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Journal of the Physical Society of Japan

The Constrained Hartree-Fock-Bogoliubov Approximation at Finite Temperature for Superconductors -Proposal of the Formalism and Its Numerical Method-

K. Tanabe, Mitsuru Ogoshi

Published Sep 15, 1990• 3 citations
Publisher

Abstract

To describe comprehensively the superconducting phase together with the other phases realized in the copper oxcides, the normal phase, the antiferro-magnetic and the spin-glass phases, an application of the constrained thermal Hartree-Fock-Bogoliubov (CTHFB) approximation is proposed. A set of the electron number- and the magnetization-constrained formalism is exemplified by employing the one-band Hubbard model. The CTHFB equation is applicable to the microscopic interaction whether it includes the phonon-mediated pairing interaction, or it provides a self-sustaining mechanism of the Cooper pair formation. A new type of the order parameter is introduced by extending the generalized Bogoliubov transformation to mix electron spin projections onto the quantization axis (i.e. \(\hat{S}_{z}\)). This yields the lowered minimum of the grand potential, which corresponds to the physical solution violating time-reversal invariance. The gradient method is proposed to solve the CTHFB equation, the non-linear eigenvalue broblem with constraints.

DOI
10.1143/jpsj.59.3307

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•Jun 24, 2011•1 citation
To describe comprehensively the superconducting phase together with the other phases realized in the copper oxcides, the normal phase, the antiferro-magnetic and the spin-glass phases, an application of the constrained thermal Hartree-Fock-Bogoliubov (CTHFB) approximation is proposed. A set of the electron number- and the magnetization-constrained formalism is exemplified by employing the one-band Hubbard model. The CTHFB equation is applicable to the microscopic interaction whether it includes the phonon-mediated pairing interaction, or it provides a self-sustaining mechanism of the Cooper pair formation. A new type of the order parameter is introduced by extending the generalized Bogoliubov transformation to mix electron spin projections onto the quantization axis (i.e. (\hat{S}_{z})). This yields the lowered minimum of the grand potential, which corresponds to the physical solution violating time-reversal invariance. The gradient method is proposed to solve the CTHFB equation, the non-linear eigenvalue broblem with constraints.
The Hartree-Fock-Bogoliubov equations that govern the quasi-particle states of space-dependent superconductivity are formulated in terms of the pure state density matrices ρn(′, )  Φn() Φ (). If is the Hartree-Fock-Bogoliubov Hamiltonian where HH is the normal metal part and Δ() the pairing potential, then the density operator equations are The pairing potential Δ() = V Σnun() V() tanh (βEE/2) is a mixed state quantity and, just as in the normal metal, the equations must be solved self-consistently. We use a steepest descent (optimum gradient) technique, developed by McWeeny for Hartree-Fock equations, to derive an iterative algorithm for solving the above equations. The formulism is developed in general for nonzero temperatures. The method is applied to the T ≈︁ TT region.
The Hartree-Fock-Bogoliubov approximation is very useful for treating both long- and short-range correlations in finite quantum fermion systems, but it must be extended in order to describe detailed spectroscopic properties. One problem is the symmetry-breaking character of the HFB approximation. We present a general and systematic way to restore symmetries and to extend the configuration space using pfaffian formulas for the many-body matrix elements. The advantage of those formulas is that the sign of the matrix elements is unambiguously determined. It is also helpful to extend the space of configurations by constraining the HFB solutions in some way. A powerful method for finding these constrained solutions is the gradient method, based on the generalized Thouless transformation. The gradient method also preserves the number parity of the Bogoliubov transformation, which facilitates the application of the theory to systems with odd particle number.
Abstract On the basis of an extended attractive Hubbard model with nearest‐neighbour interaction of opposite spins two functional methods are adopted for deriving a self‐consistent equation for the superconducting order parameter. The functional‐integral formalism is exploited in saddle‐point approximation. The functional‐derivative technique using Schwinger source fields leads via the self‐energy in Hartree‐Fock‐Bogolubov approximation to a nonlinear gap equation. The superconducting transition temperature is investigated for the case of s‐wave pairing.
We consider the coexistence of the Hund's-rule-exchange induced spin-triplet paired state with the antiferromagnetic ordering by starting from the extended Hubbard model for a doubly degenerate band.We use the density of states appropriate for the square lattice and treat the problem in the HartreeFock approximation.The temperature dependences of the superconducting gaps, the magnetic moment, and the chemical potential are presented.The free energy in the considered phase is evaluated, as well as the corresponding free energies in four additional phases: paramagnetic, ferromagnetic, superconducting of type A and superconducting of type A1 coexisting with ferromagnetism; they occur in the proper range of parameters: band lling n and the interaction parameters U/W and J/W .The low temperature values of the superconducting gaps and staggered magnetic moment are also analyzed as a function of band lling.
Finite-temperature mean-field theories, such as the Hartree--Fock (HF) and Hartree--Fock--Bogoliubov (HFB) theories, are formulated in the grand-canonical ensemble, and their applications to the calculation of statistical properties of nuclei such as level densities require a reduction to the canonical ensemble. In a previous publication [Y. Alhassid et al., Phys. Rev. C 93, 044320 (2016)], it was found that ensemble-reduction methods based on the saddle-point approximation are not reliable in cases in which rotational symmetry or particle-number conservation is broken. In particular, the calculated HFB canonical entropy can be unphysical as a result of the inherent violation of particle-number conservation. In this work, we derive a general formula for exact particle-number projection after variation in the HFB approximation, assuming that the HFB Hamiltonian preserves time-reversal symmetry. This formula reduces to simpler known expressions in the HF and Bardeen--Cooper--Schrieffer (BCS) limits of the HFB. We apply this formula to calculate the thermodynamic quantities needed for level densities in the heavy nuclei 162Dy,148Sm^{162}\mathrm{Dy}, ^{148}\mathrm{Sm}162Dy,148Sm, and 150Sm^{150}\mathrm{Sm}150Sm. We find that the exact particle-number projection gives better physical results and is significantly more computationally efficient than the saddle-point methods. However, the fundamental limitations caused by broken symmetries in the mean-field approximation are still present.
High-temperature superconductivity in copper oxides (cuprates, e.g. La2−xSrxCuO4, YBa2Cu3O6+δ or Bi2Sr2CaCu2O8+δ) remains among the most spectacular phenomena in condensed matter physics. Since its discovery in 1986, an enormous number (∼ 105) of papers on the subject have appeared. So far, there is no single, commonly accepted theory of high-temperature superconductivity. However, it is widely believed that a basic description of this phenomenon can be provided by a single-band Hubbard model or its derivative, the t-J model. The latter model is regarded as a minimal microscopic model, capable of describing the essential aspects of the complex physics of the cuprates. Unfortunately, in general case, neither of those two models can be solved exactly and therefore various approximate methods are used. Among them, the so-called mean-field methods provide a simple, yet fairly reasonable description of the cuprates. In particular, some of the main qualitative features of the phase diagram and the essential features of electronic spectrum are roughly reproduced. A standard mean-field approach to the t-J model, known under the name of renormalized mean-field theory (RMFT) goes beyond the Hartree-Fock approximation. Consequently, its fully consistent treatment requires a novel theoretical approach. This has been our original motivation to develop a general approach to the mean-field models, which is based on the maximum entropy (MaxEnt) principle. The method is presented in detail in Part II of this Thesis, and in Part III it is applied to study RMFT of the t-J model. First, we compare the results obtained within our formalism with those of the frequently used non-variational approach based entirely on the self-consistent equations. Also, various versions of RMFT are compared, and the most satisfactory of them is selected. This optimal version is subsequently used to study different versions of the original t-J Hamiltonian. As a result, upper critical concentration and doping dependence of the selected physical quantities (e.g. the superconducting gap and the Fermi velocity) is determined at low temperatures and in the absence of external magnetic field. We compare our findings both with theoretical results obtained from the Variational Monte Carlo (VMC) methods, as well as with the experimental data for selected cuprates. We show that the version of RMFT approach formulated in this Thesis provides a reasonable qualitative and in some cases semiquantitative rationalization of the principal characteristics of the hole-doped high-temperature superconductors at the optimal doping and in the overdoped regime. Possible extensions of the proposed analysis are mentioned at the end.