Odd Entanglement Entropy and Logarithmic Negativity for Thermofield Double States

TL;DR

This paper studies the time evolution of odd entanglement entropy and logarithmic negativity in thermofield double states of free scalar quantum field theories, revealing linear growth, saturation, and oscillatory behaviors influenced by system size and zero modes.

Contribution

It provides analytical and numerical analysis of OEE and LN dynamics in TFD states, highlighting effects of zero modes and finite size on entanglement measures.

Findings

OEE exhibits linear growth followed by saturation and oscillations due to finite size.

LN remains zero until a threshold time then grows linearly, consistent with quasi-particle picture.

Zero modes induce logarithmic growth in OEE for single degree of freedom subsystems.

Abstract

We investigate the time evolution of odd entanglement entropy (OEE) and logarithmic negativity (LN) for the thermofield double (TFD) states in free scalar quantum field theories using the covariance matrix approach. To have mixed states, we choose non-complementary subsystems, either adjacent or disjoint intervals on each side of the TFD. We find that the time evolution pattern of OEE is a linear growth followed by saturation. On a circular lattice, for longer times the finite size effect demonstrates itself as oscillatory behavior. In the limit of vanishing mass, for a subsystem containing a single degree of freedom on each side of the TFD, we analytically find the effect of zero-mode on the time evolution of OEE which leads to logarithmic growth in the intermediate times. Moreover, for adjacent intervals we find that the LN is zero for times $t < \beta/2$ (half of the inverse temperature) and after that, it begins to grow linearly. For disjoint intervals at fixed temperature, the vanishing of LN is observed for times $t<d/2$ (half of the distance between intervals). We also find a similar delay to see linear growth of $\Delta S=S_{\text{OEE}}-S_{\text{EE}}$. All these results show that the dynamics of these measures are consistent with the quasi-particle picture, of course apart from the logarithmic growth.