BCS Ground State Research Articles

Entanglement entropy of the maximum geminal of the BCS ground state

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.

Multiply Quantized Vortices in Fermionic Superfluids: Angular Momentum, Unpaired Fermions, and Spectral Asymmetry.

We compute the orbital angular momentum L_{z} of an s-wave paired superfluid in the presence of an axisymmetric multiply quantized vortex. For vortices with a winding number |k|>1, we find that in the weak-pairing BCS regime, L_{z} is significantly reduced from its value ℏNk/2 in the Bose-Einstein condensation (BEC) regime, where N is the total number of fermions. This deviation results from the presence of unpaired fermions in the BCS ground state, which arise as a consequence of spectral flow along the vortex subgap states. We support our results analytically and numerically by solving the Bogoliubov-de Gennes equations within the weak-pairing BCS regime.

Order parameter, correlation functions, and fidelity susceptibility for the BCS model in the thermodynamic limit

The exact ground state of the reduced BCS Hamiltonian is investigated numerically for large system sizes and compared with the BCS ansatz. A ``canonical'' order parameter is found to be equal to the largest eigenvalue of Yang's reduced density matrix in the thermodynamic limit. Moreover, the limiting values of the exact analysis agree with those obtained for the BCS ground state. Exact results for the ground-state energy, level occupations, and a pseudospin-pseudospin correlation function are also found to converge to the BCS values already for relatively small system sizes. However, discrepancies persist for a pair-pair correlation function, for interlevel correlations of occupancies and for the fidelity susceptibility, even for large system sizes where these quantities have visibly converged to well-defined limits. Our results indicate that there exist nonperturbative corrections to the BCS predictions in the thermodynamic limit.

Neutrino reactions onLa138andTa180via charged and neutral currents by the quasiparticle random-phase approximation

Cosmological origins of the two heaviest odd-odd nuclei, $^{138}$La and $^{180}$Ta, are believed to be closely related to the neutrino-process. We investigate in detail neutrino-induced reactions on the nuclei. Charged current (CC) reactions, $^{138}$Ba$ (\nu_e, e^{-}) ^{138}$La and $^{180}$Hf$ (\nu_e, e^{-}) ^{180}$Ta, are calculated by the standard Quasi-particle Random Phase Approximation (QRPA) with neutron-proton pairing as well as neutron-neutron, proton-proton pairing correlations. For neutral current (NC) reactions, $^{139}$La$ (\nu \nu^{'}) ^{139}${La}$^*$ and $^{181}$Ta$ (\nu, \nu^{'}) ^{181}$Ta$^*$, we generate ground and excited states of odd-even target nuclei, $^{139}$La and $^{181}$Ta, by operating one quasi-particle to even-even nuclei, $^{138}$Ba and $^{180}$Hf, which are assumed as the BCS ground state. Numerical results for CC reactions are shown to be consistent with recent semi-empirical data deduced from the Gamow-Teller strength distributions measured in the ($^{3}$He, t) reaction. Results for NC reactions are estimated to be smaller by a factor about 4 $\sim$ 5 rather than those by CC reactions. Finally, cross sections weighted by the incident neutrino flux in the core collapsing supernova are presented for further applications to the network calculations for relevant nuclear abundances.

Possible high-temperature superconducting state with ad+idpairing symmetry in doped graphene

Graphene is at the forefront of condensed matter sciences, because of a variety of interesting phenomena it supports. If graphene could support high Tc superconductivity, after doping for example, it will make it even more valuable. Some authors have suggested possibility of superconductivity in graphite like systems. However, an early suggestion of one of us (Baskaran) was unique in the sense it combined Pauling's classic idea of resonating valence bond physics with band theory to obtain some exciting results for superconductivity. Black-Schaffer and Doniach took this approach further and found an unconventional d + id order parameter. To sharpen our theory and get more convincing and reliable results for superconductivity, we introduce a correlated variational BCS ground state wavefunction and perform extensive Monte Carlo study of the repulsive Hubbard model on the honeycomb lattice. We find that undoped graphene is not a superconductor, consistent with experiments and also mean field results. Interestingly, an appreciable superconducting order is obtained around an optimal doping. This result and a supportive slave particle analysis together suggest the possibility of high temperature superconductivity in doped graphene.

Quartet Structures of Particles and Quasiparticles in the BCS Model

We study the possibility of multiple paring for more than two particles or two quasiparticles in terms of the BCS model. We consider the multiple pairs of particles in terms of the BCS Hamiltonian for two different ground states: in a quiescent Fermi sea model and in the BCS ground state. In case of quasiparticles, we consider the multiple quasiparticle pairs in terms of the BCS ground state only. Although there is no interaction between Cooper’s type pairs in terms of the BCS Hamiltonian, yet we have shown that four particles or two/four quasiparticles can be paired and form a bound state in the singlet state with just twice the bound state energy of a single Cooper’s pair. In the case of a pair of quasiparticles, the bound state exists as a result of the large quasiparticle density of states and the residual interaction between two quasiparticles which is described by the off-diagonal terms in the BCS Hamiltonian written in terms of quasiparticles. In the case of four particles, the bound state exists only as a result of the Pauli principle and the sharp Fermi edge. In the BCS model, a quartet of bound fermions indeed represents a boson, and the many-particle system of the composite of bosons can undergo the conventional Bose condensation of a boson gas. The temperature of the Bose condensation corresponds to the conventional temperature of Bose condensation for non-interacting bosons where each boson has quadruple mass (4m) and the boson density is one-quarter (n/4) of fermions density. A similar conclusion remains valid in the case of the particle–hole resonance in a quiescent Fermi sea. We have shown that in the particle–hole channel there exists the multiple particle–hole resonance for four particles and four holes in a quiescent Fermi sea model similar to the case of two particles and two holes resonance. We have shown that there is no particle–hole resonance in the case of the BCS ground state because there is no hole-type of excitations in the BCS ground state. Nevertheless we have shown that in the BCS ground state quasiparticle pairs can form bound pairs similar to the Cooper’s pair of particles due to the off-diagonal terms in the BCS Hamiltonian. The bound pairs of quasiparticles exist as a result of extremely large quasiparticle density of states. We also discuss the formation of a quartet of quasiparticles and the Bose condensation of the multiple pairs of quasiparticles.

Angular momentum decomposition of Richardson's pairs

The angular momentum decomposition of pairs obtained using Richardson's exact solution of the pairing Hamiltonian for the deformed $^{174}\mathrm{Yb}$ nucleus are displayed. The probabilities for low angular momenta of the collective pairs are strikingly different from the ones obtained in the BCS ground state.

MULTIPLE PAIRING IN THE BCS MODEL

We study Cooper's paring for more than two particles in terms of the BCS model. We consider multiple pairing in terms of the BCS Hamiltonian in a quiescent Fermi sea model and in the BCS ground state. Although there is no interaction between Cooper's pairs in terms of the BCS Hamiltonian, yet we show that four particles are paired and form a bound state in the singlet state with just twice the bound state energy of a single Cooper's pair. Four-particle bound state - quartet only exists as a result of the Pauli principle and the sharp Fermi edge. We have also shown that the smearing of the Fermi edge due to Δ(k) as it is in the BCS ground state weakens the pairing of either two or four particles. We show that in the particle-hole channel there exists the multiple particle-hole resonance for four particles and four holes in a quiescent Fermi sea model similar to the resonance of two particles and two holes. There is no the particle-hole resonance in the BCS ground state as shown, which means that the particle-hole resonance is removed by Δ(k) due to smearing of the Fermi distribution. The wave function for the multiple pairing model is discussed as well.

Natural orbits of atomic Cooper pairs in a nonuniform Fermi gas

We examine the basic mode structure of atomic Cooper pairs in an inhomogeneous Fermi gas. Based on the properties of Bogoliubov quasiparticle vacuum, the single particle density matrix and the anomalous density matrix share the same set of eigenfunctions. These eigenfunctions correspond to natural pairing orbits associated with the BCS ground state. We investigate these orbits for a Fermi gas in a spherical harmonic trap and construct the wave function of a Cooper pair in the form of Schmidt decomposition. The issue of spatial quantum entanglement between constituent atoms in a pair is addressed.

Entanglement of Fermi gases in a harmonic trap

For both cases with and without interactions, bipartite entanglement of two-fermions from a Fermi gas in a trap is investigated. We show how the entanglement depends on the locations of the two fermions and the total number of particles. Fermions at the verge of trap have longer entanglement distance (beyond it, the entanglement disappears) than those in the center. We derive a lower limitation to the average overlapping for two entangled fermions in the BCS ground state, it is shown to be $\sqrt{Q/2M}$, a function of Cooper pair number $Q$ and total number of occupied energy levels $M$.

Mesoscopic BCS pairing in the repulsive one-dimensional Hubbard model

We study mesoscopic pairing in the one-dimensional repulsive Hubbard model and its interplay with the BCS model in the canonical ensemble. The key tool is comparing the Bethe ansatz equations of the two models in the limit of small Coulomb repulsion. For the ordinary Hubbard interaction the BCS Bethe equations with infinite pairing coupling are recovered; a finite pairing is obtained by considering a further density-dependent phase-correlation in the hopping amplitude of the Hubbard model. We find that spin degrees of freedom in the Hubbard ground state are arranged in a state of the BCS type, where the Cooper pairs form a noncondensed liquid on a ``lattice'' of single particle energies provided by the Hubbard charge degrees of freedom; the condensation in the BCS ground state corresponds to Hubbard excitations constituted by a sea of spin singlets.

Quantum antiferromagnetism and highTcsuperconductivity: A close connection between thet−Jmodel and the projected BCS Hamiltonian

A connection between quantum antiferromagnetism and high ${T}_{c}$ superconductivity is theoretically investigated by analyzing the $t\text{\ensuremath{-}}J$ model and its relationships to the Gutzwiller-projected BCS Hamiltonian. After numerical corroboration via exact diagonalization, it is analytically shown that the ground state of the $t\text{\ensuremath{-}}J$ model at half filling [i.e., the two-dimensional (2D) antiferromagnetic Heisenberg model] is entirely equivalent to the ground state of the Gutzwiller-projected BCS Hamiltonian with strong pairing. Combined with the high wave function overlap between the ground states of the $t\text{\ensuremath{-}}J$ model and the projected BCS Hamiltonian at moderate doping, this equivalence provides strong support for the existence of superconductivity in the $t\text{\ensuremath{-}}J$ model. The relationship between the ground state of the projected BCS Hamiltonian and Anderson's resonating valence bond state (i.e., the projected BCS ground state) is discussed.

Entanglement of electron spins in superconductors

We investigate entanglement of two electron spins forming Cooper pairs in an s-wave superconductor. The two-electron space-spin density matrix is obtained from the BCS ground state using a two-particle Green's function. It is demonstrated that a two spin state is not given by a spin singlet state but by a Werner state. It is found that the entanglement length, within which two spins are entangled, is not the order of the coherence length but the order of the Fermi wave length.

EFFECT OF PARTICLE-HOLE EXCITATIONS ON THE BCS GROUND STATE

We show that the well-known BCS Hamiltonian in terms of the BCS ground state represents, in fact, the generator of the particle-hole pairs and that the BCS ground state describes both the particle-hole excitations and the pairing condensate. We calculate the contribution of particle-hole excitation to the ground state energy. We show that the particle-hole pairs in the case of attractive interaction significantly lower the ground state energy, which means that the BCS ground state is not really the superconducting ground state. In contrast, for the repulsive interaction the particle-hole excitations are just to renormalize the ground state energy. Thus, in the case of repulsive interaction the condensate of bound pairs would lead to the real superconducting ground state of the system as shown.

Stable and resonant Cooper pairs

A Bethe-Salpeter treatment of Cooper pairs (CPs) based on an ideal Fermi gas (IFG) ‘sea’ yields the familiar stable negative-energy, infinite-lifetime two-particle bound-state if two-hole CPs are ignored. Otherwise, it gives purely imaginary energies and thus unphysical solutions. However, a Bogoliubov transformation to the BCS ground state yields the trivial sound solution plus a nontrivial one. The latter are two-fermion ‘moving CPs’ as positive-energy stable objects for zero center-of-mass momentum (CMM), and finite-lifetime resonant ones for nonzero CMM with a linear dispersion leading term. Hence, Bose-Einstein condensation of such pairs may occur in exactly two dimensions as it cannot with quadratic dispersion.

Proton-neutron pairing in the deformed BCS approach

We examine isovector and isoscalar proton-neutron pairing correlations for the ground state of even-even Ge isotopes with mass number A=64-76 within the deformed BCS approach. For N=Z 64Ge the BCS solution with only T=0 proton-neutron pairs is found. For other nuclear systems (N>Z) a coexistence of a T=0 and T=1 pairs in the BCS wave function is observed. A problem of fixing of strengths of isoscalar and isovector pairing interactions is addressed. A dependence of number of like and unlike pairs in the BCS ground state on the difference between number of neutrons and protons is discussed. We found that for nuclei with N much bigger than Z the effect of proton-neutron pairing is small but not negligible.

Cooper Pairing and Ladder Correlations in a BCS Ground State

A Bethe–Salpeter treatment of Cooper pairs (CPs) based on an ideal Fermi gas (IFG) "sea" produces unstable CPs if holes are not ignored. Stable CPs with damping emerge when the BCS ground state replaces the IFG, and are positive-energy, finite-lifetime resonances for nonzero center-of-mass momentum with a linear dispersion leading term. Bose–Einstein condensation of such pairs may thus occur in exactly two dimensions as it cannot with quadratic dispersion.

BCS Ground State and XXZ Antiferromagnetic Model as SU(2), SU(1,1) Coherent States: An Algebraic Diagonalization Method

An algebraic diagonalization method is proposed. As two examples, the Hamiltonians of BCS ground state under mean-field approximation and XXZ antiferromagnetic model in linear spin-wave frame have been diagonalized by using SU(2), SU(1,1) Lie algebraic method, respectively. Meanwhile, the eigenstates of the above two models are revealed to be SU(2), SU(1,1) coherent states, respectively. The relation between the usual Bogoliubov–Valatin transformation and the algebraic method in a special case is also discussed.

Cooper pairs as resonances

Using the Bethe–Salpeter (BS) equation, Cooper pairing can be generalized to include contributions from holes as well as particles from the ground state of either an ideal Fermi gas (IFG) or of a BCS many-fermion state. The BCS model interfermion interaction is employed throughout. In contrast to the better-known original Cooper pair (CP) problem for either two particles or two holes, the generalized Cooper equation in the IFG case has no real-energy solutions. Rather, it possesses two complex-conjugate solutions with purely imaginary energies. This implies that the IFG ground state is unstable when an attractive interaction is switched on. However, solving the BS equation for the BCS ground state reveals two types of real solutions: one describing moving (i.e., having nonzero total, or center-of-mass, momenta) CPs as resonances (or bound composite particles with a finite lifetime), and another exhibiting superconducting collective excitations analogous to Anderson–Bogoliubov–Higgs RPA modes. A Bose–Einstein-condensation-based picture of superconductivity is addressed.

On the equivalence between sine-Gordon model and Thirring model in the chirally broken phase of the Thirring model

We investigate the equivalence between Thirring model and sine-Gordon model in the chirally broken phase of the Thirring model. This is unlike all other available approaches where the fermion fields of the Thirring model were quantized in the chiral symmetric phase. In the path integral approach we show that the bosonized version of the massless Thirring model is described by a quantum field theory of a massless scalar field and exactly solvable, and the massive Thirring model bosonizes to the sine-Gordon model with a new relation between coupling constants. We show that the non-perturbative vacuum of the chirally broken phase in the massless Thirring model can be described in complete analogy with the BCS ground state of superconductivity. The Mermin-Wagner theorem and Coleman's statement concerning the absence of Goldstone bosons in the 1+1-dimensional quantum field theories are discussed. We investigate the current algebra in the massless Thirring model and give a new value of the Schwinger term. We show that the topological current in the sine-Gordon model coincides with the Noether current responsible for the conservation of the fermion number in the Thirring model. This allows to identify the topological charge in the sine-Gordon model with the fermion number.