Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStability of the quantum Sherrington-Kirkpatrick spin glass model
171
0
0.0
(
0
)
Authors
A. P. Young
Ask ChatGPT about the research
No Arabic abstract
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).
rate research
Read More
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 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.
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.
We propose an expanded spin-glass model, called the quantum Ghatak-Sherrington model, which considers spin-1 quantum spin operators in a crystal field and in a transverse field. The analytic solutions and phase diagrams of this model are obtained by using the one-step replica symmetry-breaking ansatz under the static approximation. Our results represent the splitting within one spin-glass (SG) phase depending on the values of crystal and transverse fields. The two separated SG phases, characterized by a density of filled states, show certain differences in their shapes and phase boundaries. Such SG splitting becomes more distinctive when the degeneracy of the empty states of spins is larger than one of their filled states.
To test the stability of the Parisi solution near T=0, we study the spectrum of the Hessian of the Sherrington-Kirkpatrick model near T=0, whose eigenvalues are the masses of the bare propagators in the expansion around the mean-field solution. In the limit $Tll 1$ two regions can be identified. In the first region, for $x$ close to 0, where $x$ is the Parisi replica symmetry breaking scheme parameter, the spectrum of the Hessian is not trivial and maintains the structure of the full replica symmetry breaking state found at higher temperatures. In the second region $Tll x leq 1$ as $Tto 0$, the components of the Hessian become insensitive to changes of the overlaps and the bands typical of the full replica symmetry breaking state collapse. In this region only two eigenvalues are found: a null one and a positive one, ensuring stability for $Tll 1$. In the limit $Tto 0$ the width of the first region shrinks to zero and only the positive and null eigenvalues survive. As byproduct we enlighten the close analogy between the static Parisi replica symmetry breaking scheme and the multiple time-scales approach of dynamics, and compute the static susceptibility showing that it equals the static limit of the dynamic susceptibility computed via the modified fluctuation dissipation theorem.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
