Superfluid transition of homogeneous and trapped two-dimensional Bose gases

doi:10.1073/pnas.0609957104

Journal List > Proc Natl Acad Sci U S A > v.104(5); Jan 30, 2007

Proc Natl Acad Sci U S A. 2007 January 30; 104(5): 1476–1481.

Published online 2007 January 23. doi: 10.1073/pnas.0609957104.

PMCID: PMC1785267

Physics

Superfluid transition of homogeneous and trapped two-dimensional Bose gases

Markus Holzmann,^† Gordon Baym,^‡^§ Jean-Paul Blaizot,^¶^‖ and Franck Laloë^††

^†Laboratoire de Physique Théorique de la Matière Condensée, Unité Mixte de Recherche, Centre National de la Recherche Scientifique 7600, Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France;

^‡Department of Physics, University of Illinois at Urbana–Champaign, 1110 West Green Street, Urbana, IL 61801;

^¶European Centre for Theoretical Studies in Nuclear Physics and Related Areas, 38050 Villazzano (Trento), Italy;

^‖Service de Physique Théorique, Commissariat à l'Energie Atomique–Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France; and

^††Laboratoire Kastler Brossel, Ecole Normal Supérieure, 24 rue Lhomond, 75005 Paris, France

^§To whom correspondence should be addressed. E-mail: gbaym/at/uiuc.edu

Contributed by Gordon Baym, November 15, 2006.

Author contributions: M.H., G.B., J.-P.B., and F.L. performed research and wrote the paper.

Received August 29, 2006.

Abstract

Current experiments on atomic gases in highly anisotropic traps present the opportunity to study in detail the low temperature phases of two-dimensional inhomogeneous systems. Although, in an ideal gas, the trapping potential favors Bose–Einstein condensation at finite temperature, interactions tend to destabilize the condensate, leading to a superfluid Kosterlitz–Thouless–Berezinskii phase with a finite superfluid mass density but no long-range order, as in homogeneous fluids. The transition in homogeneous systems is conveniently described in terms of dissociation of topological defects (vortex–antivortex pairs). However, trapped two-dimensional gases are more directly approached by generalizing the microscopic theory of the homogeneous gas. In this paper, we first derive, via a diagrammatic expansion, the scaling structure near the phase transition in a homogeneous system, and then study the effects of a trapping potential in the local density approximation. We find that a weakly interacting trapped gas undergoes a Kosterlitz–Thouless–Berezinskii transition from the normal state at a temperature slightly below the Bose–Einstein transition temperature of the ideal gas. The characteristic finite superfluid mass density of a homogeneous system just below the transition becomes strongly suppressed in a trapped gas.

Keywords: two dimensions, phase transitions, trapped atoms

The ability to produce two-dimensional atomic gases trapped in optical potentials has stimulated considerable interest in the Kosterlitz–Thouless–Berezinskii (KTB) transition in such systems (1 –7); recently, the transition has been observed in a quasi two-dimensional system of trapped rubidium atoms (8, 9). A homogeneous Bose gas in two dimensions undergoes Bose–Einstein condensation (BEC) only at zero temperature, since long wavelength phase fluctuations destroy long range order (10 –13); nonetheless, interparticle interactions drive a phase transition to a superfluid state at finite temperature, as first pointed out by Berezinskii (14, 15) and by Kosterlitz and Thouless (16, 17). The phase transition is characterized by an algebraic decay of the off-diagonal one-body density matrix (or single particle Green's function) in real space below the transition temperature, T_KT. Furthermore, the superfluid mass density, ρ_s, jumps with falling temperature, T, from 0 just above T_KT to a universal value ρ_s = 2m² T_KT/π just below (18), where m is the atomic mass. (We use units ħ = k_B = 1.)

A noninteracting homogeneous Bose gas in two dimensions does not undergo Bose–Einstein condensation at finite temperature. In such a system, the density is given in terms of the chemical potential μ by

A mathematical equation, expression, or formula that is to be displayed as a block (callout) within the narrative flow. The name of referred object is zpq00507-4699-m01.jpg

where β = 1/T, and λ = (2π/mT)^1/2 is the thermal wavelength. As μ approaches zero at fixed temperature, the density grows arbitrarily, implying the absence of condensation. On the other hand, a trapped noninteracting gas does undergo a Bose–Einstein condensation at a finite temperature (19). In the semiclassical limit, the total number of particles is given by N(βμ) = g₂(−βμ)(T/ω)², where g₂(x) = Σ_{n = 1}^∞ e^−nx/n² [similar, equals]

π²/6 − x(1 − log x), for x → 0, and ω is the trapping frequency; thus, as μ approaches zero, N is bounded from above, and condensation occurs at the temperature equation i1

The question of whether an interacting trapped Bose gas when cooled from the normal state first undergoes an ordinary BEC or a KTB transition is not completely settled (20). In this paper, we show that in the thermodynamic limit, in which the trap frequency, ω, vanishes and the total particle number N grows, with the product Nω² and hence the density at the origin, remaining constant, interactions at the mean-field level destroy ordinary BEC; instead, the system undergoes a KTB transition at a temperature slightly below the ideal condensation temperature, T_BEC.

We approach the phase transition of the homogeneous two-dimensional Bose gas by carrying out a scaling analysis similar to the one we used earlier to discuss the phase transition in a dilute three-dimensional homogeneous Bose gas (21 –29). The phase below T_KT is characterized by an algebraic decay in space of the single particle Green's function, G. As in three dimensions, the phase transition occurs when the single-particle spectrum in Fourier space becomes gapless, G⁻¹(0, 0) = 0, where G(k, z_ν) is the Fourier component of G in space and (imaginary) time. Below, we first discuss the scaling structure of G as T_KT is approached from above, and rederive the relation between the temperature and density at the transition

where C is a constant (30), [similar, equals]

(380 ± 3)/2π according to numerical simulation (31 –34). We write the interparticle coupling constant in two dimensions in the form g = 2πα/m, where the parameter α is dimensionless.‡‡

Although the KTB transition was initially discussed in terms of unbinding of vortex pairs via an effective action, we present here an alternative microscopic self-consistent analysis of the transition in Fourier space, which does not explicitly introduce vortices.§§ We derive the principal thermodynamic features directly from the scaling structure of the Green's functions, obtained by summing perturbation theory to all orders. (However, such a diagrammatic analysis can never guarantee prima facie inclusion of all nonperturbative effects.)

We turn then to the structure below T_KT, working in terms of the Green's function in momentum space. For an infinite system below T_KT, we expect G(r − r′, z_ν = 0) ~ 1/|r − r′|^η as |r − r′| → ∞, where η depends on T (and equals 1/4 at T_KT; ref. 16). Thus, G(k, 0) is not well defined in an infinite system; therefore, we start with a finite size system, of characteristic dimension L, to analyze the structure. The condensate density in a finite system is nonvanishing, n₀ ~ G(r ~ L, 0) ~ 1/L^η, and vanishes in the limit L → ∞, because long-range order is prohibited in two dimensions; nonetheless, we show using Josephson's relation between the condensate density, the superfluid mass density, ρ_s, and the infrared behavior of G(k) (M.H. and G.B., unpublished, and refs. 38 and 39),

that ρ_s remains finite in the thermodynamic. Our analysis of the scaling structure is valid for all coupling strengths, and thus, going beyond perturbation theory, is an extension of the work of Popov (40, 41).

In The Transition in Trapped Gas, we apply our results to a trapped gas within the local density approximation. We show that the KTB transition temperature, for a weakly interacting system, lies below the BEC temperature of the ideal trapped gas by terms of order α log² α. Furthermore, the jump of the total superfluid mass at the transition is ~ α, and is thus highly suppressed compared with that in a homogeneous system.

Scaling Structure Above the Transition

Here, we derive the KTB transition of a two-dimensional weakly interacting homogeneous Bose gas by studying, as in ref. 21, the scaling structure of G just above the transition. For wavevector k and complex frequency z, G(k, z) is given in terms of the self-energy, Σ(k, z), by

where μ is the chemical potential. The transition is defined, for fixed density, by the temperature at which the single particle spectrum becomes gapless, and consequently the off-diagonal single particle density matrix decays algebraically; at the transition point, G⁻¹(0, 0) = 0, or μ = Σ(0, 0). In terms of G, the density is

where the z_ν = 2πiνT are the Matsubara frequencies (ν = 0, ±1, ±2, …).

The mean-field contribution, Σ_mf = 2gn, to the self-energy is independent of frequency and momentum, and can be absorbed in a shift of the chemical potential. We introduce the mean-field coherence length, ζ, by

at the transition,

In terms of ζ, the mean-field Green's function is

and the mean-field density is given by

Analyzing the structure of the self-energy in the zero Matsubara frequency sector, using the mean-field propagator above, we obtain the following scaling structure

where σ₀ is dimensionless and is ~α² as α → 0. In contrast to the case in three-dimensions, self-energy diagrams beyond mean-field are ultraviolet convergent even in the zero Matsubara frequency sector.

Using Eqs. 10 and 6, we may write the transition condition, Eq. 7, as

at the phase transition the parameter αζ²/λ² [equivalent]

J approaches a finite value, J*, determined by this equation. The contribution of nonzero Matsubara frequencies in the denominator of Eq. 8 is ~ (αζ²/λ²)/α, and thus in the limit of small α, the contributions of nonzero Matsubara frequencies to the self-energy are of relative order α and higher, and can be neglected.

To calculate the critical density at given temperature we use the mean-field density, Eq. 9, as a reference

In leading order, we can neglect nonzero Matsubara frequencies in the summation on the right, and derive

Using Eq. 10 and the fact that at the transition αζ²/λ² = J*, we see that the integral in Eq. 13 is a constant of order unity, independent of α. Thus, we arrive at the critical density

as in Eq. 2.

The true correlation length, ξ, above T_KT is given by

We rewrite this equation as

where from the previous discussion we know that the right side is a function only of J = αζ²/λ². At the transition, J goes to J*, given by M(J*) = 0. Furthermore, approaching the transition from above, we see from Eq. 13 that ΔJ = J* − J [proportional, variant]

T − T_KT. The dependence of M(J) on ΔJ near J* determines the divergence of the correlation length close to the transition. In three dimensions, ξ ~ |T − T_c|^−ν with a critical exponent ν [similar, equals]

0.67 (42), as the BEC transition temperature T_c is approached from above; therefore, M(J) ~ |ΔJ|^2ν ~ |T − T_c|^2ν in the neighborhood of the fixed point J*. However, in two dimensions, the correlation length diverges as ξ ~ exp[b/(T − T_KT)^1/2] (16), and therefore, equation i2

, where b and c are nonuniversal constants.

Scaling Structure Below the Transition

At the transition temperature, T_TK, the single-particle Green's function decays algebraically in real space. Below T_TK, the scaling structure is most readily analyzed in momentum space, as above; however, this approach is made difficult by the fact that below T_TK the single particle Green's functions continues to decay algebraically in real space sufficiently slowly that its Fourier transform is not absolutely convergent. In order to avoid this problem, and to use the same approach as above, we adopt the strategy of working in a finite size system, of characteristic dimension L, in which the condensate density, n₀, is nonzero. At the very end, we take the limit L → ∞, at fixed density, in which case n₀ goes to zero, as required by the absence of long range order in two dimensions.

In a finite size system, the condensate density is given by n₀ = n − ñ, where

and k₀ ~ π/L; in leading order, G here can be taken to be the infinite size Green's function. Because, for T ≤ T_KT,

where the constants K and η(0 ≤ η ≤ 2) in general depend on T, we find to leading order

The result (Eq. 19) is given for a circular box; in general, it is modified by a numerical factor close to unity, weakly dependent on the geometry.

Below T_KT, the system is superfluid even though the condensate density is not extensive and vanishes in the thermodynamic limit. However, in two dimensions, an algebraically decaying correlation function is sufficient to yield a nonvanishing superfluid density, ρ_s, in the thermodynamic limit, as can be seen from the Josephson relation (M.H. and G.B., unpublished, and refs. 38 and 39) between ρ_s and n₀ in a finite system¶¶

Thus from Eqs. 18 and 19, one finds

Inverting the relation, we obtain η = m² T/2πρ_s; thus, we recover the prediction of the spin-wave approximation for the critical exponent below T_KT. Because ρ_s ≤ mn, we see that

Furthermore, because η ≤ 2, the superfluid mass density can never vanish at finite temperature below T_KT, and thus it must be discontinuous across the phase transition. At T_KT, the value of the universal jump of the superfluid density predicted by Nelson and Kosterlitz (18), ρ_s = 4T_KTm²/2π, leads to η = 1/4.

Let us now discuss the detailed structure of the Green's function below the transition. We basically follow the scaling approach used in ref. 43 in three dimensions. Our strategy is to expand the self-energies formally in powers of α, n₀, and k₀. In the infinite size system, k₀ → 0, the self-energies diverge as n₀ → 0; the point, k₀ → 0, n₀ → 0 is singular. However Josephson's relation constrains the limit n₀ → 0 and k₀ → 0 in terms of ρ_s.

The particle density, n, in the condensed phase is a function of α, n₀, and T, and has the form, n(α, n₀, k₀, T) = n₀ + ñ(α, n₀, T), where ñ(α, n₀, T) is the density of noncondensed particles (with momentum k > k₀). At the transition temperature, ñ(α, 0, T_c) = n_c. We calculate ñ(α, n₀, T) in terms of the matrix Green's function

where the two component field operator is Ψ(rt) = (ψ(rt), ψ^†(rt)). The Fourier components of An external file that holds a picture, illustration, etc., usually as some form of binary object. The name of referred object is Gcurly.jpg

An external file that holds a picture, illustration, etc., usually as some form of binary object. The name of referred object is Gcurly.jpg

⁻¹ have the form,

where ε_k = k²/2m, and the Σ_ij(k, z_n) are the self-energies. The chemical potential, μ, depends on n₀ and k₀, and is specified by the Hugenholtz–Pines relation (44),

The lowest-order mean-field self-energies, Σ₁₁ = Σ₁₁^mf = 2g(n₀ + ñ), Σ₁₂ = Σ₁₂^mf = gn₀, are independent of momenta and Matsubara frequency, and, as above T_KT, we absorb them in a mean-field coherence length, [zeta] ,

Using power-counting, we can derive the scaling structure of the self-energies. As above the transition, we may neglect nonzero Matsubara contributions to leading order. Unlike in three-dimensions, there are no formal ultraviolet divergencies in the expansion beyond those in mean field, and therefore no need for renormalization. The expansion of the self-energies beyond mean field starts at order α² [zeta] ²/λ²; furthermore, Σ₁₂ is formally at least of order n₀. Diagrams of order g^κ with κ ≥ 3 in the formal expansion contain vertices with two Green's functions entering; similar to the structure at T_c, they involve the dimensionless combinations α [zeta] ²/λ² [equivalent] P and n₀λ². The latter part originates from the dependence of An external file that holds a picture, illustration, etc., usually as some form of binary object. The name of referred object is Gcurly.jpg ^mf on 2mΣ₁₂^mf ~ αn₀. Any diagram with an explicit power, p, of n₀ can be generated from a corresponding diagram of power p − 1, in which a line is replaced by equation i3 at each of its ends. Thus, each power of n₀ involves one fewer two-momentum loop to be integrated over. The explicit n₀ dependence enters in two ways. Terms involving G₁₁ − G₁₂ lead to the combination P²n₀λ², which vanishes as n₀ → 0. On the other hand, terms involving the combination G₁₁ + G₁₂, which in mean field diverges as n₀ → 0 in the infrared limit, lead to divergences which are cutoff by k₀ and thus produce an additional (k₀ [zeta] )²/(n₀λ²P) dependence. In the limit n₀ → 0, k₀ → 0, only the dependence on (k₀ [zeta] )²/n₀λ² [equivalent] Q survives.

Then, scaling all momenta k by 1/ [zeta] , we find the following scaling structure for the self-energies in this limit

where the σ_ij are dimensionless functions of dimensionless variables, and we neglect terms proportional to positive powers of n₀ and k₀. In particular, for k → 0,

where s is a dimensionless function. Comparing with Eq. 26, we see that s + 1 = 0, an equation that determines P as a function of Q, so that

where h is a dimensionless function.

We now take the limit n₀ → 0 and k₀ → 0. Eqs. 26 –28 imply that G(k₀, 0) [equivalent] G₁₁(k₀, 0) = −ζ² [ell] (Q), and thus the Josephson relation implies that the superfluid mass density has the structure

where

is a dimensionless function; this equation relates n₀ and k₀ to ρ_s in the limiting process. In this limit P as well as Q are constants dependent only on ρ_s/T. Similarly, the other dimensionless functions depend in this limit only on ρ_s. These forms are valid over the temperature range from T_KT down to T = 0 (with nonzero Matsubara frequency contributions neglected); this entire region is therefore critical.

We can gain further insight into the structure below T_KT by reformulating the analysis in terms of the correlation lengths, ζ_T and ζ_L, that control the infrared behavior of the transverse and longitudinal Green's functions, G_T = G₁₁ − G₁₂, and G_L = G₁₁ + G₁₂. We regularize the infrared divergent structure below T_KT by assuming a finite condensate density, n₀, and finite correlation lengths. In this description, the low-temperature phase of the KTB transition is characterized by the ratio of amplitude (L) and phase (T) fluctuations of the order parameter, even in the absence of long-range order in the thermodynamic limit in which n₀ → 0. In the end, we take the limit, n₀ → 0, ζ_T → ∞, and ζ_L → ∞. In this way, we do not have to introduce an explicit infrared cutoff, k₀, as we had to above. We define

and

in terms of which G_T and G_L are given by

and

The equilibrium state of the system is specified below the transition by ζ_T → ∞, as seen from the Hugenholtz–Pines relation (25), as well as n₀ → 0. From Eqs. 31 and 32, we obtain the ratio

The second term is of order n₀ζ_T². From the previous discussion, we deduce that in the limit n₀λ² → 0, 4mζ_T²Σ₁₂(0, 0) → 8παn₀ζ_T² f(ζ_T/ζ_L), where f is a dimensionless function. Eq. 35 thus implies that the ratio ζ_T/ζ_L is a function only of n₀ζ_T². Therefore, we arrive at a self-consistent solution in the limit n₀ → 0 and ζ_T → ∞ in which n₀ζ_T² remains finite. The value of this parameter is determined in terms of ρ_s by Josephson's relation, Eq. 20, which we can write as

The derivative of the self-energies with respect to k² are finite as k → 0 for any finite ζ_T, but must diverge as ζ_T² in the limit ζ_T → ∞. The right side is a function only of n₀ζ_T² in the limit ζ_T² → ∞ and n₀ → 0, thus defining n₀ζ_T² in terms of ρ_s.

The Transition in a Trapped Gas

We turn now to the behavior of a two-dimensional system trapped in an oscillator potential, of frequency ω. We consider for simplicity only the thermodynamic limit N → ∞, ω → 0, with Nω² constant, where N is the total particle number.

In the absence of interactions, the system undergoes a Bose–Einstein condensation at the critical temperature equation i4 , as mentioned. In the thermodynamic limit, this result can be obtained by a local density approximation by integrating the density profile, n_ideal(r),

where V(r) = mω²r²/2 is the trap potential. Although the density in the center diverges at condensation, the number of excited particles near the center of the trap remains finite. Explicitly, let us assume a BEC with N₀ ~ T/μ ~ N condensed particles. The number of particles in excited states near the center of the trap is N_c ~ n_ideal(r = 0)πd², where d = (mω)^−1/2 is the oscillator length. At the transition temperature, μ ~ T/N and N_c ~ (T_BEC/ω) log 1/|βμ| ~ N^1/2 log N.

Extending this argument to an interacting Bose gas, we can see, even at the mean field level, how interactions destroy simple Bose-Einstein condensation at finite temperature. The density profile calculated in the local density approximation is

where V_eff(r) = mω²r²/2 + 2gn_mf(r). Because n_mf(r) has negative curvature at the origin, interactions tend to reduce the effective trapping frequency. Expanding Eq. 38 to order r² about the origin, we find the self-consistent trap frequency at the origin

As μ → 2gn_mf(0), the limit in which BEC occurs in mean field,

and the potential becomes arbitrarily flat. In the thermodynamic limit, the effective oscillator length d_eff [equivalent]

(mω_eff)^−1/2 is ~ λ (NT/(2gn_mf(0) − μ))^1/4. Just below the point of mean-field condensation, 2gn_mf(0) − μ [equivalent]

1/2mζ(0)² is ~ T/N, and the number of particles in the center, ~ n_mf(0)πd_eff², grows as (T/ω) equation i5

(ζ(0)/λ) log(ζ(0)/λ) ~ N log N. Thus the critical temperature for BEC vanishes as ~ (α^1/2 logN)⁻¹; mean fields destroy finite temperature BEC in a interacting trapped system in the thermodynamic limit.

However, the system does undergo a KTB transition in the thermodynamic limit. Using the mean-field density profile, Eq. 38, we can calculate the transition temperature to leading order in α. The KTB transition occurs when the chemical potential reaches the critical value Σ(0, 0), calculated for a homogeneous system of the same density as in the center of the trap, or equivalently, when the central density reaches the critical value, Eq. 14. As we see from Eqs. 12 and 13 in the homogeneous case, the critical density is given to logarithmic accuracy by the mean-field density evaluated at the critical ζ given by Eq. 7; critical fluctuations produce corrections, which are, however, important only inside the critical region at small distances where 1/2mω²r² [less, similar] |gn(0) − μ|, or

In the thermodynamic limit, the critical region produces corrections to the total number, ΔN ~ Nα² log α, which can be neglected; thus, to leading order in α, we can calculate the transition from density profile, Eq. 38. Expanding n_mf(r) to first order in α, we have,

where 1/2mζ² = 2gn(0) − μ, and

is the ideal gas density. Integration over space yields the correction to the total number of particles at given μ

At the KTB transition, ζ² ~ J*/α. To within logarithmic corrections, we may replace n₀(0) by −λ⁻² log(α), and g₂(−βμ) by g₂(0), so that at given T,

where N_BEC(T) = (π²/6)(T/ω)² relates the particle number and temperature at the ideal Bose gas transition in the trap. Thus, we find that the KTB transition occurs at a temperature of order α log² α below the ideal BEC transition temperature

In a KTB transition in a homogeneous system, the superfluid mass density, ρ_s, jumps discontinuously from zero to 2m²T_KT/π as the temperature drops through T_KT. In a trap, however, the transition first occurs in the center, extending over a region of size r_c, Eq. 41. The total superfluid mass, M_s, is therefore

so that

where M = mN, and J* = αζ²/λ² at T_KT is the solution of Eq. 11. The jump in M_s is thus highly suppressed for α [double less-than sign]

J*, going to zero in the limit of an ideal gas.

A key indicator of superfluid behavior below the transition would be creation of a vortex at the center of the trap, where the system first becomes superfluid, e.g., by cooling a rotating system through T_KT. To create a vortex at T_KT, it is necessary that the vortex core, of radius ~ ζ, fit within the critical region of size ~ r_c at the transition. From Eq. 41, ζ/r_c ~ mωζ², so that in the thermodynamic limit, equation i6 . Thus, vortex formation within the critical region is possible for large N ~ α⁻². The critical rotation frequency, Ω_c, for creation of a vortex is of order (1/mr_c²) log(r_c/ζ), and therefore equation i7

A further probe of the state of the system below T_KT would be determination of the density correlations, as have been recently measured in Bose gases trapped in optical lattices (45). These correlations will depend strongly on the amplitude fluctuations, described by ζ_T (M.H. and G.B., unpublished data).

Acknowledgments

M.H. and G.B. are grateful to the Aspen Center for Physics, where this work was completed. We thank Giuliano Orso for helpful comments. This research was supported in part by National Science Foundation Grants PHY0355014 and PHY0500914, and facilitated by the Projet de Collaboration Centre National de la Recherche Scientifique (CNRS)/University of Illinois at Urbana-Champaign. Laboratoire Kastler Brossel and Laboratoire de Physique Théorique de Matière Condensée are Unités Associées au CNRS Unité Mixte de Recherche (UMR) 8552 and UMR 7600.

Abbreviations

KTB	Kosterlitz–Thouless–Berezinskii
BEC	Bose–Einstein condensate.

Footnotes

The authors declare no conflict of interest.

This article is a PNAS direct submission.

^‡‡The scattering length in two dimensions is not a well defined concept because the scattering cross-section vanishes in the limit of low energies and low momentum. Nonetheless, for a singular potential such as a hard core, it is sufficient to sum particle–particle scattering processes in the T matrix and to replace the bare potential by the T matrix in the Hamiltonian (36, 37). Because the T matrix depends only logarithmically on energy and momentum, we can still work to leading order with a momentum- and energy-independent coupling constant, g = 2π/(m log 1/na²), where a corresponds to the hard core diameter. We further neglect the density dependence of the coupling constant in the following, because it does not enter in an essential way, and write g = 2πα/m.

^§§The expected phase transition has been calculated by path integral simulations (35) without introduction of explicit vortex degrees of freedom; the vorticity correlation function in these calculations showed no direct evidence of vortex unbinding at the transition.

^¶¶Josephson's relation remains valid inside the critical region of a finite size system, with the limit of zero wavevector replaced by k → k₀.

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