Negativity and momentum entanglement in Bardeen-Cooper-Schrieffer states at finite temperature

We analyze the momentum entanglement of homogeneous Bardeen–Cooper–Schrieffer (BCS) states between spin-up and spin-down particles at finite temperature. This is achieved by studying the structure of the partial transposition of the two-particle density matrix (denoted by ${\rho }_2^{T_A}$) in momentum space, as well as the corresponding negativity. We show that ${\rho }_2^{T_A}$ consists of infinitely many decoupled $2\times 2$ submatrices, and momentum entanglement is a necessary and sufficient condition for a nonzero order parameter $\Delta$. From the distribution of eigenvalues of ${\rho }_2^{T_A}$, we discover that momentum entanglement is most significantly contributed by pairs with momenta slightly above a surface related to the Fermi energy. In addition, we propose an entanglement witness operator for detecting momentum entanglement in BCS states.