Effect of long-range hopping on dynamic quantum phase transitions of an exactly solvable free-fermion model: Nonanalyticities at almost all times

TL;DR

This paper studies how long-range hopping affects dynamic quantum phase transitions in a solvable free-fermion model, revealing non-analyticities in the free energy at almost all times under certain conditions.

Contribution

It provides an exact analysis of non-analyticities in dynamic free energy for a long-range hopping fermion model, showing dependence on hopping decay and range, and characterizing critical times.

Findings

Non-analyticities occur at all times for small decay exponent and large range.

Critical times are determined explicitly for the spatially uniform case.

First derivatives of free energy are discontinuous at non-degenerate zeros, but all derivatives are finite at degenerate zeros.

Abstract

In this work, we investigate quenches in a free-fermion chain with long-range hopping which decay with the distance with an exponent $\nu$ and has range $D$. By exploring the exact solution of the model, we found that the dynamic free energy is non-analytical, in the thermodynamic limit, whenever the sudden quench crosses the equilibrium quantum critical point. We were able to determine the non-analyticities of dynamic free energy $f(t)$ at some critical times $t^{c}$ by solving nonlinear equations. We also show that the Yang-Lee-Fisher (YLF) zeros cross the real-time axis at those critical times. We found that the number of nontrivial critical times, $N_{s},$ depends on $\nu$ and $D$. In particular, we show that for small $\nu$ and large $D$ the dynamic free energy presents non-analyticities in any time interval $\Delta t\sim1/D\ll1$, i.e., there are \emph{non-analyticities at almost all times}. For the spacial case $\nu=0$, we obtain the critical times in terms of a simple expression of the model parameters and also show that $f(t)$ is non-analytical even for finite system under anti-periodic boundary condition, when we consider some special values of quench parameters. We also show that, generically, the first derivative of the dynamic free energy is discontinuous at the critical time instant when the YLF zeros are non-degenerate. On the other hand, when they become degenerate, all derivatives of $f(t)$ exist at the associated critical instant.