# Static structure factor of a strongly correlated Fermi gas at large momenta

Strongly interacting fermions are an important feature of many physical systems in condensed matter physics, nuclear physics and astrophysics. The recent achievements of trapping and cooling ultracold fermionic atoms have made them significant in atomic physics as well [1]. With unprecedented controllability of interactions and geometries [1], these atomic systems are prototypes of many important theoretical models. The strength of the interactions is governed by a single (i.e., -wave) two-body scattering length, and can be tuned by means of Feshbach resonances, allowing for a systematic exploration of the crossover from a weakly coupling BCS superfluid to a Bose-Einstein condensate (BEC) of tightly-bound molecules. Ultracold atoms therefore provide a new platform for investigating the intriguing many-body properties of fermions. A rich variety of theoretical predictions has emerged, many of which await verification. Theoretical challenges arise especially when the scattering length is comparable to or larger than the inter-particle distance [2]. In this strongly interacting regime, the system has universal scaling properties that depend on only [3]. These universal properties, however, cannot fully be understood using conventional perturbation methods, which are reliable only for [5]. Ab-initio quantum Monte Carlo (QMC) simulations are very helpful [7], but suffer from the Fermi sign problem in many cases. The challenging nature of strong interactions therefore makes *exact* results very valuable.

In this Letter, we an exact relation for the large-momentum behavior of the spin-antiparallel static structure factor , showing that it has a simple universal power-law () tail. .

to the family of exact relations obtained by Tan in 2005 [10], which link the short-range, large-momentum, and high-frequency asymptotic behavior of many-body systems to their thermodynamic properties [10]. For instance, the momentum distribution and rf spectrum fall off as and , respectively. All the Tan relations are related to each other by a single coefficient , referred to as the integrated contact density or contact. The contact measures the probability of two fermions with unlike spins being close together [15]. It also links this short-range behaviour to thermodynamics via the adiabatic relation, , which gives the change in the total energy due to adiabatic changes in the scattering length. The fundamental importance of the Tan relations is due to their wide applicability: to zero or finite temperature, superfluid or normal phase, homogeneous or trapped, few-body or many-body systems.

Tan relation for follows directly from the short-range behavior of the pair correlation function , which diverges as

A Fourier transformation of then leads to

where is the Fermi wave-vector and is the total number of atoms. On the right hand side of the above equation, we have defined . Eq. (Equation 1) holds in a scaling region of sufficiently large near the unitarity limit () so that the next-order correction in the bracket () is small compared to the leading term of .

The power-law tail of is more amenable for experimental investigation than the or tail . The fast decay due to the higher power law in these latter two cases imposes very stringent signal-to-noise requirements for studying a given range of momenta or frequency. Experimentally, the static structure factor can be readily measured using two-photon Bragg spectroscopy on a balanced two-component atomic Fermi gas near a Feshbach resonance[16]. In the large- limit, to an good approximation . One thus can directly determine the spin-antiparallel structure factor and verify a simple asymptotic behavior, .

The purpose of this Letter is to quantify the size of the universal scaling region where the Tan relation (Equation 1) is valid. For this goal, it is desirable to calculate at large momentum (i.e., ) at *arbitrary* interaction strength and temperature. This is a challenging theoretical problem. So far, the static structure factor of strongly interacting fermions has only been studied using QMC simulations at zero temperature in a homogeneous medium [8].

In the present work, we first revisit the zero-temperature problem by extending the scaling result Eq. (Equation 1) to the weak coupling BEC and BCS limits with . In the BEC regime, is mainly caused by ground state molecules. In the opposite BCS regime, we use a random-phase linear response theory to obtain first the dynamic and subsequently the static structure factor. These two limits are then interpolated smoothly to the unitarity limit. In this procedure, the zero-temperature contact is determined from a reliable theoretical prediction of the ground state energy along the BEC-BCS crossover, via the adiabatic Tan relation. As an alternative, at finite temperatures close to the Fermi temperature (), we develop a quantum virial expansion theory and calculate the structure factor quantitatively at high temperature. In this high temperature regime, Tan relation (Equation 1) can be analytically demonstrated, and the temperature dependence of the contact is extracted. The interpolation strategy adopted at zero temperature can also be strictly examined.

*Weak coupling limit* — We start by considering the weak coupling BEC limit with a small and positive scattering length, . Here the system is a dilute gas of weakly interacting bound molecules, consisting of two atoms with opposite spins, with a relative wave function . Neglecting the correlation among molecules at , the pair correlation function is given by , leading to [8],

which is essentially geometry and temperature independent. For , the first two terms in the Taylor expansion of the expression agrees exactly with Eq. (Equation 1), since [10].

In the opposite BCS limit of , we follow a mean-field picture and treat the system as a gas of quasiparticles subjected to a (dynamical) mean-field Hamiltonian [17], , where is the free particle Hamiltonian and are the space- and time-dependent density fluctuations with respect to equilibrium. Here, we consider a normal state since the result differs by an exponentially small amount from that of the BCS superfluid state. The dynamic structure factor may be calculated via the density response functions within a (random-phase) linear response theory [17], which are given by, .

The free response function can be written in a simple form in terms of a dimensionless function : , with and . The function can be found in standard textbooks [17]. To leading order in , it is straightforward to show that for a *homogeneous* Fermi gas, the static structure factor is given by

This weak-coupling equation holds for arbitrary transferred momentum. We may rewrite it into the form, , where at we have defined and have confirmed that [10]. At large momentum, asymptotically, . Thus, at the leading order of interaction strengths (), the structure factor in the BCS limit is in agreement with the Tan relation (Equation 1). We note, however, that the dominant tail in the strongly interacting regime (i.e., the factor 1 in the square bracket in Eq. (Equation 1)) is lost. The large momentum tail comes from pairing effects and therefore cannot be captured by the random-phase approximation for linear response.

*Structure factor from an interpolation strategy*. — The excellent agreement between the large- expansion of the weak coupling results, Eqs. (Equation 2) and (Equation 3), and the Tan relation (Equation 1) suggests that we may use the following extrapolation for the calculation of the structure factor at and at arbitrary interaction strengths:

In doing so, we remove the severe requirements of and for validating the Tan relation (Equation 1). In Fig. 1, we show the resulting static structure factor for a homogeneous Fermi gas along the BEC-BCS crossover. We have determined the contact by applying the adiabatic relation to the ground state energy calculated from a perturbative Gaussian pair fluctuation theory [6]. In the figure, the weak coupling results in Eqs. (Equation 2) and (Equation 3) are also reported respectively by the dashed and dot-dashed lines. By applying a local density approximation, we may also calculate the zero temperature structure factor of a Fermi gas in a harmonic trap with frequency and length . Using the ideal Fermi temperature and vector at the trap center, and , we find that the trapped structure factor differs only slightly with the uniform one (see the black line in Fig. 2).

The accuracy of the interpolation may be justified by comparing our results to QMC simulations [8], as shown in the inset for , , and . We find a large deviation occurring at on the BEC side (not shown in the inset), where one has to take into account the correlations between different bound molecules. However, this is irrelevant to our purpose of examining large momentum correlations. As we shall see, a further justification of our interpolation scheme can be obtained at finite temperature, where a quantitative calculation of the contact and structure factor is feasible by using a quantum virial expansion. In this temperature regime, a random-phase linear response theory predicts for a uniform gas,

where the function . We find that .

*Quantum virial expansion at finite temperatures*. — We now calculate the structure factor at high temperatures for a *trapped* Fermi gas, by extending a previous virial expansion theory [18]. At high temperatures, the fugacity is a small controllable parameter, even in the strongly interacting limit. We may expand the thermodynamic potential , where , in a series of powers of fugacity. By defining partition functions of clusters in the -particle subspace, , we find that . The pair correlation function can be calculated by taking a functional derivative of the thermodynamic potential with respect to the generating elements of a source term , i.e., . Tthe virial expansion form of the thermodynamic potential is particularly suitable for taking a derivative of to , and hence for applying the Hellmann–Feynman theorem.

To leading order, it is readily seen that , where the summation is over all the two-body states of a pair made of atoms with opposite spins. In a harmonic trap, the center-of-mass and relative motion parts of the wavefunction are separable. By taking the Fourier transform of the relative coordinate () and performing a spatial integral over the center-of-mass part (), we finally arrive at

where , and represents the same sum as the first term in the bracket, but for a non-interacting system. The relative energies and radial wavefunctions are known [18]: , where is given by , and , where is the normalization factor, and and are respectively the Gamma and confluent hypergeometric functions.

The accuracy of the virial expansion (Equation 5) may be examined by applying the same expansion to the equation of state [21], which turns out to be quantitatively reliable at . It is straightforward to improve the accuracy by including contributions from , , and so on. However, considerable insight can already be obtained from the leading two-body contribution shown in Eq. (Equation 5). At short distance, . Therefore, for large momentum transfer , we find from Eq. (Equation 5) that,

where and

is the contact predicted by the virial expansion. We thus explicitly confirm Tan relation (Equation 1) within second order virial expansion theory.

In the BEC limit, the two-body bound state dominates the summation in Eq. (Equation 5), with energy . The virial expansion of the number equation gives . By inserting these results into Eq. (Equation 5), we correctly recover the structure factor in the weak-coupling BEC limit, Eq. (Equation 2). At unitarity, the normalization factor is analytically known and we find that . Thus, at where , the contact is given by , and therefore rapidly decreases with temperature. The structure factor is simply . Finally, in the BCS limit, we may also work out analytically that . This leads to , agreeing exactly with our prediction from random-phase linear response theory . The resulting structure factor is given by

In Fig. 2, we present the numerical results of the leading virial expansion for the static structure factor () of a trapped Fermi gas. The fugacity is determined by the number equation, , expanding up to the second order virial coefficient. We show also the zero temperature result obtained from the local density approximation. The temperature dependence of the structure factor is evident, particularly on the BCS side. In the inset, we check explicitly the interpolation strategy with . For , the extrapolation results (symbols) are indistinguishable to the numerical virial expansion calculations (lines).

*Universal scaling region of our Tan relation*. — We are now ready to determine the scaling region in which we expect that the scaled structure factor becomes momentum independent and converges to the contact of the system. Fig. 3 shows the scaled structure factors of a trapped Fermi gas at different transferred momenta and at low and high temperatures. On the BCS side, the scaled structure factor is rather insensitive to the varying momentum. We observe that the scaling limit can be easily reached at the BEC-BCS crossover regime (), at a relatively small momentum . However, to access the scaling limit in a broader region of interaction strengths (i.e., ), a larger momentum () is necessary.

*Conclusion*. — Based on Tan relation , we present a systematic study of the static structure factor of an interacting Fermi gas near the BEC-BCS crossover at a transferred momentum . The scaling region of the Tan relation is clarified. These predictions can be readily checked in future Bragg experiments at crossover, either near the ground state or at finite temperatures.

We thank C. J. Vale, E. D. Kuhnle, M. Mark, P. Dyke, and P. Hannaford for fruitful discussions. This work was supported in part by the Australian Research Council (ARC) Centre of Excellence for Quantum-Atom Optics, ARC Discovery Project No. DP0984522 and No. DP0984637, NSFC Grant No. NSFC-10774190, and NFRPC Grant No. 2006CB921404 and No. 2006CB921306.

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