Random Transverse Field Ising model in $d=2$ : analysis via Boundary Strong Disorder Renormalization

TL;DR

This paper introduces a modified Boundary Strong Disorder RG method to analyze the 2D Random Transverse Field Ising model, confirming the robustness of the standard RG results and exploring detailed critical exponents and phase behaviors.

Contribution

The paper presents a new Boundary Strong Disorder RG procedure that simplifies topology issues and demonstrates its consistency with standard RG results for the 2D RTIM.

Findings

Critical point location matches previous RG results

Activated exponent ψ ≈ 0.5 consistent with Infinite Disorder scaling

Finite-size correlation exponent ν_FS ≈ 1.3 in agreement with prior studies

Abstract

To avoid the complicated topology of surviving clusters induced by standard Strong Disorder RG in dimension $d>1$, we introduce a modified procedure called 'Boundary Strong Disorder RG' where the order of decimations is chosen a priori. We apply numerically this modified procedure to the Random Transverse Field Ising model in dimension $d=2$. We find that the location of the critical point, the activated exponent $\psi \simeq 0.5$ of the Infinite Disorder scaling, and the finite-size correlation exponent $\nu_{FS} \simeq 1.3$ are compatible with the values obtained previously by standard Strong Disorder RG.Our conclusion is thus that Strong Disorder RG is very robust with respect to changes in the order of decimations. In addition, we analyze in more details the RG flows within the two phases to show explicitly the presence of various correlation length exponents : we measure the typical correlation exponent $\nu_{typ} \simeq 0.64$ in the disordered phase (this value is very close to the correlation exponent $\nu^Q_{pure}(d=2) \simeq 0.63$ of the {\it pure} two-dimensional quantum Ising Model), and the typical exponent $\nu_h \simeq 1$ within the ordered phase. These values satisfy the relations between critical exponents imposed by the expected finite-size scaling properties at Infinite Disorder critical points. Within the disordered phase, we also measure the fluctuation exponent $\omega \simeq 0.35$ which is compatible with the Directed Polymer exponent $\omega_{DP}(1+1)=1/3$ in $(1+1)$ dimensions.