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arXiv:1207.0706 · New Journal of Physics

Pseudogap and preformed pairs in the imbalanced Fermi gas in two dimensions

Serghei N. Klimin, Jacques Tempere, J. T. Devreese

Published Oct 29, 2012• 37 citations
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Abstract

The physics of the pseudogap state is intimately linked with the pairing mechanism that gives rise to superfluidity in quantum gases and to superconductivity in high- T c cuprates, and therefore, both in quantum gases and in superconductors, the pseudogap state and preformed pairs have been under intensive experimental scrutiny. Here, we develop a path integral treatment that provides a divergence-free description of the paired state in two-dimensional Fermi gases. Within this formalism, we derive the pseudogap temperature and the pair fluctuation spectral function, and compare these results with a recent experimental measurement of the pairing in the two-dimensional Fermi gas. The removal of the infrared divergence in the number equations is shown both numerically and analytically, through a study of the long-wavelength and low-energy limit of the pair fluctuation density. Besides the pseudogap temperature, the pair formation temperature and the critical temperature for superfluidity are also derived. The latter corresponds to the Berezinski–Kosterlitz–Thouless (BKT) temperature. The pseudogap temperature, which coincides with the pair formation temperature in the mean field, is found to be suppressed with respect to the pair formation temperature by fluctuations. This suppression is strongest for large binding energies of the pairs. Finally, we investigate how the pair formation temperature, the pseudogap temperature and the BKT temperature behave as a function of both binding energy and imbalance between the pairing partners in the Fermi gas. This allows us to set up phase diagrams for the two-dimensional Fermi gas, in which the superfluid phase, the phase-fluctuating quasicondensate and the normal state can be identified.

DOI
10.1088/1367-2630/14/10/103044

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The physics of the pseudogap state is intimately linked with the pairing mechanism that gives rise to superfluidity in quantum gases and to superconductivity in high- T c cuprates, and therefore, both in quantum gases and in superconductors, the pseudogap state and preformed pairs have been under intensive experimental scrutiny. Here, we develop a path integral treatment that provides a divergence-free description of the paired state in two-dimensional Fermi gases. Within this formalism, we derive the pseudogap temperature and the pair fluctuation spectral function, and compare these results with a recent experimental measurement of the pairing in the two-dimensional Fermi gas. The removal of the infrared divergence in the number equations is shown both numerically and analytically, through a study of the long-wavelength and low-energy limit of the pair fluctuation density. Besides the pseudogap temperature, the pair formation temperature and the critical temperature for superfluidity are also derived. The latter corresponds to the Berezinski–Kosterlitz–Thouless (BKT) temperature. The pseudogap temperature, which coincides with the pair formation temperature in the mean field, is found to be suppressed with respect to the pair formation temperature by fluctuations. This suppression is strongest for large binding energies of the pairs. Finally, we investigate how the pair formation temperature, the pseudogap temperature and the BKT temperature behave as a function of both binding energy and imbalance between the pairing partners in the Fermi gas. This allows us to set up phase diagrams for the two-dimensional Fermi gas, in which the superfluid phase, the phase-fluctuating quasicondensate and the normal state can be identified.
This paper addresses the transition from the normal to the superfluid state in strongly correlated two dimensional fermionic superconductors and Fermi gases. We arrive at the Berezinskii-Kosterlitz-Thouless (BKT) temperature TBKTT_{\text{BKT}}TBKT​ as a function of \emph{attractive} pairing strength by associating it with the onset of quasi-condensation in the normal phase. Our approach builds on a criterion for determining the BKT transition temperature for atomic gases which is based on a well established Quantum Monte Carlo analysis of the phase space density. This latter quantity, when derived from BCS-BEC crossover theory for fermions, leads to non-monotonic behavior for TBKTT_{\text{BKT}}TBKT​ as a function of the attractive interaction or inverse scattering length. In Fermi gases, this implies a robust superconducting dome followed by a long tail from the flat BEC asymptote, rather similar to what is observed experimentally. For lattice systems we find that TBKTT_{\text{BKT}}TBKT​ has an absolute maximum of the order of 0.1EF0.1 E_F0.1EF​. We discuss how our results compare with those derived from the Nelson Kosterlitz criterion based on the mean field superfluid density and the approach to the transition from below. While there is agreement in the strict mean-field BCS regime at weak coupling, we find that at moderate pairing strength bosonic excitations cause a substantial increase in TBKTT_\text{BKT}TBKT​ followed by an often dramatic decrease before the system enters the BEC regime.
Abstract The physics of two-dimensional (2D) quantum gases can be revealed in strongly confining optical lattices. Upon cooling, 2D bosonic quantum gases as well as Fermi gases where pairing is present, become superfluid. The superfluid-to-normal transition is no longer governed by the presence or absence of a condensate, but by the Berezinskii–Kosterlitz–Thouless (BKT) mechanism: above a critical temperature, vortices and antivortices proliferate and destroy phase coherence. We investigate the BKT transition for superfluid 2D Fermi gases in the whole range of the BCS–BEC crossover, from weakly bound Cooper pairing (the BCS state), up to strongly bound molecules (the BEC state). Using a path-integral description, we then focus on the case of imbalanced gases: when the number of ‘spin-up’ and ‘spin-down’ fermions that form the pair is no longer equal. When an excess of one spin species exists, pairing is frustrated and the vortex energetics is strongly affected, influencing the KT mechanism. In the present work we are concentrated on the effect of both phase and amplitude fluctuations on phase diagrams of the fermion system. The amplitude fluctuations only slightly influence the BKT phase transition temperature. However, they lead to a substantial modification of the complete phase diagram for the Fermi gas in 2D with respect to that obtained taking into account only phase fluctuations.
We investigate pseudogap phenomena in a two-dimensional Fermi gas. Including pairing fluctuations within a self-consistent TTT-matrix approximation, we determine the pseudogap temperature T∗T^*T∗ below which a dip appears in the density of states ρ(ω)ρ(ω)ρ(ω) around the Fermi level. Evaluating T∗T^*T∗, we identify the pseudogap region in the phase diagram of this system. We find that, while the observed BKT (Berezinskii-Kosterlitz-Thouless) transition temperature TBKTexpT^{\rm exp}_{\rm BKT}TBKTexp​ in a 6^66Li Fermi gas is in the pseudogap regime, the detailed pseudogap structure in ρ(ω)ρ(ω)ρ(ω) at TBKTexpT^{\rm exp}_{\rm BKT}TBKTexp​ still differs from a fully-gapped one, indicating the importance of amplitude fluctuations in the Cooper channel there. Since the observed TBKTexpT^{\rm exp}_{\rm BKT}TBKTexp​ in the weak-coupling regime cannot be explained by the recent BKT theory which only includes phase fluctuations, our results may provide a hint about how to improve this BKT theory. Although ρ(ω)ρ(ω)ρ(ω) has not been measured in this system, we show that the assessment of our results is still possible by using the observable Tan's contact.
Abstract This paper addresses the transition from the normal to the superfluid state in strongly correlated two dimensional fermionic superconductors and Fermi gases. We arrive at the Berezinskii–Kosterlitz–Thouless (BKT) temperature T BKT as a function of attractive pairing strength by associating it with the onset of ‘quasi-condensation’ in the normal phase. Our approach builds on a criterion for determining the BKT transition temperature for atomic gases which is based on a well established quantum Monte Carlo analysis of the phase space density. This latter quantity, when derived from BCS–BEC crossover theory for fermions, leads to non-monotonic behavior for T BKT as a function of the attractive interaction or inverse scattering length. In Fermi gases, this implies a robust superconducting dome followed by a long tail from the flat BEC asymptote, rather similar to what is observed experimentally. For lattice systems we find that T BKT has an absolute maximum of the order of 0.1E F. We discuss how our results compare with those derived from the Nelson–Kosterlitz criterion based on the mean field superfluid density and the approach to the transition from below. While there is agreement in the strict mean-field BCS regime at weak coupling, we find that at moderate pairing strength bosonic excitations cause a substantial increase in T BKT followed by an often dramatic decrease before the system enters the BEC regime.
We investigate single-particle excitations and strong-coupling effects in a twodimensional Fermi gas. Including pairing fluctuations within a Gaussian fluctuation theory, we calculate the density of states ρ ( ω ) near the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature T BKT . Near T BKT , we show that superfluid fluctuations induce a pseudogap in ρ (ω). The pseudogap structure is very similar to the BCS superfluid density of states, although the superfluid order parameter is absent in the present two-dimensional case. Since a twodimensional 40 K Fermi gas has recently been realized, our results would contribute to the understanding of single-particle properties near the BKT instability.
We theoretically investigate strong-coupling properties of a two-dimensional Fermi gas in the normal phase. Including pairing fluctuations within the framework of a self-consistent T -matrix approximation, we calculate the single-particle density of states (DOS) near the observed Berezinskii-Kosterlitz-Thouless phase transition temperature T BKT . We show that the pseudogap phenomenon associated with strong pairing fluctuations appears as an almost fully gapped single-particle density of states in the strong-coupling regime, indicating that amplitude fluctuations of the order parameter are almost absent and the system is dominated by phase fluctuations of the order parameter, as expected in a two-dimensional system near T BKT . On the other hand, although pairing fluctuations are enhanced by the low-dimensionality of the system, they are found to be still not strong enough to open such a full gap in the intermediate coupling (crossover) regime. In this regime, a dip structure is only seen in DOS around ω = 0, indicating that amplitude of the superfluid order parameter is also fluctuating, in addition to phase fluctuations. In theoretically determining T BKT in a two-dimensional Fermi gas, it is frequently assumed that, while phase fluctuations exist, the amplitude of the superfluid order parameter is fixed. However, our results indicate the necessity of including both amplitude and phase fluctuations of the order parameter in evaluating T BKT in the crossover region. Since the BKT phase transition has recently been reported in a two-dimensional 6 Li Fermi gas, our results would be useful for the theoretical study of this observed novel Fermi superfluid and effects of strong low-dimensional pairing fluctuations.