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

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 $d\le 3$, 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.

22 pages, v2=final version

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