Collective modes of a two-dimensional Fermi gas at finite temperature
CCollective modes of a two-dimensional Fermi gas at finite temperature
Brendan C. Mulkerin, Xia-Ji Liu, and Hui Hu Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne 3122, Australia. (Dated: September 28, 2018)In this work we examine the breathing mode of a strongly interacting two-dimensional Fermigas and the role of temperature on the anomalous breaking of scale invariance. By calculating theequation of state with different many-body T -matrix theories and the virial expansion approach, weobtain a hydrodynamic equation of the harmonically trapped Fermi gas (with trapping frequency ω ) through the local density approximation. By solving the hydrodynamic equation we determinethe breathing mode frequencies as functions of interaction strength and temperature. We find thatthe breathing mode anomaly depends sensitively on both interaction strength and temperature.In particular, in the strongly interacting regime we predict a significant down-shift of the breath-ing mode frequency, below the scale invariant value of ω for temperatures of order the Fermitemperature. PACS numbers: 03.75.Hh, 03.75.Ss, 67.85-d
I. INTRODUCTION
Fermi gases in two dimensions are of significant impor-tance in understanding many-body systems [1, 2]. Ultra-cold atomic Fermi gases near a Feshbach resonance of-fer a new type of strongly interacting quantum system,where experimentalists have precise control over almostall the physical properties of the system, such as interac-tion strength, particle number, and dimension. Interac-tions are described through a zero-range delta-potential,which in two dimensions is known to be an example of aquantum anomaly [3]. The classical symmetry of a sys-tem interacting through a delta potential in two dimen-sions is scale invariant under the transformation r → λ r ,and an analysis of the scattering properties of a quantumsystem shows that it is divergent and must be regularized[4–6]. This well known regularization introduces a newlength scale, the two-dimensional (2D) scattering length a D , and the scale invariance of the classical theory hasbeen broken. The quantum anomaly can be seen throughthe breathing mode of trapped ultracold gases [7, 8]. Har-monically trapped gases posses a hidden SO(2,1) symme-try [9], which excites a breathing mode at a frequency of ω = 2 ω , where ω is the frequency of the trapping poten-tial. The breaking of scale invariance through the delta-potential interaction will excite a breathing mode, whosefrequency is dependent on the regularized 2D scatteringlength, a D . The advent of 2D ultracold gases realizedfor fermions [10–14] and bosons [15–18] are ideal systemsfor measuring the breathing mode and the anomalousbreaking of scale invariance.In Fermi gas experiments the breathing mode andanomalous corrections have been studied by Refs. [19,20], where the breathing mode frequency shifts - awayfrom the scale invariance value of ω - were of order a fewpercent, but the errors within the experiment made theobservation of the anomaly inconclusive. It was arguedthat at the temperature of the experiment, T /T F (cid:39) . ,the anomaly was damped, and that we would expect atlower temperatures the frequency shifts of the breathing mode to be more pronounced. Theoretically the breath-ing mode of harmonically trapped 2D gases has been ex-tensively studied at zero temperature [21–23] using quan-tum Monte Carlo to determine the equation of state [24],and at finite temperature [25] with the high-temperaturevirial expansion up to the second order. In all cases,the breathing mode frequency can be calculated througha variational approach of the hydrodynamic equations[26–30]. At T = 0 the breathing mode shifts from thescale invariant value found by Refs. [21, 23] are posi-tive for all interaction strengths at the crossover froma Bose-Einstein condensate (BEC) to a Bardeen-CooperSchrieffer (BCS) superfluid, and in the strongly interact-ing regime they can reach approximately 10% of the scaleinvariant result. The virial expansion analysis at finitetemperature done by Ref. [25] focused on the comparisonto the experiment of Ref. [19] and did not examine therole of temperature on the breathing mode. To addressthe temperature effect, it necessarily requires sufficientknowledge about the equation of state of a strongly inter-acting 2D Fermi gas at finite temperature, which unfor-tunately remains a grand challenge both experimentally[12, 14] and theoretically [31].In Bose gas experiments [15, 16] the breathing modewas measured for only weakly interacting systems wherethe system appears to be scale invariant and robust totemperature, and no breathing mode anomaly has beenobserved. Theoretically, it was found that there is a tem-perature dependence of the breathing mode in the weaklyinteracting regime [32], however it should be noted thatthis deviation is not due to the breaking of scale invari-ance and is a result of a small deviations from the effectiveharmonic trapping potential.In this work, we aim to present a systematic inves-tigation of the finite-temperature breathing mode of astrongly interacting 2D Fermi gas, by gathering themost advanced knowledge of the 2D homogeneous equa-tion of state developed in some recent theoretical works[31, 33–39]. At low temperature, we consider the non-self-consistent pair-fluctuation theory by Nozières and a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug Schmitt-Rink (NSR) [40, 41] and the self-consistent T -matrix approximation [42] in the normal state, taking theeffects of pairs explicitly into account. In the high tem-perature and weakly interacting regimes we employ thevirial expansion [43] to second and third order [33, 36].Using the local density approximation with the homoge-neous equation of state we calculate the collective modesof a trapped system using a variational approach at agiven trap temperature and interaction strength [29].We find theoretically that the breathing mode anomalyis temperature dependent as well as interaction depen-dent. In the weakly interacting BCS regime, the breath-ing mode is reduced towards the scale invariant result of ω , as temperature is increased. In the strongly inter-acting regime and in the high temperature regime, wellabove the superfluid temperature T c , the breathing modelowers below the scale invariant value, indicating the im-portance of pair formation at high temperature.This paper is organized as follows. In Sect. II we intro-duce a diagrammatic approach to the pressure equationof state and the virial expansion. In Sect. III we intro-duce the variational formalism for ultracold gases andderive the thermodynamic properties for the calculationof the breathing mode. In Sect. IV we discuss the resultsof the breathing mode. We present a brief conclusion andoutlook in Sect. V. The two appendices (Appendix A andAppendix B) are devoted to the details of variationallysolving the hydrodynamic equation. II. THEORY
In this section, for self-containedness we review thetheories of a strongly interacting Fermi gas in two di-mensions [31]. At low temperature, we focus on the non-self-consistent NSR approach, since it provides the easi-est way to take into account strong pair-fluctuations in2D, with the most stable numerical outputs for the equa-tion of state. At high temperature, we introduce a Pádeapproximation to re-organize the virial expansion series,which may extend the applicability of virial expansiontowards the low-temperature regime. It is worth not-ing that for a 2D Fermi gas at finite temperature (i.e., < T (cid:46) T F ), all the strong-coupling theories so far are qualitative only. It is notoriously difficult to carry outnumerically accurate quantum Monte Carlo simulationsat finite temperature due to the Fermi sign problem, evenin the normal state [44]. A. Many-body T -matrix theories In order to explore the finite temperature behaviorof 2D Fermi gases at the BEC-BCS crossover, follow-ing NSR we consider the contribution of pairing fluctu-ations to the thermodynamic potential for a given tem-perature T and binding energy ε B = (cid:126) / ( M a D ) throughthe functional integral formulation, which has been ex- tensively studied in both two and three dimensions inthe normal and superfluid states [34, 39, 45–51]. Theequation of state is found through the thermodynamicpotential Ω = − k B T ln Z where the partition function is Z = ´ D (cid:2) ψ σ , ¯ ψ ¯ σ (cid:3) e − S [ ψ σ , ¯ ψ ¯ σ ] , and ψ σ and ¯ ψ ¯ σ are inde-pendent Grassmann fields representing fermionic speciesfor each spin degree of freedom, σ = ↑ , ↓ , of equal mass M . The partition function is defined through the action S = ˆ (cid:126) β dτ (cid:34) ˆ d r (cid:88) σ ¯ ψ σ ( x ) ∂ τ ψ σ ( x ) + H (cid:35) , (1)and the single channel Hamiltonian given by H = ¯ ψ σ ( x ) K ψ σ ( x ) − U ¯ ψ ↑ ( x ) ¯ ψ ↓ ( x ) ψ ↓ ( x ) ψ ↑ ( x ) , (2)where K = − (cid:126) ∇ / (2 M ) − µ , β = ( k B T ) − , µ is thechemical potential, and x = ( x , τ ) for position x andimaginary time τ . We take a contact interaction with U > , which has known divergences and can be fixedthrough renormalizing in terms of the bound state energyvia the relation, U = (cid:88) k (cid:15) k + ε B , (3)where (cid:15) k = (cid:126) k / (2 M ) . Using the Hubbard-Strantonivich transformation to write the action in termsof the bosonic field and integrating out the fermionicGrassmann fields, at the Gaussian fluctuation level in thenormal state we obtain the thermodynamic potential, Ω = Ω − (cid:88) q ,ν n ln (cid:2) − Γ − ( q , iν n ) (cid:3) . (4)where Ω = 2 (cid:80) k ln(1 + e − βξ k ) is the non-interactingthermodynamic potential and ξ k = (cid:15) k − µ . The many-body vertex function, Γ ( q , iν n ) , for bosonic Matsubarafrequencies ν n = 2 nπ/β , is given by Γ − ( q , iν n ) = (cid:88) k (cid:20) − f ( ξ k ) − f ( ξ k + q ) iν n − ξ k + q − ξ k + 12 (cid:15) k + ε B (cid:21) (5)where we have defined the Fermi distribution f ( x ) =1 / ( e βx + 1) . The system can be viewed as a non-interacting mixture of fermions and pairs. We can analyt-ically continue the Matsubara summation of the bosonicfrequencies and find the contribution to the pairing fluc-tuations, writing the thermodynamic potential as Ω =Ω + Ω GF , Ω GF = − (cid:88) q + ∞ ˆ −∞ dωπ b ( ω )Im ln (cid:2) − Γ − ( q , ω + i + ) (cid:3) , (6)where b ( ω ) = 1 / ( e βω − . The pressure equation of stateis found from the thermodynamic potential for a giventemperature T and binding energy ε B by Ω = − P V ,where V is the volume of the system, and the densityequation of state is found by satisfying n = − ∂ Ω /∂µ .The dimensionless pressure equation of state is defined, P λ T k B T = f p (cid:18) µk B T (cid:19) (7)where the thermal wavelength is λ T = (cid:112) π (cid:126) / ( M k B T ) ,and pressure equation of state is related to the densitythrough f n = ∂f p /∂ ( βµ ) . This derivation of the thermo-dynamic potential and calculation of the density equationof state is equivalent to taking the T -matrix approxima-tion with bare fermionic Green functions in the trun-cated Schwinger-Dyson equations [52]. For this reason,the above NSR approach is alternatively termed as thenon-self-consistent T -matrix approximation.The NSR equation of state can be considered againstother T -matrix schemes such as the Luttinger-Ward the-ory, which is a fully self-consistent calculation of themany-body Green’s function and we will refer to as GG theory [31, 37, 53]. The NSR method in two-dimensionshas been found to underestimate the density when com-pared to current experimental results and GG theory[31], for the calculation of the hydrodynamic equationsthe NSR method is advantageous over other T -matrixschemes as it allows for the calculation of the thermody-namic properties and collective modes at a fixed βµ and βε B . B. Virial expansion
In the high temperature and weakly interacting limitswe calculate the equation of state through the virial ex-pansion, where the thermodynamic potential is expandedin powers of the fugacity, z = e βµ [43]. This allows for anexact calculation of the thermodynamic properties andthe breathing mode. It is straight forward to calculatethe high temperature regime through the virial expan-sion, here we present some details of the calculation ofthe virial coefficients and densities and further readingcan be found in Refs. [36, 43, 54, 55].Let us consider the virial expansion up to third order,the pressure and density equations of state are f p = ˆ ∞ dt ln (cid:2) ze − t (cid:3) + ∆ b z + ∆ b z + . . . ,f n = ∂f p ∂βµ = ln [1 + z ] + 2∆ b z + 3∆ b z + . . . , (8)where ∆ b and ∆ b are the second and third virial coeffi-cients respectively. It is important to note that the virialcoefficients are functions of the dimensionless binding en-ergy, βε B , only. We calculate the second order virial co-efficient through the relation [36], ∆ b = e βε B − ˆ ∞ dpp e p π + ln [ p / ( βε B )] , (9) and the third order coefficient, ∆ b , has been tabulatedas a function of βε B in the range βε B = (0 . , . [12]. The third order expansion of the pressure and den-sity equations of state are divergent when z > , howeverthrough a Páde expansion we may overcome this diver-gence and expand the regime of the equation of state [56].Specifically the third order reduced pressure through thePáde expansion is, PP = 1 + (cid:16) b (1)2 + ∆ b − ∆ b / ∆ b (cid:17) e βµ + · · · (cid:16) b (1)2 − ∆ b / ∆ b (cid:17) e βµ + · · · (10)where b (1)2 = − / is the second virial coefficient of anideal 2D Fermi gas and P = − πλ − T Li ( − e βµ ) is theideal pressure with Li ( x ) being the polylogarithm. Al-though the expansion can be found for any value of thefugacity, z , and the pressure equation of state no longerdiverges, there is no a priori reason for the expansion tobe physically correct. We note that the determinationof higher order terms such as e βµ requires knowledge ofthe fourth and fifth virial coefficients.The virial expansion is valid in the BEC limit where ε B becomes large and the chemical potential approaches µ → − ε B / . In this limit the virial expansion would ap-pear to be valid in the limit T → , however the dimercontribution to the virial coefficients dominates and theexpansion of the thermodynamic potential in this regimeshould be in terms of a shifted fugacity, z (Bose) = e βε B / z ,the criterion for the validity of the expansion then be-comes µ < − ε B / . In this expansion the virial coeffi-cients correspond to b (Bose) j = e − jβε B / b j . An expansionof the Bose thermodynamic potential, taking the effectivedimer-dimer scattering length a dd (cid:39) . a D [49, 57], isalso dominated by the large binding energy and has thesame behavior in the BEC regime. C. The pressure equation of state
The pressure equation of state plays a key role in de-termining the collective modes of a strongly interactingFermi gas in the hydrodynamic regime [28]. To illustratethe structure of the equation of state before we calcu-late the collective modes we plot the reduced pressureequation of state,
P/P , in Fig. 1. We show the pres-sure equation of state for the NSR (red solid) and GG (blue dotted) T -matrix theories, the second (purple long-dashed) and third (black short dashed) virial expansions,and the Páde expansion (green dot dashed) of the thirdorder virial for interaction strengths of βε B = 0 . and βε B = 1 . , as a function of the reduced chemical po-tential, βµ . As has been discussed in Refs. [37, 51] theequation of state exhibits a non-trivial maximum as thetemperature reduces ( βµ → ∞ ), and in Fig. 1 we see thatthe pressure equations of state for the NSR and GG the-ories begin to reduce at low temperatures. The pressureis underestimated by the NSR when compared to the GG Figure 1. (color online). The pressure equation of state, inunits of the pressure of an ideal Fermi gas P , as a function ofthe chemical potential at interaction strengths βε B = 0 . and βε B = 1 . for the NSR (red solid) and GG (blue dotted) T -matrix theories, second (purple long-dashed) and third (blackdashed) order virial expansions, and the Páde expansion ofthe third order virial (green dot-dashed). and virial expansions. The virial expansions do not havethe non-trivial dependence on the temperature, and areincorrect in the low temperature regime ( βµ → ∞ ) wherethey overestimate the pressure.We see in Fig. 1 that the NSR theory breaks down athigher temperatures compared to the GG theory, mak-ing the range of temperatures in the calculation of thebreathing mode smaller. It is well known that two-dimensional T -matrix theories cannot predict a transi-tion to superfluidity due to the breakdown of long-rangeorder at any finite temperature, however the NSR theorystill suffers from a loss of accuracy as the temperature isreduced and the chemical potential approaches half thebinding energy (for a detailed analysis see Ref. [1, 38]).The GG theory is more robust and the equation of statecan be calculated to lower temperatures, however wefocus on calculating the collective modes of the two-dimensional gas through the NSR theory as we can cal-culate the numerical derivatives with respect to βε B and βµ . The equation of state for the GG theory is found fora fixed βε B and we iteratively find a chemical potentialthat satisfies the corresponding number equation, mak- ing the calculation of thermodynamic properties heavilyreliant on numerical interpolation. We expect from pre-vious studies that the breathing mode will not be toosensitive to the equation of state [58, 59]. Therefore, theuse of the equations of state from different theories willnot greatly affect the calculation of the breathing modefrequency and the qualitative behavior of the breathingmode could be captured. III. HYDRODYNAMIC FRAMEWORK
We follow the work of [27, 28] to calculate the finitetemperature collective modes of a two-dimensional Fermigas trapped in a harmonic potential V tr = M ω r .From the equation of state of the homogeneous 2D Fermigas and the local density approximation, the collectivemodes of a Fermi gas can be found for frequency ω andtemperature T by minimizing a variational action, whichin terms of the normal state displacement fields u n ( r ) takes the form [29], S (2) = 12 ˆ d r (cid:20) ω ρ u n ( r ) − ρ (cid:18) ∂P∂ρ (cid:19) ¯ s ( δρ ) − ρ (cid:18) ∂T∂ρ (cid:19) ¯ s δρδ ¯ s − ρ (cid:18) ∂T∂ ¯ s (cid:19) ¯ s ( δ ¯ s ) (cid:21) . (11)We define the total mass density at equilibrium, ρ ( r ) ≡ M n ( r ) = ρ , the local pressure P ( r ) = P , the entropyper unit mass ¯ s ( r ) ≡ s/ρ = ¯ s , and δρ ( r ) = −∇ · ( ρ u n ) and δ ¯ s ( r ) = − u n · ∇ ¯ s are the density and entropy fluc-tuations, respectively. Taking the variation of the actionwith respect to u n we arrive at ω ρ u n + ∇ (cid:20) ρ (cid:18) ∂P∂ρ (cid:19) ¯ s ( ∇ · u n ) (cid:21) + ∇ · ( ρ u n ) ∇ V tr M + ∇ [ ∇ P · u n ] = 0 , (12)which is Euler’s equation as first derived in Re. [26]. Foran untrapped gas the solution of Eq. (12) is a wave vector q , with dispersion ω = c n q and c n is the speed of sound, c n = (cid:115) m (cid:18) ∂P∂n (cid:19) ¯ s , (13)where n (cid:18) ∂P∂n (cid:19) ¯ s = n (cid:18) ∂P∂n (cid:19) s + s (cid:18) ∂P∂s (cid:19) n , = n ∂ ( P, s ) /∂ ( µ, T ) ∂ ( n, s ) /∂ ( µ, T ) + s ∂ ( P, n ) /∂ ( µ, T ) ∂ ( s, n ) /∂ ( µ, T ) , = n ( P µ s T − P T s µ ) − s ( P µ n T − P T n µ )( n µ s T − n T s µ ) . (14)For convenience we use the shorthand notation P µ ≡ ( ∂P/∂µ ) T , s T ≡ ( ∂s/∂T ) µ , etc, and in dimensionlessform we obtain (cid:18) ∂P∂n (cid:19) ¯ s = k B T ( ˜ P µ ˜ s T − ˜ P T ˜ s µ ) − ˜¯ s ( ˜ P µ ˜ n T − ˜ P T ˜ n µ )(˜ n µ ˜ s T − ˜ n T ˜ s µ ) . (15)All of the thermodynamic quantities can then be foundfrom the pressure equation of state by taking partialderivatives with respect to the reduced chemical poten-tial, βµ and interaction strength, βε B .We calculate the thermodynamic potential of a har-monically trapped gas through the local density approx-imation, µ ( r ) = µ − V tr ( r ) , and write the local pressureand number density using the universal functions, P ( r ) = k B Tλ f p (cid:20) µ ( r ) k B T (cid:21) , n ( r ) = 1 λ f n (cid:20) µ ( r ) k B T (cid:21) , (16)where the local chemical potential is µ ( r ) and the Fermitemperature is defined as, k B T F = (cid:126) (2 N ω ) / . (17)Using the number equation N = ´ d r n ( r ) we relate thereduced chemical potential βµ to the reduced tempera-ture in the trap T /T F by, TT F = (cid:20) ˆ ∞ dyf n ( βµ − y ) (cid:21) − / . (18)For a given trap temperature we need the dimensionlesschemical potential, βµ c , which satisfies Eq. (18) (as tem-perature reduces βµ c → ∞ ). To minimize the action inEq. (11) we expand the displacement field in a variational(polynomial) basis by using the following ansatz, u n ( r ) = N ⊥ (cid:88) n =0 A n r n +1 , (19)where we can increase the precision of the variationalcalculation by increasing the number of basis functions, N ⊥ . Expanding the action we get, S (2) n = 12 (cid:2) A ∗ , · · · , A ∗ N ⊥ (cid:3) A ( ω ) [ A , · · · , A N ⊥ ] T , (20)where A ( ω ) is a N ⊥ × N ⊥ matrix, A nm ( ω ) = ω M nm − K nm . (21)Here we have defined the weighted mass moments, M = M nm , and spring constants, K = K nm given in AppendixA. To find the mode frequencies we need to solve thematrix equation, A (˜ ω ) x = 0 , (22)where the vector of displacement fields are given by, x = [ A , . . . , A n , . . . ] T , and we write the matrix A (˜ ω ) = M ˜ ω − K in terms of the dimensionless frequency ˜ ω = ω/ω . The details of this calculation are given in Ap-pendix B. The breathing mode is the lowest frequencyfound from solving Eq. (22). The variational result con-verges quickly, where the n = 0 giving the most signifi-cant contribution. ��� ��� ��� ��� ��� ��� ��� - ���� - ���������������� Figure 2. (color online). The frequency shift of the breathingmode from the scale invariant value, δω = ω − ω , at tem-perature T /T F = 0 . as a function of interaction strength ln ( k F a D ) for the second order virial (red solid), dilute limitsecond order virial (blue dotted) [25], third order virial (blackdashed), Páde virial expansion (green dot-dashed), and ex-perimental results from Ref. [19] (symbols). IV. RESULTS AND DISCUSSION
We now turn to examining the breathing mode at finitetemperature. We first consider the results obtained byusing the virial expansion approach, contrasted to theexperimental data of Ref. [19] and the previous virialresults of Ref. [25], and then compare the predictionsfrom the different T -matrix theories. We finally check thebreathing mode in the high temperature regime, wherethe virial expansion is more reliable and we examine therole of pairing at high temperature. A. Virial comparison
In Fig. 2 we compare the frequency shift of the breath-ing mode from the scale invariant value, δω = ω − ω ,as a function of interaction strength ln ( k F a D ) on theBCS side for the virial expansion at a temperature of T /T F = 0 . . The use of virial expansion at this (low)temperature could be questionable. However, we presentthe comparison for the purpose of making connect to theexperimental measurement [19] and also making contactwith the previous virial expansion study [25]. We com-pute the breathing mode using the second order viral (redsolid), third order virial (black dashed), Páde expansionof the third order virial (green-dot dashed), a dilute ex-pansion of the second order virial as found in Ref. [25](blue dotted), and the experimental results of Ref. [19](symbols).The breathing mode found in Ref [25] is calculatedfrom a second order expansion of the equation of state,however the authors further approximate the trap den-sity and the speed of sound in the dilute limit, computing ��� ��� ��� ��� ��� ��� ��� - ���� - ���� - �������������������� Figure 3. (color online). The frequency shift of the breathingmode from the scale invariant value, δω = ω − ω , at tem-perature T /T F = 0 . as a function of interaction strength ln ( k F a D ) for the NSR (red solid) and GG (blue dotted) T -matrix theories, second order virial (black dashed), and ex-perimental results from Ref. [19] (symbols). the breathing mode to be ω ω = 1 − T ε B T ∂ ∆ b ∂ ( βε B ) , (23)where ∆ b ( βε B ) is given by Eq. (9). We see that oursecond order virial calculation extends to stronger inter-action strengths before lowering, however we emphasizethat at this temperature the virial expansion is not ac-curate and the results should be treated as qualitativeonly. We see that at the reduced trap temperature of T /T F = 0 . over the interaction regime the virial expan-sions of second order, third order, and Páde expansion arenot the same. This can be understood by looking at thepressure equation of state in Fig. 1, the critical chemicalpotential βµ c needed to determine the trap temperaturein Eq. (18) is in the regime where the second and thirdexpansions differ significantly, and where the third orderis diverging. Although we can calculate the thermody-namic properties for the speed of sound within the virialexpansion, for this temperature and interaction regime,the breathing mode results are only qualitative. B. T -matrix results at low temperature The T -matrix theories take into account the many-body effects and pairing fluctuations, extending the equa-tion of state found through the virial expansions to lowertemperatures. In Fig. 3 we compare the frequency shiftsof the breathing mode δω = ω − ω at a reduced tem-perature of T /T F = 0 . obtained by the NSR (redsolid) and self-consistent GG T -matrix (blue dotted) the-ories, second order virial expansion (black dashed), andalso compare them to the experimental work of Ref. [19](symbols). For each of the frequency shifts there is amaximum, and as the interaction becomes stronger the ��� ��� ��� ��� ��� ��� ��� - ���� - ���� - �������������������� Figure 4. (color online). The frequency shift of the breathingmode, δω = ω − ω , as a function of interaction strength ln ( k F a D ) for the NSR T -matrix at temperature T /T F = 0 . (red solid), T /T F = 0 . (blue dotted), T /T F = 0 . (blackdashed), and T /T F = 0 . (green dot-dashed). frequency shift reduces towards δω = 0 , as is seen inthe experimental results of Ref. [19]. The difference be-tween the NSR and GG theories could be due to the factthat the NSR approach underestimates the pressure. Al-though the second order virial expansion is not reliablein this regime, the qualitative behavior is similar to theNSR and GG theories.We see in Fig. 3 that the NSR breathing mode breaksdown for an interaction of ln ( k F a D ) (cid:39) . where thecritical chemical potential needed for the reduced traptemperature is too large within the T -matrix theory. The GG frequency shift breaks down at ln ( k F a D ) (cid:39) . , andis different to the breakdown of the NSR theory. Forstronger interactions the numerical noise in the calcula-tion of thermodynamic properties is too large to accu-rately determine the speed of sound.In Fig. 4 we consider how the frequency shift of thebreathing mode behaves as a function of temperature us-ing the NSR theory, specifically for T /T F = 0 . (redsolid), T /T F = 0 . (blue dotted), T /T F = 0 . (blackdashed), and T /T F = 0 . (green dot-dashed). As tem-perature decreases the frequency shift increases, indicat-ing that as temperature is reduced the breathing modeanomaly will be larger. We find that the range of validityof the NSR calculation reduces with decreasing temper-ature and the NSR theory also breaks down at highertemperatures as we increase the binding energy. The fre-quency shift at all temperatures considered has a maxi-mum value that lowers as interaction strength increases,a result consistent in all of the theories and temperaturesconsidered in this work. Therefore, we believe that thiswill be qualitatively true for further experimental checksat finite temperature. However, for temperatures below T /T F = 0 . it is not clear if the frequency shift will loweror be positive over the whole BEC-BCS crossover. Figure 5. (color online). The frequency shift of the breathingmode, δω = ω − ω , at temperature T /T F = 0 . as a functionof interaction strength ln ( k F a D ) for the NSR T -matrix (redsolid), second order virial expansion (blue dotted), and thirdorder Páde virial expansion (black dashed), for (a) the BCSregime and (b) the BEC-BCS crossover. C. High temperature limit
For the reduced trap temperature of
T /T F = 0 . inFig. 4 the interaction range of validity is increased andwe see that the frequency shift becomes negative beforethe NSR results break down. This increased range ofvalidity allows us to examine the breathing mode and theeffects of pairing at temperatures far above the criticaltemperature. In two dimensions the role of pairing inthe high temperature regime is a widely discussed areaof research [10, 60].The NSR and virial expansions are valid for a widerange of interactions at high temperatures and we cancover the BEC-BCS crossover, where we also assume thatthe hydrodynamic equations are still valid [61]. In Fig. 5we calculate the breathing mode shift for a temperatureof T /T F = 0 . , with the NSR theory (red solid), sec-ond order virial (blue dotted), and Páde expansion (blackdashed). Figure 5(a) shows that in the BCS limit the fre-quency shift is approaching the scale invariant result of δω = 0 . As we approach the strongly interacting regime,the frequency shift has a maximum value and then tendsnegative, and the breathing mode anomaly seems to be weakened at first glance. However, expanding the in-teraction range and looking at Fig. 5(b) we see that theNSR, virial, and Páde expansion predict a significant neg-ative shift in the strongly interacting regime. The NSRtheory finds a breathing mode shift of and the sec-ond order virial expansion . Going deeper to theBEC side the NSR and Páde theories break down for ln( k F a D ) (cid:39) − . , and the second order virial expan-sion finds that the breathing mode approaches the scaleinvariant value of ω by ln( k F a D ) (cid:39) − .We would like to argue that the significant down-shift of the breathing mode in the strongly interactingregime is due to pairing effects. On the BEC side of thestrongly interacting regime (i.e., − < ln( k F a D ) < with βµ < ), the two-body pairing is captured by thesecond order virial expansion or the Páde expansion.Here, it is known that pairing above T c is without a Fermisurface [5], and therefore is not associated with the pseu-dogap. In contrast, on the BCS side of the strongly inter-acting regime, the NSR theory predicts a chemical poten-tial of βµ (cid:39) , and pair formation is a many-body effect.Any possible experimental observations of the predicteddown-shift of the breathing mode frequency are there-fore of important for understanding the role of pairing intwo-dimensional Fermi systems. V. CONCLUSION
In summary we have investigated the behavior of thebreathing mode at finite temperature for a strongly in-teracting 2D Fermi gas at the BEC-BCS crossover. Us-ing the equation of state found from the NSR and self-consistent T -matrix theories as well as the virial expan-sion at different orders for a homogenous Fermi gas, wehave predicted the breathing mode at finite temperatureof a trapped gas through the local density approximationand a variational approach to the hydrodynamic Eulerequation. The use of different theories for a stronglyinteracting Fermi gas enable us to paint a broad andqualitative picture of the breathing mode at finite tem-perature. Both T -matrix theories and virial expansionsconsistently show the sensitivity of the quantum anomalyon the temperature and interaction strength, that is, thefrequency shift of the breathing mode is sensitively de-pendent on temperature and interaction.On the BCS side, we have predicted that the breath-ing mode frequency reduces towards the scale invariantvalue of ω as temperature increases and, the quan-tum anomaly is more prominent at low temperatures, asone may anticipate. At the typical interaction strength ln( k F a D ) (cid:39) , the frequency shift predicted by the NSRapproach is at the level of 1% for temperature up to . T F . Considering the high experimental resolution forfrequency measurements with cold-atoms, which is about0.1% [62], this shift is significant enough to be resolvedin future experiments.In the strongly interacting regime, we have confirmedthat a significant negative frequency shift at high tem-peratures, peaking near ln ( k F a D ) (cid:39) − . , as predictedby both NSR and virial expansion theories. This may bedue to the strong pairing effects included in the calcu-lations of the homogeneous equation of state. We notethat, a down-shift of the breathing mode frequency, be-low the scale invariant value, was also predicted by Chafinand Schäfer using the second-order virial expansion at T = 0 . T F [25]. Our more systematic virial expansionstudies, beyond the second order, unambiguously con-firm their finding and establish the qualitative behaviorof the breathing mode frequency at high temperature atthe whole BEC-BCS crossover. ACKNOWLEDGMENTS
We would like to thank T. Peppler, P. Dyke, and C.Vale for their useful discussions. This research was sup-ported under Australian Research Council’s DiscoveryProjects funding scheme (project numbers DP140100637,DP140103231, and DP170104008) and Future Fellow-ships funding scheme (project numbers FT130100815 andFT140100003).
Appendix A: Variational approach
Here we consider in more detail the expressions forthe weighted mass moments, M nm , and spring constants, K nm . We present a detailed derivation of the weightedmass moments as this will be instructive to the choice ofunits. The weighted mass moments, M nm , arise from thefollowing action term ω ˆ d r ρ u n = A n A m ω ˆ d r ρ ( r )˜ r n + m +2 , (A1)where we have denoted ˜ r ≡ r/R TF and R =2 k B T F / ( mω ) is the Thomas-Fermi temperature for azero-temperature noninteracting Fermi gas. Recallingthat within the local density approximation, we have, ρ ( r ) = M n ( r ) = M λ − T f n [( µ − V tr ( r )) /k B T ] , thus ˆ d r ρ ( r )˜ r k = R Mλ T ˆ d ˜ r f n (cid:18) βµ − M ω r k B T (cid:19) ˜ r k , = R M πλ T ˆ d ˜ rf n (cid:18) βµ − ˜ r T /T F (cid:19) ˜ r k +1 . (A2)There is a constant here that will set to 1 as it appears inall of the equations. As a result of the above calculationwe can tie the reduced temperature T /T F of the systemto a given µ/k B T by using the number equation for N atoms, M N = ´ d r ρ ( r ) , that is N λ T = ˆ d r f n (cid:18) βµ − M ω r k B T (cid:19) ,N λ T R π = ˆ d ˜ r ˜ rf n (cid:18) βµ − ˜ r T /T F (cid:19) . (A3)Using the fact that λ T = 2 π (cid:126) / ( M k B T ) and k B T F = (cid:126) (2 N ω ) / , we have T F T = ˆ d ˜ r ˜ rf n (cid:18) βµ − ˜ r T /T F (cid:19) , (A4)changing coordinates to y = ˜ r / ( T /T F ) gives in total, TT F = (cid:20) ˆ ∞ dyf n ( βµ − y ) (cid:21) − / (A5)This definition allows us to find a reduced temperaturefor the trapped system for a given βµ in the homogeneoussystem.The spring constant ( ∇ ρ · u n ) (cid:18) ∇ V tr M · u n (cid:19) = ∂ρ ∂r A n ˜ r n +1 rA m ω ⊥ ˜ r m +1 rρ ( ∇ · u n ) (cid:18) ∇ V tr M · u n (cid:19) = ρ n + 2 R TF ˜ r n A n ω ⊥ r ˜ r m +1 A m ρ (cid:18) ∇ V tr M · u n (cid:19) ( ∇ · u n ) = ρ A n ω r ˜ r n +1 A n m + 2 R TF ˜ r m c n ( ∇ · u n ) = c n n + 2 R TF m + 2 R TF r n r m A n A m , (A6)The spring constants cancel leaving only Eq. (A6) tocalculate, K nm =( n + 2)( m + 2) M ω k B T F ˆ d r (cid:20) n (cid:18) ∂P∂n (cid:19) ¯ s (cid:21) ˜ r n + m , =( n + 2)( m + 2) M ω R F k B T F λ T k B T × ˆ d r f n (cid:18) βµ − M ω r k B T (cid:19) (cid:18) ∂P∂n (cid:19) ¯ s ˜ r n + m , =( n + 2)( m + 2) M ω R F πλ T TT F × ˆ ∞ drf n (cid:18) βµ − r T /T F (cid:19) (cid:18) ∂P∂n (cid:19) ¯ s ˜ r n + m +1 . (A7) Appendix B: Solving for det A (˜ ω ) = 0 In order to solve the matrix equation A (˜ ω ) x =0 , where the vector of displacement fields is, x =[ A , . . . , A n , . . . ] T , we expand the matrix A (˜ ω ) = M ˜ ω − K to find the eigenvalues. Here, M and K are the matri-ces of the reduced weighted mass moments and the springconstants, respectively. The matrix M can be written asa product of a lower triangular matrix L , its conjugatetranspose, and a diagonal matrix D , M = L · D · L T . 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