Abstract
Initially predicted in nuclear physics, Efimov trimers are bound configurations of three quantum particles that fall apart when any one of them is removed. They open a window into a rich quantum world that has become the focus of intense experimental and theoretical research, as the region of ‘unitary’ interactions, where Efimov trimers form, is now accessible in coldatom experiments. Here we use a pathintegral Monte Carlo algorithm backed up by theoretical arguments to show that unitary bosons undergo a firstorder phase transition from a normal gas to a superfluid Efimov liquid, bound by the same effects as Efimov trimers. A triple point separates these two phases and another superfluid phase, the conventional Bose–Einstein condensate, whose coexistence line with the Efimov liquid ends in a critical point. We discuss the prospects of observing the proposed phase transitions in coldatom systems.
Introduction
A striking analogue to Borremean rings, Efimov trimers are bound configurations of three quantum particles that form near the point of ‘unitary’ interactions^{1}, where the pair potential becomes too weak to bind any two of them. Efimov trimers open a window into a largely uncharted world of quantum physics based on manyparticle bound states. They were discovered in nuclear physics^{2,3,4}, but have also been discussed in quantum magnets^{5}, biophysics of DNA^{6} and, most importantly, in ultracold atomic gases. In these systems, it has become possible to finetune both the sign and the value of the pair interactions through the Feshbach mechanism^{7}, from a scattering length a close to zero up to the unitary point a=∞. In current experiments, Efimov trimers have not been seen directly, but their presence has been traced through the variations of the rate at which the gas loses particles as the interactions are scanned through the unitary region^{8,9,10,11}.
Theoretical research on Efimov physics has unveiled its universal nature, with the threeparticle bound states that form an infinite sequence at unitarity, and that disappear with zero energy at large negative scattering lengths. Beyond the physics of threebound particles, the groundstate properties of small unitary clusters were studied numerically^{12}, and their spectrum computed with a variational ansatz^{13} in a trap. However, the macroscopic manybody properties of the unitary Bose gas have remained unknown. The understanding of its thermodynamic behaviour is of great importance, especially as the experimental stability of the unitary Bose gas of cold atoms has been reported for appreciable time scales^{11,14,15}.
In this work, we apply a dedicated PathIntegral Monte Carlo algorithm to the unitary Bose gas. This allows us to address the thermodynamics of unitary bosons at finite temperature, both above and below Bose–Einstein condensation. We obtain the phase diagram of a finite number of trapped bosons, and back up our numerical calculations by a general theoretical model that yields a homogeneous phase diagram. At high temperature, we find that the unitary Bose gas is very well described by the available virial coefficients^{16}. At lower temperature, the unitary Bose gas undergoes a firstorder phase transition to a new superfluid phase, the Efimov liquid, held together by the same effects as Efimov trimers, and whose physical quantities are given by threebody observables. From these two phases, transition lines to a third phase, the conventional Bose–Einstein condensate (BEC), start at a triple point. The coexistence line between the Efimov liquid and the conventional BEC ends in a critical point at high temperature. At a difference with the experimental systems, our model is thermodynamically stable. This is assured through a parameter, the threebody hard core R_{0}, that bounds from below the energies of the Efimov states. In coldatom systems, this parameter is on the order of the van der Waals length l_{vdW}^{17,18,19}. Experimental coldatom systems are metastable, and particles disappear from the trap into deeplybound states via the notorious threebody losses. This intricate quantum dynamics, and the description of the losses, are naturally beyond our exact Quantum Monte Carlo approach, and we discuss them on a phenomenological level in order to assess the prospects for experimental tests of our predictions.
Results
Model Hamiltonian
We describe the system of N interacting bosons by a Hamiltonian
where p_{i}, x_{i} and m are the momentum, the position and the mass of particle i, respectively. The pair interaction has zero range and may be viewed, as illustrated in Fig. 1a–c, as a square–well interaction potential whose range r_{0} and depth V_{0} are simultaneously taken to 0 and ∞ while keeping the scattering length a constant. The unitary point, where the only bound state disappears with zero energy and infinite extension, corresponds to an infinite scattering length a. The threebody interaction V_{3} implements a hardcore hyperradial cutoff condition, R_{ijk}>R_{0}, where the hyperradius R_{ijk} of particles i, j and k corresponds to their root mean square pair distance . This threebody hard core prevents the socalled Thomas collapse^{20} into a manybody state with vanishing extension and infinite negative energy by setting a fundamental trimer energy . The final term in equation (1) models an isotropic harmonic trapping potential of length as it is realized in ultracold bosons experiments. The properties of the system described by equation (1) are universal when R_{0} is much smaller than all other length scales.
Pathintegral representation and Efimov trimers
In thermal equilibrium, within the pathintegral representation of quantum systems that we use for our computations, position variables x_{i}(τ) carry an imaginary time index τε[0, β=1/k_{B}T], where k_{B} is the Boltzmann constant and T is the temperature. The fluctuations of x_{i}(τ) along τ account for the quantum uncertainty. The bosonic nature of manyparticle systems manifests itself through the periodic boundary conditions in τ and, in particular, through the permutation structure of particles. The length of permutation cycles correlates with the degree of quantum coherence^{21,22,23}. Interactions set the statistical weights of configurations^{24,25}. Ensemble averaging, performed by a dedicated Quantum Monte Carlo algorithm, yields the complete thermodynamics of the system (see the Methods section for computational details). The Nbody simulation code is massively run on a cluster of independent processors. It succeeds in equilibrating samples with up to a few hundred bosons.
Figure 1 presents snapshots of three bosons in a shallow trap at ω~0 that illustrate the quantum fluctuations in x_{i}(τ), and characterize the Efimov trimer (see the Methods section for a full description of the three bosons simulation). Indeed, for twobody interactions without a bound state, virtually free particles fluctuate on the scale of the de Broglie thermal wavelength that diverges at low temperature (see Fig. 1a). In contrast, for positive a, a bound state with energy −E_{dimer}=−ħ^{2}/(ma^{2}) forms in the twobody interaction potential. Two particles bind into a dimer, and the third particle is free (see Fig. 1c). At unitarity, the bound state of the pair potential is at resonance E_{dimer}=0, and the scattering length a is infinite (see Fig. 1b). At this point, the twobody interaction is scalefree. While two isolated particles do not bind in the threeparticle system, pairs of particles approach each other and then dissociate, so that, between τ=0 and τ=β, the identity of closeby partners changes several times. This coherent particlepair scattering process, the hallmark of the Efimov effect^{4}, is highlighted in Fig. 1b. At small temperatures, the fluctuations of this bound state remain on a scale proportional to R_{0} and do not diverge as λ_{th}.
While the twoparticle properties are universal at unitarity, the threeboson fundamental trimer state generally depends on the details of the pair interaction. Excited trimers form a geometric sequence of asymptotically universal Efimov trimers with an asymptotic ratio of energies E_{n}/E_{n+1}≈515.0 when n→∞, where E_{n} is the energy of the nth excited trimer^{1}. Owing to this large ratio, thermal averages cannot identify individual trimer states other than the fundamental trimer at low temperature. For the model Hamiltonian of equation (1), the groundstate trimer is virtually identical to a universal Efimov trimer^{12}, and we obtain excellent agreement of the probability distribution of the hyperradius R with its analytically known distribution p_{R}(R) (see Fig. 1d)^{1}. This effectively validates our algorithm. Furthermore, the observed quadratic divergence of the pair distance distribution ρ_{r}(r), leading to an asymptotically constant r^{2}ρ_{r}(r) for r→0 (see Fig. 1e), checks with the Bethe–Peierls condition for the zerorange unitary potential^{26}.
Equation of state of the unitary Bose gas
In localdensity approximation, particles experience an effective chemical potential μ(r)=μ_{0}−m ω^{2}r^{2}/2 that depends on the distance r from the centre of the trap. This allows us to obtain the grand canonical equation of state (pressure P as a function of μ) from the doubly integrated density profile^{27} obtained from a single simulation run at temperature T. We find that the equation of state of the normal gas is described very accurately by the virial expansion up to the third order in the fugacity e^{βμ} (see Fig. 2 and the Methods section)^{16}. The thirdorder term is crucial to the description of Efimov physics as it is the first term at which threebody effects appear^{28}. It depends explicitly on T and R_{0}.
Phase transitions in the trapped unitary Bose gas
In the harmonic trap, particles can be in different thermodynamic phases depending on the distance r from its centre. We monitor the correlation between the pair distances and the position in the trap, and are able to track the creation of a drop of highdensity liquid at r~0 (see Fig. 3a–c). This drop grows as the temperature decreases. The observed behaviour corresponds to a firstorder normalgastosuperfluidliquid transition, and is fundamentally different from the secondorder free Bose–Einstein condensation (see Supplementary Fig. 1). All particles in the drop are linked through coherent closeby particle switches as in Fig. 1b, showing that the drop is superfluid. Deep inside the liquid phase, the Quantum Monte Carlo simulation drops out of equilibrium on the available simulation times. Nevertheless, at its onset and for all values of R_{0}, the peak of the pair correlation function is located around 10R_{0}, which indicates that the liquid phase is of constant density (see Fig. 3c). At larger values of R_{0}, the density difference between the trap centre and the outside vanishes continuously, and the phase transition is no longer seen (see Fig. 3d–f). Beyond this critical point, the peak of the pair correlation also stabilizes around 10R_{0}, which indicates a crossover to liquid behaviour (see Fig. 3f and the Methods section for additional details on the characterization of the firstorder phase transition).
Model for the normalgastoEfimovliquid transition
Our numerical findings suggest a theoretical model for the competition between the unitary gas in thirdorder virial expansion and an incompressible liquid of density and constant energy per particle −ε∝E_{t}, as suggested by groundstate computations for small clusters^{12} (see Supplementary Fig. 2). For simplicity, we neglect the entropic contributions to the liquidstate free energy ε≫TS, so that F_{1}≈−N ε. The phase equilibrium is due to the difference in free energy and in specific volume at the saturated vapour pressure (see the Methods section and Supplementary Fig. 3). We extend the thirdorder virial expansion to describe the gaseous phase in the region where quantum correlations become important. Because the conventional Bose–Einstein condensation is continuous, it still conveys qualitative information about the transition into the superfluid Efimov liquid in that region. At small R_{0}→0, the coexistence line approaches infinite temperatures as the fundamental trimer energy diverges. For larger values of R_{0}, the density of the liquid decreases and approaches the one of the gas. The liquid–gas transition line ends in a critical point, where both densities coincide. As the liquid is bound by quantum coherence intrinsic to the Efimov effect, this critical point must always be inside a superfluid, that is, between the Efimov liquid and the BEC, which become indistinguishable. The agreement between this approximate theory and numerical calculations for the trap centre phase diagram is remarkable (see Fig. 4a). Beyond the critical point, we no longer observe a steep drop in the density on decreasing the temperature. We also notice that quantum coherence builds up in the gaseous phase, so that only a conventional Bose–Einstein condensation takes place. Our numerical results suggest it occurs at a temperature slightly lower than that for the ideal Bose gas^{29}, .
Homogeneous phase diagram of the unitary Bose gas
Our theoretical model also yields a phase diagram for a homogeneous system of unitary bosons (see Fig. 4b) where, in addition, the conventional Bose–Einstein condensation is simply modelled by that of free bosons^{30}. In absence of a harmonic trap, only two independent dimensionless numbers may be built, k_{B}T/ε, and . As a consequence, the phase diagram in these two dimensionless numbers is independent of the choice of ε, that is, of R_{0}. The scaling of the dimensionless pressure with and of the dimensionless temperature with explains that the triple point appears much farther from the critical point in the homogeneous phase diagram than in the trap. We expect modeldependent nonuniversal effects in two regions. At high temperature λ_{th}<<R_{0}, only a classical gas should exist as the quantum fluctuations are too small to build up quantum coherence and, in particular, permutations between particles. At high pressure P∝T, we expect a classical solid phase driven by entropic effects, as for conventional hardsphere melting.
Discussion
To situate the theoretical results presented in this work in an experimental context, we present the data of Fig. 4b in terms of R_{0}n^{1/3} and λ_{th}n^{1/3} (see Fig. 5). In this diagram, systems with identical values of T and R_{0} and different densities correspond to straight lines passing through the origin. In the unstable region, we expect a homogeneous system described by the Hamiltonian of equation (1) to phaseseparate on the same such line into the superfluid Efimov liquid and the normal gas or the BEC.
In coldatom systems, the atomic interactions have deeplybound states not present in our model, and the strict hyperradial cutoff is absent. Nevertheless, an effective threebody barrier at a universal value ~2l_{vdW}^{18} induces a universal relation between the fundamental trimer energy E_{t} and the van der Waals length l_{dvW}^{17,19,31,32}. In Fig. 5, a vertical line marks the experimentally realistic value R_{0}n^{1/3}~0.023 obtained for ^{133}Cs atoms at a density n=10^{13}cm^{−3}, using R_{0}~2l_{vdW} and l_{vdW}=101a_{0}^{7}, where a_{0} is the Bohr radius. Other atomic species yield smaller values of R_{0}n^{1/3}, for which this discussion is also valid. An experimental system may be quenched along this line into the unstable region by suddenly increasing the scattering length up to unitarity via the Feshbach mechanism. If this quench starts from a weakly interacting BEC at very low temperature, the fourbody recombination process (that saturates at a rate ) will dominate the threebody recombination process (that saturates at a rate )^{11,33,34,35}, a condition that may be written as nK_{4}≳K_{3} (cf. refs 36, 37). Whereas the latter is responsible for the threebody losses into deeplybound dimer states, the former may represent one possible strategy for creating Efimov trimers.
In the unstable region, the recombination process would create liquid droplets of ever increasing size. Deep inside the unstable region, far away from the coexistence line, we expect this growth to be a fast, barrierfree, runaway process involving spinodal decomposition^{38,39}, rather than a slow activation process of nucleation over a freeenergy barrier produced by the competition of bulk and surface energies. In a thermodynamically stable system, complete separation on macroscopic length scales proceeds through a coarsening process on length scales that slowly increase with time^{38,40}. At the Bose–Einstein condensation temperature of the noninteracting gas, this would lead to a large liquid fraction 1−n_{g}/n, where n_{g} is the density of the gas at coexistence, and n the density of the system before phase separation. However, this will certainly not be observable in current coldatom experiments, as sufficiently large droplets of dense liquid are unstable towards decay into deeplybound atomic states. Nevertheless, the instability of the gas and the creation of microscopic liquid droplets might be quite realistic. These could be observed as they would lead to an important increase of the threebody losses, proportional to in the liquid phase.
The time scales for the creation of a minority liquid phase after a quench is a classic problem of nonequilibrium statistical mechanics. In the present context, it is rendered even richer by the fact that the initial lowtemperature phase is a weakly interacting superfluid, and that even the normal gas phase is thermodynamically unstable. These questions are closely related to the very existence of the unitary Bose gas on time scales larger than its thermalization time. Very recent experimental works indicate that the ultracold unitary Bose gas can indeed be stabilized on appreciable time scales^{11,14,15}.
Methods
Pathintegral Monte Carlo algorithm
In our pathintegral quantum Monte Carlo simulation, the contribution of the threebody hardcore interaction to the statistical weights of discretized path configurations are computed using Trotter’s approximation, which consists in simply rejecting configurations with R<R_{0}. The contribution of the zerorange unitary interaction is computed using the pairproduct approximation, which estimates the weight of two nearby particles without taking other particles into consideration. Both approximations are valid when the discretization step is small^{23}.
In the simulation, new configurations are built from existing configurations according to several possible update moves. A new one, the compressiondilation move, was introduced to specifically address the divergence of the pair correlation function at small distances (see Fig. 1e; Supplementary Fig. 4). For each set of parameters, the simulation was run on up to 16 independent processors for up to 10 days to reduce the statistical error.
Simulation of three bosons
In our simulations of unitary bosons, the system is contained in a harmonic trap. This regulates the available configuration space. For the same purpose, in the threebody calculations at ω~0 presented in Fig. 1, we impose that the three bosons are on a single permutation cycle: in Fig. 1a–c the blue, red and green bosons are, respectively, exchanged with the red, green and blue bosons at imaginary time β. This condition does not modify the properties of the fundamental trimers at unitarity, as other permutations could be sampled at no cost at points of close encounter such as those highlighted in Fig. 1b.
Figure 1a–c presents fourdimensional cocyclic pathintegral configurations in threedimensional plots. In this graphic representation, the centre of mass, whose motion is decoupled from the effect of the interactions, is set to zero at all τ. The three spatial dimensions are then reduced to two dimensions by rotating the triangle formed by the three particles at each imaginary time to the same plane in a way that does not favour any of the three spatial dimensions while conserving the permutation cycle structure and the pair distances.
Hightemperature equation of state
Within the localdensity approximation, the grand canonical equation of state P(μ) may be obtained from the numerical doubly integrated density profile ^{27} as
where μ(x)=μ_{0}−m ω^{2}x^{2}/2 is the local chemical potential along direction x, and μ_{0} the chemical potential at the centre of the trap, measured from a fit of the equation of state to that of an ideal gas in the outer region of the trap.
We compare this numerical equation of state to the cluster expansion that expresses the pressure in terms of the fugacity e^{βμ} (see Fig. 2):
The lth cluster integral b_{l} follows from the virial coefficients of smaller order. It represents lbody effects that cannot be reduced to smaller noninteracting groups of interacting particles^{28}. We use the analytical expressions of b_{2} and b_{3} at unitarity that have become available^{16}.
Monitoring the phase transitions
When the densities of the gas and the superfluid Efimov liquid approach each other, observing directly the twodimensional histogram of pair distances and centreofmass positions does not allow to distinguish between a weakly firstorder phase transition and a crossover (see Supplementary Fig. 5). In this regime, we monitor the normalgastosuperfluidliquid phase transition more accurately by following the evolution of the first peak of the pair correlation function (obtained by ensemble averaging) with temperature (see Supplementary Fig. 6). In Fig. 4, Bose–Einstein condensation is assumed when particles lie on a permutation cycle of length >10 with probability 0.05 (ref. 22).
Analytical model for the transition into the Efimov liquid
Firstorder phase transitions take place when the free energy F of a homogeneous physical system is not a convex function of its volume V. Splitting the system into two phases is then favourable over keeping it homogeneous (see Supplementary Fig. 3). In this situation, at coexistence and in absence of interface energy, the chemical potentials and the pressures of both phases are equal.
The virial expansion is an excellent approximation to describe the normal gas phase of unitary bosons far from the superfluid transition. Although this expansion becomes irrelevent in the superfluid gas, its analytic continuation conveys important qualitative features because of the continuous nature of conventional Bose–Einstein condensation, and is therefore a suitable approximation for the conventional BEC.
The theoretical model for the superfluid liquid is that of an incompressible liquid of specific volume ν_{1} and negligible entropic contribution to the free energy F_{1}=−N ε. Simulations yield , and the negligible contribution of the entropy to the free energy is ensured for nonpathological systems at low temperature. The results in ref. 12 may be extrapolated to ε=10.1E_{t} (see Supplementary Fig. 2), a value adjusted to ε=8E_{t} in Fig. 4a to account for finitesize effects.
In practice, we draw the transition line into the superfluid liquid by finding the smallest chemical potential at which the presssures of the incompressible liquid and of the normal gas coincide at a temperature T:
As n=1/ν=∂_{μ}P, the crossing to the regime where this equation has no solution corresponds to the critical point, where both densities are equal.
To draw the coexistence line for the trap centre with N=100 particles in Fig. 4a, its chemical potential μ_{0} is found from integrating the density throughout the trap:
where the local chemical potential μ(r)=μ_{0}−m ω^{2}r^{2}/2 is computed within the local density approximation.
Additional information
How to cite this article: Piatecki, S. and Krauth, W. Efimovdriven phase transitions of the unitary Bose gas. Nat. Commun. 5:3503 doi: 10.1038/ncomms4503 (2014).
References
 1
Braaten, E. & Hammer, H.W. Universality in fewbody systems with large scattering length. Phys. Rep. 428, 259–390 (2006).
 2
Efimov, V. Energy levels arising from resonant twobody forces in a threebody system. Phys. Lett. B 33, 563–564 (1970).
 3
Efimov, V. Weaklybound states of three resonantlyinteracting particles. Sov. J. Nucl. Phys. 12, 589–595 (1971).
 4
Efimov, V. Lowenergy property of three resonantly interacting particles. Sov. J. Nucl. Phys 29, 1058–1069 (1979).
 5
Nishida, Y., Kato, Y. & Batista, C. D. Efimov effect in quantum magnets. Nat. Phys. 9, 93–97 (2013).
 6
Maji, J., Bhattacharjee, S. M., Seno, F. & Trovato, A. When a DNA triple helix melts: an analogue of the Efimov state. New J. Phys. 12, 083057 (2010).
 7
Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225–1286 (2010).
 8
Gross, N., Shotan, Z., Kokkelmans, S. & Khaykovich, L. Observation of universality in ultracold 7Li threebody recombination. Phys. Rev. Lett. 103, 163202 (2009).
 9
Pollack, S. E., Dries, D. & Hulet, R. G. Universality in three and fourbody bound states of ultracold atoms. Science 326, 1683–1685 (2009).
 10
Zaccanti, M. et al. Observation of an Efimov spectrum in an atomic system. Nat. Phys. 5, 586–591 (2009).
 11
Rem, B. S. et al. Lifetime of the Bose gas with resonant interactions. Phys. Rev. Lett. 110, 163202 (2013).
 12
von Stecher, J. Weakly bound cluster states of Efimov character. J. Phys. B 43, 101002 (2010).
 13
Thøgersen, M., Fedorov, D. V. & Jensen, A. S. Nbody Efimov states of trapped bosons. Europhys. Lett. 83, 30012 (2008).
 14
Fletcher, R. J., Gaunt, A. L., Navon, N., Smith, R. P. & Hadzibabic, Z. Stability of a unitary Bose gas. Phys. Rev. Lett. 111, 125303 (2013).
 15
Makotyn, P., Klauss, C. E., Goldberger, D. L., Cornell, E. A. & Jin, D. S. Universal dynamics of a degenerate unitary Bose gas. Nat. Phys. 10, 116–119 (2014).
 16
Castin, Y. & Werner, F. Le troisiéme coefficient du viriel du gaz de Bose unitaire. Can. J. Phys. 91, 382–389 (2013).
 17
Berninger, M. et al. Universality of the threebody parameter for Efimov states in ultracold cesium. Phys. Rev. Lett. 107, 120401 (2011).
 18
Wang, J., D’Incao, J. P., Esry, B. D. & Greene, C. H. Origin of the threebody parameter universality in Efimov physics. Phys. Rev. Lett. 108, 263001 (2012).
 19
Wild, R. J., Makotyn, P., Pino, J. M., Cornell, E. A. & Jin, D. S. Measurements of Tan’s contact in an atomic BoseEinstein condensate. Phys. Rev. Lett. 108, 145305 (2012).
 20
Thomas, L. H. The interaction between a neutron and a proton and the structure of H3. Phys. Rev. 47, 903–909 (1935).
 21
Feynman, R. P. Statistical Mechanics: a Set of Lectures AddisonWesley (1982).
 22
Krauth, W. Quantum Monte Carlo calculations for a large number of bosons in a harmonic trap. Phys. Rev. Lett. 77, 3695–3699 (1996).
 23
Krauth, W. Statistical Mechanics: Algorithms and Computations Oxford Univ. Press (2006).
 24
Pollock, E. L. & Ceperley, D. M. Simulation of quantum manybody systems by pathintegral methods. Phys. Rev. B 30, 2555–2568 (1984).
 25
Ceperley, D. M. Path integrals in the theory of condensed helium. Rev. Mod. Phys. 67, 279–355 (1995).
 26
Bethe, H. A. & Peierls, R. The scattering of neutrons by protons. Proc. R. Soc. Lond. A 149, 176–183 (1935).
 27
Ho, T.L. & Zhou, Q. Obtaining the phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases. Nat. Phys. 6, 131–134 (2010).
 28
Huang, K. Statistical Mechanics 2nd ed. Wiley (1987).
 29
Dalfovo, F., Giorgini, S., Pitaevskii, L. P. & Stringari, S. Theory of BoseEinstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999).
 30
Landau, L. D. & Lifshitz, L. M. Statistical Physics 3rd ed. Course of theoretical physics No. 5ButterworthHeinemann (1980).
 31
Schmidt, R., Rath, S. P. & Zwerger, W. Efimov physics beyond universality. Eur. Phys. J. B 85, 386 (2012).
 32
Roy, S. et al. Test of the universality of the threebody Efimov parameter at narrow Feshbach resonances. Phys. Rev. Lett. 111, 053202 (2013).
 33
D’Incao, J. P., Suno, H. & Esry, B. D. Limits on universality in ultracold threeboson recombination. Phys. Rev. Lett. 93, 123201 (2004).
 34
Mehta, N. P., Rittenhouse, Seth T., D’Incao, J. P., von Stecher, J. & Greene, Chris H. General theoretical description of nbody recombination. Phys. Rev. Lett. 103, 153201 (2009).
 35
Li, W. & Ho, T.L. Bose gases near unitarity. Phys. Rev. Lett. 108, 195301 (2012).
 36
Ferlaino, F. et al. Evidence for universal fourbody states tied to an Efimov trimer. Phys. Rev. Lett. 102, 140401 (2009).
 37
Greene, C. H. Universal insights from fewbody land. Phys. Today 63, 40 (2010).
 38
Binder, K. Theory of firstorder phase transitions. Rep. Prog. Phys. 50, 783–859 (1987).
 39
Cahn, J. W. & Hilliard, J. E. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (2004).
 40
Lifshitz, I. M. & Slyozov, V. V. The kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids 19, 35–50 (1961).
Acknowledgements
We thank S. Balibar, Y. Castin, F. Chevy, I. FerrierBarbut, A.T. Grier, B. Rem, C. Salomon and F. Werner for very helpful discussions. W.K. acknowledges the hospitality of the Aspen Center for Physics, which is supported by the National Science Foundation Grant No. PHY1066293.
Author information
Affiliations
Contributions
Both authors contributed equally to this work.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures 16 (PDF 318 kb)
Rights and permissions
About this article
Cite this article
Piatecki, S., Krauth, W. Efimovdriven phase transitions of the unitary Bose gas. Nat Commun 5, 3503 (2014). https://doi.org/10.1038/ncomms4503
Received:
Accepted:
Published:
Further reading

Universal prethermal dynamics of Bose gases quenched to unitarity
Nature (2018)

Unitarity and Discrete Scale Invariance
FewBody Systems (2017)

Universal dynamics of a degenerate unitary Bose gas
Nature Physics (2014)

Energy and Structure of FewBoson Systems
FewBody Systems (2014)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.