Aspects of Pseudo Entropy in Field Theories

TL;DR

This work systematically analyzes pseudo entropy, a two-state generalization of entanglement entropy, across free field theories, spin models, and holographic CFTs. It develops and unifies computational methods (correlator and operator approaches) for Gaussian states, demonstrates area-law behavior, and establishes universal properties like entropy saturation and the non-positivity of the difference when states share a quantum phase. The XY spin model and Lifshitz scalar analyses reveal both the robustness of these properties and interesting violations (e.g., strong subadditivity) when phases differ, while global quenches and perturbations uncover time-dependent and perturbative structures, including a first-law-like relation. Holographic results with Janus interfaces and gapped/grand phases corroborate and extend the field-theoretic findings, showing that the sign of the pseudo entropy difference depends on the interface structure and phase topology. Overall, the paper provides a comprehensive framework linking field theory, many-body physics, and gravity duals to illuminate pseudo entropy’s behavior under quenches, perturbations, and phase transitions.

Abstract

In this article, we explore properties of pseudo entropy [1] in quantum field theories and spin systems from several approaches. Pseudo entropy is a generalization of entanglement entropy such that it depends on both an initial and final state and has a clear gravity dual via the AdS/CFT. We numerically analyze a class of free scalar field theories and the XY spin model. This reveals the basic properties of pseudo entropy in many-body systems, namely, the area law behavior, the saturation behavior, and the non-positivity of difference between the pseudo entropy and averaged entanglement entropy in the same quantum phase. In addition, our numerical analysis finds an example where the strong subadditivity of pseudo entropy gets violated. Interestingly we find that the non-positivity of the difference can be violated only if the initial and final states belong to different quantum phases. We also present analytical arguments which support these properties by both conformal field theoretic and holographic calculations. When the initial and final states belong to different topological phases, we expect a gapless mode localized along an interface, which enhances the pseudo entropy leading to the violation of the non-positivity of the difference. Moreover, we also compute the time evolution of pseudo entropy after a global quench, were we observe that the imaginary part of pseudo entropy shows interesting characteristc behavior.

Figures (25)

• Figure 1: $S(\tau^{1|2}_A)$ as a function of the size of the subsystem $\ell$. We set $L=100$ and $z_1=z_2=1$. The solid curves are $(1/3) \log[(L/\pi)\sin(\pi \ell/L)]+\textrm{const.}$, where the constant term is defined in \ref{['eq:PEscalar']}.
• Figure 2: Left: $\Delta S_{12}$ for free scalar with $z_1=z_2=1$. Here we set $m_1=10^{-5}$ and $m_2=10^{-4}$. We have small $\ell$ and $L$-dependence but it is negligible up to $3$ or $4$ digit. Therefore, the second term of \ref{['eq:PEscalar']} essentially explains this negative value. Right: The $z$-dependence of $\Delta S_{12}$ for $z>1$. Here we set $L=2000, m_1= 10^{-7}, m_2=10^{-8}$ and $z_1=z_2\equiv z$. It can be perfectly explained by the equation \ref{['eq:liflifzz']}. Note that we can also produce the same results from smaller total systems and see the perfect agreement in the whole subregions.
• Figure 3: The upper plot shows the pseudo entropy as a function of the subsystem size $\ell$ 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 $L=100$.
• Figure 4: Dynamical exponent dependence of the pseudo entropy. We set $m_1=m_2=10^{-5}$ and $\ell=30$ on an infinite lattice. The pseudo entropy saturates at the value of the larger $z$.
• Figure 5: The pseudo entropy and regularized pseudo entropy for fixed $m_1=10^{-3}$ with various mass $m_2$. As a reference, we also plot the entanglement entropy for vacuum with $c=1$ (orange curve). Note that this formula is valid only in the small subsystem such that $m_i\ell\ll1$. Out of this regime, as we can see from the right-top figure, there is a small deviation.