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The linear response of electron systems with the pairing instabilities - the flow equation study

M. Curie

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

We address the linear response of the electron system with the pairing instabilities using nonperturbative framework of the continuous unitary transformation technique. For the case of the Bose-Einstein condensed pairs we have analytically derived the BCS result, which in the static and long wavelength limit accounts for the Meissner effect. We next apply our treatment to the mixture of the non-condensed (preformed) pairs interacting with the mobile electrons through the charge-exchange Andreev scattering. We determine the leading contributions to the current-current response function, expressing the vertices by the set of the corresponding flow equations. Our study indicates that the residual diamagnetic behavior detected by the torque magnetometry well above the transition temperature [L. Li et al, Phys. Rev. B 81, 054510 (2010)] can originate from the noncondensed preformed pairs.

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We address the linear response of the electron system with the pairing instabilities using nonperturbative framework of the continuous unitary transformation technique. For the case of the Bose-Einstein condensed pairs we have analytically derived the BCS result, which in the static and long wavelength limit accounts for the Meissner effect. We next apply our treatment to the mixture of the non-condensed (preformed) pairs interacting with the mobile electrons through the charge-exchange Andreev scattering. We determine the leading contributions to the current-current response function, expressing the vertices by the set of the corresponding flow equations. Our study indicates that the residual diamagnetic behavior detected by the torque magnetometry well above the transition temperature [L. Li et al, Phys. Rev. B 81, 054510 (2010)] can originate from the noncondensed preformed pairs.
We apply the flow equation method for studying the current-current response function of electron systems with the pairing instability. To illustrate the specific scheme in which the flow equation procedure determines the two-particle Green's functions, we reproduce the standard response kernel of the BCS superconductor. We next generalize this nonperturbative treatment considering the pairing field fluctuations. Our study indicates that the residual diamagnetic behavior detected above the transition temperature in the cuprate superconductors can originate from the noncondensed preformed pairs.
We apply the flow equation method to the study of the fermion systems with pairing interactions which lead to the BCS instability signalled by the appearance of the off-diagonal order parameter.For this purpose we rederive the continuous Bogoliubov transformation in a fashion of renormalization group procedure where the low and high energy sectors are treated subsequently.We further generalize this procedure to the case of fermions interacting with the discrete boson mode.Andreev-type interactions are responsible for developing a gap in the excitation spectrum.However, the long-range coherence is destroyed due to strong quantum fluctuations.
We apply the flow equation method for studying the fermion systems where pairing interactions can either trigger the BCS instability with the symmetry breaking manifested by the off-diagonal order parameter or lead to the gaped single particle spectrum without any symmetry breaking. We construct the continuous Bogoliubov transformation in a scheme resembling the renormalization group procedure. We further extend this continuous transformation to a case where fermion pairs interact with the boson field. Due to temporal quantum fluctuations the single particle excitation spectrum develops a gap which is centered around the renormalized boson energy. When bosons undergo the Bose Einstein condensation this structure evolves into the BCS spectrum.
In the context of a simple model featuring an explicit, central interaction potential, and using a standard functional-integral technique, we study superconductivity with angular momentum quantum number l=2l=2l=2 as an emergent property of the many-body system. Our interaction potential is attractive at a finite distance r0,{r}_{0},r0​, and the breaking of the rotational symmetry is the result of an interplay between r0{r}_{0}r0​ and the interparticle distance rs.{r}_{s}.rs​. This interplay is generic to interactions of this type and is responsible for the existence of d-wave pairing for a range of densities. However, we find that l=2l=2l=2 pairing takes place only in the BCS limit. In contrast, as the Bose-Einstein (BE) limit is approached the internal energy of the ``preformed pairs'' becomes the dominant contribution and there is a quantum phase transition in which the s-wave symmetry is restored. We also find that the limiting value of the critical temperature is ${k}_{B}{T}_{c}\ensuremath{\rightarrow}3.315{\ensuremath{\Elzxh}}^{2}{/2m}^{*}[n/2(2l+1){]}^{2/3},$ which coincides with the usual result only for l=0;l=0;l=0; for l>0,l>0,l>0, it differs in the degeneracy factor 1/(2l+1),1/(2l+1),1/(2l+1), which lowers Tc.{T}_{c}.Tc​. Our results thus place constraints on exotic pairing in the BE limit, while at the same time indicating a particularly interesting route to pairing with l>0l>0l>0 in a BCS superconductor.
A realistic description of the Josephson effect at finite temperature with ultracold Fermi gases embedded in nontrivial geometrical constraints (typically, a trap plus a barrier) requires appropriate consideration of pairing fluctuations that arise in inhomogeneous environments. Here, we apply the theoretical approach developed in a companion paper [Pisani et al., Phys. Rev. B 108, 214503 (2023)], where the inclusion of pairing fluctuations beyond mean field across the BCS--Bose-Einstein-condensate (BEC) crossover at finite temperature is combined with a detailed description of the gap parameter in a nontrivial geometry. In this way, we are able to account for the experimental results on the Josephson critical current, reported both at low temperature for various couplings across the BCS-BEC crossover and as a function of temperature at unitarity. Besides validating the theoretical approach of the companion paper, our numerical results reveal generic features of the Josephson effect which may not readily emerge from an analysis of corresponding experiments with condensed-matter samples owing to the unique intrinsic flexibility of experiments with ultracold gases.