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arXiv 1205.4512cond-mat.dis-nn
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Cayley树上随机横场伊辛模型:边界强无序重整化分析

Random Transverse Field Ising model on the Cayley Tree : analysis via Boundary Strong Disorder Renormalization

Cecile Monthus, Thomas Garel

摘要:

随机横场伊辛模型的强无序重整化导致了幸存团簇的复杂拓扑结构,只要$d>1$。即使从凯莱树开始,幸存的重整化簇的网络将包含循环,因此无法获得解析解。在这里,我们介绍了一个修改后的程序,称为“边界强无序重整化”,保留了树结构,使人们可以写简单的递归的代数。我们首先表明,这种修改后的程序允许恢复正是大多数的临界指数的一维链。在这一重要的检查之后,我们研究了Cayley树上的量子伊辛模型的RG方程,该模型具有均匀铁磁耦合$J$和具有支撑$[h_{min},h_{max}]$的随机横场。我们发现下面的图片(i)对于$J>h_{max}$,只有键被抽取,使得整个树是量子铁磁簇(ii)对于$J<h_{min}$,只有位点被抽取,使得没有量子铁磁簇形成,并且到边界的铁磁耦合与随机介质中的定向聚合物模型的配分函数一致(iii)对于$h_{min}<J<h_{max}$,位点和键都可以被减少:量子铁磁簇可以保持有限(物理学类似于(ii),具有到修改的定向聚合物模型的定量映射)或者出现无限量子铁磁簇。我们发现量子跃迁可以有两种类型:(a)量子跃迁发生在量子铁磁团簇保持有限的区域,并且铁磁耦合到边界的奇异性涉及典型的相关长度指数$ν_{typ}=1$;(b)量子跃迁发生在广泛的量子铁磁团簇出现的点。

英文摘要:

Strong Disorder Renormalization for the Random Transverse Field Ising model leads to a complicated topology of surviving clusters as soon as $d>1$. Even if one starts from a Cayley tree, the network of surviving renormalized clusters will contain loops, so that no analytical solution can been obtained. Here we introduce a modified procedure called 'Boundary Strong Disorder Renormalization' that preserves the tree structure, so that one can write simple recursions with respect to the number of generations. We first show that this modified procedure allows to recover exactly most of the critical exponents for the one-dimensional chain. After this important check, we study the RG equations for the quantum Ising model on a Cayley tree with a uniform ferromagnetic coupling $J$ and random transverse fields with support $[h_{min},h_{max}]$. We find the following picture (i) for $J>h_{max}$, only bonds are decimated, so that the whole tree is a quantum ferromagnetic cluster (ii) for $J<h_{min}$, only sites are decimated, so that no quantum ferromagnetic cluster is formed, and the ferromagnetic coupling to the boundary coincides with the partition function of a Directed Polymer model in a random medium (iii) for $h_{min}<J<h_{max}$, both sites and bonds can be decimated : the quantum ferromagnetic clusters can either remain finite (the physics is then similar to (ii), with a quantitative mapping to a modified Directed Polymer model) or an infinite quantum ferromagnetic cluster appears. We find that the quantum transition can be of two types : (a) either the quantum transition takes place in the region where quantum ferromagnetic clusters remain finite, and the singularity of the ferromagnetic coupling to the boundary involves the typical correlation length exponent $ν_{typ}=1$ (b) or the quantum transition takes place at the point where an extensive quantum ferromagnetic cluster appears.

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