Entanglement Entropy from String Field Theory (and a Higher-Spin Example)
arXiv:1905.06708
The paper investigates new solutions in linearized open string field theory that describe reduced density matrices for higher‑spin modes and computes the resulting entanglement entropy using series involving weighted partition numbers.
Abstract
We study the new class of solutions in linearized open string field theory (OSFT) involving higher-spin modes. Unlike the elementary OSFT solutions (on-shell vertex operators) that, acting on a vacuum, define wavefunctions of pure states (e.g. a tachyon), the solutions that we describe correspond to the reduced density matrices which eigenvalues describe the entanglement between higher-spin modes with different spin values. We compute the entanglement entropy on these OSFT solutions, and the answer is expressed in terms of converging series in inverse weighted partition numbers. In the case of $D$-dimensional bosonic string theory, the entanglement entropy of spin $1$ subsystem and the system of all the spin values is given by $D{\log{λ_0}}+{D\over{λ_0}}\sum_{N=3}^\infty{{|β(N)|}\over{λ(N)}} {\log{({{λ(N)}\over{|β(N)|}})}}$, where $λ(N)$ is the weighted number of partitions of $N$, $β(N)={{(N-1)ζ(3)-ζ(2)}\over{(N-1)^4}}$ and $λ_0=\sum_{N=1}^{\infty}{{β(N)}\over{λ(N)}}$ ($ζ$ is Riemann's zeta-function). The first term, $D{\log{λ_0}}$, represents the entanglement swapping between string vacuum and string excitations. We generalize this result to obtain the entanglement for a subsystem of a given spin $s$ in a given space-time dimension. We also discuss how open string field theory may be used to study the entanglement of systems other than higher spin excitations in string theory.
23 pages, SFT solution is revisited and modified to regularization-free expression to define a normalizable mized state; final answer for entanglement entropy and discussion section are modified accordingly