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arXiv:1311.4614 · International Journal of Modern Physics B

Thermal resonating Hartree–Bogoliubov theory based on the projection method

Seiya Nishiyama, João da Providência, Hiromasa Ohnishi

Published May 30, 2014• 1 citation
arXivHTMLPDF

Abstract

We propose a rigorous thermal resonating mean-field theory (Res-MFT). A state is approximated by superposition of multiple MF wavefunctions (WFs) composed of non-orthogonal Hartree–Bogoliubov (HB) WFs. We adopt a Res-HB subspace spanned by Res-HB ground and excited states. A partition function (PF) in a SO (2N) coherent state representation (CS Reps) |g 〉(N: Number of single-particle states) is expressed as Tr (e-βH) = 2N-1∫〈g|e-βH|g〉dg (β = 1/kBT). Introducing a projection operator P to the Res-HB subspace, the PF in the Res-HB subspace is given as Tr (Pe-βH), which is calculated within the Res-HB subspace by using the Laplace transform of e-βH and the projection method. The variation of the Res-HB free energy is made, which leads to a thermal HB density matrix [Formula: see text] expressed in terms of a thermal Res-FB operator [Formula: see text] as [Formula: see text]. A calculation of the PF by an infinite matrix continued fraction (IMCF) is cumbersome and a procedure of tractable optimization is too complicated. Instead, we seek for another possible and more practical way of computing the PF and the Res-HB free energy within the Res-MFT.

DOI
10.1142/s0217979214501318

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We propose a rigorous thermal resonating mean-field theory (Res-MFT). A state is approximated by superposition of multiple MF wavefunctions (WFs) composed of non-orthogonal Hartree–Bogoliubov (HB) WFs. We adopt a Res-HB subspace spanned by Res-HB ground and excited states. A partition function (PF) in a SO (2N) coherent state representation (CS Reps) |g 〉(N: Number of single-particle states) is expressed as Tr (e-βH) = 2N-1∫〈g|e-βH|g〉dg (β = 1/kBT). Introducing a projection operator P to the Res-HB subspace, the PF in the Res-HB subspace is given as Tr (Pe-βH), which is calculated within the Res-HB subspace by using the Laplace transform of e-βH and the projection method. The variation of the Res-HB free energy is made, which leads to a thermal HB density matrix [Formula: see text] expressed in terms of a thermal Res-FB operator [Formula: see text] as [Formula: see text]. A calculation of the PF by an infinite matrix continued fraction (IMCF) is cumbersome and a procedure of tractable optimization is too complicated. Instead, we seek for another possible and more practical way of computing the PF and the Res-HB free energy within the Res-MFT.
The finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation often breaks symmetries of the underlying many-body Hamiltonian. Restricting the calculation of the HFB partition function to a subspace with good quantum numbers through projection after variation restores some of the correlations lost in breaking these symmetries, although effects of the broken symmetries such as sharp kinks at phase transitions remain. However, the most general projection after variation formula in the finite-temperature HFB approximation is limited by a sign ambiguity. Here, I extend the Pfaffian formula for the many-body traces of HFB density operators introduced by Robledo [L. M. Robledo, Phys. Rev. C. 79, 021302(R) (2009)] to eliminate this sign ambiguity and evaluate the more complicated many-body traces required in projection after variation in the most general HFB case. The method is validated through a proof-of-principle calculation of the particle-number-projected HFB thermal energy in a simple model.
The finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation often breaks symmetries of the underlying many-body Hamiltonian. Restricting the calculation of the HFB partition function to a subspace with good quantum numbers through projection after variation restores some of the correlations lost in breaking these symmetries, although effects of the broken symmetries such as sharp kinks at phase transitions remain. However, the most general projection after variation formula in the finite-temperature HFB approximation is limited by a sign ambiguity. Here, we extend the Pfaffian formula for the many-body traces of HFB density operators introduced by L. M. Robledo in Ref. [1] to eliminate this sign ambiguity and evaluate the more complicated many-body traces required in projection after variation in the most general HFB case. We validate our method through a proof-of-principle calculation of the particle-number-projected HFB thermal energy in a simple model.
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.
Applying thermofield dynamics, we reformulate the exact quantum number projection in the finite-temperature Hartree-Fock-Bogoliubov theory. Explicit formulas are derived for the simultaneous projection of particle number and angular momentum, in parallel to the zero-temperature case. We also propose a practical method for the variation-after-projection calculation, by approximating entropy consistently with the Peierls inequality. Using quantum number projection in finite-temperature mean-field theory will be useful for studying the effects of quantum fluctuations associated with the conservation laws on thermal properties of nuclei.
A resonating Hartree-Fock (Res-HF) method is revisited and improved by the complete spin projection for the interacting Fermion system. This method approximates a many-body wave function by the superposition of nonorthogonal Slater determinants (S-dets). The nonorthogonality of the S-dets makes it possible to describe the large quantum fluctuations efficiently, since each S-det naturally includes the full-electron-excitation effects from other S-dets. The molecular orbitals in every S-det, as well as the superposition coefficients, are variationally determined. So far, however, the spin contamination, caused by the conventional half-projection, has been a serious obstacle to obtain the accurate wave functions for large-size fermion systems, especially in the intermediate and large correlation regimes. In this paper, we apply the complete spin projection method to the Res-HF calculations. As an example, the improved Res-HF method is applied to the one-dimensional Hubbard model. It will be shown that the complete spin projection improves the Res-HF wave functions significantly. In fact, the correlation energies explained by the improved Res-HF method are better than those by the variational Monte Carlo method in all the correlation regimes. The correlation structures at arbitrary fillings are also well described by the improved Res-HF method.
We develop a general theory to describe a superconducting Fermion system with large quantum fluctuations. The theory is a direct extension of the resonating Hartree-Fock (HF) theory to the Hartree-Bogoliubov case including pair correlations. We start with an exact coherent state representation of a superconducting Fermion system on a special orthogonal group. A state with large quantum fluctuations is approximated by a superposition of non-orthogonal Hartree-Bogoliubov (HB) wave functions with different correlation structures. We obtain the variational equations to determine the coefficients of configuration mixing and the orbitals in the resonating HB wave functions.