Several authors have considered the possibility of a generalized
Bose-Einstein condensation (BEC) in which a band of low states is occupied so
that the total occupation number is macroscopic, even if the occupation number
of each state is not extensive. The Hohenberg theorem (HT) states that there is
no BEC into a single state in 2D; we consider its validity for the case of a
generalized condensation and find that, under certain conditions, the HT does
not forbid a BEC in 2D. We discuss whether this situation actually occurs in
any theoretical model system.Comment: 6 pages, Latex, JLTP class, accepted by Jour. Low Temp. Phys.,
  Quantum Fluids and Solids Conference QFS200

Holzmann, M.

Laloe, F.

Mullin, W. J.

English

arXiv

Several authors have considered the possibility of a generalized Bose-Einstein condensation (BEC) in which a band of low states is occupied so that the total occupation number is macroscopic, even if the occupation number of each state is not extensive. The Hohenberg theorem (HT) states that there is no BEC into a single state in 2D; we consider its validity for the case of a generalized condensation and find that, under certain conditions, the HT does not forbid a BEC in 2D. We discuss whether this situation actually occurs in any theoretical model system

Lalö, F.

ScholarWorks@UMass Amherst

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics2000Validity of the Hohenberg Theorem for aGeneralized Bose-Einstein Condensation in TwoDimensionsW. J. MullinUniversity of Massachusetts - AmherstM. HolzmannF. LalöFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationMullin, W. J.; Holzmann, M.; and Lalö, F., "Validity of the Hohenberg Theorem for a Generalized Bose-Einstein Condensation in TwoDimensions" (2000). Journal of Low Temperature Physics. 7.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/7arXiv:cond-mat/0009070v1  [cond-mat.stat-mech]  5 Sep 2000Validity of the Hohenberg Theorem for aGeneralized Bose-Einstein Condensation in TwoDimensionsW. J. Mullina, M. Holzmannb, and F. Laloe¨baPhysics Department, Hasbrouck LaboratoryUniversity of Massachusetts Amherst, MA 01003 USAbLKB, De´partement de Physique de l’ENS24 rue Lhomond 75005 Paris, FranceSeveral authors have considered the possibility of a generalized Bose-Einsteincondensation (BEC) in which a band of low states is occupied so that thetotal occupation number is macroscopic, even if the occupation number ofeach state is not extensive. The Hohenberg theorem (HT) states that thereis no BEC into a single state in 2D; we consider its validity for the case ofa generalized condensation and find that, under certain conditions, the HTdoes not forbid a BEC in 2D. We discuss whether this situation actuallyoccurs in any theoretical model system.PACS numbers: 03.75.Fi,05.30.Jp,05.70.Fh,67.40.Db,68.35.Rh.1. INTRODUCTIONThe Hohenberg theorem1 (HT) provides the general statement thatBose-Einstein condensation (BEC) cannot occur in a two-dimensional sys-tem. In this analysis a condensation implies extensive occupation of a singlestate of the system, that is, a density of particles of order N/V, (whereN is the number of particles in the system and V the volume of the sys-tem) in the thermodynamic limit. Recently however there has been renewedinterest in the possibility of a “smeared,” “fragmented,” or “generalized”BEC, in which some finite band of states, rather than a single state, is oc-cupied. Nozie`res and Saint James2 and more recently Nozie`res3, using aHartree-Fock approximation at zero temperature, showed that repulsive in-teractions favor single-state occupancy. There have been earlier discussionsof generalized BEC in the literature, for example, by Girardeau4, Luban5W. J. Mullin, M. Holzmann, and F. Laloe¨and Van den Berg et al6. Van den Berg and Lewis7 have even presented anon-interacting model in which an extensive BEC occurs in each state of aband of momentum states. Ho and Yip8 discuss fragmentation in a recentpaper and claim that the spin-1 Bose gas9 is an example of the occurrenceof this phenomenon.In two dimensions (2D), where the HT applies, there is no single-stateBEC, but there is a Kosterlitz-Thouless10 transition to a superfluid state.However, one can ask whether there could be a generalized transition, and,if so, whether it is related to the KT transition, or whether the generalizedtransition is also forbidden by the HT.If the generalized BEC is to be possible, we might envision differentforms of it. One possibility (which we call “fragmentation”) is that each ofa finite number of states is extensively occupied (occupation proportionalto N) and that the sum of all the particles in such a condensate is still oforder N . Another interesting possibility (here termed “smearing”) is thatno single state is extensively occupied, but that the sum of the occupationnumbers of all the states in a band is extensive. For example, one might haveO(√N) states each occupied by O(√N) particles so that the total numberof particles is of order N. The question is whether the HT forbids either ofthese kinds of generalized condensation in 2D.The generalized condensation studied in Ref. 7 occurs in 2D, whichseems to violate the HT, but there are some subtle aspects that need to beconsidered to see how this case fits into the general picture. The examplegiven there consists of particles moving freely in one dimension and boundharmonically in the other. The condensation occurs in the lowest harmonicband of states. In a similar problem, free particles trapped in a 2D harmonicpotential are known to have a BEC, but this arises because there is a “loop-hole” in the HT, due to of the inhomogeneous potential. One can show11that the HT does indeed apply to the case of trapped inhomogenous flu-ids, but that its application requires that the particle density be everywherebounded. When Bose condensed, the ideal gas in a trap has an integrablesingularity in its density in the thermodynamic limit, and thus does notfall under the conditions covered by the HT. However, as soon as repulsivehard-core interactions are turned on, this singularity disappears and the HTapplies.11 Similarly, the example of Ref. 7 is not a case of a violation of theHT because of fragmenting of the condensate, but rather a case where theHT does not apply because one has an ideal gas in a trap.The situation we will consider here is a homogeneous system (particlesin a box) in which we suppose that a generalized BEC occurs as mentionedabove, namely, a narrow band of momentum states occupied either exten-sively or non-extensively, but with the sum of their occupation numbersHohenberg Theorem for a Generalized Bose-Einstein Condensationextensive.2. DERIVATIONWe consider a homogenous system of N particles. We follow the deriva-tion of the Hohenberg theorem due to Chester.12 Start with the Bogoliubovinequality given by 〈12{A,A†}〉>kT |〈[C,A]〉|2〈[[C,H] , C†]〉 , (1)where 〈...〉 means thermal average, A and C are operators, and H is thesystem Hamiltonian. Choose A = apa†p+k, and C =∑q a†qaq+p, where a†pcreates a particle with momentum p. C, being the Fourier transform of thedensity operator, commutes with the interaction potential energy inH, so weneed only to consider commutators with the kinetic energy. Upon carryingout these commutators we find〈12{A,A†}〉= npnp+k +12(np + nk) + δk,0(np + 1), (2)〈[C,A]〉 = np − np+k, (3)〈[[C,H] , C†]〉=~22m∑q[(k+ q)2 + (k− q)2 − 2q2]nq=~2k2m∑qnq = N~2k2m, (4)where np is the thermal average number of particles in state p. The inequal-ity becomesnpnp+k +12(np + nk) + δk,0(np + 1) >kTmN~2k2(np − np+k)2. (5)We assume that there areMc condensed states (our generalized conden-sate band) containing N0 particles with N0 = O(N), and that these statesare clustered in momentum space around k = 0 in a circle out to radiuspc. The non-condensed states are in the region beyond pc. We assume thatthe condensed states each have occupation that, while not necessarily ex-tensive, still greatly exceeds that of any non-condensed state. We have then∑p<pc1 =Mc,∑p<pcnp = N0. If we change the sum to an integral, thefirst of these equations becomesMc =pc∑p=01 =L22pip2c2, (6)W. J. Mullin, M. Holzmann, and F. Laloe¨where L is the dimension of the box and n = N/L2 is the density. Thuspc = c1(McN)1/2n1/2, (7)where c1 is a constant of order unity.Sum both sides of Eq.(5) on k over the range 2pc < k < pm, where theupper limit km satisfies km = c2n1/2 with c2 a constant. Below we will seewhy we take the minimum k value as twice pc rather than just pc. We havenp∑knp+k +12∑k(np + nk) >kTmN~2∑k(np − np+k)2k2. (8)We assume that p is in the range of condensed states so that the secondterm in the second sum on the left is much smaller than the first and can beneglected. The remaining term in that sum isnp∑k1 =L22pi∫ pm2pcdkk = npNn(4pi)(p2m − 4p2c) = Nnpγ, (9)where γ = (c2 − 4c1McN )/(4pi) is a constant of order unity. The first sum onthe left of Eq.(8) is over only a portion of momentum space and results in afraction of the total particle number. Increasing the sum to the full numberof particles just amplifies the inequality. We then can write npN(1 + γ) forthe left side.The sum on the right side of Eq.(8) has the vector k + p extendingoutside the condensate range because of the restriction that k > 2pc. If wehad defined the minimum value of k as just pc then, by putting k and p inopposite directions, the sum could extend back into the condensate circle.As it is we can neglect np+k relative to np on the right side of the equationto give simply n2p∑k 1/k2. The sum is evaluated by doing the appropriateintegral: ∑k1k2=Nn(2pi)∫ pm2pcdk1k= c3Nnln(pm2pc). (10)The result isnpN(1 + γ) > n2pkTmN~2c3Nnln(pm2pc)(11)ornpN6~2nmkTc31 + γln(pm2pc) . (12)Hohenberg Theorem for a Generalized Bose-Einstein CondensationNow sum this over all states in the condensate circle to giveN0N6~2nmkTc3(1 + γ)ln(c4√NMc)Mc. (13)where we have used the fact thatpm2pc=c2√n2c1√nMcN= c4√NMc. (14)If Mc = 1, as in the usual case, then, in the thermodynamic limit,the right side goes to zero as 1/ ln(N) and there is no BEC. However, ifMc = O(Nν), where ν is any power not equal to zero, then the inequalitybecomesN0N6~2nmkTNνln(N (1−ν)/2) . (15)Now the right side diverges as N →∞. For example, if ν = 12 then the rightside is of order√N/ ln(N), which clearly diverges. In this case we can nolonger rule out the possibility of BEC by this argument.3. DISCUSSIONOur derivation does not prove that BEC actually happens in a general-ized mode. Moreover, it also does not insure that there might not be anothermore powerful derivation of the HT that outlaws a generalized condensationin 2D. In the case of a fragmented condensate (as in Ref. 7), one can haveonly a finite number, Mc, of states each containing O(N) particles so thatMc is O(1) and there can be no fragmented BEC in a 2D homogeneous sys-tem. The fragmented case of Ref. 7 is “tainted” by the fact that the HTconditions for the inhomgeneous case are violated by a divergent density ofparticles so that it is not a counter-example to the HT case that we discussin this paper. A 2D version of the spin-1 gas, claimed in Ref. 8 to be anexample of fragmentation in 3D, would not fall directly under our derivationbecause it is a condensation in spin space, not in the momentum space of ourderivation. (The spin-1 Bose system appears to be a case that violates theclaim of Refs. 2 and 3 that a single-state BEC was favored over generalizedBEC. However, that discussion was based on use of mean-field theory andthe spin-1 system analyses go beyond mean-field approximations.)On the other hand, the analysis above leaves open the possibility of asmeared BEC in 2D. In this case, Nν(ν < 1) states each containing only aW. J. Mullin, M. Holzmann, and F. Laloe¨non-extensive number of particles constitute a condensate as a unit. Notethat the width of this band of states, as given in Eq.(7), tends to zero asN increases, so one still gets a δ-function occupation in the thermodynamiclimit.A question yet to be answered is whether the KT transition is relatedin some way to a generalized BEC. The analysis of Popov13 of the KT stateat low temperature involves what he calls a “bare” condensate spread overa set of states up to some cut-off k0. He says “the particles with momentasmall compared with the faster particles behave like a condensate.” However,it is not clear to us at this time whether this situation would qualify as ageneralized condensate in the sense of our theorem above.4. ACKNOWLEDGMENTSThe Laboratoire Kastler-Brossel is Unite´ Associe´e au CNRS (UMR8552)et a` l’Universite´ Pierre et Marie Curie. WJM would like to thank EcoleNormale Supe´rieure, where some of this research was carried out, for a travelgrant and for its excellent hospitality.REFERENCES1. P.C. Hohenberg, Phys. Rev. 158, 383 (1967).2. P. Nozie`res and D. Saint James, J. Physique 43, 1133 (1982).3. P. Nozie`res, in Bose-Einstein Condensation, edited by A. Griffin, D. W. Snoke,and S. Stringari (Cambridge University Press, Cambridge, England, 1996).4. M. Girardeau, Phys. Fluids 5, 1468 (1962); J. Math Phys. 11, 684 (1970).5. M. Luban, Phys. Rev. 128, 965 (1962).6. M. Van den Berg, J. T. Lewis, P. de Smedt, J. Stat. Phys. 37, 697 (1984).7. M. Van den Berg and J. T. Lewis, Commun. Math. Phys. 81, 475 (1981).8. T.-L. Ho and S. K. Yip, Phys. Rev. Lett. 84, 4031 (2000).9. C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett. 81, 5257 (1998).10. J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973); V. L. Berezinskii,Sov. Phys. JETP 32, 493 (1971).11. W. J. Mullin, J. Low Temp. Phys. 106, 615 (1997).12. C. V. Chester, in Lectures in Theoretical Physics, edited by K. T. Mahanthappa(Gordon and Breach Science Publishers, Inc., New York, 1968), Vol 11B, P. 253.13. V. N. Popov, Functional Integrals in Quantum Field Theory and StatisticalPhysics, (Reidel Publishing Company, Dordrecht, 1983).

Validity of the Hohenberg Theorem for a Generalized Bose-Einstein Condensation in Two Dimensions

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