Generalized Replica Manifolds I: Surgery and Averaging

TL;DR

Generalized Replica Manifolds I: Surgery and Averaging develops a framework to implement replica-trick calculations by correlated averaging over codimension-one defects, enabling a controlled path-integral surgery that yields generalized replica manifolds for non-spatial entanglement. The approach hinges on averaging over operator insertions forming a representation of the Heisenberg group, and it is extended from simple scalar theories to gauge theories via an extended Hilbert space, with a clear route to color entanglement and quiver gauge theories. The work demonstrates concrete procedures in simple models (two coupled oscillators and scalar fields) and discusses how partial tracing corresponds to constraints on auxiliary fields, offering a gauge-invariant perspective and connecting to large-$N$ holography. It culminates in an explicit color-entanglement construction in matrix models and outlines holographic implications and future directions, including extending operator averaging beyond the Heisenberg case.

Abstract

We develop a simple framework for implementing a type of path integral "surgery" via correlated averaging over codimension-one defects/extended operators. This technique is used to construct replica manifolds by effectively cutting and gluing the path integral without explicitly modifying the underlying manifold. We argue that restricted forms of this averaging can be used to calculate Rényi entanglement entropy corresponding to a wide range of choices of subsystem partitioning. When the entanglement entropy being calculated in this way does not simply correspond to entanglement between subregions, we call the resulting objects from this surgery "generalized replica manifolds". We show how this framework extends to gauge theories and, in particular, how in non-Abelian gauge theories it establishes a connection between replica calculations of a gauge-invariant notion of entanglement between color degrees of freedom and a quiver gauge-theory structure. Finally, we discuss how this framework appears in the context of large-$N$ theories and holography, with a bird's-eye view of potential future directions. This paper focuses on averaging over operators that form a representation of the Heisenberg group; a subsequent paper will extend the framework to more general operator averaging.

Figures (10)

• Figure 1: (a) A representation of a Euclidean path integral relevant for an evaluation of the half-space entanglement entropy in the ground state of a quantum field theory in $1+1$-dimensional Minkowski space. The vertical direction can be understood as Euclidean time evolution, while the double-sided arrows represent identifications of the color-coded cuts they are pointing to. The path integral can be interpreted as introducing a conical singularity in the Euclidean manifold at the entangling point separating the two halves of space. This in turn allows us to understand the boundary conditions we need to impose in the corresponding bulk calculation. (b) A representation of the analogous path integral for the entanglement entropy between two interacting fields. The shading pattern here represents interactions between the two fields. Our goal will be to try to replace the question mark with something more suitable for an application of the AdS/CFT dictionary.
• Figure 2: A representation of the Euclidean path integral that evaluates $Z_4$. The $x_2$ oscillator lives on the outer Euclidean time circle which has length $4\beta$, while $x_1$ lives on the inner arcs each with length $\beta$. The blue lines represent that for each arc the endpoints are identified. The shading pattern between the inner arcs and the outer circle represents the fact that the two oscillators are interacting.
• Figure 3: Schematic representation of the path integral surgery we wish to perform. By cutting the Euclidean time circle into four pieces and then gluing each piece to itself we go from a path integral evaluating $Z(4\beta)$ to one evaluating $Z(\beta)^4$.
• Figure 4: A source distribution along the Euclidean time circle that, after being integrated over, takes us from $Z(4\beta)$ to $Z(\beta)^4$. Sources labeled by $J_j$ are linearly coupled to the position operator while sources labeled by $K_j$ are linearly coupled to the momentum operator in the precise sense given in \ref{['result1']}.
• Figure 5: Illustration of how using the same techniques of integrating over external sources can take us from $Z(\beta)^n$ to $Z(4\beta)$. Both the parameter $\epsilon$, used to separate sources in Euclidean time, and the limit $\epsilon \rightarrow 0$ are being suppressed here for illustration clarity.