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 < β/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 ∆S = SOEE− SEE. All these results show that the dynamics of these measures are consistent with the quasi-particle picture, of course apart from the logarithmic growth.
Highlights
Covariance matrix for Gaussian thermofield double (TFD) stateAs we have mentioned in the introduction, we would like to study the dynamics of odd entanglement entropy (OEE) and logarithmic negativity (LN)
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
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
Summary
As we have mentioned in the introduction, we would like to study the dynamics of OEE and LN. We will focus on the TFD state of a free real scalar QFT. This TFD state is a Gaussian state and we can use the power of the covariance matrix to calculate the OEE and LN.. In the normal mode decomposition, the discretized QFT takes the form of N decoupled simple harmonic oscillators. We will first consider the construction of time-independent as well as time-dependent TFD states for two copies of simple harmonic oscillators. The OEE and LN for the TFD state of discretized QFT will be constructed by a sum on the contribution of each normal mode (simple harmonic oscillator)
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