Cluster property and entanglement in the superconducting state

Abstract

We present the conditions for satisfying the cluster property in the superconducting state. Using these conditions, we show that the cluster property employed when deriving the order parameter of the superconducting state (OPSS) holds true, and that the OPSS is given by ${}_S\langle Ψ| \hatψ(x)\hatψ(x') | Ψ\rangle_S$, where $|Ψ\rangle_S$ is the superconducting state, and where $\hatψ(x)$ is the field operator of superconducting electrons. This provides a resolution to the long-standing concern raised by Leggett et al.\ regarding particle-number fluctuation in the superconducting state. In addition, by considering the entanglement between two particles composed of the maximum geminal of the BCS ground state [J.\ Phys. Commun. 5, 095003 (2021)], it is confirmed that the larger the deviation from the conditions for satisfying the cluster property becomes, the greater the magnitude of the entanglement becomes. This shows not only the effectiveness of the conditions for satisfying the cluster property but also the justification of the OPSS that has been commonly used.