Scaling at quantum phase transitions above the upper critical dimension

TL;DR

This paper develops a new finite-size scaling formalism for quantum phase transitions above the upper critical dimension, revealing the influence of dangerous irrelevant variables on critical behavior and providing comprehensive critical exponents for the long-range transverse-field Ising chain.

Contribution

It introduces a coherent FSS formalism for quantum phase transitions above the upper critical dimension, extending the Q-FSS approach from thermal to quantum systems.

Findings

Generalized hyperscaling relation derived.

Full set of critical exponents determined for the long-range transverse-field Ising chain.

Confirmed the impact of dangerous irrelevant variables on characteristic length scales.

Abstract

The hyperscaling relation and standard finite-size scaling (FSS) are known to break down above the upper critical dimension due to dangerous irrelevant variables. We establish a coherent formalism for FSS at quantum phase transitions above the upper critical dimension following the recently introduced Q-FSS formalism for thermal phase transitions. Contrary to long-standing belief, the correlation sector is affected by dangerous irrelevant variables. The presented formalism recovers a generalized hyperscaling relation and FSS form. Using this new FSS formalism, we determine the full set of critical exponents for the long-range transverse-field Ising chain in all criticality regimes ranging from the nearest-neighbor to the long-range mean field regime. For the same model, we also explicitly confirm the effect of dangerous irrelevant variables on the characteristic length scale.