A study of momentum entanglement and negativity in Bardeen-Cooper-Schrieffer states at finite temperature

We study the momentum entanglement between the spin-up and spin- down particles of the homogeneous Bardeen-Cooper-Schrieffer (BCS) state at finite temperature. To achieve this, we construct from the BCS state the partial transposition $\rho_2^{\mathrm{T} _A}$ of the two particle density matrix in momentum space. The structure of $\rho_2^{\mathrm{T}_A}$ and its corresponding negativity ${\cal N}_2$ are examined. We show that $\rho_2^ {\mathrm{T}_A}$ consists of infinitely many decoupled $2 \times 2 $ submatrices, and momentum entanglement coexists with the pairing order parameter $\Delta$. It is found that pairs with momenta slightly above a surface related to the Fermi energy contribute this entanglement most significantly. We propose an entanglement witness operator as a measurable quantity to detect momentum entanglement in BCS states.