Entwinement in discretely gauged theories

Abstract

We develop the notion of entwinement to characterize the amount of quantum entanglement between internal, discretely gauged degrees of freedom in a quantum field theory. This concept originated in the program of reconstructing spacetime from entanglement in holographic duality. We define entwinement formally in terms of a novel replica method which uses twist operators charged in a representation of the discrete gauge group. In terms of these twist operators we define a non-local, gauge-invariant object whose expectation value computes entwinement in a standard replica limit. We apply our method to the computation of entwinement in symmetric orbifold conformal field theories in 1+1 dimensions, which have an $S_N$ gauging. Such a theory appears in the weak coupling limit of the D1-D5 string theory which is dual to AdS$_3$ at strong coupling. In this context, we show how certain kinds of entwinement measure the lengths, in units of the AdS scale, of non-minimal geodesics present in certain excited states of the system which are gravitationally described as conical defects and the $M=0$ BTZ black hole. The possible types of entwinement that can be computed define a very large new class of quantities characterizing the fine structure of quantum wavefunctions.

Figures (4)

• Figure 1: Path integral in radial quantization. The path integral is left open on the cut (indicated in full blue lines) with boundary conditions $\phi$ on the lower cut and $\phi '$ on the upper cut. The dashed lines define the complementary interval along which boundary conditions are matched $\phi=\phi '$. The operator $\sigma$ prepares the state.
• Figure 2: (Left) The $n$-sheeted Riemann surface from cyclically gluing $n$ copies of the plane. The dashed arrows denote how to sew fields across the cuts. (Right) Correlator in the plane. The $\Sigma$ insertions represent Rényi twist operators, while the $\sigma$ insertions define the replicated state.
• Figure 3: Multiwound strings, each consisting of $m$ strands. There are $N_m$$m$-wound strings such that the total number of strands is $N$. Here we depict $N=7$, $N_2=2$, $N_3=1$. (Left) The entanglement entropy is computed by inserting Rényi twist operators at the endpoints of the interval on every strand. The entangling region can be visualized in the long string picture as a union of disjoint intervals on all strands. (Right) Configuration of twists corresponding to the bilocal operator of single interval entwinement. The entangling region extends across different strands of the 3-cycle.
• Figure 4: Representation of the branching structure of a correlator of the form \ref{['eq:correlator']} in the simple case of a single strand entwinement, $\ell=0$, on a $3$-cycle factor, $m=3$.