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arXiv:1508.04256 · Journal of the Physical Society of Japan

Ginzburg–Landau Formalism for 2n-Body Condensation

Atsuhi Tsuruta, Shinya Imamura, Kazumasa Miyake

Published Aug 8, 2014• 3 citations
arXivHTMLPDF

Abstract

The Ginzburg-Landau formalism is constructed for Fermi superfluidity based on the 2n-body condensation in parallel with the usual Ginzburg-Landau formalism for the Cooper pair condensation. By this formalism, the transition temperature of the 2n-body condensation is given compactly by the zero of coefficient of the quadratic term in the 2n-body condensed order parameter without counting complicated Feynman diagrams for the 2n-body condensation susceptibility. It is shown that the 2n-body condensed state is stabilized in the intermediate- or strong-coupling regime against the BCS state based on the Cooper pair condensation. This theory is applied to the case of the possible superfluid state of cold atomic gases such as $^9$Be and $^{173}$Yb; the quartet state is possible in the former case, while the sextet state is expected in the latter case. It is predicted that the $^{173}$Yb atomic gas so far attained satisfies the condition for the sextet condensed state to be realized.

DOI
10.7566/jpsj.83.094603

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The Ginzburg-Landau formalism is constructed for Fermi superfluidity based on the 2n-body condensation in parallel with the usual Ginzburg-Landau formalism for the Cooper pair condensation. By this formalism, the transition temperature of the 2n-body condensation is given compactly by the zero of coefficient of the quadratic term in the 2n-body condensed order parameter without counting complicated Feynman diagrams for the 2n-body condensation susceptibility. It is shown that the 2n-body condensed state is stabilized in the intermediate- or strong-coupling regime against the BCS state based on the Cooper pair condensation. This theory is applied to the case of the possible superfluid state of cold atomic gases such as 9^99Be and 173^{173}173Yb; the quartet state is possible in the former case, while the sextet state is expected in the latter case. It is predicted that the 173^{173}173Yb atomic gas so far attained satisfies the condition for the sextet condensed state to be realized.
The standard Ginzburg-Landau theory is valid only near the critical temperature, where the order parameter is small. As a consequence it is not straightforward to develop a Ginzburg-Landau framework for two-bandgap superconductors such as magnesium diboride, where two order parameters exist with different intrinsic critical temperatures. We derive an extended Ginzburg-Landau functional, capable of describing both single-band and two-bandgap superfluids at all temperatures below the critical temperature. In the limit of small temperatures, we retrieve the ground state energy obtained from the microscopic path-integral theory for Fermi superfluids, and for temperatures near the critical temperature, we obtain the standard Ginzburg-Landau expressions. The present formalism is applied to reveal hidden criticality and the presence of two healing length scales in two-bandgap superfluids. The developed method is focused on strong-coupling superfluid systems, in particular, the ultracold atomic Fermi gases in the BCS-BEC crossover regime.
A possibility of the condensation of excitations with non-zero momentum in moving superfluid media is considered in terms of the Ginzburg-Landau model. The results might be applicable to the superfluid 4^44He, ultracold atomic Bose gases, various superconducting and neutral systems with pairing, like ultracold atomic Fermi gases and the neutron component in compact stars. The order parameters, the energy gain, and critical velocities are found.
Using the Boson-Fermion formalism of superconductivity we calculate the condensation energy for several superconductors ranging from conventional to unconventional, or high temperature superconductors. It is calculated as the difference between the Helmholtz free energies of the superconducting and the normal state, which is a gas of N attractive electron gas, while the superconducting state is formed by the condensed Cooper pairs taken as composite bosons, coming from a fraction of the electrons inside the Debye shell, plus those electrons inside and outside the Debye shell that are unable to pair. After giving the analytic expressions for the internal energy U and the entropy S we obtain the Helmholtz free energy F = U -TS for both the superconducting and the normal states as functions of temperature. In the search for universalities, we calculate the ratio of the condensation energy at T=0 to the Sommerfeld constant (the normal state electronic specific heat over the temperature when T it's almost zero) using two different methods: the Boson-Fermion formalism developed here, as well as an analytical expression deduced from a combination of the BCS and Ginzburg-Landau theories. We find for the Boson-Fermion formalism Econd/γ0=0.252 Tc1.997E_{cond}/γ_0= 0.252\,T_{c}^{1.997}Econd​/γ0​=0.252Tc1.997​, which is the same behavior described by the experimental fit of Kim et al Econd/γ0=0.2 Tc2.06E_{cond}/γ_0= 0.2\,T_{c}^{2.06}Econd​/γ0​=0.2Tc2.06​ and by the one recently reported by Tallon et al for overdoped cuprate superconductors; while for the Ginzburg-Landau-BCS we get the expression Econd/γ0=0.236 Tc2E_{cond}/γ_0= 0.236\,T_{c}^{2}Econd​/γ0​=0.236Tc2​, also in very good agreement with the partifirst method.
The crossover between the Cooper-pair condensation and the Bose-Einstein condensation of ``di-electronic'' molecules in two-dimensional superconductors is investigated in detail on the basis of the Nozi\`eres and Schmitt-Rink formalism. It is shown that temperature dependence of the chemical potential \ensuremath{\mu} so calculated is classified into two classes as decreasing temperatures; i.e., class (a) where \ensuremath{\mu} approaches the point of Bose-Einstein condensation of two-dimensional ideal Bose gas of ``di-electronic'' molecules, and class (b) where \ensuremath{\mu} diverges positively along the line of BCS-type mean-field pair condensation. This feature is rather universal irrespective of strength V of the attractive interaction of the s-wave type. While the former class (a) has been found by Schmitt-Rink, Varma, and Ruckenstein, existence of the latter class (b) is recognized here. In the case where V is fixed, class (a) is realized for electron number density N smaller than Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​, which is an increasing function of V, and class (b) is realized for N larger than Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​. If N\ensuremath{\gg}Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​ in particular, there exists a regime, where the Fermi-liquid-like description is valid, between the BCS-type mean-field transition temperature and the Fermi temperature. In the situation where V is changed with N being fixed, low-temperature states for the strong-coupling case belong to class (a) while those for the weak-coupling case belong to class (b). Therefore, with decreasing V, the chemical potential \ensuremath{\mu}(T), at temperatures far below the Fermi temperature, shows a discontinuous jump at V=Vcr{\mathit{V}}_{\mathrm{cr}}Vcr​(N) corresponding to the transition from class (a) to (b). However, this is in contradiction to a physical picture that the chemical potential should smoothly cross over between the above two limits unless the liquid-gas transition occurs. This shows in turn a necessity of improving the Nozi\`eres and Schmitt-Rink formalism itself especially in two dimensions. A preliminary approach beyond their formalism is briefly discussed.
Various BCS-like pairing theories are still invariably used to describe the unusual superconducting/superfluid states and properties of condensed matters without clarifying the fermionic or bosonic nature of Cooper pairs and the relevant mechanisms of superconductivity and superfluidity. We show that the bosonization of Cooper pairs and the novel superconductivity and superfluidity resulting from the condensation of an attractive Bose gas of Cooper pairs into a superfluid state in high-Tc cuprates and other exotic systems occur under the condition εF≲2εA (where εF is the Fermi energy, εA is the energy of the effective attraction between fermions). We argue that the simple BCS criterion (i.e. formation of an energy gap on the Fermi surface) is insufficient for the occurrence of unconventional superconductivity (superfluidity) and the superconducting/superfluid transition in attractive Bose systems is more λ-like than the usual Bose–Einstein condensation. The new and specific criteria for unconventional superconductivity and superfluidity are formulated. Mean-field theory of the three- and two-dimensional superfluid Bose condensates of Cooper pairs describes reasonably well the novel superconducting states (i.e., two distinct superconducting phases below the bulk superconducting transition temperature Tc and a vortex-like state above Tc) and properties (such as λ-like transition at Tc, first-order phase transition and gapless bosonic excitations somewhat below Tc or even far below Tc) of high-Tc cuprates in full agreement with the experimental findings. The unusual superconducting/superfluid states and properties of heavy-fermion and organic compounds, Sr2RuO4, 3He, 4He and atomic Fermi gases are explained more clearly by this theory.
We consider the evolution of superfluid properties of a three-dimensional p-wave Fermi gas from a weak coupling Bardeen-Cooper-Schrieffer (BCS) to strong coupling Bose-Einstein condensation (BEC) limit as a function of scattering volume. At zero temperature, we show that a quantum phase transition occurs for p-wave systems, unlike the s-wave case where the BCS to BEC evolution is just a crossover. Near the critical temperature, we derive a time-dependent Ginzburg-Landau (GL) theory and show that the GL coherence length is generally anisotropic due to the p-wave nature of the order parameter, and becomes isotropic only in the BEC limit.