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arXiv:1306.3189 · Physical Review A

Compressibility in strongly correlated superconductors and superfluids: From the BCS regime to Bose-Einstein condensates

Hao Guo, Yan He, Chih-Chun Chien, Kathryn Levin

Published Oct 31, 2013• 11 citations
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Abstract

We present a theoretical study of the compressibility $\ensuremath{\kappa}$ in a Fermi gas with attractive contact interactions, providing predictions for the strongly attractive regime and the superfluid phase. Our work emphasizes the compressibility sum rule and gauge invariance as constraints on $\ensuremath{\kappa}$, and we show how within a Gor'kov-based $t$-matrix approach, these can be satisfied in the normal phase when no approximations are made. For tractability, approximations must be introduced, and it is believed that thermodynamical approaches to $\ensuremath{\kappa}$ are more reliable than correlation-function-based schemes. Contrasting with other studies in the literature, we present thermodynamic calculations of $\ensuremath{\kappa}$. As expected of an approximate approach, ours do not satisfy the compressibility sum rule. However, without adjusting parameters, they yield semiquantitative agreement with experiment and provide physical insight into similar results obtained via quantum Monte Carlo simulations.

DOI
10.1103/physreva.88.043644

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We present a theoretical study of the compressibility \ensuremathκ\ensuremath{\kappa}\ensuremathκ in a Fermi gas with attractive contact interactions, providing predictions for the strongly attractive regime and the superfluid phase. Our work emphasizes the compressibility sum rule and gauge invariance as constraints on \ensuremathκ\ensuremath{\kappa}\ensuremathκ, and we show how within a Gor'kov-based ttt-matrix approach, these can be satisfied in the normal phase when no approximations are made. For tractability, approximations must be introduced, and it is believed that thermodynamical approaches to \ensuremathκ\ensuremath{\kappa}\ensuremathκ are more reliable than correlation-function-based schemes. Contrasting with other studies in the literature, we present thermodynamic calculations of \ensuremathκ\ensuremath{\kappa}\ensuremathκ. As expected of an approximate approach, ours do not satisfy the compressibility sum rule. However, without adjusting parameters, they yield semiquantitative agreement with experiment and provide physical insight into similar results obtained via quantum Monte Carlo simulations.
We present fundamental constraints required for a consistent linear response theory of fermionic superfluids and address temperatures both above and below the transition temperature Tc. We emphasize two independent constraints, one associated with gauge invariance (and the related Ward identity) and another associated with the compressibility sum rule, both of which are satisfied in strict BCS theory. However, we point out that it is the rare many body theory which satisfies both of these. Indeed, well studied quantum Hall systems and random-phase approximations to the electron gas are found to have difficulties with meeting these constraints. We summarize two distinct theoretical approaches which are, however, demonstrably compatible with gauge invariance and the compressibility sum rule. The first of these involves an extension of BCS theory to a mean field description of the BCS-Bose Einstein condensation crossover. The second is the simplest Nozieres Schmitt–Rink (NSR) treatment of pairing correlations in the normal state. As a point of comparison we focus on the compressibility κ of each and contrast the predictions above Tc. We note here that despite the compliance with sum rules, this NSR based scheme leads to an unphysical divergence in κ at the transition. Because of the delicacy of the various consistency requirements, the results of this paper suggest that avoiding this divergence may repair one problem while at the same time introducing others.
We theoretically investigate the isothermal compressibility ${\ensuremath{\kappa}}_{T}$ in the normal state of an ultracold Fermi gas. Including pairing fluctuations, as well as preformed-pair formations, within the framework of the self-consistent TTT-matrix approximation, we evaluate the temperature dependence of this thermodynamic quantity over the entire BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover region. While ${\ensuremath{\kappa}}_{T}$ in the weak-coupling BCS regime is dominated by Fermi atoms near the Fermi surface, correlations between tightly bound Cooper-pair molecules are found to play crucial roles in the strong-coupling BEC regime. In the latter region, besides a two-body molecular interaction, a three-body one is shown to sizably affect ${\ensuremath{\kappa}}_{T}$ near the superfluid phase-transition temperature. Our results indicate that the strong-coupling BEC regime of an ultracold Fermi gas would provide a unique opportunity to study multibody correlations between Cooper-pair molecules.
We theoretically present an economical and convenient way to study ground-state properties of a strongly interacting superfluid Fermi gas. Our strategy is that complicated strong-coupling calculations are used only to evaluate quantum fluctuation corrections to the chemical potential $\ensuremath{\mu}$. Then, without any further strong-coupling calculations, we calculate the compressibility, sound velocity, internal energy, pressure, and Tan's contact, from the calculated $\ensuremath{\mu}$ without loss of accuracy, by using exact thermodynamic identities. Using a recent precise measurement of $\ensuremath{\mu}$ in a superfluid 6Li^{6}\mathrm{Li}6Li Fermi gas, we show that an extended TTT-matrix approximation (ETMA) is suitable for our purpose, especially in the BCS-unitary regime, where our results indicate that many-body corrections are dominated by superfluid fluctuations. Since precise determinations of physical quantities are not always easy in cold Fermi gas physics, our approach would greatly reduce experimental and theoretical efforts toward the understanding of ground-state properties of this strongly interacting Fermi system.
The two-species Fermi gas with attractive short-range interactions in two spatial dimensions provides a paradigmatic system for the understanding of strongly correlated Fermi superfluids in two dimensions. It is known to exhibit a BEC to BCS crossover as a function of ln(k_{F}a), where a is the scattering length, and to undergo a Berezinskii-Kosterlitz-Thouless superfluid transition below a critical temperature T_{c}. However, the extent of a pseudogap regime in the strongly correlated regime of ln(k_{F}a)∼1, in which pairing correlations persist above T_{c}, remains largely unexplored with controlled theoretical methods. Here, we use finite-temperature auxiliary-field quantum Monte Carlo methods on discrete lattices in the canonical ensemble formalism to calculate thermodynamical observables in the strongly correlated regime. We extrapolate to continuous time and the continuum limit to eliminate systematic errors and present results for particle numbers ranging from N=42 to N=162. We estimate T_{c} by a finite-size scaling analysis, and observe clear pseudogap signatures above T_{c} and below a temperature T^{*} in both the spin susceptibility and free-energy gap. We also present results for the contact, a fundamental thermodynamic property of quantum many-body systems with short-range interactions.
We present a diagrammatic theory for determining consistent electromagnetic response functions in strongly correlated fermionic superfluids. While a gauge-invariant electromagnetic response is well understood at the BCS level, a treatment of correlations beyond BCS theory requires extending this theoretical formalism. The challenge in such systems is to maintain gauge invariance, while simultaneously incorporating additional self-energy terms arising from strong correlation effects. Central to our approach is the application of the Ward-Takahashi identity, which introduces collective mode contributions in the response functions and guarantees that the fff-sum rule is satisfied. We outline a powerful method, which determines these collective modes in the presence of correlation effects and in a manner compatible with gauge invariance. Since this method is based on fundamental aspects of quantum field theory, the underlying principles are broadly applicable to strongly correlated superfluids. As an illustration of the technique, we apply it to a simple class of theoretical models that contain a frequency-independent order parameter. These models include BCS-BEC crossover theories of the ultracold Fermi gases, along with models specifically associated with the high-Tc{T}_{c}Tc​ cuprates. Finally, as an alternative approach, we contrast with the path integral formalism. Here, the calculation of gauge-invariant response appears more straightforward. However, the collective modes introduced are those of strict BCS theory, without any modification from additional correlations. As the path integral simultaneously addresses electrodynamics and thermodynamics, we emphasize that it should be subjected to a consistency test beyond gauge invariance, namely that of the compressibility sum rule. We show how this sum rule fails in the conventional path integral approach.
We calculate the pair correlation function and the order parameter correlation function, which probe, respectively, the intrapair and interpair correlations of a Fermi gas with attractive interparticle interaction, in terms of a diagrammatic approach as a function of coupling throughout the BCS-Bose-Einstein condensation (BEC) crossover and of temperature, both in the superfluid and normal phase across the critical temperature Tc{T}_{c}Tc​. Several physical quantities are obtained from this calculation, including the pair coherence and healing lengths, the Tan's contact, the crossover temperature T∗{T}^{*}T∗ below which interpair correlations begin to build up in the normal phase, and the signature for the disappearance of the underlying Fermi surface which tends to survive in spite of pairing correlations. A connection is also made with recent experimental data on the temperature dependence of the normal coherence length as extracted from the proximity effect measured in high-temperature (cuprate) superconductors.