T HE MAT HEMAT ICS OF T HE BOSE 6 0 0 2 t GAS AND IT S CONDENSAT ION c O 4 1 v 7 1 1 0 1 6 0 / t a m - Elliott H. Lieb, Robert Seiringer, d n o Jan Philip Solovej and Jakob Yngvason c : v i X r a ii Elliott H. Lieb Robert Seiringer Departments of Mathematics and Physics Department of Physics Princeton University Princeton University Jadwin Hall, P.O. Box 708 Jadwin Hall, P.O. Box 708 Princeton, NJ 08544 Princeton, NJ 08544 USA USA [email protected] [email protected] Jan Philip Solovej Jakob Yngvason Department of Mathematics Institut fu¨r Theoretische Physik University of Copenhagen Universit¨at Wien Universitetsparken 5 Boltzmanngasse 5 2100 Copenhagen 1090 Wien Denmark Austria [email protected] [email protected] This bookwasfirstpublished by Birkh¨auserVerlag(Basel-Boston-Berlin)in 2005 underthe title “TheMathematicsoftheBoseGasanditsCondensation”asnum- ber 34 of its “OberwolfachSeminar” series. Thepresentversiondiffersfromtheoriginallypublishedversioninseveralrespects. Various minor errors have been corrected, Chapter 9 has been slightly expanded, references have been updated, and Chapter 12 has been added in which results obtained after the publication are noted. ISBN 3-7643-7336-9 c 2005ElliottH.Lieb,RobertSeiringer,JanPhilipSolovejandJakobYngvason. (cid:13) This material may be copied for non-commercialpurposes, provided the source is properly acknowledged. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Introduction 1 1.1 The Ideal Bose Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Concept of Bose-Einstein Condensation . . . . . . . . . . . . . 4 1.3 Overview and Outline . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 The Dilute Bose Gas in 3D 9 2.1 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 The Dilute Bose Gas in 2D 27 4 Generalized Poincar´e Inequalities 35 5 BEC and Superfluidity for Homogeneous Gases 41 5.1 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . 41 5.2 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6 Gross-Pitaevskii Equation for Trapped Bosons 49 6.1 Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.2 Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7 BEC and Superfluidity for Dilute Trapped Gases 67 8 One-Dimensional Behavior of Dilute Bose Gases in Traps 75 8.1 Discussion of the Results. . . . . . . . . . . . . . . . . . . . . . . . 79 8.2 The 1D Limit of 3D GP Theory. . . . . . . . . . . . . . . . . . . . 82 8.3 Outline of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 iv Contents 9 Two-Dimensional Behavior in Disc-Shaped Traps 91 9.1 The 2D Limit of 3D GP Theory. . . . . . . . . . . . . . . . . . . . 96 9.2 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 9.3 Scattering Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.4 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 10 The Charged Bose Gas, the One- and Two-Component Cases 115 10.1 The One-Component Gas . . . . . . . . . . . . . . . . . . . . . . . 115 10.2 The Two-Component Gas . . . . . . . . . . . . . . . . . . . . . . . 118 10.3 The Bogoliubov Approximation . . . . . . . . . . . . . . . . . . . . 119 10.4 The Rigorous Lower Bounds . . . . . . . . . . . . . . . . . . . . . . 122 10.5 The Rigorous Upper Bounds . . . . . . . . . . . . . . . . . . . . . 130 11 BE Quantum Phase Transition in an Optical Lattice Model 137 11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 11.2 Reflection Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . 142 11.3 Proof of BEC for Small λ and T . . . . . . . . . . . . . . . . . . . 143 11.4 Absence of BEC and Mott Insulator Phase . . . . . . . . . . . . . 148 11.5 The Non-Interacting Gas . . . . . . . . . . . . . . . . . . . . . . . 153 11.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 12 New Developments 155 A Elements of Bogoliubov Theory 157 B An Exactly Soluble Model 173 C Definition and Properties of Scattering Length 179 D c-Number Substitutions and Gauge Symmetry Breaking 185 Bibliography 195 Index 209 Preface The mathematical study of the Bose gas goes back to the first quarter of the twentieth century, with the invention of quantum mechanics. The name refers to the Indian physicist S.N. Bose who realized in 1924 that the statistics governing photons(essentiallyinventedbyMaxPlanckin1900)isdetermined(usingmodern terminology)by restricting the physical Hilbert space to be the symmetric tensor product of single photon states. Shortly afterwards, Einstein applied this idea to massive particles, such as a gas of atoms, and discovered the phenomenon that we now call Bose-Einstein condensation. At that time this was viewed as a mathematical curiosity with little experimental interest, however. The peculiar properties of liquid Helium (first liquefied by Kammerlingh Onnes in 1908) were eventually viewed as an experimental realization of Bose- Einstein statistics applied to Helium atoms. The unresolved mathematical prob- lem was that the atoms in liquid Helium are far from the kind of non-interacting particles envisaged in Einstein’s theory, and the question that needed to be re- solved was whether Bose-Einstein condensation really takes place in a strongly interacting system — or even in a weakly interacting system. That question is still with us, three quarters of a century later! The first systematic and semi-rigorousmathematical treatment of the prob- lem was due to Bogoliubov in 1947, but that theory, while intuitively appealing and undoubtedly correct in many aspects, has major gaps and some flaws. The 1950’s and 1960’s brought a renewed flurry of interest in the question, but while theoreticalintuitionbenefitedhugelyfromthisactivitythemathematicalstructure did not significantly improve. The subject was largely quiescent until the 1990’s when experiments on low density (and, therefore, weakly interacting instead of strongly interacting, as in the case of liquid Helium) gases showed for the first time an unambiguous man- ifestation of Bose-Einstein condensation. This created an explosion of activity in the physics community as can be seen from the web site http://bec01.phy.georgiasouthern.edu/bec.html/bibliography.html, which contains a bibliography of several thousand papers related to BEC written in the last 10 years. At moreorlessthe sametime some progresswasmade inobtainingrigorous mathematical proofs of some of the properties proposed in the 50’s and 60’s. A general proof of Bose-Einstein condensation for interacting gases still eludes us, but we are now in a much stronger position to attack this problem rigorously. These notes, which are an extension of our 2004 Oberwolfach course, summarize vi Preface some rigorous results that have been obtained by us in the past decade. Most of themareaboutthegroundstateenergyinvariousmodelsanddimensions,butwe dohave afew results aboutthe occurrence(andnon-occurrence)ofcondensation. This pedagogicalsummary has severalantecedents.It has grownorganically as new results emerged. The first one was [LY3], followed by [LSeY3], [Se2], [L7], [LSeY4], [LSSY], and [LSSY2]. Apart from this stream, there was another peda- gogical survey going back to the 60’s [L3] that dealt with Bogoliubov theory and other things. Some of that material is reproduced in Appendices A and B. There is, of course, a large body of rigorous work by other people on var- ious aspects of BEC that was not covered in the Oberwolfach course and is not mentioned in these notes. The subject can be approached from many an- gles and our aim was not to give a complete overview of the subject but to fo- cus on themes where we have been able to make some contributions. The recent Physics Reports article [ZB] on the Bogoliubov model is a good source of refer- ences to some other approachesand results. There exist also severalreviews,e.g., [DGPS, ISW, Leg, C, Yu] and even monographs [PS, PiSt2] on the fascinating physics of the Bose gas and its condensation. Acknowledgments OurthanksgotoLetizziaWastavinoforproducingthe twofiguresandtoMichael Aizenman and Kai Schnee for permission to include the material in Chapters 11 and 9, respectively. We are also grateful to Daniel Ueltschi, Bruno Nachtergaele andValentin Zagrebnovfor very useful comments onvarious parts of these notes. Finally,wethanktheOberwolfachInstitutefortheopportunitytogivethiscourse and we thank the participants for pointing out numerous typos in a previous version of the manuscript. The work was supported in part by US NSF grants PHY 0139984-A01 (EHL), PHY 0353181 (RS) and DMS-0111298 (JPS); by an A.P. Sloan Fellowship (RS); by EU grant HPRN-CT-2002-00277 (JPS and JY); by the Alexander von Humboldt Foundation (EHL); by FWF grant P17176-N02 (JY);byMaPhySto–ANetworkinMathematicalPhysicsandStochasticsfunded by The Danish National Research Foundation (JPS), and by grants from the Danish research council (JPS). Chapter 1 Introduction 1.1 The Ideal Bose Gas Schr¨odinger’s equation of 1926 defined a new mechanics whose Hamiltonian is basedon classicalmechanics.The “ideal”gas ofparticles consists ofthe following ingredients:AcollectionofN 1non-interactingparticlesina largeboxΛ R3 ≫ ⊂ andvolumeV =L3.Weareinterestedinthe“thermodynamiclimit”,whichmeans that we will take N and L in such a way that the density ρ=N/V is →∞ →∞ held fixed. Thefactthattheparticlesarenon-interactingmeansthattheclassicalenergy, or Hamiltonian H, is entirely kinetic energy and this, in turn, is N 1 H = p2 (1.1) 2m i i=1 X wheremisthe particlemassandp isthe momentumofparticlei.Thelowest(or i ground state) energy of this classical system is, of course, 0. The thermodynamic properties are determined from the partition function 1 N N VN 2m 3N/2 ZN = h3NN!ZΛN i=1dxi ZR3N i=1dpie−βH = h3NN!(cid:18) β (cid:19) (1.2) Y Y (withβ =1/k T,k isBoltzmann’sconstantandT isthetemperature,andwith B B h an arbitrary constant with the dimension of momentum times length) in terms of which the free energy is given as 1 F = lnZ . (1.3) N −β 2 Chapter 1. Introduction The pressure p is p= ∂F/∂V =ρk T (1.4) B − whereStirling’sapproximation,lnN! NlnN,hasbeenused.Theaverageenergy ≈ is ∂ 3 E = lnZ = Nk T. (1.5) B −∂β 2 Inquantummechanics,theHamiltonianisanoperatorobtainedbyreplacing eachp by i~ , acting on the Hilbert space L2(Λ), with appropriateboundary j j − ∇ conditions. The eigenvalues of p2 = ~2∆ (with ∆ =Laplacian = 2), for a box − ∇ with periodic boundary conditions, are (2π~)2n2/L2, where n is a vector with integer components. If the statistics of the particles is disregarded, the partition function, which in the quantum case is given by 1 Z = tr e βH, (1.6) N − N! factorizes as Z = ZN/N!. This equals the classical expression in the thermody- N 1 namic limit, if one takes h=2π~. Taking the statistics of the particles into account, we have to restrict the trace in (1.6) to the symmetric or anti-symmetric subspace of the total Hilbert space NL2(Λ),dependingonwhetherweintendtodescribebosonsorfermions. This makes the prefactor 1/N! superfluous, but one has to face the problem that N Z is no longer determined by Z . For this reason, it is more convenient to pass N 1 to the grand-canonicalpartition function (or generating function) Ξ= Z zN, (1.7) N N 0 X≥ where z = eβµ is the fugacity for chemical potential µ. The chemical potential is then determined by the averageparticle number, ∂ N =z lnΞ. (1.8) h i ∂z The grand-canonical partition function Ξ can be calculated because it factorizes into the contributions from the single particle energy levels. For bosons the result is 1 Ξ= , (1.9) 1 exp( β(ε µ)) i i 0 − − − Y≥ where ε ε ... denote the single-particle energy levels (given, in this case, 0 1 ≤ ≤ by (2π~)2n2/(2mL2), with n Z3). Note that in this “free particle” bosonic case ∈ it is necessary that µ<ε . In the thermodynamic limit ε 0. 0 0 → 1.1. The Ideal Bose Gas 3 For fixed µ<0, the average particle number, is given by 1 N = . (1.10) h i exp(β(ε µ)) 1 i i 0 − − X≥ In the thermodynamic limit, the sum becomes an integral (more precisely, L 3 (2π~) 3 ), and we have − p → − R3 P R N 1 lim h i ρ=h−3 dp . (1.11) L L3 ≡ exp(β(p2/(2m) µ)) 1 →∞ Z − − This is a monotonously increasing function of µ, which is bounded as µ 0, → however,by the critical density ρ (β)=g (1)(2π~2β/m) 3/2. (1.12) c 3/2 − Here g3/2(1)= ∞ℓ=1ℓ−3/2 ≈2.612. That is, the density seems to be bounded by thisvalue.Thisisabsurd,ofcourse.ThisphenomenonwasdiscoveredbyEinstein P [E], and the resolution is that the particles exceeding the critical number all go into the lowest energy state. In mathematical terms, this means that we have to let µ 0 simultaneously with L to fix the density at some number >ρ . In c → →∞ thiscase,wehavetobe morecarefulinreplacingthesumin(1.10)byanintegral. It turns out to be sufficient to separate the contribution from the lowest energy level, and approximate the contribution from the remaining terms by an integral. The result is that, for ρ>ρ , c ρ=ρ (β)+ρ , (1.13) c 0 where 1 1 ρ = lim (1.14) 0 L→∞V exp(β(ε0−µ))−1 is the density of the “condensate”. The dependence of µ on L is determined by (1.10), writing N =L3ρ with fixed ρ. h i The phenomenon that a single particle level has a macroscopic occupation, i.e., a non-zero density in the thermodynamic limit, is called Bose-Einstein con- densation (BEC).Notethatinthemodelconsideredthereisnocondensationinto the excited energy levels, and one always has 1 1 lim =0 (1.15) L→∞V exp(β(εi−µ))−1 for i 1, since ε µ ε ε =const.L 2. i i 0 − ≥ − ≥ − 4 Chapter 1. Introduction Note that in the case of zero temperature, i.e., the ground state, all the particles are in the condensate, i.e., ρ = ρ . In a sector of fixed particle number, 0 thegroundstatewavefunctionissimplyaproductofsingleparticlewave-functions in the lowest energy state. 1.2 The Concept of Bose-Einstein Condensation So far we have merely reproduced the standard textbook discussion of BEC for non-interacting particles. The situation changes drastically if one considers inter- actingsystems,however.Forparticlesinteractingviaapairpotentialv(x x ), i j | − | the Hamiltonian takes the form ~2 N H = ∆ + v(x x ). (1.16) N i i j −2m | − | i=1 1 i<j N X ≤X≤ (We could also include three- and higher body potentials, but we exclude them for simplicity. These do exist among real atoms, but for understanding the basic physics it is presumably sufficient to consider only pair potentials.) Even at zero temperature, it is not entirely obvious what is meant by a macroscopic occupation of a one-particle state, because the eigenfunctions of H N are not products of single particle states. The concept of a macroscopic occupation of a single one-particle state ac- quires a precise meaning through the one-particle density matrix. Given the nor- malized ground state wave function of H (or any other wave function, for that N matter), Ψ , this is the operator on L2(R3) given by the kernel 0 γ(x,x)=N Ψ (x,X)Ψ (x,X)dX , (1.17) ′ 0 0 ′ Z where we introduced the short hand notation N X=(x ,...,x ) and dX= dx . (1.18) 2 N j j=2 Y Then γ(x,x)dx = tr[γ] = N. BEC in the ground state means, by definition, that this operator has an eigenvalue of order N in the thermodynamic limit. This R formulation was first stated in [PO] by Penrose and Onsager. For the ground state Ψ of H , the kernel γ is positive and, hopefully, translation invariant in 0 N the thermodynamic limit, and hence the eigenfunction belonging to the largest