Finite-Size Scaling for Quantum Criticality above the Upper Critical Dimension: Superfluid-Mott-Insulator Transition in Three Dimensions

TL;DR

This paper demonstrates the validity of modified finite-size scaling for quantum critical points above the upper critical dimension, specifically in the superfluid-Mott-insulator transition in three dimensions, using theoretical analysis and quantum Monte Carlo simulations.

Contribution

It shows that modified finite-size scaling applies above the upper critical dimension for quantum phase transitions, validated through exact solutions and simulations.

Findings

Modified finite-size scaling holds exactly in the large-N limit.

Standard large-system-size limit does not reproduce correct critical behavior.

Quantum Monte Carlo confirms the applicability of modified scaling for N=1.

Abstract

Validity of modified finite-size scaling above the upper critical dimension is demonstrated for the quantum phase transition whose dynamical critical exponent is $z=2$. We consider the $N$-component Bose-Hubbard model, which is exactly solvable and exhibits mean-field type critical phenomena in the large-$N$ limit. The modified finite-size scaling holds exactly in that limit. However, the usual procedure, taking the large system-size limit with fixed temperature, does not lead to the expected (and correct) mean-field critical behavior due to the limited range of applicability of the finite-size scaling form. By quantum Monte Carlo simulation, it is shown that the same holds in the case of N=1.