Abstract
Vortices are commonly observed in the context of classical hydrodynamics: from whirlpools after stirring the coffee in a cup to a violent atmospheric phenomenon such as a tornado, all classical vortices are characterized by an arbitrary circulation value of the local velocity field. On the other hand the appearance of vortices with quantized circulation represents one of the fundamental signatures of macroscopic quantum phenomena. In twodimensional superfluids quantized vortices play a key role in determining finitetemperature properties, as the superfluid phase and the normal state are separated by a vortex unbinding transition, the BerezinskiiKosterlitzThouless transition. Very recent experiments with twodimensional superfluid fermions motivate the present work: we present theoretical results based on the renormalization group showing that the universal jump of the superfluid density and the critical temperature crucially depend on the interaction strength, providing a strong benchmark for forthcoming investigations.
Introduction
Quantized vortices are characterized by a circulation of the velocity field quantized in multiples of ħ/m^{*}, where ħ is Planck’s constant and m^{*} is the mass of a superfluid particle, in the case of a bosonic superfluid, or the mass of a Cooper pair, in the case of a fermionic superfluid. Quantized vortices are a fundamental feature of superfluid and superconducting systems^{1} and have been observed in a wide variety of systems, including typeII superconductors^{2,3,4}, superfluid liquid Helium^{5,6}, superfluid liquid Helium nanodroplets^{7,8}, ultracold gases^{9,10}, and excitonpolaritons inside semiconductor microcavities^{11,12}.
From a phenomenological standpoint quantized vortices resemble nonquantized vortices in classical hydrodynamical systems. The quantization of circulation is a peculiar consequence of the existence of an underlying compact real field, whose spatial gradient determines the local superfluid velocity of the system^{13,14}. This compact real field, the socalled NambuGoldstone field, is the phase angle of the complex bosonic field which describes, in the case of attractive fermions, stronglycorrelated Cooper pairs of fermions with opposite spins^{14}.
In twodimensional (2D) superfluid systems there can not be BoseEinstein condensation and offdiagonal longrange order at finite temperature, as a consequence of the MerminWagnerHohenberg (MWH) theorem^{15,16,17}. Nevertheless a vortexdriven phase transition at a finite temperature T_{BKT} is still present due to the BerezinskiiKosterlitzThouless (BKT) mechanism^{18,19}. Below the critical temperature T_{BKT} the system is superfluid and characterized by bound vortexantivortex pairs and algebraic longrange order. Above T_{BKT}, on the other hand, vortexantivortex pairs unbind, free quantized vortices proliferate, and the system loses its superfluid properties with exponential decay of coherence. Within this scenario it is clear that quantized vortices play a key role in determining the finitetemperature properties of a 2D superfluid.
The rapid developments in the realization and manipulation of ultracold gases allow for the observation of dilute atomic vapors trapped in quasitwodimensional configurations. In 2006 the BKT transition and the associated unbinding of vortices has been observed in an atomic Bose gas by Hadzibabic et al.^{9}; in this experiment, the proliferation of free vortices is directly imaged by letting two 2D clouds expand and interfere with each other; the free vortices can then be counted individually by looking at the number of defects in the interference pattern. The same transition was also observed by Schweikhard et al.^{10} in an optical lattice, using the usual absorption imaging technique of the vortex cores. Recent experiments^{20,21,22,23} deal with 2D attractive Fermi gases in the crossover from the weakcoupling BCS regime of largely overlapping Cooper pairs to the strongcoupling BEC regime of composite bosons and provide motivation for the present theoretical investigation.
Results
Singleparticle and collective excitations in ultracold Fermi superfluids
In a fermionic superfluid with tunable swave interaction the meanfield theory predicts the existence of fermionic singleparticle excitations, whose lowenergy spectrum is
where m is the mass of a fermion, μ is the chemical potential of the system, and Δ_{0} is the pairing energy gap. The inclusion of beyondmeanfield effects, namely quantum fluctuations of the pairing field, gives rise to bosonic collective excitations^{24}, whose lowenergy spectrum across the BCSBEC crossover is^{25,26}
These collective excitations are density waves reducing to the BogoliubovGoldstoneAnderson mode E_{col}(k) = c_{s}ħk in the limit of small momenta. Here c_{s} is the speed of sound, while λ and γ are parameters taking into account the increase of kinetic energy due to the spatial variation of the density and depend on the strength of the attractive interaction: in the deep BEC regime one finds λ = 1/4 and γ = 0 such that E_{col}(k) = ħ^{2}k^{2}/(4m) for large momenta. It has been demonstrated that the inclusion of collective excitations in the equation of state, as briefly outlined in the Methods and derived in refs 27 and 28, recovers the correct composite boson limit at zero temperature^{28}, also providing qualitatively good results for many observable quantities across the whole crossover^{27,29}; we follow this approach in the present work.
The superfluid (number) density n_{s} of the twodimensional (2D) fermionic system can be written as
where n is the 2D total number density and n_{n} = n_{n,sp} + n_{n,col} is the 2D normal density due to both singleparticle and collective elementary excitations^{30}. For a uniform superfluid system at zero temperature n_{n} = 0 and n_{s} = n. As the temperature is increased the normal density n_{n} increases monotonically and, correspondingly, the superfluid density n_{s} decreases. According to Landau’s approach^{30,31}, the two contributions to the normal density read
and
where β = 1/(k_{B}T), k_{B} the Boltzmann constant and T the absolute temperature. The superfluid density n_{s} can also be inferred from the coefficient governing phase fluctuations in an effective action for the system^{32}; it turns out that for a Gaussianlevel action this approach is equivalent to setting n_{s} = n − n_{n,sp}, ignoring the contribution from collective excitations to the superfluid density; this contribution, however, will turn out to be fundamental in the strong coupling regimes that have become recently accessible^{21}.
More generally, in the extreme BCS (BEC) limit only the fermionic (bosonic) excitations contribute to the total superfluid density. As already discussed in ref. 27, the present approximation, considering the fermionic and bosonic excitations as separate, neglects the Landau damping that hybridizes the collective modes with the singleparticle excitations^{33}. It should be stressed, however, that the Landau damping is absent at T = 0, making our approximation reliable in the lowtemperature limit. Moreover we also discussed^{27} that Landau damping would affect the bosonic contribution n_{b} in the BCS region, where the physics is dominated by the fermionic contribution. This interplay makes the Landau damping less relevant as far as the present work is concerned, justifying the present choice of approximation.
The effective lowenergy Hamiltonian of a fermionic superfluid can be recast as that of an effective 2D XY model^{34,35,36}:
having introduced the pairing field Δ(r) = Δ(r)e^{iθ(r)} with θ(r) the socalled NambuGoldstone field^{13}. The phase stiffness J is a function of the fermionfermion attractive strength and of the temperature; it measures the energy cost associated to space variation in the phase angle θ(r) of the pairing field. Moreover the phase stiffness J is proportional to the superfluid number density n_{s}, namely^{37}
The compactness of the phase angle field θ(r) implies that for any closed contour . Here q = 0, ±1, ±2, … is the integer number associated to the corresponding quantum vortex (positive q) or antivortex (negative q). Consequently the circulation of the superfluid velocity v(r) = (ħ/m^{*})∇θ(r) is quantized according to where m* = 2m is the mass of a Cooper pair. Formally, one can rewrite the phase angle as follows
where θ_{0}(r) has zero circulation (no vortices) while θ_{v}(r) encodes the contribution of quantized vortices. Consequently, the Hamiltonian in Eq. (6) can be rewritten^{37} as H = H_{0} + H_{v} where H_{0} = J/2∫d^{2}r(∇θ_{0}(r))^{2} is the Hamiltonian of density oscillations, while
is the Hamiltonian of quantized vortices located at position r_{i} with quantum numbers q_{i}, interacting through a 2D Coulomblike potential
where ξ is healing length, i.e. the cutoff length defining the vortex core size, and μ_{c} the energy associated to the creation of a vortex^{38,39}.
Renormalization group analysis for a Fermi superfluid
The total number of quantized vortices varies as a function of the temperature: at zero temperature there are no vortices, however as the temperature increases vortices start to appear in vortexantivortex pairs. Due to the logarithmic energy cost the pairs are bound at low temperature, until at the critical temperature T_{BKT} an unbinding transition occurs above which a proliferation of free vortices and antivortices is observed^{18}. Vortexantivortex pairs with small separation distance can screen the potential in Eq. (10) between a vortexantivortex pair with larger distance r; as a consequence, the phase stiffness J and the vortex energy μ_{c} are renormalized^{40}. In particular analyzing the effect of increasing the spatial cutoff ξ, thereby excluding vortexantivortex configurations with distance smaller than ξ, Nelson and Kosterlitz obtained the renormalization group equations^{38,39,40}
subsequently extended by Amit^{41} and Timm^{42}, including nexttoleading order terms, in order to describe higher vortex densities
for the running variables and , as a function of the adimensional scale subjected to the initial conditions K(0) = βJ = βħ^{2}n_{s}/(4m) and y(0) = exp(−βμ_{c}). As discussed in ref. 39, the choice of μ_{c}, slightly affecting the final results, is still an open problem. The 2D XY model on a lattice with a finite difference approximation of spatial derivatives implies μ_{c} = π^{2}J/2^{38}. However, for the 2D XY model in the continuum it has been suggested within the GinzburgLandau theory of superconducting films^{43,44,45} and, more recently, within a phenomenological BCS approximation^{39}. In our study of the 2D BCSBEC crossover with Eqs (11) we adopt μ_{c} = π^{2}J/4, that is currently the most rigorous choice for superconductors and superfluids^{43,44,45}. The renormalized phase rigidity J^{(R)} and the renormalized vortex energy^{38,44} are then derived from K(∞) and y(∞). Finally, one obtains the renormalized superfluid density as
The renormalized superfluid density is a monotonically decreasing function of the temperature, as is the bare (unrenormalized) superfluid density n_{s}; however, while n_{s} is continuous, jumps discontinuously from a finite value to zero as the temperature reaches the BKT critical temperature T_{BKT}, implicitly defined by the KosterlitzNelson condition^{40}:
Let us verify the validity of the perturbative treatment of the renormalization group analysis. Combining Eq. (7), Eq. (14) and the definition of μ_{c} one readily sees that the expansion parameter y is a monotonically increasing function of the temperature, increasing from at T = 0, to at T = T_{BKT}. This fact suggests that even the leadingorder renormalization group in Eq. (11) could give accurate results for the present problem, and in fact including the nextorder correction as in Eq. (12) modifies our estimates of the critical temperature T_{BKT} by at most 1.5% over the whole crossover (see below), confirming the validity of the renormalization group analysis.
In Fig. 1 we report the renormalized and bare superfluid densities for three different values of the interacting strength, in the BCS, intermediate and BEC regimes. The renormalization of superfluid density as analyzed in Eq. (13) is more evident at higher temperatures, as the universal jump defined by Eq. (14) is approached. We also note that, although always a monotonically decreasing function of the temperature, the superfluid density exhibits different behaviors across the BCSBEC crossover, as it can be dominated either by fermionic, singleparticle excitations, in the weaklycoupled regime, or by bosonic, collective excitations, in the stronglycoupled regime.
Phase diagram
The finitetemperature phase diagram in the present 2D case is profoundly different with respect to a threedimensional Fermi gas as a result of the BKT mechanism just analyzed and also as a result of the MWH theorem^{15,16,17} prohibiting symmetry breaking at any finite temperature. These striking qualitative differences render a complete analysis of the 2D Fermi gas compelling both from the theoretical and experimental point of view. Let us briefly discuss the three possible phases^{14}:
Condensation
A 2D superfluid system exhibits condensation and offdiagonal longrange order (ODLRO) only strictly at T = 0: this zerotemperature regime is characterized by a nondecaying phasephase correlator , where C is independent of r, and by a finite condensate density^{46}.
Quasicondensation
The intermediate phase from T = 0^{+} to T_{BKT} is characterized by the phasephase correlator showing algebraic quasilongrange order for an opportune exponent α > 0. Although the condensate density is strictly zero, a finite superfluid density is still present.
Normal state
Finally for T > T_{BKT} the system enters the normal phase, characterized by the exponential decay of the phasephase correlator, and by the absence of both superfluid and condensate.
The gray dashed line in Fig. 1 corresponds to the KosterlitzNelson condition in Eq. (14), identifying the critical temperature T_{BKT}, separating the normal state from the phase characterized by quasicondensation. A determination of the critical temperature across the whole crossover is reported in the upper panel of Fig. 2, black solid line. The rapid decrease of T_{BKT} approaching both the BCS and the BEC limit is a consequence of the fermionic singleparticle excitations and bosonic collective excitations dominating the superfluid density, respectively, rapidly decreasing the normal density as either limit is approached. A consequence of this interplay is that the critical temperature is higher in the intermediate regime (), where the superfluid density is neither fermiondominated nor bosondominated.
The current approach, involving the inclusion of Gaussian fluctuations in the equation of state, the inclusion of bosonic collective excitations in the superfluid density along with a renormalization group analysis is able to reproduce the downward trend as the interaction get stronger; the renormalization group analysis on top of a meanfield theory would not have been sufficient to reproduce the correct trend, as shown by the gray dashed line in the upper panel of Fig. 2. In other words, as also observed elsewhere^{27,28,29}, Gaussian fluctuations are required in order to correctly describe the physics of an interacting Fermi gas in the stronglycoupled limit.
The underestimation of experimental data^{21}, as observed in Fig. 2 may have different causes:
In the experiment there is a harmonic trap also in the planar direction. The effect of the trap can enhance the critical temperature with respect to the uniform system, as found in the 3D case by Perali et al.^{47,48}.
It has been argued^{49} that the algebraic decay of the firstorder correlation function, presented in ref. 21 as the signature of the superfluid state, could be interpreted in terms of the strongcoupling properties of a normalstate. Experimental data in ref. 21 would then overestimate T_{BKT}.
The determination of the critical temperature may be affected by threedimensional effects, the superfluid not being trapped in a strictly 2D configuration.
On more general grounds one may argue that T_{BKT} > 0.125ε_{F}, as experimentally observed in the BCS regime, is not compatible with the KosterlitzNelson condition, signaling different mechanisms at work^{27}.
For the sake of completeness, in the lower panel of Fig. 2 we plot the BKT critical temperature T_{BKT} obtained with the KosterlitzThouless renormalization group equations (11) and the generalized renormalization group equations (12), starting with the bare superfluid density derived from the Gaussian theory. As previously stressed the relative difference in the determination of T_{BKT} is below 1.5% in the whole crossover. Moreover, the figure shows that this very small difference is larger in the intermediate coupling regime. ().
Discussion
In the present work we have analyzed the role of vortex proliferation in determining the finitetemperature properties of a 2D interacting Fermi gas, throughout the BCSBEC crossover, as the fermionfermion interaction strength is varied. Using the Kosterlitz renormalization group equations we have shown that the bare superfluid density is renormalized as the vortexvortex potential is screened at large distances. The renormalization of superfluid density lowers the BKT critical temperature, correctly reproducing the trend observed in experimental data through a nontrivial interplay between the singleparticle and collective excitations. As previously pointed out, and analyzed in ref. 49, currently available experimental data may overestimate the BKT critical temperature of the uniform system and our theoretical predictions are providing a benchmark for forthcoming experiments.
Methods
Equation of state
The pairing gap Δ_{0} and the chemical potential μ are calculated selfconsistently by jointly solving the gap and number equation, as done e.g. in refs 29 and 27. The Gaussian pair fluctuations scheme^{50,51} has been adopted which, as opposed as the NozièresSchmittRink^{52} approach, leads to finite, converging results in 2D. The spectrum of fermionic and collective excitations, E_{sp}(k) and E_{col}(q) as introduced in Eqs (1) and (2), are calculated by looking at the poles of the respective Green’s functions, as analyzed e.g. in ref. 24. Accordingly, the corresponding thermodynamical grand potential has two contributions, namely the meanfield, fermionic part
and the bosonic part
We stress that Ω_{F} accounts for the meanfield description of a tunable Fermi gas, whereas Ω_{B} includes the contribution of density waves on top of the meanfield picture.
Data availability
The data is available upon request. Requests should be addressed to either author.
Additional Information
How to cite this article: Bighin, G. and Salasnich, L. Vortices and antivortices in twodimensional ultracold Fermi gases. Sci. Rep. 7, 45702; doi: 10.1038/srep45702 (2017).
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Acknowledgements
The authors thank L. Benfatto, P. A. Marchetti, C. Sà de Melo, I. Nandori G. Strinati, F. Toigo, and A. Trombettoni for fruitful discussions.
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G.B. and L.S. jointly defined the project and derived analytical formulas. G.B. carried out the numerical calculations. G.B. and L.S. analyzed the data and wrote the paper.
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Bighin, G., Salasnich, L. Vortices and antivortices in twodimensional ultracold Fermi gases. Sci Rep 7, 45702 (2017). https://doi.org/10.1038/srep45702
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