Finite-size scaling of the random-field Ising model above the upper critical dimension

TL;DR

This paper investigates the finite-size scaling behavior of the random-field Ising model in seven dimensions, confirming that an effective length scale replaces the linear size above the upper critical dimension, and provides critical exponents.

Contribution

It demonstrates that finite-size scaling above the upper critical dimension applies to disordered systems using an effective length scale, supported by extensive numerical simulations.

Findings

Confirmed the effective length scale $L_{eff} = L^{D/D_u}$ for $D > D_u$

Estimated the critical point and critical exponents for the 7D random-field Ising model

Validated the modified finite-size scaling approach in disordered systems

Abstract

Finite-size scaling above the upper critical dimension is a long-standing puzzle in the field of Statistical Physics. Even for pure systems various scaling theories have been suggested, partially corroborated by numerical simulations. In the present manuscript we address this problem in the even more complicated case of disordered systems. In particular, we investigate the scaling behavior of the random-field Ising model at dimension $D = 7$, i.e., above its upper critical dimension $D_{\rm u} = 6$, by employing extensive ground-state numerical simulations. Our results confirm the hypothesis that at dimensions $D > D_{\rm u}$, linear length scale $L$ should be replaced in finite-size scaling expressions by the effective scale $L_{\rm eff} = L^{D / D_{\rm u}}$. Via a fitted version of the quotients method that takes this modification, but also subleading scaling corrections into account, we compute the critical point of the transition for Gaussian random fields and provide estimates for the full set of critical exponents. Thus, our analysis indicates that this modified version of finite-size scaling is successful also in the context of the random-field problem.