The ferromagnetic Ising model on an $n\times n$ square lattice region $Λ$ with mixed boundary conditions can exhibit a phase transition as temperature varies. For this spin system, if we fix the spins on the top and bottom sides of the square to be $+$ and the left and right sides to be~$-$, a standard Peierls argument based on energy shows that below some critical temperature~$t_c$, any local Markov chain $\mathcal{M}$ requires time exponential in $n$ to mix. Spin glasses are magnetic alloys that generalize the Ising model by specifying the strength of nearest neighbor interactions on the lattice, including whether they are ferromagnetic or antiferromagnetic. Whenever a face of the lattice is bounded by an odd number of edges with ferromagnetic interactions, the face is considered {\it frustrated} because the local competing objectives cannot be simultaneously satisfied. We consider spin glasses with exactly four well-separated frustrated faces that are symmetric around the center of the lattice region under $90$ degree rotations. We show that local Markov chains require exponential time for all spin glasses in this class. This argument extends to the ferromagnetic Ising model with mixed boundary conditions described above, which behaves like spin glasses with frustrated faces on the boundary. The standard Peierls argument breaks down when the frustrated faces are on the interior of $Λ$ and yields weaker results when they are on the boundary of $Λ$ but not near the corners. We show that there is a universal temperature $T$ below which $\mathcal{M}$ will be slow for all spin glasses with four well-separated frustrated faces. Our argument shows that there is an exponentially small cut indicated by the {\it free energy}, carefully exploiting both entropy and energy to establish a small bottleneck in the state space to establish slow mixing.

翻译：在具有混合边界条件的$n\times n$方格区域$\Lambda$上，铁磁伊辛模型随温度变化可呈现相变现象。对于该自旋系统，若将正方形上下两侧的自旋固定为$+$、左右两侧固定为$-$, 基于能量的标准佩尔斯论证表明：在某一临界温度$t_c$以下，任何局部马尔可夫链$\mathcal{M}$均需指数时间（关于$n$）才能达到混合。自旋玻璃是广义伊辛模型的磁性合金，通过指定格点上最近邻相互作用的强度（包括铁磁或反铁磁性质）来定义。当晶格面的边界被奇数条具有铁磁相互作用的边所包围时，该面被视为“受挫”态——因局部竞争目标无法同时满足。我们考虑具有四个良好分离且绕格子区域中心呈90度旋转对称的受挫面的自旋玻璃模型，证明此类所有自旋玻璃的局部马尔可夫链均需要指数时间。该论证可推广至上述混合边界条件的铁磁伊辛模型，其行为类似于边界处存在受挫面的自旋玻璃。当受挫面位于$\Lambda$内部时，标准佩尔斯论证失效；而当其位于$\Lambda$边界（但非角落附近）时，该论证仅能得出较弱结果。我们证明存在普适温度$T$：对于所有具有四个良好分离受挫面的自旋玻璃，在该温度以下$\mathcal{M}$将保持慢收敛性。本论证通过巧妙利用熵与能量，揭示由自由能指示的指数级小割集，从而在状态空间中建立微小瓶颈，证明慢混合性。