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Register a new userSpin-glass phase transition and behavior of nonlinear susceptibility in the Sherrington-Kirkpatrick model with random fields
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The behavior of the nonlinear susceptibility $chi_3$ and its relation to the spin-glass transition temperature $T_f$, 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 $lambda_{rm AT}$ (replicon) on the random fields is analyzed. Particularly, in absence of random fields, the temperature $T_f$ can be traced by a divergence in the spin-glass susceptibility $chi_{rm SG}$, which presents a term inversely proportional to the replicon $lambda_{rm AT}$. As a result of a relation between $chi_{rm SG}$ and $chi_3$, the latter also presents a divergence at $T_f$, which comes as a direct consequence of $lambda_{rm AT}=0$ at $T_f$. However, our results show that, in the presence of random fields, $chi_3$ presents a rounded maximum at a temperature $T^{*}$, which does not coincide with the spin-glass transition temperature $T_f$ (i.e., $T^* > T_f$ for a given applied random field). Thus, the maximum value of $chi_3$ at $T^*$ reflects the effects of the random fields in the paramagnetic phase, instead of the non-trivial ergodicity breaking associated with the spin-glass phase transition. It is also shown that $chi_3$ still maintains a dependence on the replicon $lambda_{rm AT}$, 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 LiHo$_x$Y$_{1-x}$F$_4$ compound.
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We 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. We find that the replica symmetric (RS) 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).
We investigate generalized Sherrington--Kirkpatrick glassy systems without reflection symmetry. In the neighbourhood of the transition temperature we in general uncover the structure of the glass state building the full-replica-symmetry breaking solution. Physical example of explicitly constructed solution is given.
The interplay between quantum fluctuations and disorder is investigated in a spin-glass model, in the presence of a uniform transverse field $Gamma$, and a longitudinal random field following a Gaussian distribution with width $Delta$. The model is studied through the replica formalism. This study is motivated by experimental investigations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, where the application of a transverse magnetic field yields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic susceptibility $chi_3$, which have led to a considerable experimental and theoretical debate. We analyzed two situations, namely, $Delta$ and $Gamma$ considered as independent, as well as these two quantities related as proposed recently by some authors. In both cases, a spin-glass phase transition is found at a temperature $T_f$; moreover, $T_f$ decreases by increasing $Gamma$ towards a quantum critical point at zero temperature. The situation where $Delta$ and $Gamma$ are related appears to reproduce better the experimental observations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, with the theoretical results coinciding qualitatively with measurements of the nonlinear susceptibility. In this later case, by increasing $Gamma$, $chi_3$ becomes progressively rounded, presenting a maximum at a temperature $T^*$ ($T^*>T_f$). Moreover, we also show that the random field is the main responsible for the smearing of the nonlinear susceptibility, acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temperature $T^*$, one for $T_f<T<T^*$, and another one for $T>T^*$. It is argued that the conventional paramagnetic state corresponds to $T>T^*$, whereas the temperature region $T_f<T<T^*$ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities.
We develop a simple method to study the high temperature, or high external field, behavior of the Sherrington-Kirkpatrick mean field spin glass model. The basic idea is to couple two different replicas with a quadratic term, trying to push out the two replica overlap from its replica symmetric value. In the case of zero external field, our results reproduce the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, we prove the validity of the Sherrington-Kirkpatrick replica symmetric solution up to a line, which falls short of the Almeida-Thouless line, associated to the onset of the spontaneous replica symmetry breaking, in the Parisi Ansatz. The main difference with the method, recently developed by Michel Talagrand, is that we employ a quadratic coupling, and not a linear one. The resulting flow equations, with respect to the parameters of the model, turn out to be much simpler, and more tractable. By applying the cavity method, we show also how to determine free energy and overlap fluctuations, in the region where replica symmetry has been shown to hold.
In a region above the Almeida-Thouless line, where we are able to control the thermodynamic limit of the Sherrington-Kirkpatrick model and to prove replica symmetry, we show that the fluctuations of the overlaps and of the free energy are Gaussian, on the scale N^{-1/2}, for N large. The method we employ is based on the idea, we recently developed, of introducing quadratic coupling between two replicas. The proof makes use of the cavity equations and of concentration of measure inequalities for the free energy.
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