Random Transverse Field Ising Model in dimension $d>1$ : scaling analysis in the disordered phase from the Directed Polymer model

TL;DR

This paper investigates the disordered phase of the quantum Ising model in dimensions greater than one, revealing its connection to the Directed Polymer model and identifying conditions for Infinite-Disorder fixed points.

Contribution

It establishes a scaling analysis linking the quantum Ising model's disordered phase to the Directed Polymer model and characterizes the nature of fixed points across different dimensions.

Findings

Correlation length scales with the Directed Polymer droplet exponent.

Infinite-Disorder fixed point governs the phase for d ≤ 3.

Finite disorder strength needed for fixed points in d > 3.

Abstract

For the quantum Ising model with ferromagnetic random couplings $J_{i,j}>0$ and random transverse fields $h_i>0$ at zero temperature in finite dimensions $d>1$, we consider the lowest-order contributions in perturbation theory in $(J_{i,j}/h_i)$ to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as : $\ln C(r) \sim - \frac{r}{\xi_{typ}} +r^{\omega} u$, where $\xi_{typ} $ is the typical correlation length, $u$ is a random variable, and $\omega$ coincides with the droplet exponent $\omega_{DP}(D=d-1)$ of the Directed Polymer with $D=(d-1)$ transverse directions. Our main conclusions are (i) whenever $\omega>0$, the quantum model is governed by an Infinite-Disorder fixed point : there are two distinct correlation length exponents related by $\nu_{typ}=(1-\omega)\nu_{av}$ ; the distribution of the local susceptibility $\chi_{loc}$ presents the power-law tail $P(\chi_{loc}) \sim 1/\chi_{loc}^{1+\mu}$ where $\mu$ vanishes as $\xi_{av}^{-\omega} $, so that the averaged local susceptibility diverges in a finite neighborhood $0<\mu<1$ before criticality (Griffiths phase) ; the dynamical exponent $z$ diverges near criticality as $z=d/\mu \sim \xi_{av}^{\omega}$ (ii) in dimensions $d \leq 3$, any infinitesimal disorder flows towards this Infinite-Disorder fixed point with $\omega(d)>0$ (for instance $\omega(d=2)=1/3$ and $\omega(d=3) \sim 0.24$) (iii) in finite dimensions $d > 3$, a finite disorder strength is necessary to flow towards the Infinite-Disorder fixed point with $\omega(d)>0$ (for instance $\omega(d=4) \simeq 0.19$), whereas a Finite-Disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension $d=\infty$ where $\omega=0$, we discuss the similarities and differences with the case of finite dimensions.