Ensemble dependence in the Random transverse-field Ising chain

TL;DR

This paper investigates whether critical exponents differ between microcanonical and canonical ensembles in the disordered random transverse-field Ising chain, finding that exponents are generally the same but amplitudes can vanish in the microcanonical case, potentially mimicking different exponents.

Contribution

It provides a detailed analysis showing that critical exponents are ensemble-independent in the disordered quantum Ising chain, highlighting the role of integrability.

Findings

Critical exponents are the same in both ensembles.

Some critical amplitudes vanish in the microcanonical ensemble.

Apparent exponent differences may result from amplitude vanishing, not actual exponent change.

Abstract

In a disordered system one can either consider a microcanonical ensemble, where there is a precise constraint on the random variables, or a canonical ensemble where the variables are chosen according to a distribution without constraints. We address the question as to whether critical exponents in these two cases can differ through a detailed study of the random transverse-field Ising chain. We find that the exponents are the same in both ensembles, though some critical amplitudes vanish in the microcanonical ensemble for correlations which span the whole system and are particularly sensitive to the constraint. This can \textit{appear} as a different exponent. We expect that this apparent dependence of exponents on ensemble is related to the integrability of the model, and would not occur in non-integrable models.