发表机构
The University of Tokyo; Trans-Scale Quantum Science Institute, The University of Tokyo; Institute for Physics of Intelligence, The University of Tokyo（东京大学; 东京大学跨尺度量子科学研究所; 东京大学智能物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过同步蒙特卡洛动力学在离散时间下重现了SK模型的Parisi解，证明了其长时间结构与Parisi方程一致，并通过大规模模拟验证了该解的准确性。
AI 中文摘要
Sompolinsky和Zippelius通过假设广泛分离的时间尺度层次，从连续时间中软自旋的弛豫动力学获得了Sherrington-Kirkpatrick (SK)模型的Parisi解。在此，我们对离散时间动力学执行相同的方案，其中所有Ising自旋同时更新。这种动力学，即同步蒙特卡洛（SyncMC），引入高斯辅助场，使得自旋条件独立，并且对于任何具有对称成对耦合的系统，它精确地采样玻尔兹曼分布。我们证明，对于任何此类系统，SyncMC的相关函数和响应函数在平衡态下满足离散时间中的精确涨落-耗散关系。SK模型的SyncMC动力学平均场方程包含连续时间动力学中不存在的两项：每个自旋的自耦合和通过耦合相关的噪声；两者仅影响快速弛豫。因此，在与Sompolinsky-Zippelius构造相同的假设下（广泛分离的时间尺度，以及在每个慢尺度上，与随机初始条件弛豫中预期的缩减比x相同的关联），SyncMC的长时间结构重现了Parisi方程，包括Parisi偏微分方程和局域场分布方程。使用副本交换对多达N=2048个自旋的平衡模拟排除了实际局域场的副本对称分布，并与Parisi解一致。从重叠分布以及从随机构型弛豫的响应中估计的x(q)函数（对于多达N=4096个自旋）也与Parisi解一致。
英文摘要
Sompolinsky and Zippelius obtained Parisi's solution of the Sherrington-Kirkpatrick (SK) model from the relaxational dynamics of soft spins in continuous time, by assuming a hierarchy of widely separated time scales. Here we carry out the same programme for a discrete-time dynamics in which all Ising spins are updated at once. This dynamics, synchronous Monte Carlo (SyncMC), introduces Gaussian auxiliary fields so that the spins become conditionally independent, and it samples the Boltzmann distribution exactly for any system with symmetric pairwise couplings. We show that, for any such system, the correlation and response functions of SyncMC obey an exact fluctuation-dissipation relation in discrete time in equilibrium. The dynamical mean-field equations of SyncMC for the SK model contain two terms absent in continuous-time dynamics, a self-coupling of each spin and a noise correlated through the couplings; both affect only the fast relaxation. As a result, under the same assumptions as in the Sompolinsky-Zippelius construction (widely separated time scales, and on each slow scale the same relation with a reduced ratio x, as expected in relaxation from random initial conditions), the long-time structure of SyncMC reproduces the Parisi equations, including the Parisi partial differential equation and the equation for the distribution of local fields. Equilibrium simulations with replica exchange for up to N=2048 spins exclude the replica-symmetric distribution of the actual local fields and are consistent with the Parisi solution. The function x(q) estimated from the overlap distribution, and from the response in relaxation from random configurations for up to N=4096 spins, is also consistent with Parisi's.
Comments26 pages, 6 figures
