Localization in the ground state of a disordered array of quantum rotators

Abstract

We consider the zero-temperature behavior of a disordered array of quantum rotators given by the finite-volume Hamiltonian:

$$H_\Lambda = - \mathop \Sigma \limits_{x \in \Lambda } \frac{{h(x)}}{2}\frac{{\partial ^2 }}{{\partial \varphi (x)^2 }} - J\mathop \Sigma \limits_{\left\langle {x,y} \right\rangle \in \Lambda } \cos (\varphi (x) - \varphi (y))$$

, wherex,y∈Z ^d, 〈,〉 denotes nearest neighbors inZ ^d;J>0 andh={h(x)>0,x∈Z ^d} are independent identically distributed random variables with common distributiondμ(h), satisfying ∫h ^−δ dμ(h)<∞ for some δ>0. We prove that for anym>0 it is possible to chooseJ(m) sufficiently small such that, if 0<J<J(m), for almost every choice ofh and everyx∈Z ^d the ground state correlation function satisfies

$$\left\langle {\cos (\varphi (x) - \varphi (y))} \right\rangle \leqq C_{x,h,J} e^{ - m\left| {x - y} \right|} $$

for ally∈Z ^d withC _x,h,J <∞.