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Register a new userFinite size effects and loss of self-averageness in the relaxational dynamics of the spherical Sherrington-Kirkpatrick model
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Added by
Leticia Cugliandolo
Publication date
2021
fields
Physics
and research's language is
English
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We revisit the gradient descent dynamics of the spherical Sherrington-Kirkpatrick ($p=2$) model with finite number of degrees of freedom. For fully random initial conditions we confirm that the relaxation takes place in three time regimes: a first algebraic one controlled by the decay of the eigenvalue distribution of the random exchange interaction matrix at its edge in the infinite size limit; a faster algebraic one determined by the distribution of the gap between the two extreme eigenvalues; and a final exponential one determined by the minimal gap sampled in the disorder average. We also analyse the finite size effects on the relaxation from initial states which are almost projected on the saddles of the potential energy landscape, and we show that for deviations scaling as $N^{- u}$ from perfect alignment the system escapes the initial configuration in a time-scale scaling as $ln N$ after which the dynamics no longer self-averages with respect to the initial conditions. We prove these statements with a combination of analytic and numerical methods.
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We argue that when the number of spins $N$ in the SK model is finite, the Parisi scheme can be terminated after $K$ replica-symmetry breaking steps, where $K(N) propto N^{1/6}$. We have checked this idea by Monte Carlo simulations: we expect the typical number of peaks and features $R$ in the (non-bond averaged) Parisi overlap function $P_J(q)$ to be of order $2K(N)$, and our counting (for samples of size $N$ up to 4096 spins) gives results which are consistent with our arguments. We can estimate the leading finite size correction for any thermodynamic quantity by finding its $K$ dependence in the Parisi scheme and then replacing $K$ by K(N). Our predictions of how the Edwards-Anderson order parameter and the internal energy of the system approach their thermodynamic limit compare well with the results of our Monte Carlo simulations. The $N$-dependence of the sample-to-sample fluctuations of thermodynamic quantities can also be obtained; the total internal energy should have sample-to-sample fluctuations of order $N^{1/6}$, which is again consistent with the results of our numerical simulations.
A detailed analysis of the finite-size effects on the bulk critical behaviour of the $d$-dimensional mean spherical model confined to a film geometry with finite thickness $L$ is reported. Along the finite direction different kinds of boundary conditions are applied: periodic $(p)$, antiperiodic $(a)$ and free surfaces with Dirichlet $(D)$, Neumann $(N)$ and a combination of Neumann and Dirichlet $(ND)$ on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary $d$. It is found, for $2<d<4$, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for $(p)$ and $(a)$. For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At $d=3$, the critical amplitude of the singular part of the free energy (related to the so called Casimir amplitude) is estimated. We obtain $Delta^{(p)}=-2zeta(3)/(5pi)=-0.153051...$, $Delta^{(a)}=0.274543...$ and $Delta^{(ND)}=0.01922...$, implying a fluctuation--induced attraction between the surfaces for $(p)$ and repulsion in the other two cases. For $(D)$ and $(N)$ we find a logarithmic dependence on $L$.
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).
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 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.
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