Random Transverse Field Ising Model in dimension $d=2,3$ : Infinite Disorder scaling via a non-linear transfer approach

TL;DR

This paper extends the cavity-mean-field approximation to finite-dimensional random transverse field Ising models using a non-linear transfer approach, revealing infinite disorder scaling and droplet exponents consistent with directed polymer models.

Contribution

It introduces a non-linear transfer method for finite dimensions, linking surface magnetization scaling to directed polymer droplet exponents, and provides numerical evidence of infinite disorder criticality in 2D and 3D.

Findings

Critical point exhibits infinite disorder scaling.

Surface magnetization scales with droplet exponents from directed polymer models.

Distribution of surface magnetization shows power-law behavior near zero.

Abstract

The 'Cavity-Mean-Field' approximation developed for the Random Transverse Field Ising Model on the Cayley tree [L. Ioffe and M. M\'ezard, PRL 105, 037001 (2010)] has been found to reproduce the known exact result for the surface magnetization in $d=1$ [O. Dimitrova and M. M\'ezard, J. Stat. Mech. (2011) P01020]. In the present paper, we propose to extend these ideas in finite dimensions $d>1$ via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization of the transfer equations correspond to the transfer matrix for a Directed Polymer in a random medium of transverse dimension $D=d-1$, in agreement with the leading order perturbative scaling analysis [C. Monthus and T. Garel, arxiv:1110.3145]. We present numerical results of the non-linear transfer approach in dimensions $d=2$ and $d=3$. In both cases, we find that the critical point is governed by Infinite Disorder scaling. In particular exactly at criticality, the one-point surface magnetization scales as $\ln m_L^{surf} \simeq - L^{\omega_c} v$, where $\omega_c(d)$ coincides with the droplet exponent $\omega_{DP}(D=d-1)$ of the corresponding Directed Polymer model, with $\omega_c(d=2)=1/3$ and $\omega_c(d=3) \simeq 0.24$. The distribution $P(v)$ of the positive random variable $v$ of order O(1) presents a power-law singularity near the origin $P(v) \propto v^a$ with $a(d=2,3)>0$ so that all moments of the surface magnetization are governed by the same power-law decay $\bar{(m_L^{surf})^k} \propto L^{- x_S}$ with $x_S=\omega_c (1+a)$ independently of the order $k$.