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Nature of Bose gases near Feshbach resonance : the interplay between few-body and many-body physics Mashayekhi, Mohammad S. 2014

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Nature of Bose Gases Near FeshbachResonanceThe Interplay Between Few-Body and Many-BodyPhysicsbyMohammad S. MashayekhiB.Sc., Sharif University of Technology, 2006M.Sc., The University of British Columbia, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)February 2014c? Mohammad S. Mashayekhi 2013AbstractIn this thesis, we investigated the physics of two- and three-dimensional ul-tra cold Bose gases in the strongly interacting regime at zero temperature.This regime can be experimentally accessed using a Feshbach resonance. Weapplied a self-consistent diagrammatic approach to determine the chemicalpotential of three-dimensional Bose gases for a wide range of interaction val-ues. We showed that such strongly interacting Bose gases become unstabletowards the formation of molecules at a finite positive scattering length. Infact, the interaction between atoms becomes effectively attractive and thesystem looses its metastability before reaching the unitary limit. We alsofound that such systems are nearly fermionized close to the instability point.Near this critical point, the chemical potential reaches a maximum and thecontribution to the system energy due to three-body forces is estimated to beonly a few percent. We also studied the same system using a self-consistentrenormalization group method. This approach confirms the existence of aninstability point towards the formation of molecules as well as fermioniza-tion. We showed that the instability and accompanying maximum are pre-cursors of the sign change of the effective two-body interaction strength fromrepulsive to attractive near resonance. In addition, we examined the physicsof two-dimensional Bose gases near resonance using a similar self-consistentdiagrammatic approach as the one introduced for three-dimensional Bosegases. We demonstrated that a competition between three-body attractiveinteractions and two-body repulsive forces results in the chemical potentialof two-dimensional Bose gases to exhibit a maximum at a critical scatteringlength beyond which these quantum gases possess a negative compressibil-ity. For larger scattering lengths, the increasingly prominent role played bythree-body attractive interactions leads to an onset instability at a secondcritical value. The three-body effects studied for these systems are univer-sal, fully characterized by the effective two-dimensional scattering length andare, in comparison to the three-dimensional case, independent of three-bodyultraviolet physics.iiPreface? A version of first part of chapter 1 is published in Annual Review ofCold Atoms and Molecules [1]. I wrote parts of the manuscript.? A version of chapter 2 has been published in Phys. Rev. A [2]. I con-ducted part of analytical and numerical calculations in this research.? A version of chapter 3 has been published in Annals of Physics [3]. Ihave conducted part of calculations and produced all plots.? A version of chapter 4 has been published in Phys. Rev. Lett. [4]. Ihave conducted most of the analytical and numerical calculations ofthis research.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . 21.2 Three-Dimensional Bose Gases . . . . . . . . . . . . . . . . . 41.2.1 Current Experimental Status . . . . . . . . . . . . . . 51.2.2 Current Theoretical Understanding . . . . . . . . . . 71.3 Two-Dimensional Bose Gases . . . . . . . . . . . . . . . . . . 122 Nature of Three-Dimensional Bose Gases: Self-ConsistentApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Chemical Potential, Metastability and Efimov Effects . . . . 172.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Nature of Three-Dimensional Bose Gases: RenormalizationGroup Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 A Caricature . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.1 Relevance of Fermionization: A Scaling Argument . . 303.2.2 Running Two-Body Interaction Constants . . . . . . 32ivTable of Contents3.3 Dimers and Trimers in a Condensate: The Spectrum Flow . 383.4 Sign Change of g2: A Consequence of Spectrum Flow . . . . 413.5 Diagrammatic Resummation: A Self-Consistent Approach . 483.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 Nature of Two-Dimensional Bose Gases . . . . . . . . . . . 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Self-Consistent Approach for Two-Dimensional Bose Gases . 554.3 Competing Three- and Two-Body Interactions . . . . . . . . 594.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66AppendicesA Solving Self-Consistent Eq. (2.5) in the Dilute Limit . . . 71B A Comparison Between the Self-Consistent Approach andthe Dilute Gas Theory in Three Dimensions . . . . . . . . . 73C Including Three-Body Forces in the Self-Consistent Equa-tions in Three Dimensions . . . . . . . . . . . . . . . . . . . . 75D Single Parameter Limit . . . . . . . . . . . . . . . . . . . . . . 78E Two- and Three-Body Scattering Amplitudes in a Conden-sate in Three Dimensions . . . . . . . . . . . . . . . . . . . . . 80F Two- and Three-Body Scattering Amplitudes in a Conden-sate in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . 85G Numerical Method to Find The Three-Body Interaction Po-tential in Two Dimensions . . . . . . . . . . . . . . . . . . . . 92vList of Tables3.1 Comparison of different theory approaches to find the natureof three-dimensional Bose gases. . . . . . . . . . . . . . . . . 48viList of Figures1.1 Schematic picture of magnetic Feshbach resonance . . . . . . 32.1 Classification of M-body scattering processes and sample di-agrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Scattering processes showing contribution to the total energyE(n0, ?). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Profile of the chemical potential and condensate fraction of athree-dimensional Bose gas as a function of scattering lengthfound using self-consistent method. . . . . . . . . . . . . . . . 242.4 Three-body interaction potential for a three-dimensional Bosegas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Schematic of the renormalization of the low energy on-shellscattering amplitude. . . . . . . . . . . . . . . . . . . . . . . . 343.2 The solution to the self-consistent boundary condition pre-sented in Eq. (3.14). . . . . . . . . . . . . . . . . . . . . . . . 373.3 Diagrams for the calculations of dimer and trimer energy ina condensate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 (a) Effective two-body interaction strength as a function ofscattering length for a three-dimensional Bose gas. (b) Illus-tration of the dimer energy in the presence of a condensate. . 453.5 The numerical solution to the self-consistent equation pre-sented in Eq. (3.34). . . . . . . . . . . . . . . . . . . . . . . . 474.1 The chemical potential of a two-dimensional Bose gas as afunction of scattering length found using self-consistent method 574.2 Three-body interaction potential for a two-dimensional Bosegas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59viiList of Figures4.3 Ratio between the contribution of three-body and two-bodyinteractions as a function of scattering length, condensationfraction and imaginary part of the chemical potential dueto three-body recombination processes for a two-dimensionalBose gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62F.1 Different types of the interaction vertices between condensedand non-condensed atoms. . . . . . . . . . . . . . . . . . . . . 88viiiAcknowledgementsI would like to acknowledge the advice and guidance of Dr. Fei Zhou, mysupervisor. I also thank the members of my graduate committee, Dr. MonaBerciu, Dr. Gordon Semenoff and Dr. Jeff Young for their guidance andsuggestions. Special thanks go to my friend Dr. Jean-Sebestien Bernier,without whose knowledge and assistance this study would not have beensuccessful.Finally, I would express my gratitude to my parents and my sisters,whose support and constant encouragement helped me through the hardtimes of this program. My deepest appreciation is expressed to them fortheir love, understanding, and inspiration. Without their blessings and en-couragement, I would not have been able to finish this work.I also would like to thank my colleagues Dr. Jun-Liang Song, DmitryBorzov and Dr. Shizhong Zhang for their collaboration on this work.ixDedicationTo my family.xChapter 1IntroductionUltra cold gases are an excellent test bed to explore the fundamentals ofmany-body quantum physics. Since the first realization of a Bose-Einsteincondensate of ultra cold atoms in 1995 [5, 6], a large number of experimentshave been done to understand the complex behavior of these systems. Thisis achieved by reducing the of these gases to nano Kelvins in order to sup-press thermal fluctuations and reveal the quantum nature of these systems.Ultra cold gases are quite unique as experimentalists have impressive con-trol over the system Hamiltonian parameters. For example, the ability totune the strength of the atom-atom interaction has facilitated the explo-ration of strongly interacting gases. Understanding such systems is of greatimportance as strong interactions can generate complex states that cannotbe trivially inferred from the weakly interacting regime. Moreover, in thisregime the crucial role played by many-body physics requires the develop-ment of novel theoretical frameworks. Consequently, exploring this regimeis extremely exciting.While all physical properties of weakly interacting Bose gases were un-derstood theoretically in the 1950?s [7?12], the physics of strongly correlatedBose gases is still poorly understood. Such gases, where complex many-bodyeffects are important, can be produced using magnetic Feshbach resonances[13]. Applying this technique, experimentalists are able to vary the strengthof the interaction between atoms and to access a strongly interacting regimeclose to a Feshbach resonance. Although the realization of stable stronglycorrelated Bose gases is limited by the large inelastic loss of atoms due to theformation of molecules, these Bose gases can be held for about a few mili-seconds [14] before losing stability. The physics of these strongly correlatedatomic gases remains to be fully understood. This is the main objectiveof this thesis. In the next sections, we first briefly review the basic the-ory of Feshbach resonances and the theory of dilute gases before presentingour results pertaining to the unusual physics of two- and three-dimensionalstrongly interacting Bose gases. Note that this is a rapidly developing sub-ject with new data and theoretical attempts made after we finished the firstversion of this thesis toward the end of summer 2013. As a result, there are11.1. Feshbach Resonancessome recent works that are not included in our review of previous studies.1.1 Feshbach ResonancesFeshbach resonances are considered as an essential tool to control the in-teraction strength between the atoms of a quantum gas. To explain thisphenomenon, we consider two molecular potentials Vopen and Vclosed whichare named as open channel and closed channel respectively. As depicted inFig. 1.1, at large inter-atomic distance, Vopen asymptotically tends to twofree atoms in the gas. This channel is called the open channel since the en-ergy is very close to zero in collision between these two atoms. In contrast,the closed channel can have few molecular bound states which are close tothe threshold of the open channel.In magnetic Feshbach resonances, the energy difference between thesetwo channels could be controlled by applying a magnetic field if the magneticmoments of channels are different. When the energy of one of the boundstates of the closed channel is approaching the scattering state in open chan-nel, even a weak coupling between the atoms can cause the mixing betweentwo channels. On the other hand, scattering atoms spend a finite time as abound state and as a result, the effective interaction between the scatteringatoms could be very strong. Resonance happens when the energy of oneof the bound states is equal to the scattering atoms energy. The effectiveinteraction between atoms is described by a simple equation introduced inRef. [15] for s-wave scattering length a as a function of magnetic field B:a(B) = abg(1? ?B ?B0)(1.1)where abg is the scattering length defined for open channel and is known asthe off-resonant value. B0 is the magnetic field where the resonance happensand ? is the resonance width. On the positive side of the resonance, wherethe energy of bound states of the closed channel is a little lower that theenergy of the atoms in open channel, the scattering state known as ?upperbranch? is a metastable state and atoms will form bound state in sufficientlylong time. The energy of these bound states is shown in Fig. 1.1. The boundstate energy is zero at the resonance. Away from resonance, the boundstate energy varies linearly with the magnetic field with slope ??. But nearresonance, the strong mixing between two channels bends the molecularstate and the binding energy is given by:21.1. Feshbach ResonancesFigure 1.1: Schematic picture of magnetic Feshbach resonance. When theenergy of the bound state of the closed channel approaches the energy of scat-tering atoms in open channels, the effective interaction between the atomswould be very high due to strong mixing of two channels. The positiveside of the resonance labeled by a > 0 is known as upper branch and it ismetastable. More discussions could be found in Ref. [16].31.2. Three-Dimensional Bose GasesEb =?~2ma2 (1.2)which is proportional to (B?B0)2. The importance of near resonance regimeis the existence of universal properties. Here, the state can be described onlyin terms of the scattering length independent of details of the interaction.In this thesis, we are interested in physics of two- and three-dimensionalultra cold gases initially prepared in the metastable upper branch near Fes-hbach resonance. In the theoretical models presented in this thesis, theinteraction between atoms is solely contact and therefore corresponds tothe limiting case of a broad Feshbach resonance where the atomic (open)channel dominates the physics.1.2 Three-Dimensional Bose GasesThe experimental ability to vary the Hamiltonian parameters of ultra coldatomic gases provides novel opportunities to create strongly interactingsystems exhibiting phenomena previously associated only with condensedmatter. In particular, the control over the magnitude and sign of the ef-fective atom-atom interaction is achieved using Feshbach resonances [13].In recent years, taking advantage of this tuning knob, a few experimentshave been carried out to understand the properties of Bose gases near res-onance [14, 17?19]. As attractive Bose gases are unstable at low tempera-tures, the primary focus of these experiments was to explore the propertiesof a repulsive molecule-free Bose gas, commonly referred as ?upper branchphysics?. The results of these experiments suggest that when a resonanceis approached from the side of small positive scattering lengths, Bose atomsin the upper branch form a metastable condensate as they appear to equi-librate on a short timescale compared to the one set by which atoms arelost.Meanwhile, on the theoretical front, even though the properties of quan-tum degenerate gases far from resonance are well described by existing dilutegas theories [7?12, 20], near resonance, very little is known. Very few theo-retical works have tried to address the physics of Bose gases at large positivescattering lengths. In fact, early numerical Monte Carlo simulations mainlyexplored the physics of repulsive bosonic systems [21, 22]. More recently, nu-merical energy minimization studies, conducted in a truncated Hilbert spaceof experimental relevance, consistently pointed out the near fermionizationof Bose gases close to resonance [23?25]. By fermionization, we mean that41.2. Three-Dimensional Bose Gasesthe physical properties of the Bose gases, such as the momentum distribu-tion and the spatial density resemble that of a Fermi gas. However, whetherthe minimum of the restricted energy landscape still remains metastableonce the full Hilbert space is considered remains to be addressed.The complexity of this topic to a large extent appears to be two-fold:1) the role of few-body physics (two-body, three-body states etc) inmany-body systems. In other words, to what extent does the underlyingfew-body physics influence the many-body correlations and which channeldictates the many-body properties near resonance?2) the effect of many-body background (condensate) on the few-bodystructures. How are the few-body structures or multiple scatterings affectedby the presence of many other identical particles?These two issues can be treated successfully and separately in the usualdilute limit. For instance, in the leading order, one can neglect the effect offinite-density background (condensed) atoms on multiple scatterings or un-derlying few-body structures; the energy density can therefore be calculatedperturbatively by assuming the few-body scatterings are given by their prop-erties in the vacuum and applying the low density expansion. The effects offew-body structures on many-body physics can be explored perturbatively.One can also further study the leading effect of quantum gases on two- andthree-body bound states because the many-body states in the dilute limitare well-known. However, near resonance, these two issues are genericallyentangled and ideally have to be addressed self-consistently, which usuallybecomes very challenging.In light of these theoretical challenges, and motivated by the recent ex-perimental realizations of upper branch metastable condensates, we devel-oped a novel non-perturbative self-consistent approach discussed in chapters2 and 3 to explore the physics of long-lived condensates beyond the dilutelimit. Our work highlights that a quantum gas at a positive scattering lengthnear resonance is not always equivalent to a gas of effectively repulsive atoms.This idea will be further developed in this thesis.1.2.1 Current Experimental StatusDespite the clear need for a solid experimental understanding of a stronglycorrelated bosonic fluid, fulfilling this task, in a cold atom context, hasproven extremely challenging. In contrast to Fermi gases [26, 27], for Bosegases, difficulties arise as increasing the scattering length to very large pos-itive values is accompanied by a catastrophic loss of atoms [28]. Neverthe-less, using techniques requiring the gas to be probed for only a short time51.2. Three-Dimensional Bose Gasescompared to the time needed for the system to become unstable , a fewexperimental groups succeeded in exploring Bose gases beyond the dilutelimit [14, 17?19]. We distinguish below two different regimes: the dilutelimit, and the strongly interacting regime.Physically speaking the dilute limit is characterized by the inequalityna3 ? 1 where a is the s-wave scattering length and n the atomic density.In this regime, the ground state energy density isEV =4?~2an2m (12 +6415???n a3 + . . . ) (1.3)where V is the volume of the system and m is the mass of the bosonicatoms. The first term in Eq. (1.3) is the mean-field energy density while thesecond term, the Lee-Huang-Yang (LHY) correction [8], describes the effectof quantum fluctuations.To the best of our knowledge, Ref. [17] is the first experimental workwhich considered beyond-mean-field effects. The main focus of this experi-ment was to probe the excitation spectrum of a gas of 85Rb atoms using aspectroscopic technique. For a < 300a0, where a0 is the Bohr radius, theirfindings agreed well with mean-field predictions. However, for larger scat-tering lengths, their results deviated significantly from its mean-field value.These measurements agreed somewhat qualitatively with the predicted Beli-aev corrections [9]. This experiment was followed by the study of a Feshbachresonance in a gas of 7Li [18]. This study mapped the Feshbach resonancein a very large range of interaction strengths, and identified a region wheremean-field predictions were inapplicable. In turn, this experiment was fol-lowed by a third one which carried out a quantitative measurement of thethermodynamical equation of state of a strongly interacting Bose gas of7Li [14]. Using density profile measurements, and assuming that all mea-surements were done in the zero-temperature regime, this study found, forscattering lengths between 700a0 and 2150a0, the equation of state to bewell described by LHY theory. Moreover, Ref. [14] probed physics beyondthe dilute limit, these results will be further discussed in the next section.Finally, the most recent experiment used a fast-probing technique to investi-gate local many-body equilibrium in a trapped gas of 85Rb atoms [19]. Usingradio-frequency spectroscopy, they measured the two-body contact, C2, anextensive thermodynamical variable related to the derivative of the total en-ergy of the system as a function of the scattering length and the ultra-violetproperties of the momentum distribution function. These universal contactparameter has been first introduced for fermionic systems in Ref. [29] andlater on authors of Refs. [30, 31] utilized it to establish an exact expression61.2. Three-Dimensional Bose Gasesfor the energy density. In this experiment, the contact measurements werefound to be larger than the mean-field predictions but not as large as the val-ues predicted when LHY corrections were included. In addition, in regionswhere beyond mean-field effects were expected, no measurable contributioncoming from three-body physics was found.The strategy commonly used to access the strongly interacting limit con-sists in preparing an equilibrated weakly interacting gas and to later rampup the interaction strength to the desired value. Consequently, as the scat-tering length is increased, the dilute limit condition, na3 ? 1, stops beingfulfilled. Probing such a strongly interacting system is fairly complicated as,near resonance, the timescales of recombination processes become compara-ble to the equilibration time. Recombination process is a scattering event inwhich three atoms interact and form a diatomic molecule and a free atom.In this process, the binding energy is released into the kinetic energy ofthe outgoing two-body bound state and the third atom which leads to im-mediate trap loss. Hence, ramping up the scattering length cannot be doneadiabatically, and non-equilibrium effects must be taken into considerations.The study presented in Ref. [14] considered these effects in order to extractuniversal thermodynamical properties near unitarity (near Feshbach reso-nance where the physics is universal and it only depends on the scatteringlength). Under the assumption of universality, near resonance, the equationof state should be given by ? ? ~2mn2/3 as the only relevant length scaleshould be the inter-atomic spacing n?1/3 where ? is the chemical potential.This expression is identical, up to a multiplicative factor, to the equationof state for a Fermi gas and can be written as ? = ??F where ?F is theFermi energy at the same density. At large scattering length, due to thegas metastability, the authors of this work deduced a lower bound for thevalue of ? by extrapolating their data at unitarity. They found ? > 0.44(8)meaning that the system is nearly fermionized.1.2.2 Current Theoretical UnderstandingIn the last six decades, various theoretical frameworks were developed tostudy three-dimensional Bose gases in the dilute limit. In this regime,na3 ? 1, the average distance between the atoms is much larger than thescattering length (corresponding, in this limit, to the interaction strengthbetween bosons). The mean-field description for these systems was devisedby Bogoliubov in 1947 [7] and the energy density is given by the first term ofEq. (1.3). Later on, Lee, Huang and Yang [8] found the first correction to themean-field expression. In fact, the first corrections to the chemical potential,71.2. Three-Dimensional Bose Gasescondensate fraction, and energy density are all proportional to?na3, theperturbation parameter in the dilute limit. It is worth noting that Beliaev,using field theoretical methods, rederived LHY-type corrections and furtheranalyzed the excitation spectrum [9]. In addition, Wu [10], Sawada [11, 12]and later Braaten et al. [20] succeeded in calculating the next order correc-tions to the mean-field result.Unfortunately, these corrections are only valid for a dilute gas, as beyondthis limit, na3 is not a small parameter. The perturbation method presentedin the previous section is then inapplicable as all terms of the expansion arediverging. Therefore, one needs to define a whole new framework to studythe behavior of Bose gases near resonance where the scattering length isextremely large.In general, one can classify the scattering processes between atoms interms of (a) the order of perturbation in a given parameter (for example?na3) or (b) the number of virtual atoms involved in these processes. In(a), one takes into account all processes of the same perturbative order andconsequently the number of virtual particles involved is not fixed. On theother hand, using the classification explained in (b), one takes into accountall processes involving a fixed number of virtual particles and sums over allorders in the perturbative parameter of classification (a). Using this secondclassification method, one can find the contribution of irreducible M -bodyprocesses to the thermodynamical properties of a system beyond the diluteregime. For example, in this case, the total energy density of the system iswritten asEV =??M=2nM0M ! gM (n0, ?) (1.4)where n0 is condensate density and gM is the irreducible M -body interactionpotential. Consequently, classification (b) is not a perturbative approach andwe claim that the sum over the irreducible M -body processes is convergingrapidly (See Sec. 2.2 for more details). To verify this statement, we comparethe effect of the first and second terms of Eq. (1.4) on the self-consistent valueof the chemical potential. This approach is motivated by the observationthat, in the first order in perturbative parameter?na3, the dominatingpart of the LHY correction comes from irreducible two-body interaction,involving only two virtual particles, and that the combined effect of theother processes, involving more than two virtual atoms, is less than onepercent. In addition, in the recent study in Ref. [32], using ?-expansionnear four dimensions, it was found that near resonance the physics of theseultra cold gases is mainly dictated by two-body forces and the contribution81.2. Three-Dimensional Bose Gasesof the scattering processes with more than two virtual particles involved inthe interaction is negligible when ? goes to zero. This study will be explainedin more detail in chapter 2.Then, using the analytical expression of g2(n0, ?) in conjunction withthe number equation (which sets the number of particles in the system),one can find a self-consistent equation for the chemical potential.We present here the main features and predictions emerging from theself-consistent framework at zero temperature explained above, and comparethese results with other existing theories. Within this self-consistent ap-proach, we found, as a first salient result, that the system is nearly fermion-ized as the maximum of the chemical potential is almost 93% 1 of the Fermienergy (see chapter 2). This effect has also been identified in other theoreti-cal studies, although the ratios of fermionization were different. In Ref. [23],using the lowest order constrained variational method with a modified Jas-trow wave function, they found in the dense limit that the chemical potentialis equal to 2.92?F , exceeding complete fermionization. By comparison, us-ing a truncated Hilbert space variational method, Ref. [24] found that thechemical potential saturates at a value of 0.8?F , while in Ref. [33], usinga renormalization group approach, the fermionization ratio near resonancewas found to be 0.66?F . Finally, more recently, a sophisticated variationalmethod found the Bose gas to be fermionized at 83% [25].A clear advantage of the self-consistent approach discussed in chapters2 and 3 is that it agrees in the dilute limit with LHY results. In fact,99.96% of LHY corrections to the chemical potential are reproduced by onlytaking into account all the irreducible two-body processes (i.e. scatteringprocesses involving only two virtual atoms). In Fig. 3.1(a), one sees thatthe dashed line representing the chemical potential obtained from the mean-field and LHY correction overlaps with the self-consistent results for smallan1/3. Note that in the atom-molecule field theory calculations presentedin Refs. [25, 33], the sign of the correction of order?na3 is opposite to theLHY effect. However, in Ref. [25], the LHY effect is reproduced correctly intheir numerical simulation.For negative bare interactions, there are two channels on the side of theresonance where the scattering length goes to positive infinity, namely themolecule channel and the atom channel, known as the upper branch. Whilethe molecule channel is lower in energy, one can prepare the Bose gas inthe upper branch at very small positive scattering length. Such an upper1The value of ? = 0.93 found in self-consistent diagrammatic approach for three-dimensional Bose gases is for a range of three-body parameters relevant for cold atoms.91.2. Three-Dimensional Bose Gasesbranch state can remain stable during a long time before collapsing to themolecule channel. Using the self-consistent method one can show that, whenthe scattering length is increased, the system will become unstable towardsthe formation of molecules at the maximum of the chemical potential beforereaching the resonance point. Unfortunately, this approach does not offera complete picture of how molecules and atoms interact. Technically, thisinstability appears in these calculations as the solution for the chemicalpotential becomes complex. The imaginary part of the chemical potentialis interpreted as the hybridization of the molecule and atom channels. Inour study, this instability was first pointed out to be correlated with theoccurrence of a maximum in the chemical potential.These highly non-trivial results were also obtained using a renormaliza-tion group approach by looking at the running of the two-body couplingconstant at different energies (see chapter 3). This study shows that whenthe resonance is approached from small positive scattering length in the up-per branch, at a given scattering length, the effective interaction betweenthe atoms becomes attractive and the atomic and molecular channels hy-bridize. We found the effective two-body interaction as a function of thescattering length to be given byg2 =4?~2m11/a??2m?/~2. (1.5)One sees from the above expression that, when a is sufficiently large, thesecond term in the denominator dominates, and g2 becomes negative. Thisresult is strikingly different from the picture commonly accepted by thecold atom community where it is usually thought that, for negative bareinteractions, where scattering length is positive, the atoms repel each other.A key concept we are going to focus on is the effective interaction be-tween condensed atoms near resonance. The common belief is that althoughthe underlying short range resonance interaction has to be attractive, at lowenergy scales the two-body interaction is effectively repulsive when the scat-tering length a is positive. The argument runs as the following. The phaseshift due to a short range attractive potential is given as ? = ? arctan kawhich yields ? = ?ka when k is small. The phase shift approaches ??/2when ka is much bigger than unity but much smaller compared to a/R?,R?(? a) is the range of the attractive interaction. The low energy phaseshift ? = ?ka turns out also to be the phase shift of any repulsive inter-action with the same positive scattering length. And so the atoms interacteffectively repulsively if one is interested in the scatterings at small k. How-ever, the phase shifts of a short range attractive potential and a repulsive101.2. Three-Dimensional Bose Gasesinteraction can become significantly different whenever ka is bigger thanunity; for instance, for a hard-core potential with radius R = a, ? = ?kafor all momenta, differing from the value of ??/2 for attractive potentialswhen ka ? 1. Near resonance, a approaches infinity and the issue of theeffective interaction between condensed atoms becomes very subtle. Indeed,near resonance we show that condensed atoms no longer interact with aneffective repulsive interaction even when the scattering lengths are positive,a somewhat surprising conclusion to many.In fact, in chapters 2 and 3 we showed that the behavior of the system isset by the effective two-body interaction which becomes negative for suffi-ciently large positive scattering length. The emergence of this instability isa many-body effect which modifies the two-body physics. The effect of thebackground (condensed) atoms on two-body bound state energy is to shiftthe bound state channel by an amount proportional to the chemical poten-tial for a range of scattering lengths we are interested in. In vacuum, theenergy of the bound state touches zero at resonance. However, adding theeffect of the background, one sees that the energy of the bound state nowtouches zero at a finite and positive scattering length far before resonance.At this point, the system becomes unstable and hybridization between themolecule and atom channels occurs.Finally, in chapter 2, the effects of the three-body interactions on thechemical potential are also considered. These effects are obtained by takinginto account the second term, g3, in the energy expansion (Eq. (1.4)) 2. Forthree-dimensional gases, including this three-body potential does not changequalitatively the form of the solution. In fact, it only shifts the position of theinstability point. The effects of the three-body interaction were calculatedto be around a few percent near the instability point. compared with theother studies cited earlier, the study presented in chapter 2 is the only workthat investigated the effect of Efimov physics beyond the dilute limit.By Efimov physics, we mean the three-body physics of three-dimensionalBose gases studied by Efimov. He showed that there are three-body boundstates on the both sides of the resonance. He predicted that near resonance,there would be an infinite sequence of weakly three-body bound states with auniversal scaling behavior. Each successive Efimov state is weaker in bindingenergy by a universal factor of 515 [34].The self-consistent approach produces various interesting predictions. To2In chapters 2 and 3, three-body recombination effects have been excluded in orderto explore the thermodynamics of a quasi-static upper branch condensate on a relativelyshort timescale.111.3. Two-Dimensional Bose Gasessummarize, using this method, one finds that an instability sets in at a posi-tive critical scattering length beyond which the near-resonance Bose atomicgas becomes strongly coupled to molecules, and loses its metastability. Nearthis point of instability, the chemical potential reaches a maximum whosevalue is interpreted as the fermionization ratio. In addition, where near-resonance physics sets in, the effect of the three-body forces were estimatedin three dimensions to vary the value of the chemical potential by only a fewpercent. These three-body forces were considered for a range of parametersrelevant for cold atoms.Two of these predictions are consistent with current experimental find-ings. In Ref. [14], the lower bound of the chemical potential was deduced tobe 0.44(8)?F . This result is in agreement with the self-consistent approachwhich predicts a fermionization ratio of 93% near the instability point (seechapter 2). Additionally, in a recent attempt to understand Bose-Einsteincondensates close to resonance, Ref. [19] used radio-frequency spectroscopyto probe the effects of Efimov physics. In the accessible region beyond thedilute limit, three-body effects were found to play no significant role. Thisobservation is consistent with the theoretical prediction obtained in chapter2. In this study, three-body effects were estimated to be negligible comparedto two-body contributions in the dilute limit as well as at the point wherenear-resonance physics fully sets in.1.3 Two-Dimensional Bose GasesTwo-dimensional quantum many-body systems have been an interestingtopic for both condensed matter and nuclear physicists for many years. Af-ter realization of confined quantum Bose gases to two-dimensional geome-tries [35?38], the cold atom community also got attracted to this subject.However, to this day, most of these experimental studies have investigatedthese cold gases near the Berezinskii-Kosterlitz-Thouless phase transitionwhere quasi-condensates are destroyed by thermal fluctuations [39?41]. Nev-ertheless, the fundamental properties of two-dimensional Bose gases nearabsolute zero, where quantum fluctuations prevail thermal effects, have notbeen completely explored yet. To be more specific, there have been a verylimited number of studies conducted on properties of 2D Bose gases nearthe Feshbach resonance, both on theoretical and experimental aspects.Two-dimensional Bose gases have two important advantages in compar-ison with 3D Bose gases near resonance. The first advantage is the fact thatthe ratio between elastic and inelastic collision cross sections could be in-121.3. Two-Dimensional Bose Gasescreased by a significant amount when atoms are confined to two-dimensionalgeometries and as a result, the atom loss could be reduced [42]. The secondadvantage is the universal nature of trimers (three-body bound states) andfew-body physics in these gases as two-body binding energy is the only rel-evant energy scale of the spectrum. This universality implies that physicsof trimers in 2D is independent of short distance property of three-bodyscatterings [43?46], in contrast to the physics of Efimov states in 3D wherethe absolute energy scale is set by the ultraviolet physics of three-body in-teractions [34].Most of the previous studies on two-dimensional Bose gases are carriedout in the regime where the range of the repulsive interactions or the coresize of the hard-core bosons, a0, were much smaller than the average distancebetween the particles [47?49]. Accordingly, the results of these works areonly applicable when???1ln(na20)???(n is the density of bosons) is much smallerthan unity. This regime is known as the dilute limit of the two-dimensionalBose gases. In our work presented in chapter 4, we focused on the physicsbeyond the dilute limit for a 2D Bose gas prepared on the upper branchand interacting via strong contact interaction. This setup could be obtainedexperimentally by using Feshbach resonance and optical confinement (trapgeometry) [50?52]. To study 2D Bose gases near resonance, we introduceda 2D effective scattering length a2D that is defined as the position of thenode in the wave function for two scattering particles and it is also identifiedas the size of two-body bound state. In principle, this parameter could betuned to be larger than inter-atomic spaces and even be infinite.We applied the same self-consistent approach introduced in Sec. 1.2 toexamine the contribution of two- and three-body interactions to physics of2D Bose gases beyond the dilute limit. Our studies show that there is a com-petition between three-body attractive interactions and two-body repulsiveinteractions that determines the behavior of a Bose gas near resonance. Wealso found that the energetics of these gases in large scattering lengths areuniversal as they are fully characterized by the effective 2D scattering length.Finally, we looked into the behavior of the chemical potential for a widerange of interaction strengths. We found that the chemical potential firstincreases with a2D but very quickly reaches a maximum at 1ln(na22D) = ?0.135beyond which the Bose gas develops a negative compressibility, where thegas is unstable in size and finally shrinks to a droplet phase. Increasing a2Dfurther brings about an onset instability at 1ln(na22D) = ?0.175. We identifiedboth critical values to be a direct consequence of the significant role playedby three-body attractive interactions.131.3. Two-Dimensional Bose GasesOne might wonder how good is our truncation of diagrams by only keep-ing two- and three-body interactions in our calculations. In Ref. [53], usingvariational quantum Monte Carlo method, the physics of two-dimensionalBose gases beyond dilute limit is investigated. In this study, the inverse com-pressibility of the system is calculated for a wide range of???1ln(na20)???and it isshown that the compressibility becomes negative beyond 1ln(na22D) ? ?0.31.The inverse compressibility becoming negative is interpreted as the appear-ance of an instability in the system. This result is consistent with ourobservation of the maximum point in the chemical potential beyond whichthe compressibility is negative. The ratio of the contribution to the chemicalpotential due to three-body interactions to the one due to two-body inter-actions is negligible for very small scattering lengths, deep in dilute limit,when the physics is mainly dictated by two-body interactions. This ratiogrows fast when the gas approaches the resonance, identifying the increas-ingly prominent role played by three-body attractive interactions. Withinour approach, we can estimate the contributions from three-body interac-tions to the two-body ones to be around 27% near the maximum of chemicalpotential and 73% in the vicinity of the onset instability.14Chapter 2Nature of Three-DimensionalBose Gases: Self-ConsistentApproach2.1 IntroductionRecently, impressive experimental attempts have been made to explore theproperties of Bose gases near the Feshbach resonance [14, 17?19]. In theseexperiments, it has been suggested that when approaching the resonancefrom the side of small positive scattering lengths in the upper branch, Boseatoms appear to be thermalized within a reasonably short time, well beforethe recombination processes set in, and so to form a quasi-static condensate.Furthermore, the life-time due to the recombination processes is much longerthan the many-body time scale set by the degeneracy temperature. Thisproperty of Bose gases near resonance and the recent measurement of thechemical potentials for a long-lived condensate by Navon et al. [14] motivateus to make further theoretical investigations on the fundamental propertiesof Bose gases at large scattering lengths.The theory of dilute Bose gases has a long history, starting with theBogoliubov theory of weakly interacting Bose gases [7]. A properly regu-larized theory of dilute gases of bosons with contact interactions was firstput forward by Lee, Huang, and Yang [8] and later by Beliaev [9, 54], whodeveloped a field-theoretical approach. Higher-order corrections were fur-ther examined in later years [10, 12, 20]. Since these results were obtainedby applying an expansion in terms of the small parameter?na3 (here n isthe density and a is the scattering length), it is not surprising that, formallyspeaking, each of the terms appearing in the dilute-gas theory diverges whenthe scattering lengths are extrapolated to infinity. As far as we know, re-summation of these contributions, even in an approximate way, has beenlacking 3. This aspect, to a large extent, is the main reason why a quali-3Resummation is possible for two scattering atoms in a box of size L. The interaction152.1. Introductiontative understanding of Bose gases near resonance has been missing for solong.There have been a few theoretical efforts to understand the Bose gases atlarge positive scattering lengths. The numerical efforts have been focused onthe energy minimum in truncated Hilbert spaces, which have been argued tobe relevant to Bose gases studied in experiments [23?25]. These efforts areconsistent in pointing out that the Bose gases are nearly fermionized nearresonance. However, there are two important unanswered questions in theprevious studies. One is whether the energy minimum found in a restrictedsubspace is indeed metastable in the whole Hilbert space. The other equallyimportant issue is what is the role of three-body Efimov physics in the Bosegases near resonance.Below we outline a non-perturbative approach to study the long-livedcondensates near resonance. We have applied this approach to explore thenature of Bose gases near resonance and to address the above issues. Oneconcept emerging from this study is that a quantum gas (either fermionicor bosonic) at a positive scattering length does not always appear to beequivalent to a gas of effectively repulsive atoms; this idea, which we believehas been overlooked in many recent studies, plays a critical role in ouranalysis of Bose gases near resonance.Our main conclusions are fourfold: (a) energetically, the Bose gases closeto unitarity are nearly fermionized, i.e., the chemical potentials of the Bosegases approach the Fermi energy of a Fermi gas with equal mass and den-sity; (b) an onset instability sets in at a positive critical scattering length, be-yond which the Bose gases appear to lose the metastability as a consequenceof the sign change of effective interactions at large scattering lengths; (c)because of a strong coupling with molecules near resonance, the chemicalpotential reaches a maximum in the vicinity of the instability point; (d) atthe point of instability, we estimate, via summation of loop diagrams, theeffect of three-body forces to be around a few percent.Feature (a) is consistent with previous numerical calculations [23?25];both (b) and (c) are surprising features, not anticipated in the previousnumerical calculations or in the standard dilute-gas theory [8, 54]. Ourattempt here is mainly intended to reach an in-depth understanding of theenergetics, metastability of Bose gases beyond the usual dilute limit as wellas the contributions of three-body effects. The approach also reproducesquantitative features of the dilute-gas theory. In Sec. 2.2 and Appendicesenergy should scale like 4piamL3 (1 + AaL + ...) when a is much less than L (A is a constant)but generically saturate at a value of the order of 1/2mL2 when a becomes infinite.162.2. Chemical Potential, Metastability and Efimov EffectsA-C, we outline our main calculations and arguments. In Sec. 2.3, wepresent the conclusion of our studies.2.2 Chemical Potential, Metastability andEfimov EffectsThe Hamiltonian we apply to study this problem isH =?k(?k ? ?)b?kbk + 2U0n0?kb?kbk+ 12U0n0?kb?kb??k +12U0n0?kbkb?k+ U0???n0?k?,qb?qbk?+q2 b?k?+q2 +H.c.+ U02??k,k?,qb?k+q2 b??k+q2bk?+q2 b?k?+q2 +H.c. (2.1)Here ?k = |k|2/2m, and the sum is over nonzero momentum states. U0 isthe strength of the contact interaction related to the scattering length a viaU?10 = m(4?a)?1 ???1?k(2?k)?1, and ? is the volume. n0 is the numberdensity of the condensed atoms and ? is the chemical potential, both ofwhich are functions of a and are to be determined self-consistently. AboveHamiltonian is generated by explicitly putting the momenta of some of thecreation and annihilation operators equal to zero. For some of the terms,there are more than one way to set the momenta equal to zero. These choicescause different numerical prefactors in front of the terms in the Hamiltonian.For example, the second term in the Hamiltonian is produced by setting themomentum of one of the creation and one of the annihilation operators equalto zero. This term represents the interaction between one condensed atomand one non-condensed atom as incoming particles and one condensed atomand one non-condensed atom as outgoing particles. There are four differentways to produce this term and therefore the numerical prefactor in front ofthis term is 2 instead of 1/2.The chemical potential ? can be expressed in terms of E(n0, ?), theenergy density for the Hamiltonian in Eq. (2.1), with n0 fixed [9, 55];? = ?E(n0, ?)?n0, E(n0, ?) =??M=2gM (n0, ?)nM0M ! , (2.2)172.2. Chemical Potential, Metastability and Efimov Effectswhere gM (M = 2, 3, ...) are the irreducible M -body potentials that we willfocus on below (See Fig. 2.1). The density of condensed atoms n0 is furtherconstrained by the total number density n asn = n0 ??E(n0, ?)?? , (2.3)In the dilute limit, the Hartree-Fock energy density is given by Eq. (2.2),with g2 = 4?a/m and the rest of the potentials gM ,M = 3, 4... set to zero.The one-loop contributions to gM for M = 3, 4, ... in Figs. 2.2(c) and 2.2(d)all scale like g2?na3, and their sum yields the well-known Lee-Huang-Yang(LHY) correction to the energy density [8]. When evaluated in the usualdilute-gas expansion, g2 as well as one-loop contributions formally divergeas a becomes infinite. Below we regroup these contributions into effectivepotentials g2,3... at a finite density n0 via resummation of a set of diagramsin the perturbation theory. The approximation produces a convergent resultfor ?.Before proceeding further, we make the following general remark. In thestandard diagrammatic approach [9, 55], the chemical potentials can havecontributions from diagrams with L internal lines, S interaction vertices, andX incoming or outgoing zero momentum lines, and X = 2S?L. For the nor-mal self-energy (?114 ) and the anomalous counterpart (?02) introduced byBeliaev, by classifying the diagrams Hugenholtz and Pines had shown that,in general, the following identity holds [55] in the limit of zero energy andmomentum: ? = ?11??02. Normal self-energy describes processes in whichthe number of particles out of the condensate is conserved (one incoming andone outgoing non-condensed particle) and anomalous self-energy describesthe absorption of two non-condensed particles to condensate. Similarly, mudescribes the processes from condensate to condensate. Hugenholtz-Pinestheorem proves that for a general scattering process with different types ofinternal vertices, zero-energy normal and anomalous self-energies and chem-ical potential differ by the numerical factors and this numerical factor comesfrom the number of different ways to connect the external lines of the di-agram. For normal self-energy there are two ways to connect the externallines to body of the diagram and for anomalous self-energy and the chemicalpotential there is only one way to connect these lines. As a result, at lowenergies ?11 ? ?. Following a very similar calculation, we further find that?11(n0, ?) = ?+ n0???n0, (2.4)4To simplify the notation, in chapters 3, 4 and Appendices we denote ?11 by ?.182.2. Chemical Potential, Metastability and Efimov EffectsFigure 2.1: (a) Classification of M-body scattering processes: A diagramgive contribution to M-body potential if at most M virtual lines of thediagram are cut at time t (dashed red line). All scattering processes aretime-ordered from left to right. (b) Sample of diagrams give contribution totwo- and three-body potentials.192.2. Chemical Potential, Metastability and Efimov Effectswhere ? = ?E(n0, ?)/?n0. The equality in Eq. (2.4) is effectively of a hydro-dynamic origin. Following Eq. (2.4), the speed of Bogoliubov phonons [7]vs can be directly related to an effective compressibility ?n0/?? via mv2s =?11 ? ? = n0 ???n0 , where the first equality is due to the Hugenholtz-Pinestheorem on the phonon spectrum 5. Note that hydrodynamic considerationshad also been employed previously by Haldane to construct the Luttinger-liquid formulation for one-dimensional (1D) Bose fluids [56]. When na3 issmall, Eq. (2.4) leads to the well-known result, ?11 = 2? (see Appendix D).The self-consistent approach outlined below is mainly suggested by anobservation that a subclass of one-loop diagrams [shown in Fig. 2.2(c)] yieldsalmost all contributions in the LHY correction (see below and AppendicesA and B). Resummation of these and their N -loop counterparts can be con-veniently carried out by introducing the renormalized or effective potentialsg2,3 as shown in Figs. 2.2(a) and 2.2(b), where all internal lines represent,instead of the noninteracting Green?s function G?10 (?,k) = ???k+?+i?, theinteracting Hartree-Fock Green?s function, G?1(?,k) = ?? ?k??11+?+ i?.This approximation captures the main contributions to the chemical poten-tial in the dilute limit because the renormalization of two-body interactionsis mainly due to virtual states with energies higher than ? where the Hartree-Fock treatment turns out to be a good approximation (see Appendix D). Theself-consistent equation for ? can be derived by estimating g2,3,...(n0, ?) dia-grammatically (see examples in Fig. 2.2). When neglecting g3,4,... potentialsin Eq. (2.2), one obtains? = n0g2(n0, ?) +n204 g22(n0, ?)? d3k(2?)3??11/?n0(?k +?11 ? ?)2,n = n0 +n204 g22(n0, ?)? d3k(2?)31? ??11/??(?k +?11 ? ?)2,1g2= m4?a +12? dk(2?)3 (1?k +?11 ? ?? 1?k). (2.5)Eqs. (2.4) and (2.5) can be solved self-consistently.We first benchmark our results with the LHY correction or Beliaev?sresults for ? by solving the equations in the limit of small na3. We find? = 4pim n0a(1 + 3?2??n0a3 + ...), and the number equation yields an es-5If we attribute the energy density to the zero-point energy of Bogoliubov phonons, foran arbitrary scattering length, one can, using Eqs. (2.3) and (2.4), express the condensa-tion density as n0 = n ? 1/3?2(??/? lnn0)3/2m3/2. The long-wavelength dynamics thussets an upper bound on the value of ??/? lnn0. The upper bound can be estimated to be21/3?F , where ?F = (6?2)2/3n2/3/2m is the Fermi energy defined for a gas of density n.202.2. Chemical Potential, Metastability and Efimov EffectsFigure 2.2: Diagrams showing contributions to the total energy E(n0, ?).The dashed lines are for k = 0 condensed atoms, thick solid internallines in (a) and (b) are for interacting Green?s functions G?1(?,k) =? ? ?k ? ?11 + ? + i?, and thin solid lines in (c) and (d) are for nonin-teracting Green?s function G?10 (?,k) = ?? ?k+?+ i?. (a) The blue circle isfor g2(n0, ?); vertices here represent the bare interaction U0 in Eq. (2.1). (b)(N = 1, 2, ...)-loop diagrams that lead to the integral equation for G3(?3?, p)in Eq. (2.6). Note that the usual tree-level diagram violates the momentumconservation and does not exist; the one-loop diagram has already been in-cluded in g2(n0, ?) and therefore needs to be subtracted when calculatingg3(n0, ?). Arrowed dashed lines here as well as in (c) and (d) stand for out-going condensed atoms, and the remaining dashed lines stand for incomingones. (c) and (d) The tree level and examples of one-loop diagrams thatyield the usual Lee-Huang-Yang corrections in the limit of small na3. Theself-consistent approach contains contributions from (c)-type diagrams butnot (d)-type ones (see further discussion in the text). Patterned green circlesalso represent the sum of diagrams in (a), but with thin internal lines, orthe noninteracting Green?s function G0 lines. All vertices are time orderedfrom left to right. 212.2. Chemical Potential, Metastability and Efimov Effectstimate n0/n = (1 ??2pi2?na3 + ...). The second terms in the parenthesesare of the same nature as the LHY correction. Comparing to Beliaev?s per-turbative result for chemical potential, ? = 4pim n0a(1 + 403?pi?n0a3 + ...) [9],and for the condensation fraction n0/n = 1? 83?pi?na3 + ..., one finds thatthe self-consistent solution reproduces 99.96%(= 9??2/40) of the Beliaev?scorrection for the chemical potential, and 83.30%(= 3??2/16) of the de-pletion fraction in the dilute limit. Technically, one can further examineg2(n0, ?) by expanding it in terms of a and ?11 and then compare with theusual diagrams in the dilute gas theory [9]. One indeed finds that g2(n0, ?)in Eq. (2.5) effectively includes all one-loop diagrams with X = 3, 4, 5, ...incoming or outgoing zero-momentum lines that involve a single pair of vir-tually excited atoms [between any two consecutive scattering vertices; Fig.2.2(c)]. The one-loop diagrams with X = 4, 5, ... incoming or outgoing zero-momentum lines that involve multiple pairs of virtual atoms [Fig. 2.2(d)]have been left out, but they only count for less than 0.04% of Beliaev?sresult 6.Following the same line of thought, one can also verify that g2(n0, ?)further contains (N = 2, 3, 4, ..)-loop contributions that only involve one pairof virtual atoms; each two adjacent loops only share one interaction vertexand are reducible. g3(n0, ?) included below, on the other hand, includes(N = 2, 3, 4, ..)-loop contributions with S = 4, 5... interaction vertices thatonly involve three virtual atoms; two adjacent loops share one internal lineinstead of a single vertex [see Fig. 2.2(b)] and are irreducible, i.e., cannotbe expressed as a simple product of individual loops. Effectively, we takeinto account all the virtual processes involving either two or three dressedexcited atoms in the calculation of the chemical potential ? by includingthe effective g2,3 (defined in Fig. 2.2) in Eq. (2.2). The processes involvingfour or more excited atoms only appear in g4,5... and are not included here;at the one-loop level following the above calculations, the correspondingcontributions from the processes involving multiple pairs of virtual atomsare indeed negligible.A solution to Eq. (2.5) is shown in Fig. 2.3. An interesting featureof Eq. (2.5) is that it no longer has a real solution once n1/3a exceeds thecritical value of 0.18, implying an onset instability; this is not anticipatedin the dilute-gas theory [8]. So as a approaches infinity, condensed atomswith a chemical potential ? typically see each other as attractive rather6Two diagrams in Fig. 2.2(d) correspond to the lowest-order contribution to the irre-ducible renormalized g4.222.2. Chemical Potential, Metastability and Efimov Effectsthan repulsive, resulting in molecules 7. Thus, beyond the critical pointthe upper branch atomic gases become strongly coupled to the moleculeswith a strength proportional to the imaginary part of ?. Consequently, weanticipate that ? decreases quickly beyond the critical scattering length dueto the formation of molecules, leading to a maximum in ? in the vicinity ofthe critical point 8.A renormalization group approach based on atom-molecule fields wasalso applied in a previous study to understand Bose gases near resonance9 [33].Our results differ from theirs in two aspects. First, in our approach, an onsetinstability sets in near resonance even when the scattering length is positive,a key feature that is absent in that previous study. Second, when extrapo-lated to the limit of small na3, the results in Ref. [33] imply a correction ofthe order of?na3 to the usual Hartree-Fock chemical potential but with anegative sign, opposite to the sign of LHY corrections. In a recent study [25],a self-consistent mean-field equation was employed, leading to a similar con-clusion as the approach in Ref. [33]; the approach does not yield the correctsign of the LHY corrections. And so the onset instability pointed out in ourstudy, which is surprising from the point of view of dilute-gas theory, is alsoabsent there.The chemical potential near the critical point can be estimated usingEq. (2.5) and is close to 0.93?F , where ?F = (6?2)2/3n2/3/2m is the Fermienergy defined for a gas of density n. This is consistent with the pictureof nearly fermionized Bose gases suggested by the previous calculations andexperiments [14, 23?25, 33].We now turn to the effect of g3(n0, ?) on the chemical potential by in-cluding it in Eq. (2.2). We estimate g3 by summing up all N -loop diagramswith X = 3 incoming or outgoing zero momentum lines, which are repre-7One should thus expect an instability in a near-resonance Fermi gas as well. Thepairing dynamics previously emphasized in D. Pekker, M. Babadi, R. Sensarma, N. Zinner,L. Pollet, M. W. Zwierlein, and E. Demler, Phys. Rev. Lett. 106, 050402 (2011) isconsistent with this conclusion. Molecule dynamics in a Bose-Einstein condensate wasalso studied in L. Yin, Phys. Rev. A 77, 043630(2008). Although LHY corrections andthree-body forces were not taken into account in that random phase approximation, themolecule formation discussed there appears to be consistent with our conclusion on theloss of metastability.8Interestingly, a maximum in the Bragg line shift at a finite wave vector was foundwhen the LHY correction is 0.22 in Ref. [17]. The maximum in ? in this paper occurswhen the LHY correction is 0.45.9For field-theory-based approaches to the lower-branch unitary gases, see Y. Nishidaand D. T. Son, Phys. Rev. Lett. 97, 050403 (2006), P. Nikolic and S. Sachdev, Phys.Rev. A 75, 033608 (2007) and M. Y. Veillette, D. E. Sheehy, and L. Radzihovsky, Phys.Rev. A 75, 043614 (2007).232.2. Chemical Potential, Metastability and Efimov Effects0 0.1 0.200.10.20.30.40.50.60.70.80.91(a)n1/3a?/E F102 106?0.0200.02?n?1/30 0.1 0.20.70.750.80.850.90.951(b)n1/3anc/nFigure 2.3: (a) Chemical potential ? in units of the Fermi energy ?F and(b) condensation fraction as a function of n1/3a. Beyond a critical valueof 0.18 (shown as circles), the solutions become complex, and only the realpart of ? is plotted; the imaginary part of ? scales like ?F (a/acr ? 1)1/2near acr. (However, the sharp transition would be smeared out if the smallimaginary part of G3 is included.) Dashed lines are the result of the Lee-Huang-Yang theory, thin solid blue lines are the solution without three-bodyeffects (i.e. g3 = 0). Thick solid red lines are the solution with g3 included;the momentum cutoff is ? = 100n1/3. The inset is the relative weight ofthree-body effects in the chemical potential as a function of ?n?1/3 at thecritical point.242.2. Chemical Potential, Metastability and Efimov Effects10?6 10?4 10?2 100?2?1.5?1?0.500.511.523?/E?Re(G3)?  ?a = ??a = 102Figure 2.4: ReG3(?3?, 0)? as a function of ? = ?11(n0)? ?. E? = ?2/2mand ? is the momentum cut-off. The imaginary part of G3 (not shown) iszero once 3? > 1/(ma2).252.2. Chemical Potential, Metastability and Efimov Effectssented in Fig. 2.2(b). All diagrams have three incoming or outgoing zeromomentum lines but with N = 2, 3, .. loops. The effect of three-body forcesdue to Efimov states [34] was previously studied in the dilute limit [20]. Thedeviation of the energy density from the usual universal structures (i.e., onlydepends on na3 ) was obtained by studying the Efimov forces in the zero-density limit. The contribution obtained there scales like a4, apart froma log-periodic modulation [57], and again formally diverges as other termswhen approaching a resonance.It is necessary to regularize the usual a4 behavior at resonance in thethree-body forces by further taking into account the interacting Green?sfunction when calculating the N -Loop six-point correlators. Including theself-energy in the calculation, we remove the a4 dependence that usuallyappears in the Bedaque-Hammer-Van Kolck theory for the three-body in-teractions [57]; when setting ?,?11 to zero, the equation collapses into thecorresponding equation for three Bose atoms in vacuum, which was previ-ously employed to obtain the ? function for the renormalization flow in anatom-dimer field-theory model. The sum of loop diagrams in Fig. 2.2(b),G3(?3?, p), satisfies a simple integral equation (m set to be unity; see Ap-pendix C):G3(?3?, p) =2??dqK(?3?; p, q)? q2?3q24 + 3? ? 1a[G3(?3?, q) ?1q2 + 3? ],K(?3?; p, q) = 1pq lnp2 + q2 + pq + 3?p2 + q2 ? pq + 3? , (2.6)where we have introduced ? = ?11(n0) ? ?. G3(?3?, 0) is plotted numeri-cally in Fig. 2.4. Three-body potential g3(n0, ?) is related to G3(?3?, 0) viag3(n0, ?) = 6g22ReG?3(?3?, 0) where G?3 is obtained by further subtractingfrom G3 the one-loop diagram in Fig. 2.2(b) because its contribution hasalready been included in g2(n0, ?). The structure of G3(?3?, 0) is particu-larly simple at a = +?, as shown in Fig. 2.4: It has a desired log-periodicbehavior reflecting the underlying Efimov states [34]. When 3? is close to anEfimov eigenvalue Bn = B0 exp(?2?n/s0) [n = 1, 2, 3....,exp(2?/s0) = 515]that corresponds to a divergence point in Fig. 2.4, the three-body forces arethe most significant. When 3? is in the close vicinity of zeros in Fig. 2.4,the three-body forces are the negligible and Bose gases near resonance aredictated by the g2 potential.262.3. SummaryWhen including the real part of g3(n0, ?) in the calculation of E(n0, ?),we further get an estimate of three-body contributions to the energy densityand chemical potential ?. The contribution is non-universal and dependson the momentum cutoff in the problem. For typical cold Bose gases, itis reasonable to assume the momentum cutoff ? in the integral equationEq. (2.6) to be 100n1/3 or even larger. Quantitative effects on the chemicalpotential are presented in Fig. 2.3.Note that G3(?3?, 0) also has an imaginary part even at small scatter-ing lengths; this corresponds to the well-known contribution of three-bodyrecombination. The onset instability discussed here will be further roundedoff if the imaginary part of G3 is included. However, for the range of pa-rameters we studied, both the real and imaginary parts of G3 appear to benumerically small (see also Fig. 2.3); the energetics and instabilities near acrare found to be mainly determined by the renormalized two-body interactiong2(n0, ?).2.3 SummaryIn conclusion, we have investigated the energetics of Bose gases near reso-nance beyond the Lee-Huang-Yang dilute limit via a simple resummationscheme. We have also pointed out an onset instability and estimated three-body Efimov effects that had been left out in recent theoretical studies ofBose gases near resonance [23?25, 33]. In addition, we showed that theBose gases are nearly fermionized before an onset instability sets in nearresonance. The non-perturbative method presented above becomes exact inthe dilute limit if one only keeps the virtual processes involving two atoms.In this limit, this approach reproduces 99.96% of the LHY result for thechemical potential. Near resonance, it was established that the three-bodyEfimov effect only contributes a few percent to the overall chemical po-tential. Unless the contributions coming from M -body interactions are anon-monotonic function of M , we believe the contributions from four-, five-, etc. body interaction processes should be even smaller and hence negligible.The recent study presented in Ref. [32] supports the above argument. Inthis study, the authors combined ?-expansion method near four dimensionswith the self-consistent approach introduced in this chapter to investigatethe physics of Bose gases in higher dimensions. Using this technique, theycould estimate the full expression of the total energy density of the systemincluding all the M -body potentials. They found that near resonance, onlycontribution from the one-loop diagrams are important and higher order272.3. Summaryloops contributions are suppressed by higher powers of ?. Moreover, theyshowed that the two-body interaction potential has the main contributionto the chemical potential of a Bose gas found by solving self-consistent equa-tions, indicating the dominant role of two-body scattering events in higherdimensions. This study implies that the contribution of M -body potentialswith M > 2 becomes less and less important when one goes to higher di-mensions. On the other hand, in our study on two-dimensional Bose gasespresented in chapter 4 we have shown that three-body interaction plays animportant role in the physics of these gases beyond the dilute limit. But,as mentioned in this chapter in three dimensions the contribution of thesescattering events is approximated to be around a few percent near the in-stability point. So, we predict that the non-trivial results we found in thischapter would remain qualitatively unchanged by adding the contributionof M -body interaction potentials with M > 3.28Chapter 3Nature of Three-DimensionalBose Gases: RenormalizationGroup Approach3.1 IntroductionIn this chapter, we make an attempt to understand the fundamental prop-erties of Bose gases near Feshbach resonance via examining the intriguinginterplay between the few- and many-body physics in Bose gases at largepositive scattering lengths. For this purpose, we introduce a simple self-consistent renormalization-group-equation approach to address both sidesof the coin. Many-body properties of a quantum gas are shown to influencethe renormalization flow of few-body running coupling constants resultingin the change of the sign of the effective two-body interaction constants.That in return completely dictates the many-body physics near resonanceand leads to peculiar features in the chemical potentials. We limit ourselvesto the resonances with a very short effective range.The approach outlined in this chapter is an alternative to the more elab-orated diagrammatic resummation mentioned in chapter 2. The two ap-proaches yield almost identical results. In Sec. 3.2, we first carry out asimple scaling argument, as a caricature of resonating Bose gases, illustrat-ing the relevance of fermionization in Bose gases near resonance, and discussthe limitation of the coarse grain procedure. We also remark on a close rela-tion between the effective two-body interactions and Lee-Huang-Yang cor-rections in Ref. [8]. In Secs. 3.3 and 3.4, we discuss the energetics of dimersand trimers in condensates and explore the implications on the many-bodyphysics. Especially, we point out that in addition to the fermionization phe-nomenon, an instability sets in at a positive critical scattering length as asignature of formation of dimers in condensates. We also show this aspect isa consequence of the sign-change of the renormalized two-body interactionsbetween the condensed atoms; the effect of the condensate on the two-body293.2. A Caricatureinteraction constant is investigated via taking into account the self-energyof dimers and via imposing an infrared boundary condition for the renor-malization flow. In Sec. 3.5, we summarize the results of diagrammaticresummation presented in chapter 2. In Sec. 3.6, we conclude our studies.3.2 A Caricature3.2.1 Relevance of Fermionization: A Scaling ArgumentThe energetics of Bose gases near resonance can be qualitatively understoodvia a coarse grain procedure which is more or less equivalent to the realspace renormalization transformation. The simplest implementation of thisis to first divide a quantum gas into N blocks each of which is of the size of? ? ? ? ? where ? = 1/?2m? is the coherence length and ? is the chemicalpotential. Because the chemical potential is a non-additive thermodynamicquantity, it is natural to define it as the change of energy when adding anadditional atom to a particular block and the effect of other blocks is toset an appropriate boundary condition. Therefore, the chemical potentialcan be considered as the interaction energy between the added atom andexisting atoms in the block. If we further assume the interaction energy isdictated by a pairwise one, then? = ?2(?, a;n)n?3 (3.1)where ?2(L = ?, a;n) is the characteristic interaction energy between twoatoms in the block and n?3 is the number of atoms in the block with nbeing the number density. This is a standard coarse grain procedure whichrelates a microscopic quantity ?2(?, a;n) and a thermodynamic quantity,the chemical potential ?. The estimate of ?2(?, a;n) itself is a full many-body problem that is usually very difficult to carry out. In the dilute limit,however, one can show that when two atoms interact in a box of size ?,the probability of being scattered by the third particle is negligible becausethe mean free path l is proportional to 1/n4?a2 (a is the scattering length)much longer than ?. In fact,?l ?a? ??na3 (3.2)which is small in the low density dilute limit. So at least in this limit,we can approximate ?2(L = ?, a;n) as the energy of two interacting atoms303.2. A Caricature?2(L = ?, a;n = 0) in an empty box of the size of the block. If we assume thisis also qualitatively correct even in the unitary limit, then we have a verysimple self-consistent equation; the only knowledge we need to solve thisequation is how two atoms interact in a box of size ? at arbitrary scatteringlength a. ?2(?, a; 0) for a contact resonance interaction can be worked out,either by assuming two atoms are in a harmonic trap of harmonic length ? orin a block of size ?. The asymptotic behaviors are universal up to numericalprefactors. For two atoms in a block of size L,?2(L, a; 0) ={4piamL3 (1 + C1aL + ...) when a ? L;C22mL2 when a ? L.(3.3)C1,2 are two positive prefactors depending on the details of the block andare of little importance for our qualitative discussions here. It is importantto notice that at resonance, ?2(L = ?, a = ?; 0) is finite and scales like thekinetic energy of an atom moving in an empty box of size ?.Substituting the results in Eq. (3.3) into Eq. (3.1), one obtains theestimate of chemical potential. In the dilute limit,? = 4?anm (1 + C1?8?na3 + ...); (3.4)the correction to the first Hartree-Fock term is the leading finite size cor-rection to the interaction energy and belongs to the well-known Lee-Huang-Yang effect. When a approaches infinity on the other hand, this simpleprocedure leads to the prediction of fermionization. That is? = 12m?2 =C2n?2m , or ? ?n2/32m , (3.5)which scales as the Fermi energy of a Fermionic quantum gas with the samedensity and mass. Although crude, the coarse grain shown here points toa phenomenon that was previously seen in a few numerical calculations.Given that it is very simple, we consider it quite a success. The relevance offermionization to Bose gases near resonance was observed in a few theoreticalstudies [23?25, 33].This aspect of Bose gases near resonance is also an essential feature ofTonks-Girardeau gases or hard-core bosons in one dimension [58, 59]. Theone-dimensional Bose fluids were later further studied using the Luttingerliquid formulation [56].313.2. A Caricature3.2.2 Running Two-Body Interaction ConstantsBut how good is the starting point that near resonance we can approximate?2(L, a;n) as the two-particle interaction in an empty box completely ne-glecting the effect of many other identical particles? To address this, weestimate ?2(L, a;n), the interaction of two atoms in a box of size L viaemploying a more sophisticated approach, the real space renormalizationtransformation (RSRT) which further takes into account the many-body ef-fect on ?2(L, a;n). This approach indicates that fermionization cannot bethe whole story.Consider, instead of ?2(L, a; 0) discussed above,g2(L, a;n) = ?2(L, a;n)L3 (3.6)which is the effective strength of the short range two-body interaction. Againwe divide the length scales in RSRT into two regimes that are separated by?: the short distance regime in which two- and few-body physics dominatesand the long wavelength regime where the many-body collective effect dom-inates. ? defines the interface where the microscopic few-body parameter g2at shorter distance needs to match the macroscopic coarse grain condition.So at scales smaller than ?, we can employ the RSRT of the two-bodyrunning coupling constant g2(L, a; 0) in vacuum to monitor the effectiveinteraction. This approximation could be done because the finite densityhas very little effect in this regime i.e. g2(L, a;n) = g2(L, a; 0) + O(L/?).At larger distances, because g2 defined here is subject to a thermodynamicconstraint of the chemical potential at L ? ?, the effect of the condensateon g2 or ?2 is to impose a boundary condition on the flow of g2(L, a;n) viathe coarse-grain condition in Eq. (3.1). And in this approach, the collectivephysics at scales larger than ? influences the flow solely through a simpleboundary condition.Practically, since L defines the size of micro-blocks in the renormaliza-tion procedure, it therefore defines the momentum cut-off via ? = L?1. Thetransformation from L to L? is equivalent to rescaling the momentum from? to ?? = L??1. To obtain the running coupling constant, one can use thestandard momentum-shell renormalization procedure to track the transfor-mation from the original g2 to the new g?2 when the Hilbert space or themomentum cut-off is rescaled from ?? to ? = ?? ? ?? (see Fig. 3.1).The reduced two-body Hamiltonian we use for this illustration is323.2. A CaricatureH2?body = H< +H> +H><H< =?k?kb?kbk +g22??k,k?(b?kb??kbk?b?k? + h.c.)H> =?p?pb?pbp +g22??p,p?(b?pb??pbp?b?p? + h.c.)H>< =g22??k,pb?kb??kbpb?p + h.c.. (3.7)Here b?k(bk) is the creation (annihilation) operator for a Bose atom withmomentum k and ? is the volume. The sum in H< is over momenta |k|, |k?|smaller than ? and the sum in H> is over the states within the shell in Fig.3.1, i.e. |p|, |p?| ? [?,??]. H>< describes the off-shell scattering from lowmomentum states with |k| < ? into the high energy states with momentap within the shell and vice versa. This interaction can induce an effectivescattering between low momentum states (k,?k) and (k?,?k?), |k|, |k?| ? ?,via virtual states (p,?p) within the shell.When rescaling, the states within the shell thus lead to an additionalcontribution to the two-body interaction in H<. One can obtain the beta-function for the renormalization equation diagrammatically. This calcula-tion is very similar to the T-matrix calculation except one should restrictto the virtual states within the shell between ?? and ? = ?? ? ??. The dia-gram in Fig. 3.1 represents such an additional contribution to the effectivetwo-body interaction,?i?g2(??) = i4g2(??)? ? dp(2?)3? d?2?G0(?,p)G0(??,?p),G0(?,p) = 1?? p22m + i?+. (3.8)Here the momentum integral? ?is over the states within the shell shown inFig. 3.1, i.e. ?? > |p| > ?.After carrying out the energy and momentum integrals, one can easilyfind the transformation for the two-body interaction constantg2(?? ? ??) = g2(??)?m2?2 g22(??)?? +O(??2). (3.9)333.2. A CaricatureFigure 3.1: a) Schematic of the renormalization of the low energy on-shellscattering amplitude. An initial state (k,?k) on the inner sphere is firstscattered into off-shell high energy states (p,?p) represented by the shellregion between two outer momentum spherical surfaces (dashed) before fi-nally being scattered back into (k?,?k?) on the inner sphere. Here the outerdashed spherical surface is defined by ?? and the inner dashed one by ?;? ? |p| ? ??. b) The one-loop diagram that leads to the renormalizationequation below. The internal lines are for states within the shell regiondescribed in a). Each vertex stands for the two-body interaction g2(??).343.2. A CaricatureThe transformation of g2 under the real space rescaling can be obtainedby converting ? to L?1. For a quantum gas with a finite density, we thereforehave (g?2 = g2/L)?g?2(L)? lnL?1 =m2?2 g?22(L) + g?2(L),g?2(L = R?) =U0R? , g?2(L = ?) =?n? . (3.10)The boundary condition at L = ? is exactly the condition in Eq. (3.1),i.e. at scale ? the microscopic running coupling constant has to match thethermodynamic constraint suggested by ?, assuming the main contributionto ? is from the two-body interaction g2(L, a;n). At a very short distanceR?,the boundary condition is set by U0, the strength of the bare two-body shortrange attractive interaction with range R?. For the resonance phenomenawe are interested in, R? is always much smaller than a.By contrast, in a vacuum the coupling constant g2 should flow to thevalue of 4?a/m, the standard form of the two-body effective interaction, or?g?2(L)? lnL?1 =m2?2 g?22(L) + g?2(L),g?2(L = R?) =U0R? , g?2(L ??, a;n = 0) =4?amL. (3.11)For a bare attractive two-body interaction with strength U0(< 0) and rangeR?, the boundary condition at L = ? in Eq. (3.11) establishes a well-known relation between U0 and the scattering lengths a. g2 = Lg?2(L) as thesolution to Eq. (3.11) can be expressed in terms of a,g2(L, a;n = 0) =4?am11? 2apiL. (3.12)Obviously, g2 appears to be repulsive only in the limit of long wavelengthwhen L ? a. At short distances R? < L ? a,g2(L, a;n = 0) ? ?2?2Lm (3.13)is negative and independent of R? or a. Eq. (3.13) indicates a universal formof the two-body running coupling constant that induces resonance scatter-ings. This crossover from repulsive to attractive interactions happens atL? ? a.353.2. A CaricatureOne can further show that for a repulsive interaction that leads to thesame zero energy scattering length a, g2 also flows toward the value of 4?a/mwhen L is much longer than the range of interaction. For instance for ahard-core potential with a = R? where R? is the radius of the hard-core,one obtains the same expression as Eq. (3.12) except that the range of L is2a/? < L < ?; and not surprisingly, g2 in this case is repulsive for arbitrarylength scales.So only in the long wavelength limit, the attractive interaction withpositive scattering lengths yields the same physics as the repulsive oneseven though at short distances they are distinctly different. At the zeroenergy, the effective interaction is 4?a/m, repulsive as long as a is positivedisregarding whether the bare interactions are repulsive or attractive. For along time, this has been a common belief in the field of cold atom physics.As we will see below, this no longer holds near resonance when themany-body renormalization effects due to condensed atoms are further takeninto account. The reason for this is that the low energy window where wecan approximate the resonance interaction as a repulsive one (which is oforder 1/ma2) gets so narrow that the effect of condensates on the two-bodycoupling becomes particularly pronounced near resonance.The renormalization-equation approach had been previously applied toanalyze the effective field theories for few-body scattering phenomena [60].They were also successfully employed to identify the coupling constants andquantum phases in the field theory models for the lower branch unitary gases[61?63]. It was later employed to explore the physics of geometric resonancesand confinement induced scattering phenomena [64]. Our application toBose gases is perhaps another excellent example to demonstrate that thesimple and generic approach of renormalization can lead to some surprisingbreakthroughs.Eq. (3.10) is a RSRT equation which satisfies Eq. (3.1) and yieldsan estimate of ?2(?, a;n) or g2. The boundary condition leads to a self-consistent equation for ?. When expressing in terms of a using the solutionto Eq. (3.11), one finds? = n4?am11? 2pi?2m?a. (3.14)Again in the limit where a is much less than ?, the equation yields theHartree-Fock energy plus the correction of Lee-Huang-Yang character;363.2. A Caricature 0 40 0  2  4  6  8  10Y(x)X8piFigure 3.2: The solution to the self-consistent boundary condition,Eq. (3.14). The solution is obtained by solving Y (x) = x2/(n1/3a) ?(2/?)x3 = 8?, where x2 = 2m?/n2/3. From the top to bottom are Y (x) fora < acr, a = acr and a > acr. At acr, the equation has only one solutionand above acr there are no real solutions.373.3. Dimers and Trimers in a Condensate: The Spectrum Flow? = 4?anm (1 +4?2???na3 + ...). (3.15)Another solution of ? scales as 1/a2 consistent with the binding energyof lower branch molecules and we do not consider here. In the unitary limithowever, the equation not only indicates fermionization but also suggests acritical point beyond which there are no real solutions to the equation. Thisis most obvious when a is infinity and g2 becomes negative. This propertyof Eq. (3.14) is illustrated in Fig. 3.2. One can show that at the criticalpoint,n1/3acr =16?1/3, ?cr = 2?4/3n2/3m ; (3.16)two real solutions merge into a single one. Beyond this point, the equationyields a complex solution to the chemical potential.The RSRT suggests an important feature that is absent in the sim-plest coarse grain approach (Eqs. (3.1) and (3.3)). It turns out that nearresonance, there is a substantial modification of the underlying two-bodyphysics, i.e. dimer energetics and therefore the interaction energy betweencondensed atoms ?2; it can no longer be justified to approximate ?2(?, a;n)as the interaction energy in an empty box or at zero density. In fact asshown below, the uplifted dimers (towards condensates) cause an instabilityof atomic condensates when approaching the resonance from the molecularside. The emergency of the imaginary part of the chemical potential be-yond the critical point signifies a hybridization between atoms and moleculeswhich is missing in the simplest coarse grain argument. In the next section,we further elaborate on this fascinating aspect of Bose gases near resonance.3.3 Dimers and Trimers in a Condensate: TheSpectrum FlowHow are the dimers or trimers formed in the presence of a condensate orof a quantum gas? In the context of quantum mixtures, there have beena few attempts to answer this question: how are the few-body structuresaffected by the presence of a Fermi surface [65?68]? Surprisingly, so far littleeffort has been made to understand the dimers and trimers in the presenceof a condensate, partially because the background of a condensate is moredynamical compared to that of a Fermi sea. Since this plays a critical role383.3. Dimers and Trimers in a Condensate: The Spectrum Flowin the interplay between few- and many-body physics that interests us, herewe make an effort to estimate the effect.It is possible to solve the two-body and three-body S-matrices in thepresence of the many-body effect due to the self-energy. Assuming the self-energy of quasi-particles is ?, 10 one finds that for two incoming atoms withmomentum p,?p scattered into p?,?p? and with total frequency E,G2(E;p,p?) = U0 + U0? d3q(2?)31E ? 2? ? 2?q + i?+G2(E;q,p?) (3.17)where ? = ?? ?. A diagrammatic representation is given in Fig. 3.3(a). Inthe dilute limit, ? = ? = 4?an/m; in general, ?, ? and ? are unknown andneed to be determined self-consistently later on. For now, we simply assumethat ? is a given parameter (see Appendix D).G2(E;p,p?) = G2(E) as a result of the short range interaction and notethat when ? = 0 or in vacuum, G02(E = 0) = 4?a/m (superscript 0 indicatesthe case of vacuum) and G02(E) = 4?a/m(1? ia?mE). One can then showthat in the presence of the condensate,G2(E) = G02(E ? 2?); (3.18)the pole of G2(E) is shifted from the pole in vacuum by 2?. The pole definesthe dimer binding energy and so?D = ?0D + 2? (3.19)where ?D and ?0D are the binding energy of dimers in the presence of a con-densate and in vacuum, respectively. At a given positive scattering length,the dimer spectrum flows (in the energy space) towards the zero energywhere the condensate lives as one increases the ?.One can also calculate the amplitude of three-body scatterings corre-sponding to the processes described in Fig. 3.3(b). We consider a generalcase where three incoming momenta are k1 = p/2 ? q, k2 = p/2 + q andk3 = ?p, and outgoing ones are k?1 = p?/2 ? q?, k?2 = p?/2 + q? andk?3 = ?p?. The scattering amplitude between theses states are then givenby A3(E;p,p?) where q and q? do not enter explicitly; it represents the sumof diagrams identical to Fig. 3.3(b).It is more convenient to work with the reduced amplitude G3(E;p) =A3(E;p, 0) where p? is already taken to be zero. G3(E;p) itself obeys a10As explained in chapter 2, ? stands for ?11393.3. Dimers and Trimers in a Condensate: The Spectrum Flowsimple integral equation as can be seen by listing the terms in the summationexplicitly. The diagrams in Fig. 3.3(b) yield (see Appendix E; the mass isset to be one, i.e. m = 1),14G3(E, p) = ?12K(E ? 3?; p, 0)+ 2??dq K(E ? 3?; p, q)q2?34q2 + 3? ? E + i?+ ? 1a?1q2 + 3? ? E+( 2?)2 ?dqdq?(K(E ? 3?; p, q)q2?34q2 + 3? ? E ? 1aK(E ? 3?; q, q?)q?2?34q?2 + 3? ? E ? 1a? ?1q?2 + 3? ? E)+ ? ? ? (3.20)where K(E ? 3?; p, q) is the kernel defined asK(E ? 3?; p, q) = 1pq lnp2 + q2 + pq + 3? ? Ep2 + q2 ? pq + 3? ? E , (3.21)The sum of the above infinite series leads to the following integral equationof G3 asG3(E, p) = ?2K(E ? 3?; p, 0) +2??dq K(E ? 3?; p, q)q2?34q2 + 3? ? E ? 1aG3(E, q).(3.22)When ? = 0 as in vacuum, this equation is identical to an integral equa-tion previously obtained in an atom-dimer model to study the renormalizedthree-body forces [57]. Comparing to G3(E, p) in vacuum when ? = 0, againone finds that the energy of a trimer in a condensate ?T is related to ?0T , itsvacuum value via?T = ?0T + 3?. (3.23)What Eq. (3.23) shows is a simple fact of a condensate. If all the finitemomentum atoms have a mean-field energy shift ? ? ? with respect tocondensed atoms, the energy of few-body bound states (with finite k com-ponents) experiences the corresponding energy shifts. As a consequence,403.4. Sign Change of g2: A Consequence of Spectrum Flowwhen ?T = 0, we should expect that the three-body forces in a condensateshould be divergent. This was observed numerically in our study presentedin chapter 2; the three-body potential is divergent when 3? = ??n where ?nare the Efimov eigenvalues with n = 1, 2, 3, ....What is the consequences of the spectrum flow or the energy shift dueto the condensate? The main consequence is that in a condensate, a dimercrosses the zero energy, or the energy of condensed atoms at a positivecritical scattering length aD or ?D = 0 when2?(aD) =1ma2D, (3.24)where ? itself is a function of aD. By contrast, in vacuum a dimer crossesthe zero energy or the scattering threshold at resonance or a = ?. If wesimply apply the Hartree-Fock approximation ? = 2? = 8?an/m, we find? = ? andn1/3aD = (1/8?)1/3. (3.25)Beyond this point, one has to take into account the hybridization betweenatoms and molecules. The dimer formation in condensates was previouslystudied in a random-phase approximation; those results are qualitativelyconsistent with the picture painted here [69]. Pairing instability and for-mation of molecules in the upper branch Fermi gas was emphasized in Ref.[70].Since it is necessary to have molecules below condensates for the con-densed atoms to have effective repulsive interactions, the penetration ofdimers into the condensate implies a change of the sign of interactions, fromrepulsive to attractive ones that can lead to a potential instability. Belowwe further amplify this aspect.3.4 Sign Change of g2: A Consequence ofSpectrum FlowIn a condensate, the low energy two-body interaction constant is renormal-ized not only by the virtual scattering states as in vacuum but also by theinteractions with the condensed atoms. The latter effect is many-body innature. Below we focus on the particular many-body effect related to theself-energy of virtual states and include this in the renormalization proce-dure. The self-energy of non-condensed atoms is due to scatterings by thecondensate and depends on the interaction strength and density of atoms.413.4. Sign Change of g2: A Consequence of Spectrum Flow= + + +...a)b)+ + +...+Figure 3.3: Diagrams for the calculations of dimer and trimer energy in acondensate. a) is for two-atom channels. Each solid internal line here is fora Green?s function of non-condensed atoms with the self-energy effect takeninto account, G?1(?,k) = ?? ?k ? ?+ ?+ i?+. b) is the loop diagrams forthe three-atom channel.We now apply a self-consistent renormalization group equation (RGE)to investigate this issue. To study the coupling constant, we start with theassumption that the self-energy and the chemical potential of non-condensedparticles are already given as ? and ?. The simplest Hartree-Fock Green?sfunction for virtual atoms is of the formG(?,p) = 1?? p22m ? ?+ ?+ i?+. (3.26)We then calculate g02(?, ?), the zero energy effective interaction betweencondensed particles for a given ? = ? ? ? using a very similar procedureas that in Sec. 3.2 except G0 in Eq. (3.8) now should be replaced with Gdefined here. This replacement effectively takes in the multiple scatteringsby a condensate in two-body scattering processes shown explicitly in Fig.3.3(c). The corresponding renormalization group equation (RGE) for therunning couple constant g?2 = g2(?)?h can be found to be,?g?2(?h)? ln ?h= m2?2 g?22(?h) + g?2(?h),?h = ???2m? arctan ??2m? , (3.27)423.4. Sign Change of g2: A Consequence of Spectrum Flowwhere ?h is the dynamical length relevant to the renormalization transfor-mation and depends on the many-body parameter ?. When ? = 0 as invacuum, g2 flows to the desired value of 4?a/m. With a finite ?, we findthat g2 runs to the following valueg02 = lim?h?0g?2(?h)?h= 4?am11??2m?a (3.28)as ?h becomes zero and all the k 6= 0 virtual states are included in therenormalization transformation. The resultant g02 is the effective interactionbetween condensed atoms after all non-condensed or virtual states are inte-grated out. Eq. (3.28) was proposed in chapter 2 as an effective interactionfor condensed atoms. This is also fully consistent with the RSRT resultpresented in Sec. II.At first sight, the structure of Eq. (3.28) appears to be very similar to thezero-density expression for the two-body running coupling constant g?2(L) inEq. (3.12). However, the physical implication is entirely surprising. Firstof all, g02 , the effective interaction between condensed atoms, now dependson k? =?2m?; so it is now a function of ? ? ?, or the density of the gas,reflecting a many-body effect. In the dilute limit,g02 =4?am (1 +?8?na3 + ...); (3.29)the first term stands for the Hartree-Fock energy and the second one yieldsthe Lee-Huang-Yang type correction to the energy density of Bose gases.Most importantly, unlike in vacuum where the zero energy effective inter-action constant 4?a/m is always positive as far as a, the scattering length,remains positive, in condensates g02(? = 0) is positive only in the dilute limitwhen ?m?a ??na3 ? 1. When approaching the resonance, for a given?? ?, the effective interaction between condensed atoms becomes negativebefore a becomes infinity as indicated in Fig. 3.4. In other words, the pres-ence of a condensate completely alters the flow of the coupling constant atthe low energy limit; it changes the sign of the effective interaction constantnear resonance.The property of Bose gases near resonance is dictated by this change ofthe sign of interactions. In fact as a precursor of this, pure atomic conden-sates lose metastability as seen in chapter 2. Microscopically, the change ofsign of g02 is correlated with and driven by the molecules entering the conden-sate. In the approximation employed here, the sign change occurs exactly433.4. Sign Change of g2: A Consequence of Spectrum Flowwhen the molecules penetrate into the condensates at scattering length aD(see Eq. (3.24)).To further determine ? or ? and ? and understand the effect of the signchange of g02 on the condensate, we should specify a boundary condition inthe RGE. The following steps have to be carried out. Once g02 is found asa function of ? and ?, one can apply it to calculate E(n0, ?), the energydensity of the system with n0 condensed atoms and non-condensed particlesat chemical potential ?. Following the general thermodynamic relations, thechemical potential for the condensed particles ?c should be?c =?E(n0, ?)?n0, E = 12g02n20. (3.30)For the ground state we further require that the condensed atoms are inequilibrium with the non-condensed reservoir at chemical potential ?:? = ?c (3.31)as first suggested by Pines and Hugenholtz [55]. One can verify that Eqs.(3.28),(3.30) and (3.31) are identical to the corresponding self-consistentdiagrammatic equations employed in chapter 2. More explicitly, one findsthat for g02 ,g02n0 +n202?g02?????n0= ?c (3.32)One can view Eq. (3.30) and (3.31) as a boundary condition for g?2(?h)in the RGE in Eq. (3.27) when ?h = 0 and if ??/?n0 is given. To fi-nally solve the equation self-consistently, one needs to supply Eq. (2.4) inprevious chapter to further determine that ??/?n0 = 2?/n0 and the set ofequations produced in this way are identical to the set in chapter 2 deriveddiagrammatically.To illustrate the main features, we now make a few further simplificationswithout losing the generality. One is that we neglect the n0-dependence in? so that ?c = g02n0. Second is that we further approximate n0 as n becausethey are of the same order in the regime of our interest. We then havea single parameter renormalization equation Eq. (3.27) with the followingboundary conditiong02 = lim?h?0g?2(?h)?h= ?n. (3.33)443.4. Sign Change of g2: A Consequence of Spectrum Flow-5 0 5 0  1  5g 2ak?(a)-20 0 20 0  5? d1/an1/3(b)? d? dFigure 3.4: a) g2 (in units of 4?a/m) as a function of k? =?2m? at agiven scattering length a; ? = ? ? ? is a function of density and is equalto 4?na/m in the dilute limit. Note that when ak? =?8?na3 ? 1 or inthe dilute limit, g2 approaches its vacuum value of 4?a/m but deviates fromit substantially once k?a is of order of unity. At resonance when 1/a = 0,g2 is negative for any arbitrary ? implying attractive interactions betweencondensed atoms. b) Illustration of the dimer energy (in units of n2/3/2m)in the presence of a condensate (the upper curve). The dashed line indicatesdimers are no longer well defined because of the coupling to the continuum.As a reference we also show the dimer energy in vacuum (the lower curve).Note that in vacuum, the dimers reach zero energy right at resonance.453.4. Sign Change of g2: A Consequence of Spectrum FlowLast, although ? = ? ? ? in general should be ?? with ? being anunknown but smooth function of a, n0 and ?, in the dilute limit ? = 1 (seeAppendix D). Eq. (2.4) in previous chapter implies that ? varies between 1in the dilute limit and 2/3 in the fermionized limit that interests us. So wecan neglect its variation by simply setting ? = 1 for this part of discussion.Eq. (3.27) and the boundary condition for the RGE in Eq. (3.33) now leadthe following single parameter self-consistent equation for ?,?n =4?am(1??2m?a) (3.34)which, apart from a numerical prefactor in the denominator, is identical toEq. (3.14) which was obtained empirically. The numerical solution of thisis presented in Fig. 3.5.Two essential features are shown in Fig. 3.5. First, the chemical poten-tial reaches a maximum at acr as a precursor of the sign-change of two-bodyinteraction g02 near resonance. The value of the maximum is around 89%?F ,very close to the values obtained in a constrained variational approach [24]and in a diagrammatic resummation approach in chapter 2 (see Table 3.1for details). Here ?F = (6?2)2/3n2/3/2m is the Fermi energy for a gas withthe number density n.Above the critical scattering length, the chemical potential develops animaginary part.Im? = 8(3?)2/3 ?F (aacr? 1)1/2 (3.35)when the scattering length a is increased slightly beyond the critical pointacr = 0.18n?1/3.The drop in the chemical potential beyond the critical scattering lengthacr implies a negative compressibility and hence an energetic instability.This occurs at the same time as the chemical potential becomes complexand an onset dynamic instability sets in. Therefore, a quantum gas com-pletely loses its metastability beyond this critical point. The maximum inthe chemical potential and the correlated emergent dynamic instability re-main to be probed within the current experimental time scales.A different renormalization group approach based on an atom-moleculemodel was also applied in a previous study to understand Bose gases nearresonance [33]. Our results differ from theirs in two aspects. First, in our ap-proach, an onset instability sets in near resonance even when the scatteringlength is positive, a feature that is absent in that previous study. Second,463.4. Sign Change of g2: A Consequence of Spectrum Flow 0 0.2 0.4 0.6 0.8 1 0  0.1  0.2  0.3  0.4?/? Fan1/3Figure 3.5: The numerical solution to the self-consistent equation Eq. (3.34).The chemical potential (the real part) reaches the maximum (blue circle)when n1/3a = 0.18; beyond this point the chemical potential develops animaginary part (dotted line). The dashed line is the chemical potential inthe Lee-Huang-Yang theory. The smooth contributions from the three-bodypotential g3 (not shown here) were studied in the previous diagrammaticcalculations explained in chapter 2 and turn out to be around a few percentof the effect shown here.473.5. Diagrammatic Resummation: A Self-Consistent Approachwhen extrapolated to the limit of small na3, the results in Ref. [33] imply acorrection of the order of?na3 to the usual Hartree-Fock chemical potentialbut with a negative sign, opposite to the sign of LHY corrections and/or ourresults. The results of the self-consistent approach in Ref. [25] are similarto the ones in Ref. [33] but differ from ours. In table 3.1, we make furthercomparisons by listing the main features in different approaches.Fermionized? LHY Efimov Max. in ?,effect physics instabilityCowell et al., Yes No No No2002[23] (2.92?F )?Song et al., Yes No No No2009[24] (0.80?F )Lee et al., Yes No?? No No2010[33] (0.66?F )Diederix et al., Yes No?? No No2011[25] (0.83?F )Borzov et al., Yes Yes Yes Yes???2012[2] (0.93?F )Table 3.1: Comparison of different theory approaches.* The lower bound of ? was measured to be around 0.44?F in the ENSexperiment [14].? The value in the bracket indicates the estimated chemical potential; samebelow. The estimated chemical potential 2.92?F exceeds the result for acompletely fermionized gas.?? In the field theory approaches there, the signs of the correction of theorder of?na3 are opposite to the LHY effect. However, the LHY effect wasreproduced in the numerical program in Ref. [25].??? This is seen both in the diagrammatic resummation and the RG approachoutlined here. Note that 0.93?F is for a range of three-body parametersrelevant to cold atoms.3.5 Diagrammatic Resummation: ASelf-Consistent ApproachFrom a phenomenological point of view, it is quite appealing to generalizethe self-consistent coarse grain relation in Eq. (3.1) by further taking intoaccount the three-body effective interaction g3;483.5. Diagrammatic Resummation: A Self-Consistent Approach? = ng2(?, a;n) +n22 g3(?, a;n) (3.36)where g2,3(L, a;n) are the renormalized two- and three-body interaction con-stants respectively at length scale L and ??1 = ?2m?. If one can calculatethese renormalized quantities, then one is able to obtain ? which includesthe effect of g3. We have proceeded further from here using the renormaliza-tion group equations similar to what was discussed in Secs. 3.2 and 3.3; theyyield qualitatively the same results as the diagrammatic approach presentedin chapter 2. However, when benchmarking against the dilute gas theory,the diagrammatics turn out to be numerically superior; the diagrammaticresummation used in previous chapter reproduces 99.96% of Lee-Huang-Yang corrections in the dilute limit. Here, we review the framework of thediagrammatic calculations and briefly comment on the results.In the diagrammatic approach, we first define the chemical potential ofnon-condensed particles as ? and the number density of condensed atomsas n0. The energy density can be calculated as E(n0, ?) (see below). Thenone should have the following set of self-consistent equations for a gas withtotal number density n,?c =?E(n0, ?)?n0n = n0 ??E(n0, ?)??? = ?c (3.37)where ?c is the chemical potential for the condensed atoms and has to beequal to the chemical potential ? in equilibrium. We further introduce theself-energy ? of non-condensed atoms or virtual particles to facilitate thecalculation of E(n0, ?) that now explicitly depends on ?(n0, ?). Thus, Eq.(3.37) has to be further supplemented by?(n0, ?) = ?c(n0, ?) +??c? lnn0(3.38)which can be proven in the same fashion as the Pines-Hugenholtz theorem[55]. Eq. (3.37) and (3.38) have been applied to obtain the chemical poten-tial in 3D Bose gases near resonance in chapter 2.493.5. Diagrammatic Resummation: A Self-Consistent ApproachCalculations of E(n0, ?) for a given ?(= ? + ?) can be carried out dia-grammatically. If we restrict ourselves to the virtual processes involving onlytwo or three excited atoms and truncate the Hilbert space accordingly, thendiagrammatically we only need to collect the diagrams which contribute tothe effective two- and three-body interaction constants g2,3. As far as thechemical potential is concerned, this truncation turns out to be highly pre-cise in the dilute limit (refer to Sec. 2.3). The result is listed below. Themass m is set to be one:E(n0, ?) =12n20g2(2?) +13!n30g3(3?)g2(2?) = 4?a11? 2pi?2?ag3(3?) = 6g22(3?)Re2??dq K(?2?; 0, q)q2?34q2 + 3? ? 1aG?3(?3?, q)(3.39)where G?3(?3?, p) is a solution of the following integral equationG?3(?3?, p) =2??dq K(?3?; p, q)q2?34q2 + 3? ? 1a[ ?1q2 + 2? +G?3(?3?, q)]. (3.40)?12K(?3?; p, q) is again the one-particle Green?s function projected to theS-wave channel; it is defined asK(?3?; p, q) = 1pq lnp2 + q2 + 3? + pqp2 + q2 + 3? ? pq . (3.41)The numerical solution to these self-consistent equations was shown inchapter 2 and they are qualitatively the same as the solution to the self-consistent RGE for g2 and we are not going to repeat here 11. Here we wantto make a few further comments on the resummation technique.First, g2 defined this way is an effective two-body interaction renor-malized by the condensate and includes a subset of N -body interactions11There are two ways of estimating G3 which slightly differ from each other; the dif-ference is due to singular behavior of the Green?s function at k = 0 in the Hatree-Fockapproximation as commented on in Appendix C. Here we evaluate G3 by first setting allexternal lines to be the condensed atoms.503.6. Summarydefined in the vacuum. At the one-loop level, it yields the most dominatingcontribution; the residue effects are from the irreducible N = 4, 6, ...-bodyinteractions which contains less than one thousandth of the total contribu-tion.Second, the three-body contribution in our self-consistent approach ap-pears to be around a few percent and numerically insignificant. Since whencompared to g2, the contribution from g3 in the dilute limit as well as nearresonance is small, it is reasonable to conjecture that further inclusion g4,5,...would not change our result presented here in a substantial way (refer toSec. 2.3). The truncation of the energy density expression at g3 shouldbe accurate enough for all the practical purposes of studying Bose gasesnear resonance. We hope these statements can be tested in precision mea-surements of chemical potentials as well as in future quantum Monte Carlosimulations.Third, the energy density expression in Eq. (3.39) becomes exact inthe limit where only the processes involving two or three virtual atoms areallowed. Effectively, this is equivalent to truncating the Hilbert space andincluding the correlations up to the trimer channel.3.6 SummaryThe RGE approach is instrumental to our understanding of the emergentphenomena in quantum few- and many-body systems. The application toBose gases near resonance perhaps is another example of what a simple RGEtransformation can lead to. We have applied this approach to understand thenature of Bose gases near resonance and found that energetically, the Bosegases close to unitarity are nearly fermionized before an onset instabilitysets in, i.e. the chemical potentials of the Bose gases approach that ofthe Fermi energy of a Fermi gas with equal mass and density. Beyond theinstability point, the chemical potential has an imaginary part indicatingstrong hybridization with molecules.The model we have employed to study the Bose gases near resonance isa short range attractive potential which has a range much shorter than theinter-atomic distance of the gases or effectively a contact potential. This isa very good approximation of real physical interactions between cold atoms.If the potential is a short range but repulsive , then Bose gases are always inthe dilute limit because the scattering lengths are bounded by the range ofinteractions, disregarding the strength of potential. For bosons interactingwith a repulsive potential but with a range comparable to the inter-particle513.6. Summarydistance, we should anticipate the physics in this limit to be very similarto what happens in liquid 4He [54, 71?75]. The excitation spectrum shoulddevelop roton minima that imply strong short range crystal correlations.When the range of interactions is further increased, eventually there shouldbe a quantum transition to a crystal where all bosons are depleted fromthe condensate. The physics of repulsive bosons and liquid 4He belongto a different universality class which fundamentally differs from what wedescribed in this thesis, i.e. the properties of nearly fermionized Bose gasesnear resonances with a contact interaction.52Chapter 4Nature of Two-DimensionalBose Gases4.1 IntroductionTwo-dimensional quantum many-body systems have been, for many years,a subject of fascination for condensed matter and nuclear physicists alike.More recently, this topic also caught the attention of the cold atom commu-nity with the realization of quantum Bose gases confined to two-dimensionalgeometries [35?38]. These experimental studies have so far explored thesesystems at temperatures close to the Berezinskii-Kosterlitz-Thouless phasetransition [39?41]. They highlighted the loss of long-range order due to theproliferation of vortices above the transition temperature, and the existenceof two-dimensional quasi-condensates with algebraic long-range order andlong wavelength thermal fluctuations below the transition. However, thefundamental properties of two-dimensional Bose gases near absolute zero,where quantum effects are dominant, have yet to be addressed. In partic-ular, on both theoretical and experimental sides, very little work has beencarried out to study two-dimensional Bose gases near resonance. The mainpurpose of this chapter is to provide new light on the properties of two-dimensional Bose gases in this limit.Compared to three-dimensional Bose gases near resonance, which re-ceived more attention in recent years [14, 17?19], two-dimensional gasespossess important advantages. First, the ratio between elastic and inelas-tic collision cross sections can be significantly enhanced when atoms areconfined to two-dimensional traps [42]. Second, in two dimensions, trimersand few-body structures are all universal as the absolute energy scale of thespectrum is uniquely set by the two-body binding energy and is indepen-dent of the short distance property of three-body interactions [43?46]. Thisis distinctly different from the physics of Efimov states in three dimensionsas, in this case, the absolute energy scale is set by the ultraviolet physics ofthree-boson scatterings [34].534.1. IntroductionThese advantages are related to the dramatic suppression of the lowenergy effective interactions and phase shifts by coherent interference intwo-dimensional Bose gases. In fact, for an arbitrary repulsive interaction,the low energy two-body scattering phase shifts are logarithmically small in-dicating an asymptotically free limit. This aspect of scattering theory playsa critical role in the physics of two-dimensional dilute Bose gases. Most pre-vious works on two-dimensional Bose gases considered systems where therange of the repulsive interactions or the core size of the hard-core bosons,a0, were much smaller than the inter-particle distances [47?49]. Conse-quently, the results of these studies are only applicable when 1ln(na20) (n isthe density of bosons) is much smaller than unity, a limit corresponding todilute gases in two dimensions. Here, we focus on the physics beyond the di-lute limit to study two-dimensional Bose gases prepared on the upper branchand interacting via a resonating contact interaction. Such a setup can beachieved experimentally through a combination of Feshbach resonance andoptical confinement [50?52]. Theoretically, to study two-dimensional near-resonance Bose gases, we introduce a two-dimensional effective scatteringlength a2D. This new tuning parameter is formally defined as the positionof the node in the wave function for two scattering particles and is alsoidentified as the size of the two-body bound state. In general, a2D can betuned to values larger than the averaged inter-atomic distance and can evenbe infinite.Our study of two-dimensional Bose gases at large scattering lengths un-veils that near resonance the properties of these gases are primarily dictatedby the competition between three-body attractive interactions and two-bodyrepulsive forces. We also show that the energetics of two-dimensional Bosegases near resonance are universal as they only depend on the parameterna22D. Finally, we investigate the behavior of the chemical potential for awide range of scattering lengths. We find that the chemical potential first in-creases with a2D but very quickly reaches a maximum at 1ln(na22D) = ?0.135beyond which the Bose gas develops a negative compressibility. Increasinga2D further brings about an onset instability at 1ln(na22D) = ?0.175. We iden-tify both critical values to result from the important role played by three-body attractive interactions. In Ref. [53], using variational quantum MonteCarlo method, the physics of two-dimensional Bose gases beyond dilute limitwas investigated. In this study, the inverse compressibility of the system iscalculated for large range of two-dimensional scattering lengths and it isshown that the compressibility becomes negative beyond 1ln(na22D) ? ?0.31,for large gas parameters. The vanishing of inverse of compressibility is in-544.2. Self-Consistent Approach for Two-Dimensional Bose Gasesterpreted as the onset of instability against cluster formation. This resultis consistent with our observation of the maximum point in the chemicalpotential beyond which the compressibility is negative and the instability ofthe system at the second critical point. Within our approach, we can esti-mate the contributions from three-body interactions to the two-body onesto be around 0.27 near the maximum of chemical potential and 0.73 in thevicinity of the onset instability.4.2 Self-Consistent Approach forTwo-Dimensional Bose GasesTo carry out this study of two-dimensional Bose gases, we employ a methodwe previously developed to understand the physics of three-dimensional Bosegases near resonance [2, 3] explained in chapter 2. In this approach, thechemical potential of non-condensed particles, ?, and the density of con-densed atoms, n0, are first introduced as given parameters. The Hamiltoniandescribing such a condensate interacting with non-condensed atoms througha short-range interaction is the same the three-dimensional Hamiltonian inEq.(2.1) except that three-dimensional volume, ?, is now replaced by two-dimensional area, S. Later, we will evaluate n0 and ? self-consistently asa function of the two-dimensional scattering length, a2D, and of the totaldensity n.Once the full system energy density E(n0, ?) is known, one can calcu-late ?c, the chemical potential for the condensed atoms, and n ? n0, thedensity of non-condensed atoms using the thremodynamical relations in-troduced in chapter 2. In addition, in the ground state, one requires ?c,the chemical potential for the condensed atoms, to be equal to the chem-ical potential ?. This equilibrium condition, first emphasized in Ref. [55],yields a self-consistent equation. The evaluation of E(n0, ?) for a given ?and n0 is usually carried out diagrammatically [9, 55]. To capture the roleof three-body interactions and to compare it with two-body contributions,we restrict ourselves to the virtual processes involving only two or threeexcited atoms. Truncating the Hilbert space accordingly, we can then sumup all connected diagrams contributing to the energy density. Within thistruncation scheme, only the irreducible two- and three-body effective inter-action potentials g2,3 appear in the final expression for E(n0, ?). In orderto implement the self-consistency condition and simplify the computationof E(n0, ?), we introduce for the non-condensed or virtual atoms an addi-tional parameter ? = ? ? ? where ?(n0, ?) is the self-energy. Physically, ?554.2. Self-Consistent Approach for Two-Dimensional Bose Gasescan be understood as an energy shift due to the interaction between con-densed and non-condensed atoms. Using the same series of diagrams as inour study of three-dimensional Bose gases near-resonance, but carrying outthe calculations in two spatial dimensions, we obtain for the energy densityE(n0, ?) =12n20g2(2?) +13!n30Re (g3(3?))with g2(2?) =~2m4?ln B22?, g3(3?) = 6g22(2?)g?3(3?)where g?3(3?) =~2m? 4qdq2? + q2G?3(?3?, q)ln B23q2/4+3?. (4.1)g2,3 stand for, respectively, the renormalized two- and three-body interac-tions in a condensate. We will discuss this point in more details below.G?3(?3?, p) represents the three-atom off-shell scattering amplitude (corre-sponding to the sum of all N-loop contributions with N = 1, 2, 3, ...). G?3 isthe solution to the following integral equation (see Appendix F for more de-tails), where ~ and m were intentionally set to unity to improve readability,G?3(?3?, p) =? 4qdqln B23q2/4+3?1?(3? + p2 + q2)2 ? (pq)2?( ?12? + q2 ?G?3(?3?, q)). (4.2)Note that in Eqs. (4.1) and (4.2), B2 = ?exp(4pi~2U0m)where ? is an energycutoff related to the effective interaction range, R?, via ? = ~2mR?2 . AsB2 = ~2ma22D, g2,3 are uniquely determined by the parameter n~2mB2 or na22D.For repulsive interactions (or positive U0), B2 is larger than ? and soa2D is bounded from above by the interaction range R?. When U0 is infi-nite (hard-core potential), a2D is equal to the core size a0. For attractiveinteractions (or negative U0), the case we focus on here, B2 is precisely thedimer binding energy, and a2D is the size of the bound state and can wellexceed R?. As a consequence, na22D, the fundamental tuning parameter forE(n0, ?), can take values larger than unity. The gas can hence be tunedaway from the dilute limit12.12For a quasi two-dimensional cold gas near Feshbach resonance, a2D is a function of l0,the confinement radius along the perpendicular direction, and of the free space scatteringlength a3D. For shallow 2D bound states, a2D =? pi0.915 l0 exp(??pi2l0a3D)[50?52].564.2. Self-Consistent Approach for Two-Dimensional Bose Gases0.02 0.06 0.1 0.14 0.18 0.22 0.260.3246810?1/ ln(na22D)?/n[h?2 /m]Figure 4.1: The chemical potential, in units of ~2nm , as a function of na22D.The dashed (red) line is the solution of the self-consistent equation when onlytwo-body interactions are included. The full (blue) line is the solution whenboth two- and three-body interactions are included. This figure highlightsthat the behavior of the chemical potential is drastically altered by three-body physics.574.2. Self-Consistent Approach for Two-Dimensional Bose GasesThree-dimensional self-consistent equations were used in chapter 2 toobtain the chemical potential of 3D Bose gases near resonance. These self-consistent equations provided highly precise estimates for the chemical po-tential in the dilute limit. Near resonance, this approach predicted a maxi-mum in the chemical potential and an accompanied onset instability. Thesefeatures were fully consistent with the conclusions drawn from a renormaliza-tion group equation approach in chapter 3. This first study concluded thatin three dimensions the dominating contribution to the chemical potentialcame from irreducible two-body interactions; for cold atoms, the three-bodycontribution was negligible. For two-dimensional Bose gases, the story isvery different: three-body interactions play here a much more importantrole as can be seen on Fig. 4.1.To analyze the contribution coming from the three-body effect, we firstsolve self-consistent equations excluding the contribution of g3, and obtainthe chemical potential solely due to two-body interactions (see Fig. 4.1dashed red line). Here, g2 is defined as the effective two-body interactionrenormalized by scattering events off condensed atoms and includes a subsetof N -body interactions defined in the vacuum13. Neglecting g3 interactions,the self-consistent equations take the simple form?? = 4?ln 12???+ 8?2?? ln3 12???, 1n?0= 1 + 2???1ln2 12???(4.3)where ?? = m?~2n0 , n?0 =n0n and ? = n0a22D14. The solution of Eq. (4.3) in thelimit of small ? is? = nm4?~2ln 1?(1? 1ln 1?[ln | ln?| ? ln 4? + C] + ...)(4.4)where C = ln 12 within this self-consistent approach. This solution, validin the dilute limit, agrees well with previous studies [47, 48, 76]. Anothersolution with ? approaching ~2ma22D exists in this limit but is unstable. As ?or na22D is increased, the dilute gas solution approaches this higher energyunstable solution, and at the critical value na22D = 1.42 ? 10?2 the two13In the dilute limit, g2 reproduces the most dominating contribution; the residue effectsare from the irreducible N = 4, 6, ...-body interactions, and are parametrically smaller thanthe contributions from g2, i.e. smaller by a factor of 1ln(na22D) . See also similar discussionson 3D cases in chapter 2.14In obtaining this equation, we take into account Eq. (2.4) and set ??/?? = 1,??/?n0 = g2 and ? = ? to simplify the structure.584.3. Competing Three- and Two-Body Interactions100 101?120?80?40040B2g? 3(3?/B2)3?/B210?20 10?10 10?2?1020?1010?100B2g? 3(3?/B2)Figure 4.2: Three-body interaction g?3 (defined in Eq. (4.1)) as a function ofthe energy shift ? = ? ? ?; ? is determined self-consistently together with?. Inset: full and two-loop behavior of g?3 for small ? values (respectively,full (blue) and dashed (red) lines). For 3?B2 < 1, the numerical integrationover the momentum was done from 0 to 50?B2m~ .solutions coalesce into one. Beyond this point, no real solution to Eq. (4.3)exists revealing the presence of an instability. The basic structure sketchedhere, when three-body contributions are neglected, is qualitatively the sameas that of 3D Bose gases: ? is maximum when an onset instability sets in,and for larger na22D develops an imaginary part implying the formation ofmolecules.4.3 Competing Three- and Two-BodyInteractionsWe now turn our attention to the contribution of g3(3?). g3(3?) is obtainedby first numerically solving Eq. (4.2) for G?3(?3?, p) and then by carrying594.3. Competing Three- and Two-Body Interactionsout the integral involving G?3(?3?, q) in Eq. (4.1) (see Appendix G for moredetails). The result of this procedure is shown in Fig. 4.2 where we plotg?3(3?). We chose to plot g?3(3?) and not g3(3?) as the former is not clutteredby trivial effects due to g22(2?). We identify two kinds of resonant scatteringprocesses defining the basic structure of g3(3?). The first one is a three-body resonance between three condensed atoms with zero energy and adimer plus a non-condensed atom with total energy 3? ? B2. Here 3? isthe mean-field energy shift due to the exchange interaction between thenon-condensed atom-dimer structure and the condensate. This leads tothe first peak (from left to right) at 3? = B2. The second process is athree-body resonance between three condensed atoms and a trimer witheither binding energy B(1)3 or B(2)3 (or total energies 3??B(1)3 or 3??B(2)3 ).This process produces the second and third peaks at 3? = B(1,2)3 . We findnumerically that B(1)3 = 1.296B2 and B(2)3 = 16.643B2. These energies arefully consistent with the results of two previous few-body studies [43, 45].Unlike in three dimensions where a logarithmically large number of Efimovstates exist, in two dimensions there are only two trimer states. Remarkably,their energies are uniquely determined by B2 without involving an additionalthree-body parameter, a fascinating feature emphasized in Refs. [43, 45].The effect of three-body scatterings on the quantum gas is mainly de-termined by the property of g3 when ? is relatively small. We checkednumerically that in the limit of very small ?, g3 can be well fitted by anattractive interaction of the scaling form ~42m2?1ln2 B22? ln2 B23?, capturing thedominant two-loop contribution15 (see Fig. 4.2). Including the contributiondue to three-body physics in the evaluation of the chemical potential resultsin two main effects. First, due to the attractive tail of g3 in the small ? limit,as shown in Fig. 4.3, the instability is shifted away from na22D = 1.42?10?2and occurs at a much smaller value of na22D = 3.26? 10?3. At this new in-stability point, the chemical potential is dramatically reduced, from 9.82~2nmto 0.601~2nm when g3 is included. In other words, the three-body effectiveinteraction further destabilizes the quantum gas. The second and equallyimportant effect is that the inclusion of three-body interactions results in theappearance of a maximum in the chemical potential at na22D = 0.604?10?3before the onset instability occurs. The maximum value of the chemical15A diagrammatic calculation similar to the one presented in chapter 2 suggeststhat the leading N-loop contributions to g3 for small values of ?/B2 are g(N-loop)3 =CN ~42m2?1ln2 B22? lnN B23?with N = 2, 3, 4...; the prefactor CN can be computed numerically.For the most dominating two-loop contribution, C2 = ?6.3? 103.604.4. Summarypotential is ?max = 1.45~2nm and the condensation fraction at the maximumis 91%.Between the maximum and instability points, the quantum gas exhibitsa negative compressibility and can potentially collapse into a high densityphase. Although the fate of the Bose gases with negative compressibilitiesand the details of the corresponding dynamics are beyond the scope of ourinvestigation, we speculate that in this regime a quantum gas eventuallyevolves into the droplet matter discussed in Ref. [45]. In three dimensions,the instability originated from a shift of the dimers due to scatterings off con-densates and was a precursor of the sign change of the effective two-bodyinteraction g2 [3]. For two-dimensional Bose gases, the situation is com-pletely different. Here, the instability is a consequence of the competitionbetween the repulsive two-body interaction (positive g2) and the attractivethree-body interaction (negative g3) in the low energy limit. For a two-dimensional Fermi gas, the Pauli blocking effect was recently demonstratedto lead to an instability at a finite scattering length [52].We also plot in Fig. 4.3 the relative weight of the three-body to two-body contributions to the chemical potential. As anticipated, the three-bodycontribution is negligible in the dilute limit when na22D ? 1 but quicklybecomes important as na22D is increased. The prominent role played bythree-body scattering leads to a maximum in the chemical potential beforethe instability point. At this maximum, the ratio between the three-bodyand two-body contributions reaches 0.27. The shift of the instability is alsocaused by the attractive three-body interactions.4.4 SummaryIn conclusion, we demonstrated that the properties of 2D Bose gases atlarge scattering lengths or near resonance are dictated by three-body effects.We showed that the contributions of trimer states are universal as theyonly depend on the effective two-body scattering length a2D and not on theshort distance properties of three-body interactions; an aspect unique totwo-dimensional Bose gases. Our results also suggest the existence of strongcorrelations in the three-atom channel near resonance. This feature remainsto be probed experimentally.The important point is that, in this study, we only investigated the effectsof two- and three-body potentials and since three-body physics plays animportant role in two dimensions, we expect the contributions from n-bodypotentials (n > 3) to be also important. However, since the similar physics614.4. Summary0.02 0.06 0.1 0.14 0.18012?1/ ln(na22D)Re[?/n][h?2 /m]00.51ratio0.02 0.1 0.180.70.80.91?1/ ln(na22D)n 0/n0.02 0.1 0.1800.050.1?1/ ln(na22D)|Im[?/n]|[h?2 /m]Figure 4.3: Top panel: ratio between the contributions of three-body andtwo-body interactions as a function of na22D (full red line), chemical potentialtwo-dimensional 2D Bose gases (full blue line). An additional metastablesolution (dashed blue line) also exists when g3 is included. The maximumvalue of ? is 1.45~2nm and occurs at na22D = 0.604 ? 10?3. Bottom leftpanel: condensation fraction n0/n as a function of na22D. Bottom rightpanel: imaginary part of the chemical potential when taking into account thecontribution of all three-body recombination processes. Note that |Im ?| ?Re ? for all considered na22D, indicating the quasi-static nature of the Bosegases. Hence, three-body recombination plays very little role in our energeticanalysis and can be safely neglected for the range of parameters considered.624.4. Summaryhave been predicted using variational quantum Monte Carlo method in Ref.[53], we expect that considering the contributions from n-body (n > 3)scattring processes does not change the physics qualitatively.63Chapter 5ConclusionIn conclusion, we have studied the physics of two- and three-dimensionalultra cold Bose gases near Feshbach resonance using a self-consistent frame-work. Within this framework, once the full energy density of the systemas a function of chemical potential of non-condensed atoms and conden-sate density is known, one can calculate the chemical potential of condensedatoms and density of non-condensed particles. The self-consistent equationis formed by satisfying the equilibrium condition in the ground state inwhich the chemical potential of condensed atoms is equal to to the chemicalpotential of non-condensed atoms.In chapter 2, we estimated the full energy density of a three-dimensionalBose gas using a diagrammatic method. In this method, we classified scat-tering processes in terms of the number of virtual particles involved in theprocess. We pointed out an onset instability toward formation of moleculesbeyond the dilute limit and fermionization of the Bose gas near resonance.This instability originates from a shift of the dimers due to scatterings offcondensates and is a precursor of the sign change of the effective two-bodyinteraction. This sign change is a strikingly different result from the picturecommonly accepted by the cold atom community where it is usually thoughtthat, for negative bare interactions where the scattering length is positive,the atoms repel each other. In addition, we found that the effect of three-body scattering processes is only a few percent to the chemical potentialclose to instability point. In chapter 3, these highly non-trivial results wereobtained using a renormalization group approach by looking at the runningof the two-body coupling constant at different energies.In chapter 4, we investigated the properties of two-dimensional ultra coldgases at large scattering lengths emphasizing the role played by three-bodyscattering processes. Within this diagrammatic approach, we showed thatthe physics of these gases near resonance is primarily dictated by the com-petition between three-body attractive interactions and two-body repulsiveforces. This competition results in the chemical potential of Bose gasesto exhibit a maximum at a critical scattering length beyond which thesequantum gases have a negative compressibility. Furthermore, we showed64Chapter 5. Conclusionthat for larger scattering lengths, the increasingly prominent role played bythree-body forces leads to an onset instability at a second critical point.We also showed that the contributions of trimer states are universal as theyonly depend on the effective two-body scattering length and not on theshort distance properties of three-body interactions; an aspect unique totwo-dimensional Bose gases.In the future, it would be interesting to answer this obvious question onhow it is possible to experimentally detect the peculiar behavior of the chem-ical potential in these two- and three-dimensional Bose gases, i.e. whetherthere exists a spectrometry that one can apply to accurately map out thevalue of the chemical potential near resonance. Another question is whetherthe behavior of Fermi gases close to Feshbach resonances [70, 77] can alsobe understood within this novel formalism. 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Beane, Phys. Rev. A 82, 063610 (2010).[77] G.-B. Jo, Y.-R. , J.-H. Choi, C. A. Christensen, T. H. Kim, J. H.Thywissen, D. E. Pritchard and W. Ketterle, Science 325 (5947), 1521-1524 (2009).70Appendix ASolving Self-Consistent Eq.(2.5) in the Dilute LimitWe apply Eq. (2.5) to calculate the leading-order correction beyond themean-field theory. We notice that the equations for g2 and ? are arrangedin such a way that the next-order correction can be obtained by applyingthe results from the lowest-order approximation to the right-hand side. Inthe lowest-order approximation, we find ? = 8?n0a and ? = 4?n0a; thisleads to a correction to g2 asg2 = 4?a +(4?a)22? d3k(2?)3( 1?k? 1?k + ?)= 4?a(1 +?8?n0a3). (A.1)Similarly, from the relation ???n0 = 8?a and???? = 0, we can get the correctionfor the chemical potential ? as,? = 4?an0 +(4?a)3 n202? d3k(2?)31(?k + ?)2= 4?an0(1 + 3?2?n0a3)(A.2)and the depletion fractionnpn =n04 g22? d3k(2?)31(?k + ?)2=??2n0a3. (A.3)For a comparison we list the results from the dilute-gas theory,?Beliaev = 4?n0a[1 + 403?1?n0a3], (A.4)(npn)Beliaev= 83?1?n0a3. (A.5)71Appendix A. Solving Self-Consistent Eq. (2.5) in the Dilute LimitOur self-consistent approach produces 9?2pi40 (= 99.96%) of Beliaev?s resultfor the chemical potential, and 3?2pi16 (= 83%) for the depletion fraction.72Appendix BA Comparison Between theSelf-Consistent Approachand the Dilute Gas Theoryin Three DimensionsIn the following, we show explicitly that our self-consistent equation corre-sponds to a subgroup of diagrams [in Fig. 2.2(c)] in the usual dilute gastheory. The two-body T -matrix used in the dilute-gas theory [representedby the green circles in Figs. 2.2(c) and 2.2(d)] are obtained using the non-interacting Green?s function G?1(?, k) = ?? ?k +?+ i0+; in the dilute limit,we can expand the T -matrix ast(?,Q) = 4?a[1 + 4?a? d3k(2?)3(1? ? Q24 ? k2 + 2? + i0++ 1k2)+ ? ? ?],(B.1)where ? and Q are the total energy and momentum of the incoming atoms.The contribution from the first two diagrams in Fig. 2.2(c) areE(c1) ?t(0, 0)n202 ? 2?an20[1 + 4?a? d3k(2?)3( 1?k2 + 2? + i0+ +1k2)]E(c2) ? 2t2(0, 0)n202? d3k(2?)3( 1?k2 + 2?+ i0+)22n0t(?? ?k, 0) (B.2)? 2?an20[(4?a) (16?n0a)? d3k(2?)3( 1?k2 + 2? + i0+)2]. (B.3)For the leading-order correction beyond the mean-field theory, it suffices toset t(?? ?k, 0) ? 4?a in Eq. (B.3) and in higher-order diagrams. Similarly,we can get the contributions from the higher-order diagrams in this series,73Appendix B. A Comparison Between the Self-Consistent Approach and the Dilute Gas Theory in Three Dimensionsand the sum isE(c) ? 2?an20[1 + 4?a? d3k(2?)3( 1?k2 + 2?+ i0+ +1k2)]+ 2?an20(4?a)??m=1(16?an0)m? d3k(2?)3( 1?k2 + 2? + i0+)m+1(B.4)? 2?an20[1 + 4?a? d3k(2?)3( 1?k2 + 2?? 16?n0a+ 1k2)]. (B.5)We see that the energy given by the diagrams in Fig. 2.2(c) is exactly thesame as the one used in our self-consistent equation, e.g., g2n20/2, where g2should be expanded as Eq. (A.1) in the dilute limit.Next, we can sum up the rest of the one-loop diagrams that are notincluded in the self-consistent equations; they represent the lowest-ordercontributions to four- and six-body forces and so on. In the dilute limit,these diagrams [as shown in Fig. 2.2(d)] can be summed asE(d) = ? (4?an0)? d3k(2?)3??m=212(2m? 2)!m! (m? 1)!( 4?an02?k ? 2? + 16n0?a)2m?1.(B.6)Indeed, we can recover Beliaev?s result by summing up one-loop diagramsin Figs. 2.2(c) and 2.2(d) as??n0(E(c) + E(d))= 4?n0a+ 4?a? d3k(2?)3??(?k ? ?+ 6?n0a)?(?k ? ?+ 8?an0)2 ? (4?an0)2? 1 + 4?n0ak2??= 4?n0a[1 + 403?1?n0a3]= ?Beliaev (B.7)74Appendix CIncluding Three-Body Forcesin the Self-ConsistentEquations in ThreeDimensionsWe now calculate the amplitude of three-body scatterings corresponding tothe processes described in Fig. 2.2(b). First, we consider a general casewhere the three incoming momenta are k1 = p/2 ? q, k2 = p/2 + q, andk3 = ?p, and the outgoing ones are k?1 = p?/2 ? q?, k?2 = p?/2 + q?, andk?3 = ?p?. The scattering amplitude between these states is then givenby A(E ? 3?;p,p?), which represents the sum of diagrams identical to Fig.2.2(b) except that the external lines carry finite momenta.For the estimate of three-body contributions of g3, we first treat the sumof diagrams in Fig. 2.2(b) as the limit of A(E?3?;p,p?) when p and p? ap-proach zero and the total frequency E is set to zero. It is therefore more con-venient to work with the reduced amplitude G3(E?3?;p) = A(E?3?;p, 0),where p? is already taken to be zero. G3(E ? 3?;p) itself obeys a simpleintegral equation, as can be seen by listing the terms in the summation ex-plicitly. Indeed, when E is further set to zero, we find that the diagrams inFig. 2.2(b) yieldG3(?3?, p) =2??dq K(?3?; p, q)q2?34q2 + 3? ? 1a?1q2 + 3? (C.1)+( 2?)2 ?dqdq? K(?3?; p, q)q2?34q2 + 3? ? 1aK(?3?; q, q?)q?2?34q?2 + 3? ? 1a?1q?2 + 3? + ? ? ? ,(C.2)where K(?3?; p, q) is the kernel defined in chapter 2. The sum of the above75Appendix C. Including Three-Body Forces in the Self-Consistent Equations in Three Dimensionsinfinite series leads to the following integral equation of G3 :G3(?3?, p) =2??dq K(?3?; p, q)q2?34q2 + 3? ? 1a?1q2 + 3?+ 2??dq K(?3?; p, q)q2?34q2 + 3? ? 1aG3(?3?, q). (C.3)Note that G3(?3?, 0) defined above includes a diagram [the leftmost one inFig. 2.2(b)] that has already been included in g2. To avoid overcounting,we subtract the first diagram in Fig. 2.2(b) from G3 asg3 = 6g22Re??G3(?3?, 0) ?2??dq K(?3?; 0, q)q2?34q2 + 3? ? 1a?1q2 + 3??? . (C.4)Alternatively, one can also carry out a direct summation of the diagramsin Fig. 2.2(b). It leads to a result that numerically differs very little from theestimation obtained above via an asymptotic extrapolation. For instance, adirect evaluation of those diagrams yieldsG3(?3?, 0) =2??dq K(?2?; 0, q)q2?34q2 + 3? ? 1a?1q2 + 2? (C.5)+( 2?)2 ?dqdq? K(?2?; 0, q)q2?34q2 + 3? ? 1aK(?3?; q, q?)q?2?34q?2 + 3? ? 1a?1q?2 + 2? + ? ? ?(C.6)The only difference between Eqs. (C.6) and (C.2) is that the frequenciesappearing in the first kernel K(E; 0, q) in the integrands and in the lastdenominators are now ?2? instead of ?3?.One can easily verify that the sum can be written in the following com-pact form:G3(?3?, 0) =2??dq K(?2?; 0, q)q2?34q2 + 3? ? 1a[ ?1q2 + 2? +G?3(?3?, q)], (C.7)where G?3(?3?, p) is a solution of the following integral equation:G?3(?3?, p) =2??dq K(?3?; p, q)q2?34q2 + 3? ? 1a[ ?1q2 + 2? +G?3(?3?, q)] (C.8)76Appendix C. Including Three-Body Forces in the Self-Consistent Equations in Three DimensionsNote that G?3(?3?, p) defined here describes an off-shell scattering betweenthree incoming atoms with momenta p/2 ? q, p/2 + q, and ?p and threecondensed atoms. As a consequence of the Hartree-Fock approximation wehave employed here, G3(?3?, 0) is not equal to G?3(?3?, 0). G3(?3?, 0) andG?3(?3?, 0) can be obtained numerically.Finally, after subtracting the leftmost one-loop diagram in Fig. 2.2(b)we again find the three-body contribution to be:g3 = 6g22Re??G3(?3?, 0) ?2??dq K(?2?; 0, q)q2?34q2 + 3? ? 1a?1q2 + 2??? . (C.9)Now we can include the three-body forces g3n306 in a set of differentialself-consistent equations similar to Eq. (2.5). We solve the equation numer-ically, and the results are shown in Fig. 2.3, where in the inset we show themomentum cutoff ? dependence in the chemical potential. In our numer-ical program, we further use the approximation ???n0 = 2g2,???? = 0, and?11 = ?? (? = 2) to simplify the numerical calculations (see Appendix D).We have tested other types of approximation schemes for the self-energy,such as ?11 = 8?an0 or ?11 = 2g2n0. We find that the chemical poten-tial and the value of the critical point na3cr are insensitive to approximationschemes.77Appendix DSingle Parameter LimitHere we explain the single parameter limit, where the physics of two- andthree-dimensional Bose gases could be described by a single parameter, ?.We determine this limit by looking at the behavior of the two-body interac-tion potential. Three-dimensional g2 is written as:g?12 = U?10 ? i? d?2?d3k(2?)3G(?,k)G(??,?k), (D.1)where G(?,k)?1 = ? ? ?k ? ? + ? + i?+ is the interacting Green?s func-tion. Here, ? is the self-energy of non-condensed particles which in generalis a function of momentum and frequency. The self-energy is almost flat formomenta smaller than 1/a while after this point it decays to zero. Further-more, we can supplement the above equation with the following relation toregulate ultraviolet divergency in three dimensions:1U0= 14?a ?1??k12?k. (D.2)And after a little bit calculation, we can write:g?12 =14?a ?1?2?dk ?k2 + 2?? 2? +1?2?dk ?k2 + 2? ? 2?. (D.3)One can look at the two integrals in the above equation separately. Thefirst integral does not have ultra-violet divergency, since ? decays to zerofor large momenta. So, the main contribution comes from low energy limitof the integral and since ? is equal to 2? for small momenta, this integralcould be considered as a single variable function of ?. Similarly, there is noultraviolet divergency for second integral and again the physics is determinedby low energy limit. Above calculation shows that two-body interaction isa function of only a and ?. In chapter 2, we found that the instability pointhappens for relatively small value of scattering length. This results the self-energy to be independent of momenta for a wider range which make aboveestimation of integrals more reasonable. In addition, we assume ? to be78Appendix D. Single Parameter Limitindependent of momentum in our calculations in this thesis. In this regime,the above equation could be evaluated exactly and the result is:g2(2?) =4?1a ??2m?. (D.4)One can also look at three-body interactions in three dimensions and asit mentioned before in chapter 2, g3 is a function of the ultra-violet cut off aswell as a and ?. But since the contribution of the three-body forces to theenergy of the system is approximated to be around few percent, our resultsremain robust.Similar arguments could be used for a two-dimensional Bose gas. Thesingle parameter approximation is still valid in this dimension. In addition,since three-body forces are universal in two-dimensions, our calculation isrobust for such systems.79Appendix ETwo- and Three-BodyScattering Amplitudes in aCondensate in ThreeDimensionsIn the following, we show the explicit calculation of the three-atom scat-tering amplitude G3, and E(n0, ?) for a given ?(= ? + ?) by adding thediagrams with a minimum number of virtual particles involved. The modelHamiltonian could be written as:H =?k(?k ? ?)b?kbk + 2U0n0?kb?kbk+ 12U0n0?kb?kb??k +12U0n0?kbkb?k+ U0???n0?k?,qb?qbk?+q2 b?k?+q2 + h.c.+ U02??k,k?,qb?k+q2 b??k+q2bk?+q2 b?k?+q2 + h.c., (E.1)where the sum is over non-zero momentum states. U0 is the strength of thecontact interaction which is related to scattering length a as:1U0= m4?a ?1??k12?k, (E.2)where ? is the volume. Taking into account only two- and three-body in-teractions, the energy density could be written as:E(n0, ?) =12n20g2(2?) +13!n30g3(3?), (E.3)80Appendix E. Two- and Three-Body Scattering Amplitudes in a Condensate in Three Dimensionswhere g2 and g3 are irreducible two- and three-body potentials respectively.g2 could be found by writing the Bethe-Salpeter equation as:g2(2?)?1 = U?10 ? i? d?2?d3k(2?)3G(?,k)G(??,?k), (E.4)where G(?,k)?1 = ?? ?k? ? + i?+ is the interacting Green?s function. So,the two-body potential could be obtained as:g2(2?) =4?1a ??2m? (E.5)Similarly, g3 could be estimated by summing up all N-loop diagrams with3 incoming and outgoing lines which are depicted in Fig. 3.3(b). We considera general case where three incoming momenta are k1 = p/2?q, k2 = p/2+qand k3 = ?p, and outgoing ones are k?1 = p?/2 ? q?, k?2 = p?/2 + q? andk?3 = ?p?. The scattering amplitude between these states is then given byA3(E;p,p?). At the tree-level, the effective three-particle interaction is:?(0) = f0E ? ?in ? ?out ? ?p+p? ? ? + i?+, (E.6)where ?in and ?out are frequencies of lines with momenta k3 and k?3 respec-tively. Furthermore, f0 is the product of the perturbation factor fp, vertexfactor fv and symmetry factor fs which will be explained later.To keep the notation simple, we set m = 1 from now on. We consider on-shell limit and substitute ?in and ?out by p2/2+? and p?2/2+? respectively.To project into the S-wave channel, we take the average over all directions,?(0) = ?f02pp? ln(p2 + p?2 + pp? + 3? ? Ep2 + p?2 ? pp? + 3? ? E)? ?f02 K(E ? 3?; p, p?), (E.7)where we have defined the kernel K as:K(E ? 3?; p, p?) = 1pp? ln(p2 + p?2 + pp? + 3? ? Ep2 + p?2 ? pp? + 3? ? E)(E.8)The perturbation factor, fp, comes from the expansion, in perturbationtheory, of the evolution operator exp(?iHintt). The diagrams with l verticescould be written as:1l! (VA + VB)l, (E.9)81Appendix E. Two- and Three-Body Scattering Amplitudes in a Condensate in Three Dimensionswhere VA and VB stand for interaction terms corresponding to different typesof vertices namely A and B. For example, for a diagram with 2 vertices oftype A and one vertex of type B, fp is the factor in front of the V 2AVBterm in the numerator of the above equation divided by l!. In the vacuumcase, where all the vertices are the same and all the lines can have non-zeromomenta, fp is simply equal to (1/l!).The vertex factor, fv, is defined as the product of the factors in front ofg2 for different vertices shown in the Hamiltonian. For the vacuum case, allthe vertices have 1/2 factor and fv is equal to (1/2)l .The last factor is the symmetry factor, fs which shows the number ofidentical diagrams generated for a given number of vertices. For the vacuumcase, fs = l!? 4l where l! shows the number of permutations of vertices and4l is the number of different ways of connecting vertices (2 for incoming linesand 2 for outgoing lines).So, in general the prefactor appearing in ?(n) where n is the number ofloops (n = l ? 2) would befn = 2(n+2); (E.10)and f0 = 4.?(1) could be written in terms of the kernel defined above as:?(1) = 8? d3k(2?)3 (?12K(E?3?; p, k))g2(E?3??k22 ; k)(?12K(E?3?; k, p?)),(E.11)where the integral over internal frequency has been taken. g2(?;Q) has thefollowing form in 3D:g2(?;Q) =4?1a ??Q24 ? ?, (E.12)where the above equation reduces to Eq. (E.5) in the limit of zero energy andmomentum. The effective three-body interaction then could be obtained bysumming over ?(n)s:82Appendix E. Two- and Three-Body Scattering Amplitudes in a Condensate in Three Dimensions?eff = ?2K(E ? 3?; p, p?)+ 4??dkk2 K(E ? 3?; p, k)1a ??3k24 ? E + 3??K(E ? 3?; k, p?)? 8?2?dkk2?dk?k?2 K(E ? 3?; p, k)1a ??3k24 ? E + 3?? K(E ? 3?; k, k?)1a ??3k?24 ?E + 3?? K(E ? 3?; k?, p?) + . . . (E.13)The sum of the above infinite series leads to the following integral equa-tion for scattering amplitude A3:A3(E; p, p?) = ?2K(E?3?; p, p?)?2??dkk2 K(E ? 3?; p, k)1a ??3k24 ? E + 3?A3(E; k, p?)(E.14)One then obtains the reduced scattering amplitude G3(E;p) = A3(E;p, 0)in Eq. (3.22).For the calculation of g3 for condensates, one has to exclude the tree leveldiagram that no longer exists because of momentum conservation. The sumof the rest of the infinite series leads to the following integral equation forthe scattering amplitude A3(E; p, p?),A3(E; p, p?) =2??dk K(E ? 3?; p, k)k21a ??3k24 ? E + 3?(2K(E?3?; k, p?)?A3(E; k, p?))(E.15)The above scattering amplitude could then be applied to calculate thescatterings between condensed atoms when setting E, p and p? to be zeroin the above equation but with two further modifications. The first changeis the numerical factor in front of the effective interactions. This factor inthe condensate case is 1/4 of the factor in the vacuum case, because thereis a 2 ? 2 factor for changing the external legs of the external vertices fornon-zero incoming momenta. So we will get the same integral equation,but the first term in the bracket of the integrand of Eq. (E.15) would besubstituted by 1/2K.The second change would be in the shift of the energy. If we set themomentum of external legs to zero from the beginning, in the on-shell limit83Appendix E. Two- and Three-Body Scattering Amplitudes in a Condensate in Three Dimensionsthere is no shift of the energy for ?in and ?out in our diagrammatic calcula-tions. So, the energy of the first and the last kernel in all the terms of theEq. (E.13) other than the tree-level term would be E?2? in the condensatecase. Subtracting the one-loop contribution which has already been countedin the renormalized g2 and taking into account the above two modifications,we obtain g316.16As mentioned before in chapter 3, there are two ways of estimating G3 which slightlydiffer from each other; the difference is due to singular behavior of the Green?s functionat k = 0 in the Hartree-Fock approximation as commented on in Appendix C. Here weevaluate G3 by first setting all external lines to be the condensed atoms.84Appendix FTwo- and Three-BodyScattering Amplitudes in aCondensate in TwoDimensionsTwo-body effective interaction could be obtained by using Dyson?s equationin the following closed form:?ig2(E;Q) = ?iU0+? d?2?d2k(2?)2 g2(E;Q)U0G(?,Q/2+ k)G(E ? ?,Q/2? k).(F.1)where G(?,k)?1 = ? ? ?k ? ? + i?+ is the interacting Green?s function, U0is the attractive bare interaction strength and ? = ?? ? .So, two-body scattering amplitude could be written as:g2(E;Q)?1 = U?10? i? d?2?d2k(2?)2G(?,Q/2 + k)G(E ? ?,Q/2? k), (F.2)After integration over frequency and momentum space one obtains:g2(E;Q)?1 = U?10 +14? log(|?Q2/4? E ? 2? |) +i4? 14? log(|B2Q2/4? E ? 2? |) +i4 , (F.3)where ? is the ultra-violet energy cutoff. Here B2, two-body bound stateenergy is defined to be:85Appendix F. Two- and Three-Body Scattering Amplitudes in a Condensate in Two DimensionsB2 = ?e4piU0 . (F.4)a2D =?~2/mB2, the effective two-dimensional scattering length is in-troduced as the size of two-body bound state. ? is set by the interactionrange R?, via ? = ~2mR?2 .For repulsive interactions (U0 > 0), B2 is larger than ?. As a result, a2Dcan not exceed form range of interaction, and for a short range interactionthe system is always in the dilute regime. By increasing U0, the range ofa2D becomes more restricted from below and at an extreme limit when U0goes to infinity, a2D is exactly equal to the range of interaction R?. Thislimit is known as the hard-core limit. For attractive interactions (U0 < 0),which is the case we are interested in, B2 is smaller than ?. This conditionsets the lower-bound of the 2D scattering length equal to the range of theinteraction. So, a2D could be arbitrary large and potentially the system cango beyond the dilute limit and approach resonance.For two-dimensional Bose gases, the imaginary part of the two-bodyeffective interaction is independent of the total energy of the incoming scat-tering particles. As a result, the imaginary part exists even for zero totalenergy. This unique property of the two-dimensional gases is due to the factthat two-dimensional density of states is independent of energy. The realpart of the g2 is:Re(g2(E;Q)) =4?log(| B2Q2/4?E?2? |). (F.5)Three-body scattering amplitude, g3, could be estimated by summing upall N-loop diagrams with three incoming and outgoing lines. We consider ageneral case where three incoming momenta are k1 = p/2?q, k2 = p/2+qand k3 = ?p, and outgoing ones are k?1 = p?/2 ? q?, k?2 = p?/2 + q? andk?3 = ?p?. The scattering amplitude between theses states is then given byA3(E;p,p?). For tree level diagram with no loop, the effective three-particleinteraction is:?(0) = f0E ? ?in ? ?out ? ?p+p? ? ? + i?+, (F.6)where ?in and ?out are frequencies of lines with momentum k3 and k?3 re-spectively. Furthermore, as in Appendix E f0 is the product of perturbationfactor fp, vertex factor fv and symmetry factor fs which will be definedlater.86Appendix F. Two- and Three-Body Scattering Amplitudes in a Condensate in Two DimensionsAlthough this diagram does not exist when we set the momenta of ex-ternal lines to zero (the reason is that in this case, the momentum of theinternal line also has to be zero due to the conservation of the momentumat each vertex which makes this diagram trivial), it is the building blockof the scattering processes with more number of loops. Since eventually weset the momenta of external lines equal to zero in order to find the effectiveinteraction between condensed particles, the angle between these momentais not well-defined. So, we project ?(0) to s-wave channel by taking the av-erage over all solid angles. In addition, for on-shell particles, ?in and ?outare substituted by p2/2 + ? and p?2/2 + ? respectively. ? could be under-stood as the energy shift due to interaction of condensed and non-condensedparticles.?(0) = 12?? ?0d? f0E ? p2 ? p?2 ? 3? ? pp?cos? + i?+ (F.7)after doing the integration over angle we have:?(0) = ?f0?(E ? p2 ? p?2)2 ? (pp?)2? ?f02 K(E ? 3?; p, p?), (F.8)where the kernel K is defined as:K(E; p, p?) = 2?(E ? p2 ? p?2)2 ? (pp?)2. (F.9)To obtain the numerical factor in front of each diagram, it is importantto remark that each of the terms in the Hamiltonian (2.1) corresponds todifferent vertices in the scattering diagrams (See Fig. F.1). These verticesdiffer by the number of the condensed atoms (dashed lines) involved in theinteraction. In addition, each diagram is the depiction of different termsproduced by expanding the exponential of the interacting Hamiltonian inperturbation theory. For example, considering only two kinds of vertices,the l ? th order term could be written as:1l! (HA +HB)l, (F.10)where HA and HB are interaction terms corresponding to different typesof vertices A and B. The perturbation factor, fp is the numerical factor infront of each term of the above equation. As an example, consider a diagramwith s vertices of type A and l ? s vertices of type B. fp is the factor in87Appendix F. Two- and Three-Body Scattering Amplitudes in a Condensate in Two DimensionsFigure F.1: Different types of the interaction vertices between condensed andnon-condensed atoms which corresponds to different terms of the Hamilto-nian (2.1).88Appendix F. Two- and Three-Body Scattering Amplitudes in a Condensate in Two Dimensionsfront of HsAH l?sB term. The perturbation factor for this example could bewritten in terms of combination factors as following:fp =1l!ClsC l?sl?s =1s!(l ? s)! . (F.11)In general, if we have s1 vertices of type A1, s2 vertices of type A2 andso on until sm vertices of type Am, the perturbation factor would be:1s1!? s2!? ...? sm!(F.12)In the vacuum case, where all the momenta could be non-zero, all thevertices are the same and the perturbation factor is simply equal to 1/l!.As mentioned before in chapter 2, Hamiltonian (2.1) is generated byexplicitly putting the momenta of some of the creation and annihilation op-erators equal to zero. For some of the terms, there are more than one wayto set the momenta equal to zero. These choices cause different numericalprefactors in front of the terms in the Hamiltonian. For example, the secondterm in the Hamiltonian is produced by setting the momentum of one of thecreation and one of the annihilation operators equal to zero. This term rep-resents the interaction between one condensed atom and one non-condensedatom as incoming particles and one condensed atom and one non-condensedatom as outgoing particles. There are four different ways to produce thisterm and therefore the numerical prefactor in front of this term is 2 insteadof 1/2. The vertex factor, fv, is defined as the product of these prefactorsfor all the vertices involved in the scattering processes. Since for the vacuumcase, all the vertices have the same prefactor of 1/2, fv is equal to (1/2)l.The last factor is the symmetry factor, fs which shows the number ofdifferent ways of connecting legs of these vertices. For the vacuum case,fs = l!? 4l where l! is the number of permutations of vertices and 4l is thedifferent ways of choosing lines of each vertex (2 for incoming lines and 2 foroutgoing lines). For the condensate case, finding the symmetry factor is alittle tricky. Since only solid lines corresponding to non-condensed particlesare connecting the vertices, the symmetry factor in this case depends on thenumber of the solid lines in each vertex. Again, suppose we have s1 verticesof type A1, s2 vertices of type A2 and so on until sm vertices of type Am. Thenumber of permutation of the similar vertices would be s1!? s2!? ...? sm!which cancels out fp for the condensate case. In addition, different verticeshave different ways of connecting their legs to other vertices. This numbersfor vertex type A is one and for vertex types of B, C, D and E is two andfor vertex type F is four (see Fig. F.1). Note that if you multiply these89Appendix F. Two- and Three-Body Scattering Amplitudes in a Condensate in Two Dimensionsfactors by the prefactors in front of each vertex, you would get 2 except forvertex types of B and C when you get 1. Since there is only one vertexof type B and one vertex of type C in any three-body diagram, fn for thecondensate case is equal to:fn = fp ? fv ? fs = 2l?2 = 2n (condensate case) (F.13)where l is the number of vertices and n is the number of the loops in each dia-gram. Similarly, for the vacuum case the numerical factor could be obtainedas:fn = fp ? fv ? fs = 2l = 2n+2 (vacuum case). (F.14)Note that numerical factor found for the vacuum case is bigger than thenumerical factor for the condensate case by the factor of 4. This differenceis due to the exchange of the external legs of the diagrams. In the vacuumcase, the external legs has non-zero momentum and could be interchanged.This produces the extra factor. Finally, f0 = 4 in the vacuum case.The contribution from the 1-loop diagram, ?(1), to the effective interac-tion between condensed atoms in vacuum case could be written in terms ofkernel defined above as:?(1) = 8? d2k(2?)2 (?12K(E?3?; p, k))g2(E?3??k22 ; k)(?12K(E?3?; k, p?)),(F.15)where the integral over internal frequency has taken and g2(?;Q) is obtainedbefore in Eq. (F.5) in two dimensions. Here, 8 is the numerical factor foundin the vacuum case. Note that the shift of energy is 3? everywhere whichcorresponds to the case where we set the momenta p and p? to zero at theend, vacuum case. The effective three-body interaction in the vacuum caseis obtained by summing over ?(n)s:A3(E; p, p?) =? d2q(2?)2K(E; p, q)g2(E ? ?q; q)(2K(E ? ?q; q, p?)?A3(q, p?))(F.16)The reduced amplitude is defined by setting the outgoing momentum tozero, G3(E; p) = A3(E; p; p? = 0). So, the integral equation for G3 wouldbe:90Appendix F. Two- and Three-Body Scattering Amplitudes in a Condensate in Two DimensionsG3(E; p) =?dq 4qlog(| B23q2/4?E |)1?(E ? p2 ? q2)2 ? (pq)2( 4|E ? q2| ?G3(q))(F.17)The three-body irreducible potential could be found as:g3 = (6/4)g22(0; 0)Re(G?3(0; 0)) (F.18)where G?3(0; 0) has obtained by subtracting the one-loop contribution fromG3(0; 0) defined above to prevent over-counting this diagram. The divisionby 4 in the last equation is due to the fact that the factors in the vacuumcase are 4 times bigger than the factors in the condensate case because ofexchange factor of 2 external legs in the vacuum case.The alternative scheme is to set the external momenta p and p? to zerofrom the beginning. In this case, the shift of the energy for the first andthe last kernel in any order of the loops would be 2?. In this scheme, thethree-body irreducible potential isg3 = 6g22(0; 0)Re(? 4qdq2? + q21log(| B23q2/4+3? |)G?3(?3?; q)) (F.19)where G?3(?3?; p) is obtained from following integral equation:G?3(?3?; p) = 4?dq qlog(| B23q2/4+3? |)1?(3? + p2 + q2)2 ? (pq)2?( ?12? + q2 ?G?3(?3?; q))(F.20)Note that in the second scheme we don?t need to subtract the one-loopcontribution because it starts from two-loop diagram. Although the solu-tions for two schemes are very close to each other, we focus on the secondscheme in our studies which is more accurate.91Appendix GNumerical Method to FindThe Three-Body InteractionPotential in Two DimensionsHere we explain the numerical method we used to solve the integral equationof three-body scattering amplitude for a two-dimensional Bose gas showedin Eq. (F.20). First, we only keep the real part of g2 and later we also takeinto account the imaginary part of the two-body interaction potentials.The integral equation for G?3 derived in Eq. (F.20) could be rewrittenas:G?3(y, z) = a(y, z) +?G?3(y, x)K(x, y, z)dx= a(y, z) +?iKi(y, z)G?3i(y)?x (G.1)where we have introduced following dimensionless variables:3?B2? y, p?B2? z, q?B2? x (G.2)and following functions:a(y, z) =?dx 4xlog(3x2/4 + y)1?(y + x2 + z2)2 ? (xz)212y/3 + x2K(x, y, z) = 4xlog(3x2/4 + y)1?(y + x2 + z2)2 ? (xz)2(G.3)Furthermore, we discretized the x direction in the second line of the Eq.(G.1). By discretizing the space in z direction, the integral equation couldbe written in discrete space:92Appendix G. Numerical Method to Find The Three-Body Interaction Potential in Two DimensionsG?3j(y) = aj(y) +?iKji(y)G?3i(y)?x?i?jiG?3i(y) = aj(y) +?iKji(y)G?3i(y)?x (G.4)So, finally we can derive aj(y) in the following equation:aj(y) =?i(?ji ?Kji(y)?x)G?3i(y)aj(y) ??iMjiG?3i(y) (G.5)This gives a matrix-form equation for G?3i as:a(y) = MG?3(y)G?3(y) = M?1a(y). (G.6)Numerically, G?3(y) is calculated by inverting the matrix M and multi-plying that by vector a(y).In general, we need to consider the imaginary part of the two-body poten-tial as well. This part could be obtained using following simple expression:1ln(34q2 + 3? ? i?+)= 1ln(34q2 + 3?)+ i??(34q2 + 3? ? 1) (G.7)This imaginary term gives correction to the real part of the integralequation and results in g3 being complex. The imaginary part of g3 isrelated to the rate of three-body recombination process. Here, we first showthe full expression for the two-loop contribution to g3 and then determinethe matrix-form equation for real and imaginary part of G?3.The full expression for the two-loop contribution to three-body interac-tion potential is:93Appendix G. Numerical Method to Find The Three-Body Interaction Potential in Two DimensionsB2 g2-loop3 = 96g22? zdz2/3y + z2?xdx2/3y + x21?(y + x2 + z2)2 ? (xz)2?(1ln(3z2/4 + y)1ln(3x2/4 + y)? ?2?(34x2 + y ? 1)?(34z2 + y ? 1) + 2i??(34 z2 + y ? 1)ln(34x2 + y))(G.8)where the correction to the real part of g2-loop3 due to imaginary term in g2is:B2 ??g2-loop3 =(23)2g22( 1?2/3y + 4/3)2 3? 96?2?(?5y + 8)2 ? (4(1 ? y))2)(G.9)and the imaginary part of g2-loop3 has the following form:B2 ??g2-loop3 = ?43 96? g221?2/3y + 4/3? xdx2/3y + x21ln(3x2/4 + y)? 1?(y + 4/3(1 ? y) + x2)2 ? x2 4/3(1 ? y)(G.10)Similarly, one can add the imaginary part of g2 in the integral equationand separate real and imaginary parts to get a system of equations for realand imaginary parts of G?3 as:?[G?3] = (M +BM?1B)?1(a?BM?1a?)?[G?3] = (M +BM?1B)?1(a? +BM?1a) (G.11)where in addition to a and M functions which have been defined earlier, weintroduce the following functions:a?(y, z) = 432??(z2 + 4/3? 1/3y)2 ? (4/3 ? 4/3y)z21?2/3y + 4/3B(x, y, z) = 4? x?(y + z2 + x2)2 ? (xz)2?(x ??4/3(1 ? y))?3(1? y)?x.(G.12)94Appendix G. Numerical Method to Find The Three-Body Interaction Potential in Two DimensionsThe system of these equations can be solved numerically. By insertingthe real and imaginary parts of G?3 and g2 in Eq. (F.19), one can obtain thereal and imaginary parts of three-body potentials as:?[g3(y)] = ?6(4?ln(2/3y))2(? 4x dx2/3y + x2(1ln(3x2/4 + y) ?[G?3(y, x)]+ ??(3x2/4 + y ? 1)?[G?3(y, x)]))(G.13)?[g3(y)] = ?6(4?ln(2/3y))2(? 4x dx2/3y + x2(1ln(3x2/4 + y) ?[G?3(y, x)]? ??(3x2/4 + y ? 1)?[G?3(y, x)]))(G.14)Eq. (G.14) gives the imaginary part of three-body potential which isrelated to the three-body recombination process rate. We have estimatedthis imaginary contribution and we found that for the range of interest,?[g3(y)] is always much smaller than ?[g3(y)] (see Fig. 4.3). As a result,we did not consider this effect in our self-consistent calculations to find thechemical potential of the Bose gas.95

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