Entanglement Entropy from Correlation Functions of Scalar Fields in and out of Equilibrium

TL;DR

This work develops a field-theoretic framework to compute entanglement entropy of interacting scalar fields directly from correlation functions, applicable to both equilibrium and non-equilibrium/open systems. The core idea is that odd Rényi entropies $S^{(2q+1)}$ equal the free energy of $2q+1$ replicas coupled by a quadratic entangling action $S_{ent}$ at the measurement time, removing the need for traditional boundary field matching and enabling a broad diagrammatic treatment. The authors derive a Wigner-characteristic-function approach, map it to a Schwinger-Keldysh partition function, and show that odd Rényi entropies are governed by a replica-entangling action with a circulant replica-space matrix, while even entropies require handling a replica-space zero mode via boundary conditions. For free theories they obtain exact non-equilibrium formulas in terms of equal-time correlators, and for interacting theories they provide explicit Feynman rules and perturbative corrections up to second order, laying a path for practical calculations and potential experimental access through correlation measurements.

Abstract

We show that odd order Rényi entropies $S^{(2q+1)}$ of a system of interacting scalar fields can be calculated as the free energy of $2q+1$ replicas of the system with additional quadratic inter-replica couplings in the subsystem at the time of measurement of the entropy. These couplings replace boundary field matching conditions. This formalism works both in and out of thermal equilibrium, for closed as well as open quantum systems, and provides a general dictionary between measurable correlation functions and entanglement entropy. $S^{(2q+1)}$ can be analytically continued to calculate the von Neumann entropy $S^{\mathrm{vN}}$. We provide an exact formula relating $S^{(2q+1)}$ and $S^{\mathrm{vN}}$ with correlation functions in a non-interacting theory. For interacting theories, we provide rules for constructing all possible Feynman diagrams for $S^{(2q+1)}$. We show that the boundary matching conditions cannot be completely eliminated while calculating Rényi entropies of even order due to presence of zero modes in replica space.

Figures (5)

• Figure 1: Schematic representation of the WCF calculation on Keldysh time contour. It consists of a forward time evolution ($"+"$ contour) extending from $-\infty$ to $+\infty$ and a backward time evolution ($"-"$ contour) returning from $+\infty$ to $-\infty$. Fields defined on the forward and backward branches are denoted as $\phi_+$ and $\phi_-$, respectively. At the measurement time $t_0$, sources $J_{\bf r}^{\phi_+}$ and $J_{\bf r}^{\pi_+}$ are coupled to the fields $\phi_+$ and $\pi_+$ respectively only within subsystem $A$ on the forward contour, while sources $J_{\bf r}^{\phi_-}$ and $J_{\bf r}^{\pi_-}$ are coupled to the fields $\phi_-$ and $\pi_-$ only within subsystem A on the backward contour. The corresponding partition function gives the WCF of the RDM.
• Figure 2: Schematic representation of inter-replica coupling in $\mathcal{S}_{ent}$ between symmetric component of the fields $(\phi_s)$ in one replica with the conjugate fields $(\pi_s)$ in other replica only within the subsystem $A$ at the time $t_0$. Note that this is an all-to-all coupling with strength $\pm 1$. There is no self-replica coupling.
• Figure 3: Feynman diagrams for evaluating $S^{(n)}_0$ in a free scalar field theory: Three non-zero $2 \times 2$ matrices of correlators $(\mathcal{G}^{pq};\,p,q\in\{s, a\})$, defined in the text, are represented by double lines in (a). Among these, $\mathcal{G}^{ss}$ is shown as fully solid double lines, $\mathcal{G}^{sa}$ as half-solid–half-dashed double lines, and $\mathcal{G}^{as}$ as half-dashed–half-solid double lines. The solid side corresponds to symmetric (s) end, while the dashed side corresponds to antisymmetric (a) end. (b) Diagrammatic representation of the entanglement vertex $S_{ent}$ which couples across replicas; both ends represent symmetric components of fields and their conjugate momenta inside the subsystem $A$ at time $t_0$. (c) Rényi entropies are expressed as the sum of connected ring diagrams containing various powers of $\mathbb{L}$ with the symmetry factor of each diagram indicated. Note that all propagators in (c) are $2 \times 2$ equal time Keldysh propagators $\mathcal{G}^{ss}({\bf r},t_0;{\bf r}',t_0)$ evaluated inside subsystem $A$ at $t_0$.
• Figure 4: (a) The symmetric component $\phi_s$ is represented by a half solid line, and the antisymmetric component $\phi_a$ by a half dashed line. (b) The Keldysh $(G_0^{ss/K})$, retarded $(G_0^{sa/R})$ and advanced $(G_0^{as/A})$ propagators in the free theory $\mathcal{S}_{0}$ are denoted by a "solid-solid", "solid-dashed" and "dashed-solid" line respectively and are replica-independent. (c) The propagators of the theory governed by $\tilde{\mathcal{S}}=\sum_{\alpha=1}^n\mathcal{S}^{(\alpha)}_{0}+\mathcal{S}_{ent}$ include contributions from both the free theory propagators and $\mathcal{S}_{ent}$. The corresponding propagators $\tilde{G}^{ss}$, $\tilde{G}^{sa}$, $\tilde{G}^{as}$, and $\tilde{G}^{aa}$ of $\tilde{\mathcal{S}}$ are represented by "solid–solid", "solid–dashed", "dashed–solid", and "dashed–dashed" lines, respectively, each marked with a red circle at the centre indicating corrections from the effective connector $(\mathcal{V})$ defined in (d), obtained by resumming contributions from $\mathcal{S}_{ent}$. These propagators are replica-dependent. (e) The interaction term $\mathcal{S}_{\mathrm{int}}$ is represented by two four-leg vertices: one with three solid and one dashed line, and the other with three dashed and one solid line. Solid and dashed lines correspond to $\phi_s$ and $\phi_a$, respectively. Correction to $S^{(n)}$ due to interaction can be expressed as a sum of connected diagrams for various powers of interaction strength. The first-order corrections to $S^{(n)}$ are shown in (f).
• Figure 5: Second-order free-energy diagrams (i) - (xix) representing all connected ${\cal O}(\lambda^2)$ contributions to $S^{(n)}$. Out of all the nineteen diagrams, (v) and (vi) vanish in the $n\to 1$ limit as they contain two factors of the replica-diagonal propagators $\tilde{G}^{aa}_{\alpha\alpha}$.