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Chinese Physics C

Thermal pairing treatment within the path integral formalism

M. Fellah, N. H. Allal, M. R. Oudih

Published Jul 17, 2024• 2 citations
Publisher

Abstract

Abstract A method for the treatment of pairing correlations at finite temperature is proposed within the path integral formalism, based on the square root extraction of the pairing term in the Hamiltonian of the system. Gap equations and expressions for the pairing gap parameter Δ, energy E, and heat capacity C are established. The formalism is first tested using the Richardson model, which enables comparison with an exact solution. The results obtained using this formalism are also compared with the finite temperature BCS (FTBCS) results. An improvement over the FTBCS model is noted, especially at low temperatures. Indeed, the agreement between the Δ values of this study and the exact values is good at low temperatures. This leads to better agreement between the values of E and C of this model and the exact values than with the FTBCS values. However, a critical value of temperature remains. Subsequently, realistic cases are considered using single-particle energies of a deformed Woods-Saxon mean-field for the nuclei Dy and Yb. In the framework of the current approach, pairing effects persist beyond the FTBCS critical temperature. Moreover, at low temperatures, a good agreement between the model and semiexperimental values of the heat capacity is observed, and a clear improvement compared to the FTBCS method is noted. This is no more the case at higher temperatures.

DOI
10.1088/1674-1137/ad641a

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Abstract A method for the treatment of pairing correlations at finite temperature is proposed within the path integral formalism, based on the square root extraction of the pairing term in the Hamiltonian of the system. Gap equations and expressions for the pairing gap parameter Δ, energy E, and heat capacity C are established. The formalism is first tested using the Richardson model, which enables comparison with an exact solution. The results obtained using this formalism are also compared with the finite temperature BCS (FTBCS) results. An improvement over the FTBCS model is noted, especially at low temperatures. Indeed, the agreement between the Δ values of this study and the exact values is good at low temperatures. This leads to better agreement between the values of E and C of this model and the exact values than with the FTBCS values. However, a critical value of temperature remains. Subsequently, realistic cases are considered using single-particle energies of a deformed Woods-Saxon mean-field for the nuclei Dy and Yb. In the framework of the current approach, pairing effects persist beyond the FTBCS critical temperature. Moreover, at low temperatures, a good agreement between the model and semiexperimental values of the heat capacity is observed, and a clear improvement compared to the FTBCS method is noted. This is no more the case at higher temperatures.
BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing interaction between pairs of electrons. The Hamiltonian may be treated using a mean-field method, which is adequate to study equilibrium properties and a variety of nonequilibrium effects. Nevertheless, in certain nonequilibrium situations, even in a macroscopic rather than a microscopic superconductor, the application of mean-field theory may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 1960s. The relevance of Richardson's solution to macroscopic nonequilibrium superconductors was pointed out recently based on the existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the Richardson approach for macroscopic systems. We undertake this task in the current paper. It should be noted that Richardson's approach yields the full set of eigenvalues of the Hamiltonian, while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS expressions (e.g., for the electron occupation or pairing correlations) to cases where the spectrum of excitations diverges from BCS theory (e.g., in cases where the spectrum exhibits multiple gaps).
The exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation suggests that this might be due to the formula itself. Furthermore the BCS approximation, when re-evaluating the gap for each excitation, does a very good job at reproducing the energy spectra as obtained for the Richardson solution of the highly excited states where the pairing effects are small. However for the low lying excitations the assessment is not so good. When keeping the gap constant for all excitations the excitation spectra of the BCS approximation broaden a lot more than that of the exact method. This procedure gives unreasonable results for the systems and low particle numbers considered.
A systematic comparison is conducted for pairing properties of finite systems at nonzero temperature as predicted by the exact solutions of the pairing problem embedded in three principal statistical ensembles, as well as the unprojected (FTBCS1+++SCQRPA) and Lipkin+++Nogami projected (FTLN1+++SCQRPA) theories that include the quasiparticle number fluctuation and coupling to pair vibrations within the self-consistent quasiparticle random-phase approximation. The numerical calculations are performed for the pairing gap, total energy, heat capacity, entropy, and microcanonical temperature within the doubly folded equidistant multilevel pairing model. The FTLN1+++SCQRPA predictions agree best with the exact grand-canonical results. In general, all approaches clearly show that the superfluid-normal phase transition is smoothed out in finite systems. A novel formula is suggested for extracting the empirical pairing gap in reasonable agreement with the exact canonical results.
The validity of microscopic thermal treatments in comparison with microcanonical results is analyzed for the case of a pairing interaction. The approximate evaluation of microcanonical averages by means of saddle point and moment expansion methods is also examined. Results are shown for a 20-level model, where exact microcanonical averages and level densities are compared with exact thermal results and with those given by the BCS and static path approximations. \textcopyright{} 1996 The American Physical Society.
The self-consistent random-phase approximation (SCRPA) is extended to nonzero temperature. Within the SCRPA, the collective dynamics of nucleon pairs is built on the occupation numbers that deviate from the Fermi-Dirac distribution as a result of temperature-dependent correlations beyond the RPA. The equations for single-particle occupation numbers are derived using the method of the double-time Green's functions. The approach is applied to the Richardson model of pairing. The results of numerical calculations for energies of states and occupation numbers, as well as for the thermodynamic quantities such as the excitation energy and heat capacity of the system, are compared with those obtained within the conventional RPA and the perturbative thermal SCRPA and with the exact results. We find that, overall, the SCRPA offers a better agreement with the exact results.