Pseudo Entropy in Free Quantum Field Theories

TL;DR

This work investigates pseudo entropy, a transition-based generalization of entanglement entropy, in 2D free scalar fields with mass $m$ and dynamical exponent $z$, and in the 1D Ising chain. By extending a Gaussian covariance method, it computes $S(\tau_A^{1|2})$ from reduced transition matrices, revealing an area-law-like leading term with a parameter-dependent correction, saturation phenomena, and a universal non-positivity of $\Delta S_{12}=S(\tau_A^{1|2})-\frac{S(\rho_A^1)+S(\rho_A^2)}{2}$. It also analyzes perturbations of CFT vacua, showing $S(\tau_A^{1|2})-S(\rho_A^1) \le 0$ at quadratic order and matching holographic expectations via Janus solutions. In the Ising model, pseudo entropy can distinguish phases, acting as an order parameter for quantum phase transitions. Together, these results suggest that area-law behavior, saturation, and non-positivity are universal features of pseudo entropy and may provide a new diagnostic for quantum phases and topological order.

Abstract

Pseudo entropy is an interesting quantity with a simple gravity dual, which generalizes entanglement entropy such that it depends on both an initial and a final state. Here we reveal the basic properties of pseudo entropy in quantum field theories by numerically calculating this quantity for a set of two-dimensional free scalar field theories and the Ising spin chain. We extend the Gaussian method for pseudo entropy in free scalar theories with two parameters: mass $m$ and dynamical exponent $z$. This computation finds two novel properties of Pseudo entropy which we conjecture to be universal in field theories, in addition to an area law behavior. One is a saturation behavior and the other one is non-positivity of the difference between pseudo entropy and averaged entanglement entropy. Moreover, our numerical results for the Ising chain imply that pseudo entropy can play a role as a new quantum order parameter which detects whether two states are in the same quantum phase or not.

Figures (10)

• Figure 1: $S(\tau^{1|2}_A)$ as a function of the size of the subsytem $N_A$. We set $N=200$ and $z_1=z_2=1$. The curves are $c_1 \ln[(N/\pi)\sin[\pi N_A/N]]+c_0$, where $c_1 \simeq 0.3333$ and $6.028<c_0 <6.453$.
• Figure 1: Difference between the 2nd pseudo Renyi entropy and the averaged value of 2nd the Renyi entropy. Here we set $m_1=1.0\times 10^{-5}$ and $m_2=1.7\times 10^{-4}$. Note that the pseudo Renyi entropy is smaller than the averaged value of ordinary ones. We have small $l$-dependence but it is negligible up to $3$ or $4$ digit. It means that the second term of \ref{['eq:PEscalar']} essentially explains this negative value.
• Figure 2: The upper plot shows the pseudo entropy as a function of the subsystem size $N_A$ when we chose $m_1=10^{-3}$ and $m_2=10^{-5}$ for various values of $z_1=z_2$. The lower plot shows the pseudo entropy when we set $z_1=3$ and $m_1=m_2=10^{-5}$. We chose the total system $N=100$.
• Figure 2: The $z$-dependence of the difference between pseudo entropy and averaged entanglement entropies, $\Delta S_{12}$. Here we set $L=2000, m_1=1.0\times 10^{-7}, m_2=1.0\times 10^{-8}$ and $z_1=z_2\equiv z$. We stress that these are not evenly spaced and it can be perfectly explained by the equation \ref{['eq:liflifzz']}. We have seen this agreement up to $16$ digits. Notice that we did not see such an almost perfect coincidence for $z=1$ case.
• Figure 3: The upper graph shows the pseudo entropy as a function of $m_2$ when we set $z_1=1$ and $m_1=10^{-5}$. The lower graph depicts the pseudo entropy as a function of $z_2$ when we set $m_1=m_2=10^{-5}$.We chose $N_A=50$ and $N=\infty$.