Aspelmeier T, Moore M A
Institute for Theoretical Physics, Georg-August-Universität Göttingen, D37077 Göttingen, Germany.
Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom.
Phys Rev E. 2022 Mar;105(3-1):034138. doi: 10.1103/PhysRevE.105.034138.
The free-energy landscape of the Sherrington-Kirkpatrick (SK) Ising spin glass is simple in the framework of the Thouless-Anderson-Palmer (TAP) equations as each solution (which are minima of the free energy) has associated with it a nearby index-one saddle point. The free-energy barrier to escape the minimum is just the difference between the saddle point free energy and that at its associated minimum. This difference is calculated for the states with free energies f>f_{c}. It is very small for these states, decreasing as 1/N^{2}, where N is the number of spins in the system. These states are not marginally stable. We argue that such small barriers are why numerical studies never find these states when N is large. Instead, the states that are found are those that have marginal stability. For them the barriers are at least of O(1). f_{c} is the free energy per spin below which the states develop broken replica-symmetry-like overlaps with each other. In the regime f<f_{c} we can only offer some possibilities based around scaling arguments. One of these suggest that the barriers might become as large as N^{1/3}. That might be consistent with recent numerical studies on the Viana-Bray model, which were at variance with the expectations of Cugliandolo and Kurchan for the SK model.
在 Thouless-Anderson-Palmer（TAP）方程的框架下，Sherrington-Kirkpatrick（SK）伊辛自旋玻璃的自由能景观很简单，因为每个解（即自由能的最小值）都有一个与之相关的一阶鞍点。逃离最小值的自由能垒就是鞍点自由能与其相关最小值之间的差值。对于自由能(f > f_{c})的状态，计算出了这个差值。对于这些状态，该差值非常小，以(1/N^{2})的形式减小，其中(N)是系统中自旋的数量。这些状态并非边缘稳定的。我们认为，如此小的能垒就是为什么在(N)很大时数值研究从未发现这些状态的原因。相反，所发现的状态是那些具有边缘稳定性的状态。对于它们来说，能垒至少为(O(1))。(f_{c})是每个自旋的自由能，低于此值时，这些状态会彼此发展出类似复制对称破缺的重叠。在(f < f_{c})的区域，我们只能基于标度论证提供一些可能性。其中之一表明，能垒可能会变得高达(N^{1/3})。这可能与最近关于 Viana-Bray 模型的数值研究一致，该研究与 Cugliandolo 和 Kurchan 对 SK 模型的预期不同。
