We consider the mixed p-spin mean-field spin glass model with Ising spins and investigate its free energy in the spirit of the TAP approach, named after Thouless, Anderson, and Palmer [67]. More precisely, we define and compute the generalized TAP correction, and establish the corresponding generalized TAP representation for the free energy. In connection with physicists’ replica theory, we introduce the notion of generalized TAP states, which are the maximizers of the generalized TAP free energy, and show that their order parameters match the order parameter of the ancestor states in the Parisi ansatz. We compute the critical point equations of the TAP free energy that generalize the classical TAP equations for pure states. Furthermore, we give an exact description of the region where the generalized TAP correction is replica symmetric, in which case it coincides with the classical TAP correction, and show that Plefka's condition is necessary for this to happen. In particular, our result shows that the generalized TAP correction is not always replica symmetric on the points corresponding to the Edwards-Anderson parameter. © 2022 Wiley Periodicals LLC.

中文翻译：

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广义 TAP 自由能

我们考虑混合p-spin 具有伊辛自旋的平均场自旋玻璃模型，并本着以 Thouless、Anderson 和 Palmer [67] 命名的 TAP 方法的精神研究其自由能。更准确地说，我们定义和计算广义 TAP 校正，并为自由能建立相应的广义 TAP 表示。结合物理学家的复制理论，我们引入了广义 TAP 态的概念，它们是广义 TAP 自由能的最大化者，并表明它们的序参数与 Parisi ansatz 中祖先状态的序参数相匹配。我们计算了 TAP 自由能的临界点方程，这些方程概括了纯态的经典 TAP 方程。此外，我们给出了广义 TAP 校正是复制对称的区域的精确描述，在这种情况下，它与经典的 TAP 校正一致，并表明 Plefka 的条件对于这种情况的发生是必要的。特别是，我们的结果表明广义 TAP 校正并不总是在对应于 Edwards-Anderson 参数的点上复制对称。© 2022 Wiley Periodicals LLC。