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International Journal of Quantum Chemistry

Aspects of BCS to BEC “Crossover” as illustrated by fulleride superconductivity

Richard H. Squire, Norman Henry March

Published Jan 1, 2007• 5 citations
Publisher

Abstract

Abstract The pioneering works by Eagles, Leggett, and Nozieres/Schmitt‐Rink [Eagles, Phys Rev, 1969, 186, 456; Leggett, In: Modern Trends in the Theory of Condensed Matter, Springer‐Verlag: Berlin, 1980; Nozieres and Schmitt‐Rink, Low Temp Phys, 1985, 59, 980] (reviewed and augmented by Randeria [Randeria, Bose‐Einstein Condensation, Cambridge University Press, Cambridge, 1995, p 255]) emphasize that in the limits of the models studied both at T = 0 and T ≠ 0, the “cross‐over” from a BCS‐type to a BEC‐type superconductor is continuous. The BCS and BEC “end points” seem to be well‐established. However, in the intermediate region—home to fulleride and high temperature superconductors—considerable extrapolation of the models must be done as there still is no exact theory. Yet, considerable current literature is devoted to what appears to be more “singular‐type phenomena” such as quantum critical points, “stripe” formation, insulator to superconductor phase transitions, loss of validity of the Fermi liquid theory, etc. Using a connection we have made with “cold” atom fermion‐boson crossover theory [Chen, Stajic, Levin, 2005, e‐print cond‐mat/0508603], we can establish that the resonance previously discussed [Squire and March, Int J Quant Chem, 2006, 106, 3343] is a result of the crossing of the fermion band by the boson band. While the ground state appears to remain continuous, the paired energy gap becomes transformed. We discuss features of the resonance and the experimentally observed pre‐formed “BEC Cooper pair” formation, essential to the boson‐fermion resonance theory. In addition some of the various singular phenomena discussed above can be put more into perspective. Finally, in both limits the relation of characteristic lengths to the inverse Fermi momentum is strongly emphasized, as is the role of the chemical potential near the pseudogap regime. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

DOI
10.1002/qua.21445

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Abstract The pioneering works by Eagles, Leggett, and Nozieres/Schmitt‐Rink [Eagles, Phys Rev, 1969, 186, 456; Leggett, In: Modern Trends in the Theory of Condensed Matter, Springer‐Verlag: Berlin, 1980; Nozieres and Schmitt‐Rink, Low Temp Phys, 1985, 59, 980] (reviewed and augmented by Randeria [Randeria, Bose‐Einstein Condensation, Cambridge University Press, Cambridge, 1995, p 255]) emphasize that in the limits of the models studied both at T = 0 and T ≠ 0, the “cross‐over” from a BCS‐type to a BEC‐type superconductor is continuous. The BCS and BEC “end points” seem to be well‐established. However, in the intermediate region—home to fulleride and high temperature superconductors—considerable extrapolation of the models must be done as there still is no exact theory. Yet, considerable current literature is devoted to what appears to be more “singular‐type phenomena” such as quantum critical points, “stripe” formation, insulator to superconductor phase transitions, loss of validity of the Fermi liquid theory, etc. Using a connection we have made with “cold” atom fermion‐boson crossover theory [Chen, Stajic, Levin, 2005, e‐print cond‐mat/0508603], we can establish that the resonance previously discussed [Squire and March, Int J Quant Chem, 2006, 106, 3343] is a result of the crossing of the fermion band by the boson band. While the ground state appears to remain continuous, the paired energy gap becomes transformed. We discuss features of the resonance and the experimentally observed pre‐formed “BEC Cooper pair” formation, essential to the boson‐fermion resonance theory. In addition some of the various singular phenomena discussed above can be put more into perspective. Finally, in both limits the relation of characteristic lengths to the inverse Fermi momentum is strongly emphasized, as is the role of the chemical potential near the pseudogap regime. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
The pioneering works by Eagles, Leggett, and Nozieres/Schmitt-Rink (reviewed and augmented by Randeria) emphasize that in the limits of the models studied both at T = 0 and, the cross-over from a BCS-type to a BEC-type superconductor is continuous. The BCS and BEC end points seem to be well-established. However, in the intermediate region, home to fulleride and high temperature superconductors, considerable extrapolation of the models must be done as there still is no exact theory. Yet, considerable current literature is devoted to what appears to be more singular-type phenomena such as quantum critical points, stripe formation, insulator to superconductor phase transitions, loss of validity of the Fermi liquid theory, etc. Using a connection we have made with cold atom Fermion-Boson crossover theory, we can establish that the resonance previously discussed is a result of the crossing of the Fermion band by the Boson band. While the ground and singly excited states appear to remain continuous, the collective modes due to the resonance transform the nature of the superconductivity. We discuss features of the resonance and the experimentally observed pre-formed BEC Cooper pair formation in fullerides, essential to the Boson-Fermion resonance theory. In addition some of the various singular phenomena discussed above can be put into perspective. PACS numbers: 61.48.+c, 74.20.-z, 81.05Tp
There continues to be enormous interest in the BCS to BEC transition as there still is no exact theory. We recently reported a revealing reinterpretation of the condensed phase Boson-Fermion Model (BFM) by comparing it to a cold atom formulation [1]. While the ground and singly excited states appear to remain continuous in all models we have examined, the collective modes contain a singularity due to a Feshbach resonance (tuned by doping) causing a breakdown of the Migdal theorem. As a result of vertex corrections, there is a fundamental change in the nature of the superconductivity due to the formation of preformed pairs as the previously suggested location [1] of a quantum critical point in the fulleride phase diagram is passed. The result is a quantum phase transition (QPT) between BCS and BEC-like (or Feshbach resonance) superconductivity (SC). We discuss features of the resonance and the role of the experimentally observed preformed pair formation in fullerides, essential to the Boson-Fermion Model (BFM), and often speculated since the work of Nozieres and Schmitt-Rink [17]. Here, we present arguments to establish a model of the preformed pair which can be favorably compared to a circular charge density wave (CDW) isolated on a fulleride molecule. The binding is much larger than a Cooper pair. The CDW seems to be stabalized by splitting of the Jahn-Teller active vibrational modes to reduce Coulomb repulsions. Our conclusions are: 1) the doping of two electrons into triply degenerate orbitals results in the experimentally observed singlet state (CDW); and 2) this CDW (preformed pair) has a dual role as doping is varied: suppression of BCS SC and enabling a Feshbach resonance form of SC.
This article gives a detailed review of our current theoretical understanding of the crossover from cooperative Cooper pairing to independent bound-state formation and Bose–Einstein condensation in Fermi systems with increasing attractive interactions. Introduction There are two well-known paradigms for understanding the phenomena of superconductivity and superfluidity: (I) BCS theory [1], in which the normal state is a degenerate Fermi liquid that undergoes a pairing instability at a temperature T c ≪ ε f , the degeneracy scale. The formation of Cooper pairs and their condensation (macroscopic occupation of a single quantum state) both occur simultaneously at the transition temperature T c . (II) Bose–Einstein condensation (BEC) of bosons at a T c of the order of their degeneracy temperature. At a fundamental level these bosons (for example, 4 He or excitons) are invariably composite objects made up of an even number of fermions. The composite particles form at some very high temperature scale of the order of their dissociation temperature T dissoc , and these “pre-formed” bosons then condense at the BEC T c ≪ T dissoc . In most cases of experimental interest the system under consideration clearly falls into one category or the other. For instance, 3 He is a Fermi superfluid described by (I) whereas 4 He is a Bose superfluid (II). Essentially all cases of superconductivity in metals that we understand reasonably well are much closer to (I) than to (II), as evidenced by the existence of fermionic quasiparticles above T c .
For the Cooper/BCS model interaction in superconductors (SCs) it is shown: a) how BCS-Bose crossover picture transition temperatures Tc, defined self-consistently by both the gap and fermion-number equations, require unphysically large couplings to differ significantly from the Tc of ordinary BCS theory defined without the number equation since here the chemical potential is assumed equal to the Fermi energy; how although ignoring either hole- or electron-Cooper-pairs in the recent "complete boson-fermion model": b) one obtains the precise BCS gap equation for all temperatures T, but c) only half the zero-temperature BCS condensation energy emerges. Results (b) and (c) are also expected to hold for neutral-fermion superfluids (SFs)--such as liquid 3^33He, neutron matter and trapped ultra-cold fermion atomic gases--where the pair-forming two-fermion interaction of course differs from the Cooper/BCS one for SCs.
This article presents a comparison of two finite-temperature BCS--Bose-Einstein condensation (BEC) crossover theories above the transition temperature: Nozieres--Schmitt-Rink (NSR) theory and finite-TTT extended BCS-Leggett theory. The comparison is cast in the form of numerical studies of the behavior of the fermionic spectral function both theoretically and as constrained by (primarily) radio frequency (rf) experiments. Both theories include pair fluctuations and exhibit pseudogap effects, although the nature of this pseudogap is very different. The pseudogap in finite-TTT extended BCS-Leggett theory is found to follow a BCS-like dispersion which, in turn, is associated with a broadened BCS-like self-energy, rather more similar to what is observed in high-temperature superconductors (albeit, for a ddd-wave case). The fermionic quasiparticle dispersion is different in NSR theory and the damping is considerably larger. We argue that the two theories are appropriate in different temperature regimes with the BCS-Leggett approach being more suitable nearer to condensation. There should, in effect, be little difference at higher TTT as the pseudogap becomes weaker and where the simplifying approximations used in the BCS-Leggett approach break down. On the basis of momentum-integrated rf studies of unpolarized gases, it would be difficult to distinguish which theory is the better one. A full comparison for polarized gases is not possible since it is claimed that there are inconsistencies in the NSR approach (not found in the BCS-Leggett scheme). Future experiments along the lines of momentum-resolved experiments look to be very promising in distinguishing the two theories.
This review is written at the time of the twentieth anniversary of the discovery of high-temperature superconductors, which nearly coincides with the important discovery of the superfluid phases of ultracold trapped fermionic atoms. We show how these two subjects have much in common. Both have been addressed from the perspective of the BCS—Bose–Einstein condensation (BEC) crossover scenario, which is designed to treat short coherence length superfluids with transition temperatures which are “high” with respect to the Fermi energy. A generalized mean field treatment of BCS–BEC crossover at general temperatures T, based on the BCS–Leggett ground state, has met with remarkable success in the fermionic atomic systems. Here we summarize this success in the context of four different cold atom experiments, all of which provide indications, direct or indirect, for the existence of a pseudogap. This scenario also provides a physical picture of the pseudogap phase in the underdoped cuprates which is a central focus of high Tc research. We summarize successful applications of BCS–BEC crossover to key experiments in high Tc systems, including the phase diagram, specific heat, and vortex core STM data, along with the Nernst effect, and exciting recent data on the superfluid density in very underdoped samples.