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International Journal of Modern Physics E

BCS-TYPE THEORY IN CANONICAL ENSEMBLES

Hitoshi Nakada, K. Tanabe

Published Nov 1, 2006• 3 citations
Publisher

Abstract

To investigate the pairing phase transition in finite systems with the particle-number conservation, we formulate a new BCS-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. This theory enables us to distinguish the symmetry-restoring fluctuation (SRF) from additional quantum fluctuations. By numerical calculations, it is found that the phase transition is compatible with the conservation law in this theory, and that the SRF shifts up the critical temperature. Having correct zero-temperature limit, this theory also gives new interpretation on the Pauli blocking effect in systems with odd particle number.

DOI
10.1142/s0218301306005290

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To investigate the pairing phase transition in finite systems with the particle-number conservation, we formulate a new BCS-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. This theory enables us to distinguish the symmetry-restoring fluctuation (SRF) from additional quantum fluctuations. By numerical calculations, it is found that the phase transition is compatible with the conservation law in this theory, and that the SRF shifts up the critical temperature. Having correct zero-temperature limit, this theory also gives new interpretation on the Pauli blocking effect in systems with odd particle number.
We formulate a new Bardeen-Cooper-Schrieffer (BCS)-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. With its broad applicability, this theory can be a useful tool for investigating the pairing phase transition in finite systems with the particle-number conservation. This theory provides effects of the symmetry-restoring fluctuation (SRF) for the pairing phenomena in finite fermionic systems, distinctively from those of additional quantum fluctuations. It is shown by numerical calculations that the phase transition is compatible with the conservation in this theory, and that the SRF shifts up the critical temperature (Tcr{T}^{\mathrm{cr}}Tcr). This shift of Tcr{T}^{\mathrm{cr}}Tcr occurs due to reduction of degrees-of-freedom in canonical ensembles, and decreases only slowly as the particle-number increases (or as the level spacing narrows), in contrast to the conventional BCS theory.
By applying the particle-number projection to the finite-temperature Bardeen-Cooper-Schrieffer (BCS) theory, the S-shaped heat capacity, which has recently been claimed to be a fingerprint of the superfluid-to-normal phase transition in nuclei, is reexamined. It is found that the particle-number (or number-parity) projection gives S shapes in the heat capacity of nuclei that look qualitatively similar to the observed ones. These S shapes are accounted for as effects of the particle-number conservation on the quasiparticle excitations and occur even when we keep the superfluidity at all temperatures by assuming a constant gap in the BCS theory. The present study illustrates significance of the conservation laws in studying phase transitions of finite systems.
Extensions of the Hartree-Fock-Bogoliubov theory are worked out which are tailored for, (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. For both purposes we rely on a single variational principle which optimizes the characteristic function. The Bloch equation is used as a constraint to define the equilibrium state, and the trial quantities are a density operator and a Lagrangian multiplier matrix which is akin to an observable. The conditions of stationarity are respectively a Schrödinger-like equation and a Heisenberg-- like equation with an imaginary time running backwards. General conditions for the trial spaces are stated that warrant the preservation of thermodynamic relations. When the trial spaces are chosen to be of the independent-quasi-particle type, the ensuing coupled equations provide an extension of the Hartree-Fock-Bogoliubov approximation, which optimizes the characteristic function. Variational expressions for thermodynamic quantities or characteristic functions are also obtained with projected trial states, whether an invariance symmetry is broken or not. In particular, the projection on even or odd particle number is worked out for a pairing Hamiltonian, which leads to new equations replacing the BCS ones. Qualitative differences between even and odd systems, depending on the temperature TTT, the level density and the strength of the pairing force, are investigated analytically and numerically.
Particle-number projection is applied to the modified BCS (MBCS) theory. The resulting particle-number-projected MBCS theory, taking into account the effects due to fluctuations of particle and quasiparticle numbers at finite temperature, is tested within the exactly solvable multilevel model for pairing as well as the realistic 120Sn^{120}\mathrm{Sn}120Sn nucleus. The signature of the pseudogap in the crossover region above the critical temperature of superfluid-normal phase transition is discussed in terms of the pairing spectral function.
The finite-temperature variation after projection BCS approach (FT-VAP) is employed to study the pairing phase-transition for the 161,162 Dy and 171,172 Yb isotopes. Due to the restoration of particle number conservation, the pairing gap and the specific heat calculated in the FT-VAP approach vary smoothly with the temperature, indicating a gradual transition from the superfluid to the normal phase, as expected in finite systems. FT-VAP results are compared with those obtained in the exact canonical approach as well as with experimental data.
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.