Role of marginality in quantum fidelity and Loschmidt echo: Dirac points in 2-D

TL;DR

This paper explores how marginality influences quantum fidelity and Loschmidt echo near the Dirac point in 2-D systems, revealing unique behaviors such as the absence of sharp dips and logarithmic corrections, differing from 1-D cases.

Contribution

It demonstrates the impact of marginality on quantum critical behavior in 2-D Dirac systems, highlighting differences from lower-dimensional cases and explaining the absence of sharp fidelity and echo dips.

Findings

Absence of sharp fidelity dip near the QCP in 2-D Dirac systems.

Logarithmic corrections to fidelity scaling in the thermodynamic limit.

Explanation for the lack of sharp Loschmidt echo dips in higher dimensions.

Abstract

We investigate the effect of marginality on the ground state fidelity and Loschmidt echo. For this purpose, we study the above quantities near the quantum critical point (QCP) of the two-dimensional (2-D) Dirac Hamiltonian in the presence of a mass term which is tuned to zero at the Dirac point. An ideal example would be that of the low-energy carriers in graphene in which a mass term opens up a band gap. This happens to be a marginal situation where the behavior of the fidelity and the echo is markedly different as compared to that in the one-dimensional case. We encounter this marginal behavior near the Dirac point, which is displayed in the absence of a sharp dip in the ground state fidelity (or equivalently in the logarithmic scaling of the fidelity susceptibility). Most importantly, there is also a logarithmic correction to the proposed scaling of the fidelity in the thermodynamic limit which can not be a priori anticipated from the predicted scaling form. Interestingly, a sharp dip in the ground state Loschmidt echo is also found to be absent near this QCP, which is again a consequence of the marginality. We also explain the absence of a sharp dip in both the fidelity and the Loschmidt echo close to the QCP in dimensions greater than two.