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arXiv:0910.2693 · Physical Review A

Universal short-distance structure of the single-particle spectral function of dilute Fermi gases

William Schneider, Mohit Randeria

Published Feb 3, 2010• 82 citations
arXivHTMLPDF

Abstract

We show that the universal $1/{k}^{4}$ tail in the momentum distribution of dilute Fermi gases implies that the spectral function $A(\mathbf{k},\ensuremath{\omega})$ must have weight below the chemical potential for large momentum $k\ensuremath{\gg}{k}_{F}$, with observable consequences in radio-frequency spectroscopy experiments. We find that this incoherent spectral weight is centered around $\ensuremath{\omega}\ensuremath{\simeq}\ensuremath{-}\ensuremath{\epsilon}(\mathbf{k})$ in a range of energies of order ${v}_{F}k$. This ``bending back'' in the dispersion, while natural for superfluids, is quite surprising for normal gases. This universal structure is present in the hard-sphere gas as well as in the Fermi-liquid ground state of the highly imbalanced, attractive gas near unitarity. We argue that, even in the BCS superfluid, this bending back at large $k$ is dominated by interaction effects which do not reflect the pairing gap.

DOI
10.1103/physreva.81.021601

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We show that the universal 1/k41/{k}^{4}1/k4 tail in the momentum distribution of dilute Fermi gases implies that the spectral function A(k,\ensuremathω)A(\mathbf{k},\ensuremath{\omega})A(k,\ensuremathω) must have weight below the chemical potential for large momentum k\ensuremath≫kFk\ensuremath{\gg}{k}_{F}k\ensuremath≫kF​, with observable consequences in radio-frequency spectroscopy experiments. We find that this incoherent spectral weight is centered around \ensuremathω\ensuremath≃\ensuremath−\ensuremathϵ(k)\ensuremath{\omega}\ensuremath{\simeq}\ensuremath{-}\ensuremath{\epsilon}(\mathbf{k})\ensuremathω\ensuremath≃\ensuremath−\ensuremathϵ(k) in a range of energies of order vFk{v}_{F}kvF​k. This ``bending back'' in the dispersion, while natural for superfluids, is quite surprising for normal gases. This universal structure is present in the hard-sphere gas as well as in the Fermi-liquid ground state of the highly imbalanced, attractive gas near unitarity. We argue that, even in the BCS superfluid, this bending back at large kkk is dominated by interaction effects which do not reflect the pairing gap.
The dispersions, weights, and widths of the peaks of the single-particle spectral function in the presence of pair correlations, for a Fermi gas with either attractive ororor repulsive short-range inter-particle interaction, are determined in the normal phase over a wide range of wave vectors, with a twofold purpose. The first one is to determine how these dispersions identify both an energy scale known as the pseudogap near the Fermi wave vector as well as an additional energy scale related to the contact CCC at large wave vectors. The second one is to differentiate the behaviors of the repulsive gas from the attractive one in terms of crossing versus avoided crossing of the dispersions near the Fermi wave vector. An analogy will also be drawn between the occurrence of the pseudogap physics in a Fermi gas subject to pair fluctuations and the persistence of local spin waves in the normal phase of magnetic materials.
Ultracold atomic gases provide an ideal system with which to study fundamental many-body physics.Exhibiting universal interactions in clean and controllable environments, long-used simple models as well as more exotic models can now be realized.The interplay between theory and experiment is therefore very active, and, in this thesis, I will detail several works, both exact analytic results and numerical calculations, which have impacts on current experiments.I begin with an introduction to the field including a brief discussion of experiments, the microscopic model of two species of interacting fermions, the BCS-BEC crossover and an overview of the various phases of atomic Fermi gases.I then describe the various results of my theoretical investigations, which are divided into five chapters.First, I describe radio frequency (RF) spectroscopy experiments and how they probe the single-particle spectral function.This leads to my results on an exact feature of the spectral lineshape, a universal large-momentum structure which exists for all states of interacting Fermi systems and has been verified in recent angle-resolved RF experiments.Second, I focus on gases which have a normal Fermi liquid ground state and show that their lineshape exhibits a characteristic jump discontinuity.I illustrate this Fermi surface singularity and the previously mentioned universal large momentum tail with explicit calculations.Third, I turn to the low energy structure of the single-particle spectral function in the superfluid state.I argue that sharp low energy quasiparticle excitations exist across the BCS-BEC crossover using a general argument that includes the interaction of fermions with the low-energy collective mode.This is illustrated with an explicit calculation within an approximation scheme.Fourth, I address the trap-induced inhomogeneity and use a Bogoliubov-deGennes analysis to test if a simple local density approximation (LDA) can provide an adequate description of a spin imbalanced Fermi system.Finally, I discuss dynamics of atomic gases motivated by a recent experiment in which two spin-polarized clouds are allowed to collide with each other.Despite the underlying attractive interactions, the clouds are seen to bounce off of each other.Using a hydrodynamic description, I argue that the short and intermediate time dynamics reflects a metastable state in which the effective interaction is repulsive.It is my hope that the work presented in this thesis can be used to gain key insights which inspire future endeavors.6.2 Time dependence of the center-of-mass positions zσ of the two spin clouds at unitarity for an initial separation z↑ (t = 0) = R z , z↓ (t = 0) = -R z using the upper branch energy functional and damping γ = 1.Red (upper) and blue (lower) curves denote the different spin species. . . . . . . . . . . . . . . .71 6.3 The upper branch energy functional obtained from lowest order constraint variational method.(a), energy as a function of the polarization P for various values of k F a s as indicated in the figure.(b), energy as a function of the interaction parameter k F a s for various values of polarization P as indicated in the figure.(c), phase diagram of the interacting two-component Fermi gases.The two vertical red dashed lines indicates the separation between three phases.For 1/(k F a s ) > 1.124, the system is in the Fermi liquid (FL) phase.For 1.124 > 1/(k F a s ) > 0.91, the system is in the partially polarized phase (PP) and for 1/(k F a s ) < 0.91, the system is fully polarized. . . . . .74 6.4 (Color online) Time dependence of the center-of-mass positions zσ of the two spin clouds for k F (0)a s = 2. . . . . . . . . . . . . . . . . . . . . . . . . . .75 6.5 (Color online) Time dependence of the center-of-mass positions zσ of the two spin clouds for k F (0)a s = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . .
The nature of the normal state of an ultracold Fermi gas in the BCS-BEC crossover regime is an intriguing and controversial topic. While the many-body ground state remains a condensate of paired fermions, the normal state must evolve from a Fermi liquid to a Bose gas of molecules as a function of the interaction strength. How this occurs is still largely unknown. We explore this question with measurements of the distribution of single-particle energies and momenta in a nearly homogeneous gas above T(c). The data fit well to a function that includes a narrow, positively dispersing peak that corresponds to quasiparticles and an "incoherent background" that can accommodate broad, asymmetric line shapes. We find that the quasiparticle's spectral weight vanishes abruptly as the strength of interactions is modified, which signals the breakdown of a Fermi liquid description. Such a sharp feature is surprising in a crossover.
We study the contacts, large-momentum tail, radio-frequency spectroscopy, and some other universal relations for an ultracold one-dimensional (1D) two-component Fermi gas with spin-orbit coupling (SOC). Different from previous studies, we find that the ${q}^{\ensuremath{-}8}$ tail in the spin-mixing (off-diagonal) terms of the momentum distribution matrix is dependent on the two SOC parameters in the laboratory frame for 1D systems, where qqq is the relative momentum. This tail can be observed through time-of-flight measurement as a direct manifestation of the SOC effects on the many-body level. Besides the traditional 1D even-wave scattering length, we find that two new physical quantities must be introduced due to the SOC. Consequently, two new adiabatic energy relations with respect to the two SOC parameters are obtained. Furthermore, we derive the pressure relation and virial theorem at short distances for this system. To find how the SOC modifies the large-momentum behavior, we take the SOC parameters as perturbations since the strength of the SOC should be much smaller than the corresponding strength scale of the interatomic interactions. In addition, by using the operator product expansion method, we derive the asymptotic behavior of the large-momentum distribution matrix up to the ${q}^{\ensuremath{-}8}$ order and find that the diagonal terms of the distribution matrix include the contact of traditional 1D even-wave scattering length as the leading term and the SOC modified terms beyond the leading term, the off-diagonal term is beyond the subleading term and is corrected by the SOC parameters. We also find that the momentum distribution matrix shows spin-dependent and anisotropic features. Furthermore, we calculate the momentum distribution matrix in the laboratory frame for the experimental implication. In addition, we calculate the high-frequency tail of the radio-frequency spectroscopy and find that the presence of the contact related to the center-of-mass momentum in the radio-frequency spectral is due to the SOC effects. This paper paves the way for exploring the profound properties of many-body quantum systems with SOC in one dimension.
It is of central importance to probe the local spectral function $A(\mathbf{k},\ensuremath{\omega})$ of a strongly interacting Fermi gas in a trap. Momentum-resolved rf spectroscopy has been demonstrated to be able to probe the trap-averaged $A(\mathbf{k},\ensuremath{\omega})$. However, the usefulness of this technique was limited by the trap inhomogeneity. Independent of a specific theory, here we propose that by studying the momentum-resolved rf spectra of the minority fermions of a phase-separated, population-imbalanced Fermi gas at low temperature, one can effectively extract $A(\mathbf{k},\ensuremath{\omega})$ of a homogeneous superfluid Fermi gas (at the trap center). In support, we present calculated spectral functions and spectral intensity maps for various cases from BCS through Bose-Einstein condensation regimes using different theories.
Experimental advances in synthesizing spin-orbit couplings in cold atomic Bose gases promise to create single-particle dispersion laws featuring energy minima that are degenerate on a ring or a sphere in momentum space. We show that for arbitrary space dimensionality the ground-state properties of a dilute system of spin-orbit-coupled Bose particles with such dispersion and short-range repulsive interactions are universal: the chemical potential exhibits a quadratic dependence on the particle density as found in a one-dimensional free Fermi gas.