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Spin entanglement entropy and number uctuations in the BCS ground state

Xavier M. Puspus, Kristian Hauser A. Villegas, Francis N. C. Paraan

Published May 9, 2014• 0 citations
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Abstract

National Institute of Physics, University of the Philippines, Diliman, Quezon City 1101, Philippines(Dated: May 15, 2014)We study the entanglement between the spin components of the Bardeen-Cooper-Schrie er (BCS)ground state by calculating the corresponding von Neumann entropy and obtaining the full entan-glement spectrum. This spin entanglement entropy is jointly proportional to the number of electronsabout the Fermi surface, the pairing energy, and the variance in particle number. The entanglementspectrum is approximated by a canonical distribution of non-interacting fermions at a temperatureequal to the BCS critical temperature.

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•May 17, 2016•0 citations
National Institute of Physics, University of the Philippines, Diliman, Quezon City 1101, Philippines(Dated: May 15, 2014)We study the entanglement between the spin components of the Bardeen-Cooper-Schrie er (BCS)ground state by calculating the corresponding von Neumann entropy and obtaining the full entan-glement spectrum. This spin entanglement entropy is jointly proportional to the number of electronsabout the Fermi surface, the pairing energy, and the variance in particle number. The entanglementspectrum is approximated by a canonical distribution of non-interacting fermions at a temperatureequal to the BCS critical temperature.
We study entanglement between the spin components of the BCS ground state by calculating the full entanglement spectrum and the corresponding von Neumann entanglement entropy. The entanglement spectrum is effectively modeled by a generalized Gibbs ensemble (GGE) of noninteracting electrons, which may be approximated by a canonical ensemble at the BCS critical temperature. We further demonstrate that the entanglement entropy is jointly proportional to the pairing energy and to the number of electrons about the Fermi surface (an area law). Furthermore, the entanglement entropy is also proportional to the number fluctuations of either spin component in the BCS state.
Abstract From the viewpoint of the Bose–Einstein condensation (BEC) of the fermion system, the maximum geminal of the second-order reduced density matrix of the superconducting state exactly corresponds to the Cooper pair. In this paper the entanglement entropy (EE) for the maximum geminal of the BCS ground state is evaluated. The EE behaves logarithmically with respect to the number of the maximum geminal. Furthermore, the disappearance point of superconductivity is defined on the basis of the fermion BEC. In the superconducting ground state, almost all electrons in the energy width of the gap parameter near the Fermi level are condensed as a maximum geminal. They suddenly change to normal electrons with a finite gap of the EE at the disappearance point like a first-order phase transition.
The entanglement entropy of two gapless noninteracting fermion subsystems is computed approximately in a way that avoids the introduction of replicas and a geometric interpretation of the reduced density matrix. We exploit the similarity between the Schmidt basis wave function and superfluid BCS wave function and compute the entropy using the BCS approximation. Within this analogy, the Cooper pairs are particle-hole pairs straddling the boundary and the effective interaction between them is induced by the projection of the Hilbert space onto the incomplete Schmidt basis. The resulting singular interaction may be thought of as ``lifting'' the degeneracy of the single-particle distribution function. For two coupled fermion systems of linear size LLL, we solve the BCS gap equation approximately to find the entropy $S\ensuremath{\approx}({w}^{2}/{t}^{2})\mathrm{ln}L$, where www is the hopping amplitude at the boundary of the subsystem and 2t2t2t is the bandwidth. We further interpret this result based upon the relationship between entanglement spectrum, entropy, and number fluctuations.
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 ${\ensuremath{\rho}}_{2}^{{T}_{A}}$) in momentum space, as well as the corresponding negativity. We show that ${\ensuremath{\rho}}_{2}^{{T}_{A}}$ consists of infinitely many decoupled $2\ifmmode\times\else\texttimes\fi{}2$ submatrices, and momentum entanglement is a necessary and sufficient condition for a nonzero order parameter $\ensuremath{\Delta}$. From the distribution of eigenvalues of ${\ensuremath{\rho}}_{2}^{{T}_{A}}$, 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.
The entanglement of the N-harmonium system (i.e., a completely-integrable model of N particles where both the confinement and the two-particle interaction are har-monic) is shown to be analytically determined in terms of N and the relative interaction strength. First, the entanglement of the N-boson system via the von Neumann entropy is computed by use of the corresponding natural occupation numbers. Below a critical value Nc of N it is found that, for positive values of the coupling constant the entan-glement grows when the number of particles is increasing; opposite situation occurs for N> Nc. Then the entanglement of the N-fermion system is investigated by means of the linear entropy, which is a linearized form of the von Neumann entropy of the one-body reduced density matrix. We have shown that the entanglement grows when the number of particles is increasing in the region of negative coupling. For positively increasing values of the coupling the entanglement as a function of N varies according to various regimes. Moreover, in the strong coupling regime the entropy of the system decreases when N is growing. For fixed N, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Finally, let us point out that the spatial and spin contributions to the total entanglement are of comparable size.
By using Bogoliubov transformations to construct the ground states of fermionic Bardeen-Cooper-Schrieffer (BCS) superfluids and weakly-interacting Bose gases supporting Bose Einstein Condensation (BEC), their algebraic structures and implications can be analyzed in detail. Both ground states are generalized squeezed coherent states saturating a generalized Heisenberg uncertainty relation, and they acquire quantized Berry phases when the corresponding systems are transported along a closed path in their parameter spaces. While the Berry phase of the BCS ground state depends on the total particle number, the Berry phase of the BEC ground state depends only on the particles outside the BEC. The Berry phases are associated with magnetic monopoles in the parameter spaces and we found that the Dirac quantization condition is satisfied. Moreover, both ground states are entangled states of the fermion or boson quanta and we found the entanglement entropy quantifying the internal correlations. A fixed particle-number approach of fermionic superfluids does not saturate the generalized uncertainty relation, exhibits internal entanglement, and gives corresponding Berry phase. In addition, the algebraic structures of the ground states can be classified by the qqq-deformed Hopf algebra, ⨁khqk(1)\bigoplus_{\mathbf{k}}h_{q_{\mathbf{k}}}(1)⨁k​hqk​​(1) for bosons and qqq-deformed Hopf superalgebra ⨁khqk(1∣1)\bigoplus_{\mathbf{k}}h_{q_{\mathbf{k}}}(1|1)⨁k​hqk​​(1∣1) for fermions, respectively.