we are able to scan the full range of possible behaviors from the infinite-range (Sherrington-Kirkpatrick) model to the short-range model. A finite-size scaling analysis of the correlation length indicates that the Almeida-Thouless line does not occur in the region with non-mean-field critical behavior in zero field. However, there is evidence that an Almeida-Thouless line does occur in the mean-field region.