Pair excitations and parameters of state of imbalanced Fermi gases at finite temperatures
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Journal of Low Temperature Physics manuscript No. (will be inserted by the editor)
S. N. Klimin , J. Tempere , , and Jeroen P. A.Devreese Pair excitations and parameters of stateof imbalanced Fermi gases at finitetemperatures
October 3, 2018
Abstract
The spectra of low-lying pair excitations for an imbalanced two-componentsuperfluid Fermi gas are analytically derived within the path-integral formalismtaking into account Gaussian fluctuations about the saddle point. The spectra areobtained for nonzero temperatures, both with and without imbalance, and for ar-bitrary interaction strength. On the basis of the pair excitation spectrum, we havecalculated the thermodynamic parameters of state of cold fermions and the firstand second sound velocities. The parameters of pair excitations show a remark-able agreement with the Monte Carlo data and with experiment.PACS numbers: 03.75.Ss, 05.30.Fk, 03.75.Lm
Recent experimental breakthroughs in the manipulation of ultracold Bose andFermi gases have opened new prospects for advancing many-body physics . Thedimensionality of these gases can be controlled with optical lattices, and the inter-action strength can be tuned using Feshbach resonances. The experimental controlover geometry and interactions in ultracold atomic gases has turned these systemsinto powerful quantum simulators that can test and generalize many-body theoriesoriginally developed for solid state systems. Recently, much attention has beenpaid to ultracold atomic gases with strong interactions because of their possiblerelation to some striking natural phenomena including high-temperature super-conductors and neutron stars .In particular, great experimental and theoretical effort has been devoted to thestudy of superfluidity arising from pairing in ultracold Fermi gases . Specifically, the effect of population imbalance (between the pairing partners) on the super-fluid pairing mechanism is a topic of current investigation. Of no less interest inthe context of high-temperature superconductivity is the study of the crossoverbetween the Bardeen-Cooper-Schrieffer (BCS) and the molecular Bose-Einsteincondensation (BEC) regimes.In order to describe the superfluid phase transition as well as the broken-symmetry phase below a critical temperature T c , several methods based on the T -matrix approach have been developed . Amongst those, the Nozi`eres–Schmitt-Rink (NSR) theory and its path-integral reformulation have been verysuccessful and remain widely used. Here, we use an improved version of the NSRscheme, that we will denote as the Gaussian pair fluctuation theory (GPF) ,and that effectively works both at low temperatures and above T c .As shown in Ref. , the thermodynamic properties of the superfluid Fermi gasat sufficiently low temperatures can be derived within a simple model using thespectrum of low-lying elementary excitations. In that model, the thermodynamicsof the superfluid Fermi gas is based on the fermion-boson model with phe-nomenological parameters whose values are determined from the Monte Carlocalculations (in the limit of zero temperature). The model exploited inRef. describes well the thermodynamic properties of a balanced unitary Fermigas at low temperatures. Moreover it was demonstrated that in the BEC limit,the imbalanced Fermi superfluid indeed reduces to a simple Bose-Fermi mixtureof Bose-condensed molecules and unpaired fermions. The goal of the presentwork is to extend these results to non-zero temperatures, and to arbitrary scat-tering lengths.For this purpose, we calculate the parameters needed for the fermion-bosonmodel from the NSR and/or GPF theories. Our results are compared with theMonte Carlo data at unitarity. We analytically derive the spectra of low-lying el-ementary excitations for an imbalanced Fermi gas in 3D at finite temperatures inthe whole range of the BCS-BEC crossover. These spectra are obtained using thepath-integral representation of the NSR theory extended to imbalanced Fermigases as well as the GPF approach . Using the obtained spectra of the el-ementary excitations, thermodynamic parameters such as the internal energy, thechemical potential, the first and second sound velocities, are calculated. of the NSR scheme has been extended in Refs. to the case of unequal ‘spin up’ and ‘spin down’ populations of fermions. In thepresent work, the treatment of the imbalanced Fermi gas is performed using theNSR scheme and its improved version, the GPF theory extended to theimbalanced case.The thermodynamic parameters of the imbalanced Fermi gas are completelydetermined by the thermodynamic potential W of the grand-canonical ensemble.The thermodynamic potential W , the same as in Refs. , is the sum of thesaddle-point thermodynamic potential W sp and the fluctuation contribution W f l . These thermodynamic potentials are provided, respectively, by the zeroth-orderand quadratic terms of the expansion of the Hubbard-Stratonovich pair-field ac-tion around the saddle point.The saddle-point thermodynamic potential for the imbalanced Fermi gas with s -wave pairing is W sp = − V Z d k ( p ) (cid:20) b ln ( bz + b E k ) − x k − D k (cid:21) − V D p a s (1)where V is the system volume, b is the inverse to the thermal energy k B T , D isthe amplitude of the gap parameter, a s is the scattering length, x k = k − m is thefermion energy, and E k = q x k + D is the Bogoliubov excitation energy. Thechemical potentials of imbalanced fermions are expressed through the averagedchemical potential m = (cid:0) m ↑ + m ↓ (cid:1) / z = (cid:0) m ↑ − m ↓ (cid:1) /
2. We choose the units with ¯ h =
1, the fermion mass m = /
2, andthe Fermi energy E F ≡ ¯ h (cid:0) p n (cid:1) / / ( m ) = n is the total fermion density).The fluctuation contribution to the thermodynamic potential W f l is the same as inRefs. .The gap parameter is found from the gap equation minimizing the saddle-pointthermodynamic potential, ¶W sp ( T , m , z ; D ) ¶D (cid:12)(cid:12)(cid:12)(cid:12) T , m , z = D ( T , m , z ) . For an imbalanced gas, the saddle-pointthermodynamic potential can have two minima: one at D = D = . With our notation, weemphasize that the thermodynamic potential is a function of T , m , z (and actually V , but this dependency drops out). However, D is treated as an additional parame-ter on which the thermodynamic potential depends. It is this treatment of D as anadditional parameter (in the broken-symmetry phase with D =
0) that leads to adistinction between the NSR approach and the GPF approach. When calculatingthe gap equation one should use n = − ¶W ( T , m , z ; D ) ¶m (cid:12)(cid:12)(cid:12)(cid:12) T , z , D − ¶W f l ( T , m , z ; D ) ¶D (cid:12)(cid:12)(cid:12)(cid:12) T , z , m ¶D ( T , m , z ) ¶m (cid:12)(cid:12)(cid:12)(cid:12) T , z , (3) d n = − ¶W ( T , m , z ; D ) ¶z (cid:12)(cid:12)(cid:12)(cid:12) T , m , D − ¶W f l ( T , m , z ; D ) ¶D (cid:12)(cid:12)(cid:12)(cid:12) T , z , m ¶D ( T , m , z ) ¶z (cid:12)(cid:12)(cid:12)(cid:12) T , m . (4)In the standard NSR approach the last terms (involving the derivatives of D ) areomitted. The GPF approach suggested in Refs. takes into account the addi-tional derivatives for the balanced case, and corrects the NSR densities for changesin D as m and z are varied. The GPF method presented here is the path-integralformulation of the GPF theory of Refs. extended to the imbalanced case. Asshown in Ref. , the GPF theory provides the best overall agreement of its ana-lytic results with experiment and with Monte Carlo data, except in close vicinityto T c . The GPF corrections are not present when one uses the saddle-point approxi-mation and calculates n sp = − ¶W sp ( T , m , z ; D ) ¶m (cid:12)(cid:12)(cid:12)(cid:12) T , z , D , (5) d n sp = − ¶W sp ( T , m , z ; D ) ¶z (cid:12)(cid:12)(cid:12)(cid:12) T , m , D . (6)However, the correction terms will be relevant for the calculation of the fluctuationcontributions, n f l = n − n sp and d n f l = d n − d n sp . These fluctuation contributionsto the density n f l and d n f l are given by the same expressions as in Ref. n f l = − Z d q ( p ) " p Z ¥ − ¥ Im J ( q , w + i g ) e b ( w + i g ) − d w + b n (cid:229) n = − n J ( q , i n n ) , (7) d n f l = − Z d q ( p ) " p Z ¥ − ¥ Im K ( q , w + i g ) e b ( w + i g ) − d w + b n (cid:229) n = − n K ( q , i n n ) . (8)Here, n is an arbitrary positive integer, and the parameter g lies between twobosonic Matsubara frequencies n n < g < n n + , n n ≡ p n / b . The spectral func-tions J ( q , z ) and K ( q , z ) of complex frequency z are J ( q , z ) = M , ( q , − z ) ¶ M , ( q , z ) ¶m − M , ( q , − z ) ¶ M , ( q , z ) ¶m M , ( q , z ) M , ( q , − z ) − M , ( q , z ) , (9) K ( q , z ) = M , ( q , − z ) ¶ M , ( q , z ) ¶z − M , ( q , − z ) ¶ M , ( q , z ) ¶z M , ( q , z ) M , ( q , − z ) − M , ( q , z ) , (10)where M j , k ( q , z ) are the matrix elements of the pair field propagator. The matrixelements M j , k ( q , z ) are given by the expressions M , ( q , z ) = Z d k ( p ) (cid:26) k + X ( E k ) E k (cid:20) ( z − E k + e k + q ) ( E k + e k )( z − E k + E k + q ) ( z − E k − E k + q ) − ( z + E k + e k + q ) ( E k − e k )( z + E k − E k + q ) ( z + E k + q + E k ) (cid:21)(cid:27) − p a s , (11) M , ( q , z ) = − D Z d k ( p ) X ( E k ) E k (cid:20) ( z − E k + E k + q ) ( z − E k − E k + q )+ ( z + E k − E k + q ) ( z + E k + E k + q ) (cid:21) . (12)Here, the distribution function X ( E k ) is X ( E k ) = sinh b E k cosh b E k + cosh bz . (13) The derivatives in (9) and (10) within the GPF scheme are determined as men-tioned above – taking into account a variation of the gap parameter. Within theNSR scheme, these derivatives are calculated assuming D to be an independentvariational parameter.The equation of state of the imbalanced Fermi gas is thus determined as a jointsolution of the saddle-point gap equation and the number equations accountingfor Gaussian fluctuations. Within both the NSR and GPF schemes, the Gaussianfluctuations do not feed back into the saddle-point gap equation.2.2 Low-lying pair excitationsIn order to obtain the spectrum of low-lying pair excitations for the imbalancedFermi gas, we perform the long-wavelength and low-energy expansion of the ma-trix elements M j , k ( q , z ) as proposed in Ref. . We take into account the terms upto quadratic order in powers of q and z , and find: M , ( q , z ) ≈ A + Bq + Cz + Qz , M , ( q , z ) ≈ D + Eq + Hz . (14)The coefficients of the expansion (14) are derived straightforwardly. Aftersome algebra, we arrive at their expression through the integrals: A = p Z k dk (cid:18) k − E k + x k E k X ( E k ) − D X ′ ( E k ) E k (cid:19) − p a s , B = p Z k dk E k k − E k x k + x k E k + E k x k k − x k k E k X ( E k )+ D p Z k dk E k x k (cid:0) E k − x k k (cid:1) E k X ′ ( E k )+ x k k − E k (cid:0) x k + k (cid:1) E k X ′′ ( E k ) − k x k X ( ) ( E k ) ! , C = − p Z k dk x k E k X ( E k ) − D p Z k dk X ′ ( E k ) x k E k , D = D p Z k dk X ( E k ) E k − D p Z k dk X ′ ( E k ) E k , E = D p Z k dk x k k − E k (cid:0) x k + k (cid:1) E k X ( E k )+ D p Z k dk E k E k (cid:0) x k + k (cid:1) − x k k E k (cid:2) X ′ ( E k ) − E k X ′′ ( E k ) (cid:3) − k x k X ( ) ( E k ) (cid:19) , Q = − p Z k dk E k + x k E k X ( E k ) , H = D p Z ¥ k dk X ( E k ) E k . (15)The low-lying pair excitations correspond to the poles of the spectral func-tions (9) and (10). Therefore the dispersion equation for the energies of the pairexcitations w = w q is M , ( q , w ) M , ( q , − w ) − M , ( q , w ) = . (16)This equation is solved expanding w q up to the terms of the order of q . We thenobtain the energies w q in a form structurally similar to the collective excitations inRef. : w q = p c q + l q , (17)where the parameters c and l are related to the coefficient of the expansion (14)as follows, c = (cid:18) A ( B − E ) C + A ( H − Q ) (cid:19) / , (18) l = C ( B − E ) (cid:0) A ( BH − EQ ) + C ( B + E ) (cid:1) ( C + A ( H − Q )) . (19)For small pair momentum q , the energy of the pair excitation becomes linearin the momentum. Thus w q at small q represents a Bogoliubov–Anderson mode,which is gapless in accordance with the Nambu–Goldstone theorem. The param-eter c has the dimensionality of velocity. In the zero-temperature limit for a bal-anced gas, all fermions are in the superfluid state, and the velocity parameter forthe pair excitations tends to the first sound velocity for the whole fermion system.The so-called gradient parameter l provides a growth of kinetic energy due to aspatial variation of the density .The pair excitation spectra obtained in the present work generalize the long-wavelength expansion of Ref. to the case of non-zero temperatures and unequal‘spin-up’ and ‘spin-down’ fermion populations.In Fig. 1, the parameters c and l characterizing pair excitations are plottedas a function of temperature for the balanced Fermi gas in the unitarity regime(1 / a s = T c b k B T/E F T c k F a s ) = 0 = 0 GPF NSR Monte Carlo c a GPF NSR Monte Carlo / E F T/T c Fig. 1 (color online) ( a ) Sound velocity parameter of the Bogoliubov–Anderson mode c for coldfermions in 3D as a function of temperature in the unitarity regime and for z =
0. The parametersof state are determined taking into account the fluctuations in different ways: within the path-integral GPF formalism (solid curve) and within the NSR scheme (dashed curve). The symbolsshow the results of the different Monte Carlo calculations . ( b ) The parameter l asa function of temperature in the same regime. The dot-dashed vertical line indicates the criticaltemperature of the superfluid phase transition. The thin dotted curves show the formal solutionsfor the parameter c and l within the path-integral GPF and NSR approaches above T c . Inset :Chemical potential calculated within the path-integral GPF and NSR approaches compared withthe Monte Carlo data from Ref. . account fluctuations in the number equation. The fluctuation contributions to thefermion density are calculated within the path-integral GPF and NSR schemes.As found in Ref. , the NSR approach becomes inaccurate in the vicinity ofthe critical temperature T c . It was also shown that the NSR scheme reveals a re-entrant behavior of the parameters in the state above T c , leading to an artificialfirst-order superfluid phase transition . The re-entrant behavior of the parame-ters c and l obtained in the NSR approach is clearly seen in Fig. 1. The criticaltemperature T c for the balanced gas , indicated by a dash-dotted line in Fig. 1, isthe same within the NSR and GPF approaches. However, the GPF method leadsto better results with respect to NSR for the broken-symmetry phase. This can beseen, for example, in the inset of Fig. 1 where we plot the chemical potential as a function of temperature below T c calculated within the NSR and path-integralGPF approaches and compared with the Monte Carlo results of Ref. .In the zero-temperature limit, the sound velocity parameter c for the pair exci-tations obtained within both the path-integral GPF and NSR approaches exhibitsan excellent agreement with the numerical results obtained using different MonteCarlo algorithms . Also the gradient parameter l at zero temperaturelies within the range of the values of l obtained in Refs. as the best fittingparameters for the ground state energy of fermions compared with Monte Carlodata. This agreement demonstrates the accuracy of the present approach for thebroken-symmetry phase of cold fermions. -0.4 -0.2 0.0 0.2 0.4 0.6 0.80.10.20.30.40.50.6 d k B T = 0.01 E F k B T = 0.05 E F k B T = 0.1 E F a s = 0.2 c c = 0.2 a c = 0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.80.11 = 0 k B T = 0.01 E F k B T = 0.05 E F k B T = 0.1 E F a s b Fig. 2 (color online) Sound velocity parameter ( a , c ) and parameter l ( b , d ) for the Bogoliubov–Anderson mode as a function of the inverse scattering length for k B T = . E F (solid curves),for k B T = . E F (dashed curves) and for k B T = . E F (dotted curves) for a balanced Fermigas ( a , b ) and at the chemical potential imbalance z = . c , d ) in the unitarity regime. In Fig. 2, we plot the parameters c and l as a function of the inverse scatteringlength 1 / a s for the balanced gas (the left-hand panels) and at the chemical poten-tial imbalance z = . k B T = . E F the sound veloc-ity parameter monotonously decreases with increasing 1 / a s . For non-zero temper-atures, c exhibits a maximum, which shifts to higher coupling strengths for highertemperatures. The parameter l at finite temperatures has a minimum, which al-most vanishes in the zero-temperature limit. In the weak-coupling regime, both c and l are sensitive to temperature. When moving towards the strong-couplingregime, c and l gradually become almost independent on T . The imbalance leadsto the appearance of a critical inverse scattering length such that for smaller 1 / a s ,there is no superfluid state (see, e. g., Ref. ). , a similar description of the thermodynamic properties wasperformed for a unitary balanced Fermi gas using the zero-temperature spectra ofelementary excitations.In Ref. , a pair excitation spectrum of the form of expression (17) is used,where the zero-temperature sound velocity c is taken from the Monte Carlo data and the gradient parameter l is determined from a fit of the thermodynamic prop-erties to the Monte Carlo results. In the present calculation, the pair excitationspectra are obtained using the analytic path-integral GPF approach without anyfit. In this section we consider the thermodynamic functions of the cold Fermi gaswithin the model of fermionic and pair excitations. The grand-canonical thermo-dynamic potential is the sum of the saddle-point and pair excitation contributions W = W sp + W p . (20)The saddle-point thermodynamic potential W sp is given by Eq. (1). The contribu-tion of pair excitations is W p = V b Z d q ( p ) ln (cid:16) − e − bw q (cid:17) . (21)The entropy S is found using its relation to the grand-canonical thermody-namic potential, S = − ¶W¶ T (cid:12)(cid:12)(cid:12)(cid:12) V , m , z . (22)Using the thermodynamic potential W with (1) and (21) we find that the entropyis expressed as S = V b p Z ¥ x k E k E k ( cosh b E k cosh bz + ) − z sinh b E k sinh bz ( cosh bz + cosh b E k ) k dk + V p Z ¥ bw q e − bw q − e − bw q − ln (cid:16) − e − bw q (cid:17)! q dq . (23)Finally, the internal energy E of cold fermions is calculated using the relationbetween E and the grand-canonical thermodynamic potential, E = W + T S + m N , where N is the total number of fermions.In Fig. 3, the internal energy per particle for a unitary Fermi gas calculatedin different approaches is plotted as a function of temperature for the broken-symmetry phase at T T c . The critical temperature determined within the path-integral GPF model is the same as within NSR, T c ≈ . E F / k B . The results ofthe present calculation within the model of fermionic and low-lying pair excita-tions with parameters determined using the path-integral GPF and NSR methods are shown with short-dashed and dot-dashed curves, respectively. The other re-sults represented in Fig. 3 are (after Ref. ): the internal energy calculated withinthe low-temperature fermion-boson (FB) model , and the result of the analyticmodel proposed by Bulgac, Drut, and Magierski (BDM) . The analytic resultsare compared with those of the Monte Carlo calculations from Ref. and with theexperimental data of Ref. for a gas of Li atoms at unitarity.
Fig. 3 (color online) Internal energy per particle for a Fermi gas at unitarity calculated within themodel of fermionic and low-lying pair excitations with parameters determined using the path-integral GPF method (short-dashed curve) and the NSR theory (dot-dashed curve). The solidcurve shows the internal energy obtained within the low-temperature fermion-boson model inRef. . The dashed curve is the result of the BDM model . The full dots represent the MonteCarlo simulations . The squares are the experimental data . As reported in Ref. , the low-temperature fermion-boson model works wellin the broken-symmetry phase where the internal energy resulting from this modelis close to the Monte Carlo data of Ref. . The present study is performed with theparameters of the elementary excitations obtained using the analytic approachesrather than a fit to the Monte Carlo simulations. The internal energy calculatedin the present work with the parameters determined using the path-integral GPFmethod is very close to the Monte Carlo results at low temperatures T / . T c .Furthermore, our result is in good agreement with the experiment in the wholetemperature range below T c .Neither the model of fermionic and pair excitations used in the present work,nor the low-temperature fermion-boson model of Ref. predicts the superfluidphase transition: the critical temperature T c is determined within the path-integralGPF method before the long-wavelength expansion is performed in Sec. 2. How-ever, the latter model can describe well the broken-symmetry phase of cold fermionicatoms. n = p Z ¥ k dk (cid:18) − x k E k X k (cid:19) + p Z ¥ q dq e bw q − . (24)The total density is a sum of the normal and superfluid densities n = n n + n s . Thesuperfluid density n s , as well as the total density, is constituted by the saddle-pointresult for an imbalanced Fermi gas and the contribution due to the pair excitations, n s = p Z ¥ (cid:18) − x k E k X k − k Y k (cid:19) k dk + p Z ¥ e bw q − − b q e − bw q (cid:0) e − bw q − (cid:1) ! q dq , (25)where the function Y k is given by Y k ≡ ¶ X k ¶ E k = b cosh b E k cosh bz + ( cosh b E k + cosh bz ) (see the analogous expression for the Fermi gas in 2D in Ref. ). In the balancedcase, the superfluid density becomes equivalent to the corresponding expression ofRef. , but with other values of the parameters of the pair excitations, as discussedabove and shown in Fig.1.In Fig. 4, the superfluid density divided by the total density calculated withinthe model of fermionic and pair excitations is plotted as a function of temperature.As seen from the figure, the superfluid density calculated using the parameters ofthe pair excitations obtained within the NSR model exhibits a re-entrant behaviorabove T c similarly to the sound velocity parameter in Fig. 1. The analogous bend-over of the superfluid density above T c was reported in the full NSR approach inRef. .3.3 Sound velocitiesWe consider the sound propagation in a superfluid Fermi gas using the approachof the two-fluid hydrodynamics in the same way as in Ref. . The first soundvelocity u in the two-fluid hydrodynamics is determined by the formula u = ¶ P ¶ n (cid:12)(cid:12)(cid:12)(cid:12) ¯ S , V ! / , (26)with the entropy per particle ¯ S = S / N . The pressure is proportional to the grand-canonical thermodynamic potential: P = − W / V . We adopt the expression ¶ P ¶ n (cid:12)(cid:12)(cid:12)(cid:12) ¯ S , V = Pn (27) k B T / E F T c GPF NSR n s / n Fig. 4 (color online) Superfluid density for a Fermi gas at unitarity calculated within themodel of fermionic and pair excitations with parameters determined using the path-integral GPFmethod (solid curve) and the NSR theory (dashed curve). Thin dotted curves correspond to thetemperature range above T c . and use the grand-canonical thermodynamic potential given by Eq. (20) with (1)and (21).The second sound velocity u characterizes the temperature waves in whichthe motion of the normal and superfluid fractions is out-of-phase. It is determinedby the formula u = S ¶ ¯ S ¶ T (cid:12)(cid:12)(cid:12) N , V n s n n / . (28)The formulae (26) and (28) are valid as far as the first-sound and second-soundmodes are decoupled. Following Ref. , we assume that the above condition isfulfilled for a cold Fermi gas.In Fig. 5, the first and second sound velocities (divided by the zero-temperatureFermi velocity v F ) are plotted as a function of temperature. They are calculatedwithin the model of fermionic and pair excitations using the parameters of pairexcitations determined by the path-integral GPF and NSR methods.The first and second sound velocities obtained using the model of fermionicand pair excitations are in a reasonable agreement with the results of the analysisbased on the full NSR thermodynamics . In the zero-temperature limit, thefirst sound velocity tends to the same limit as the sound velocity parameter c forpair excitations, which is extremely close to the Monte Carlo data . u / v F k B T/E F T c u / v F GPF NSR Monte Carlo S ound v e l o c iti e s Fig. 5 (color online) First and second sound velocities for a unitary Fermi gas calculated withinthe model of fermionic and pair excitations with parameters determined using the path-integralGPF method (solid curves) and the NSR theory (dashed curves). The symbols show the MonteCarlo results for the first sound velocity. Thin dotted curves correspond to the temperature rangeabove T c . In the present work, the GPF modification of the NSR scheme has been for-mulated in the path-integral representation and extended to the case of imbalancedfermions. Within this path-integral GPF approach, we have analytically derivedthe spectra of low-lying pair excitations of the imbalanced Fermi gas with s -wavepairing at finite temperatures and extracted the parameters c and l for the pairexcitations from these results. Using these spectra, the finite-temperature thermo-dynamics of the Fermi gas in the superfluid state has been analyzed. The obtainedinternal energy demonstrates a good agreement with the Monte Carlo results andis remarkably close to the experimental data for the Fermi gas at unitarity. Thezero-temperature value of the first sound velocity is in a good agreement with theresults of the Monte Carlo simulations. The present method allows us to obtainthe spectra of the elementary excitations and, consequently, the thermodynamicparameters of the state for an arbitrary scattering length, at non-zero temperatures,and for non-zero imbalance. Acknowledgements
The authors gratefully acknowledge discussions with L. Salasnich. Thiswork was funded by the Fonds voor Wetenschappelijk Onderzoek-Vlaanderen (FWO-V) projectsG.0356.06, G.0370.09N, G.0180.09N, and G.0365.08. JPAD acknowledges financial support inthe form of a Ph.D. Fonds voor Wetenschappelijk Onderzoek-Vlaanderen (FWO-V).4
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