AI 中文总结
研究MK分层晶格上伊辛自旋玻璃不存在自旋玻璃序的条件，通过精确重整化群步骤简化为键渗流问题，给出方格上伊辛自旋玻璃在MK近似内无自旋玻璃相的证明，还构建了分形维数接近三维却无自旋玻璃序的晶格。
AI 中文摘要
我们推导了具有偶数分支数的Migdal--Kadanoff（MK）分层晶格上具有对称二元耦合的伊辛自旋玻璃中不存在自旋玻璃序的严格充分条件。关键发现是单个精确重整化群步骤以正概率产生零有效键，从而将问题简化为相应分层晶格上的键渗流。当诱导稀释超过渗流阈值时，在所有温度包括零温度下，自旋玻璃序和刚度都不存在。因此，我们的准则严格证明了方格上的伊辛自旋玻璃在MK近似内不表现出自旋玻璃相。更出乎意料的是，通过选择足够大的比例因子和分支数，我们构建了分形维数从下方任意接近三维但仍不表现出自旋玻璃序的MK分层晶格。这与具有相对较小比例因子和分支数的MK分层晶格的数值估计形成鲜明对比，后者将下临界维数置于约2.52附近。
英文摘要
We derive a rigorous sufficient condition for the absence of spin-glass order in the Ising spin glass with symmetric binary couplings on Migdal--Kadanoff (MK) hierarchical lattices with even branching number. The key observation is that a single exact renormalization-group step creates zero effective bonds with positive probability, thereby reducing the problem to bond percolation on the corresponding hierarchical lattice. When the induced dilution exceeds the percolation threshold, both spin-glass order and stiffness are absent at all temperatures, including zero temperature. As a consequence, our criterion gives a rigorous proof that the Ising spin glass on the square lattice does not exhibit a spin-glass phase within the MK approximation. More unexpectedly, by choosing sufficiently large scale factors and branching numbers, we construct MK hierarchical lattices whose fractal dimensions are arbitrarily close to three from below but still exhibit no spin-glass order. This sharply contrasts with numerical estimates obtained for MK hierarchical lattices with relatively small scale factors and branching numbers, which placed the lower critical dimension near \(2.52\).
Comments7 pages, 1 figure
