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Non-perturbative predictions for cold atom Bose gases with tunable interactions

We derive a theoretical description for dilute Bose gases as a loop expansion in terms of composite-field propagators by rewriting the Lagrangian in terms of auxiliary fields related to the normal and anomalous densities. We demonstrate that already in leading order this non-perturbative approach describes a large interval of coupling-constant values, satisfies Goldstone's theorem, yields a Bose-Einstein transition that is second-order, and is consistent with the critical temperature predicted in the weak-coupling limit by the next-to-leading order large-N expansion.

Nearly a century after the first observation of the lambda transition in liquid helium [1], a quantitative, first-principles description of strongly-correlated bosons remains a challenge. After the transition was recognized as the onset of superfluidity [2], the connection with Bose-Einstein condensation (BEC) was proposed [3], but it was Bogoliubov's work [4] pointing out that the dispersion of the elementary BEC excitations satisfy the Landau criterion for superfluidity [5] that motivated weaklyinteracting BEC studies to investigate superfluid properties. In weakly-interacting systems, the many-body properties do not depend on the shape of the interaction potential, but only on the s-wave scattering length, a 0 , and the boson fluid acts as point-like interacting particles [6].
Unlike liquid helium, cold atoms remain point-like even when the scattering length is tuned near a Feshbach resonance. Then, strongly-correlated cold atom bosons offer the exciting prospect of studying point-like strongly interacting bosons, possibly in the universal regime where the scattering length greatly exceeds the inter-particle distance and the latter becomes the only relevant length scale [7]. This hope appeared thwarted when it was shown that the three-body loss rate in cold atom traps scales as a 4 0 near a Feshbach resonance [8]. In accordance, the universal regime was reached only in ultra-cold fermionic gases [9], where the three-body loss is reduced by virtue of the Pauli exclusion principle. However, the recent observation that three-body losses are strongly suppressed in optical lattices when the average number of bosons per site is two or less [10], rekindles the prospect of studying medium and strongly-correlated cold atom bosons. Novel cold-atom trap technologies that produce stable, flat potentials bound by a sharp edge [11], suggest the study of finite-temperature properties such as the BEC transition temperature T c and the superfluid to normal fluid ratio and depletion, at fixed density, ρ.
At finite temperature, the description of BEC's remains a challenge even in the weakly-interacting regime. Standard approximations such as the Hartree-Fock-Bogoliubov and the Popov schemes, generally fall within the Hohenberg and Martin classification [12] of conserv-ing and gapless approximations, which implies that they either violate Goldstone's theorem or general conservation laws [13]. These approximations generally predict the BEC transition to be a first-order transition, whereas we expect the transition to be second order [14].
In this paper, we present a new theoretical framework that describes a large interval of ρ 1/3 a 0 -values, satisfies Goldstone's theorem and yields a Bose-Einstein transition that is second-order, while also predicting reasonable values for the depletion. Furthermore, this framework can predict all experimentally relevant quantities within the same calculation, determining fully consistently quantities such as T c , the collective mode frequencies [15] and the compressibility (which characterizes the density profile in a shallow trap [16]). In contrast with other resummation schemes, such as the large-N expansion [17] or functional renormalization techiques [18], here we treat the normal and anomalous densities on equal footing.
In our approach, we generate a one-parameter family of equivalent Lagrangians. We choose this parameter to reproduce the one-loop result at mean-field level in the weakly-interacting limit. Thus, we identify the optimal auxiliary-field Lagrangian for the purpose of a systematic non-perturbative expansion. Then, the critical temperature variation in leading order is the same as the one found in the next-to-leading order large-N expansion.
In dilute bosonic gas systems, the classical action is given by S[ φ, φ * ] = dx L[ φ, φ * ], with dx ≡ dt d 3 x and the Lagrangian density Here, µ is the chemical potential and the coupling is λ = 4π 2 a 0 /m. To account for the contributions of the normal and anomalous densities, we use the Hubbard-Stratonovitch transformation [19] to introduce the real and complex auxiliary fields (AF), χ(x) and A(x). We arXiv:1011.1886v1 [cond-mat.quant-gas] 8 Nov 2010 add to Eq. (1) the AF Lagrangian density [20,21] where θ is the mixing parameter between the normal and anomalous densities, χ(x) and A(x). The usual large-N approximation [21] is obtained when θ = 0. Then, the action becomes Here, we introduced a two-component notation with . Performing the path integral over the fields φ a , we obtain the effective action The small parameter allows us to perform the remaining path integral over χ i using the stationary-phase approximation. As shown in Ref. 20, counts loops in the AF propagator in analogy with , and provides the loop expansion of the effective action in terms of χ propagators. Next, we expand the effective action about the stationary points, χ i 0 (x), defined by δS eff [χ; j]/δχ i (x) = 0. Hence, we obtain where we introduced the notations We emphasize that both χ 0 and A 0 include self-consistent fluctuations. Expanding the effective action about the stationary point, we write where D −1 ij (x, x ) is given by the second-order derivatives, evaluated at the stationary points. By keeping the gaussian fluctuations and Legendre transforming, the oneparticle irreducible (1-PI) graphs generating functional is the negative of the classical action plus self-consistent one-loop corrections in the φ a and χ i propagators.
To leading order in the AF loop expansion (LOAF), one sets = 0 in the right-hand-side of (6). The static part of the effective action per unit volume is Translating (7) to the imaginary time formalism, we find where ω 2 k = ( k + χ cosh θ − µ) 2 − |A| 2 sinh 2 θ and k = k 2 /(2m). At the minimum, we have Using the U (1) gauge symmetry, we choose φ 0 to be real. Then, A is real and the dispersion, ω 2 k = k ( k + 2A sinh θ), represents the Goldstone theorem. Next, we set sinh θ = 1, such that ω k reduces to the Bogoliubov dispersion, ω k = k ( k + 2λ φ 2 0 ), in the limit of vanishing quantum fluctuations in the anomalous density. We (Color online) Normal density, χ , and anomalous density, A, from the LOAF and PA approximations, for ρ 1/3 a0 = 1. Tc and T indicate vanishing condensate density, ρ0, and anomalous density, A, respectively. PA leads to a first-order phase transition, whereas LOAF predicts a second-order phase transition. We have that Tc = T in the PA, but not in LOAF. In LOAF χ and A are equal until Tc.
note that the leading-order (LO) in the large-N expansion corresponds to θ = 0. This leads to the noninteracting (NI) dispersion, ω k = k , and we conclude that the large-N expansion is not a suitable starting point, because it is incompatible with the Bogoliubov spectrum.
We compare the LOAF results with the predictions of the Popov bosonic approximation (PA) [23]. PA is generally recognized as an accurate theoretical description of experimental data in weakly-coupled dilute trapped Bose gases [24], as long as the densities of the condensed and noncondensed atoms are comparable with each other. Unfortunately, PA produces an artificial first-order phase transition at T c . Formally, PA is obtained from Eq. (9) by setting A 0 = χ 0 = λρ 0 and neglecting the quantum fluctuations in the anomalous density. With this substitution, the PA dispersion relation reads ω 2 k = k ( k +2λρ 0 ). In Fig. 1 we depict the temperature dependence of the normal density χ , and anomalous density, A, at constant ρ 1/3 a 0 , as derived using the LOAF and PA approximations. For illustrative purposes, we set ρ 1/3 a 0 = 1 and the temperature is scaled by its NI critical value, T 0 = (2π 2 /m)[ρ/ζ(3/2)] 2/3 , where ζ(x) is the Riemann zeta function. We identify two special temperatures, at T c where the condensate density vanishes, and at T where the anomalous density, A, vanishes. These temperatures are the same in the PA formalism, but they are different in LOAF. The existence of a temperature range, T c < T < T , for which the anomalous density, A, is nonzero despite a zero condensate fraction, φ, is a fundamental prediction of LOAF. In this temperature range, the dispersion relation is expected to depart from the quadratic form predicted by the Popov approximation for T > T c . Above T c the solution of the PA equations becomes multivalued, indicating that the system undergoes a first-order phase transition at T c . In contrast, LOAF predicts a second-order transition.
The temperature dependence of the condensate fraction, ρ 0 /ρ, is depicted in Fig. 2 for two constant values of the dimensionless parameter ρ 1/3 a 0 , together with the NI result, ρ 0 /ρ = 1 − (T /T 0 ) 3/2 . Again, we observe that LOAF exhibits the correct second-order BEC phase transition behavior. Moreover, PA does not change T c relative to the NI case, because in the PA case we have T c = T and the PA and NI dispersion relations are the same at T c . The LOAF approximation predicts an increase of T c compared with the NI case.
As illustrated in Fig. 2, the LOAF and PA predictions may differ greatly even for temperatures, T T c . These differences are enhanced by a strengthening of the interaction between particles in the Bose gas (a larger value of ρ 1/3 a 0 indicates stronger coupling). The leading-order AF formalism produces a more realistic set of observables away from the weak-coupling limit because of its non-perturbative character. In contrast, PA is appropriate only in the case of a weakly-interacting gas of bosons. The former is made explicit by studying the LOAF prediction for the relative change in T c with respect to T 0 , as a function of ρ 1/3 a 0 . The inset in Fig. 3 demonstrates that in the weak-coupling regime, ρ 1/3 a 0 1, LOAF produces the same slope of the linear departure derived by Baym et al. [17] using the large-N expansion, but at next-to-leading order. The LOAF corrections to the critical temperature are due to the inclusion of selfconsistent fluctuations effects in the mean-field χ and A densities. A summary of ∆T c /T 0 theoretical predictions is found in Ref.14. For ρ 1/3 a 0 1, LOAF predicts that ∆T c /T 0 → 0.396 when the system approaches the unitarity limit. Despite that most current experiments probe only the ρ 1/3 a 0 1 regime, future experiments [11] may access the medium-to-strongly interacting regime, and verify this non-perturbative prediction.
One can systematically improve upon the LOAF approximation by calculating the 1-PI action order-byorder in . The broken U (1) symmetry Ward identities guarantee Goldstone's theorem order by order in [20]. For time-dependent problems, however, this expansion is secular [25], and a further resummation is required. The latter is performed using the two-particle irreducible (2-PI) formalism [26]. A practical implementation of this approach is the bare-vertex approximation (BVA) [27]. The BVA is an energy-momentum and particle-number conserving truncation of the Schwinger-Dyson infinite hierarchy of equations obtained by ignoring the derivatives of the self-energy, similarly to the Migdal's theorem [28] approach in condensed matter physics. The BVA proved effective in the case of classical and quantum λφ 4 field theory problems [29] and can be applied to the BEC case.
To summarize, in this paper we introduce a new nonperturbative resummation formulation for the BEC problem. At mean-field level, this approach meets three important criteria for a satisfactory mean-field theory for weakly-interacting bosons [14]: i) the excitation spectrum is gapless (to preserve Goldstone's theorem), ii) LOAF reduces to the known results from Bogoliubov theory at T = 0 and weak coupling, and iii) predicts a second-order BEC phase transition. The latter suggests that a AF formulation of the Lagrangian for systems of cold fermionic atoms may also impact the study of the BEC to BCS crossover in dilute fermionic atom systems [30].
Work performed in part under the auspices of the U.S. Department of Energy. The authors would like to thank E. Mottola and P.B. Littlewood for useful discussions.