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recent advances in the controlled fabrication and manipulation of ultra - low temperature trapped bose gases @xcite has finally made it possible to experimentally investigate the existence of off - diagonal long - range order ( odlro ) , i.e. , bose - einstein condensation ( bec ) , in dimensions lower than three . this remarkable experimental achievment has rekindled an interest in the classic theoretical problem concerning the existence of bec in two - dimensions ( 2d ) . although it is well known that finite - temperature bec can never occur in a homogeneous 2d bose gas @xcite , the ideal trapped bose gas can undergo bec below some critical temperature @xmath3 . in the case of a 2d isotropic harmonic trap with confining frequency @xmath4 ( unless stated otherwise , we always implicitly assume this confinement geometry ) , bagnato and kleppner have shown that @xmath5 @xcite , where @xmath6 ; shevchenko later extened this result to include more general power law potentials @xcite . in the thermodnyamic limit , viz . , @xmath7 , @xmath8 , such that @xmath9 , bec is also known to theoretically occur with the critical temperature remaining unchanged . note that this is not the usual thermodynamic limit , which in 2d would demand that @xmath10 , resulting in no bec ( see also ref . @xcite for an alternative thermodynamic limit in the case of an interacting bose gas ) . if repulsive interactions between bosons are included , ( we only consider repulsive interactions in this paper ) , then a definitive answer to the question of bec in the trapped 2d gas is not so clear . in fact , there is no _ a priori _ reason for the existence of bec in a confined two or three - dimensional system  bec is a purely kinematical phenomenon in the sense that even the ideal bose gas can undergo the phase transition . in contrast , a transition to the superfluid state is wholey dynamical and _ can not _ occur without interactions . therefore , one could sensibly assume that interactions prevent the spatial redistribution of atoms required to achieve bose gas degeneracy , and therefore destroy the bec phase transition altogether . it is only recently that a rigorous proof has been given to show that there is no finite temperature bec in the interacting trapped 2d bose gas in the thermodynamic limit @xcite . of course , in actual experiments , the theormodyamic limit is never fully realized , and it becomes necessary to consider systems consisting of a _ finite _ number of atoms confined in a trap . in this regard , lieb and seiringer @xcite have proved that the finite , interacting trapped bose gas does have 100% bec at @xmath11 . to date however , no irrefutable case has been made ( experimentally or theoretically ) in support of finite temperature bec in an interacting finite 2d system . in a recent paper , petrov _ et al . _ @xcite have suggested that that there are in fact two bec regimes for finite , trapped , ( quasi)-2d systems . the general argument supporting their claim hinges on an approximate analytical calculation of the one - particle density martix from which information about the phase correlations of the condensate can be obtained . for large particle number and sufficiently low temperatures , petrov _ et al . _ deduce that the phase fluctuations are governed by @xmath12 @xcite . thus , at temperatures well below @xmath13 , fluctuations in the phase are supressed and one has a `` true condensate '' whereas for @xmath14 , the phase fluctuations are enhanced and one has a `` quasi - condensate '' . similar results have recently been obtained by bogliubov _ et al . , _ in strictly two - dimenions  @xcite .    however , bhaduri _ et al . _ @xcite have suggested that there is no bec phase transition in a finite , interacting 2d bose gas ( see also refs . @xcite for similar conclusions ) . in their investigation , a semiclassical approximation to the hartree - fock - bogliubov ( hfb ) mean - field theory was used to investigate the thermodynamics of the trapped 2d bose gas . although the semiclassical hfb approach could not shed any light on the question of phase fluctuations , it was shown that the hfb equations could be solved self - consistently all the way down to @xmath11 provided one _ does not _ invoke the presence of a condensate . this finding seems to indicate that that if no _ a priori _ assumption of a condensate is made ( as it should be ) , the system remains in the same _ uncondensed phase _ at all temperatures . ( note that applying the same hfb approach in 3d requires a condensate in order to obtain self - consistent solutions below @xmath0 . ) in spite of the absence of a condensate order - parameter however , the density distribution for the uncondensed phase was found to clearly exhibit a large spatial accumulation of atoms near the center of the trap for @xmath15 , similar to what is found in systems with a condensate . this suggests that there may be some sort of `` revival '' of odlro in the system , even though it does not lead to the formation of a condensate . since the off - diagonal one - particle density matrix can provide information about phase correlations and odlro in the system , an investigation of the off - diagonal density martix within the hfb approximation needs to be performed . this is indeed the central motivation for the present work . the rest of our paper is organized as follows . in sec . [ noninteracting ] we investigte the noninteracting 2d gas in a harmonic trap where an exact expression for the finite temperature correlation function can be obtained . in sec . [ interacting ] , we calculate the correlation function for the interacting system within the semiclassical hfb approximation and investigate its temperature dependence both above and below @xmath0 . finally , in sec . [ conclusions ] , we present our concluding remarks . in this section , we wish to pay special attention to the off - diagonal , one - particle density matrix @xmath16 of the noninteracting , confined 2d bose gas . our motivation lies in the fact that odlro in the one - particle density matrix and bec are interrelated @xcite ( see below for details ) . for the isotropic ho , @xmath16 can be calculated analytically , and we have an exact result from which we can quantitatively test the validity of the semiclassical approximation . we will be particularly interested how well the seimclassical approximation describes the phase correlations of the gas below the bec critical temperature @xmath17 . to the best of our knowledge , this is the first such study in two - dimensions .    by definition , the off - diagonal one - particle density matrix is given by @xcite ^(0)(,*r* ; ) = _ j ^(0)_j ( ) ^(0)_j()^ n^(0)_j  , [ green ] where @xmath18 are exact eigenfunctions of the @xmath19-dimensional harmonic oscillator ( ho ) and @xmath20 is the bose distribution function . it is most convenient to evaluate ( [ green ] ) for the 1d case , and then use the separability of the ho potential to obtain the general result . explicitly , we have ^(0)(x , x ; ) = _ n=0^e^- ( x^2 + x^2 )  , [ green2 ] where @xmath21 are the energy levels of the ho , @xmath22 is the chemical potential , and we have scaled all energies and lengths by @xmath23 and @xmath24 , respectively . we now set the zero of energy to correspond to the lowest trap level , viz . , @xmath25 , and obtain from ( [ green2 ] ) ^(0)(x , x ; ) = _ j=1^e^j _ n=0^e^- ( x^2 + x^2)e^-jn  , where we have used @xcite _ j=1^e^je^-jn =  . the summation over trap levels can be perfomed exactly @xcite , and we finally obtain in any dimension ^(0)(, ; ) = _ j=1^ e^-1/4[|+|^2(j/2)+|-|^2(j/2 ) ]  . putting @xmath26 in the above equation reproduces the result obtained by barnett _ et al_. ,  @xcite . the chemical potential is obtained by enforcing that the total particle number @xmath27 , remain fixed , viz . , n = _ j=1^  . the diagonal part of the one - particle density matrix yields the exact quantum - mechanical finite temperature density , @xmath28 , of the trapped gas . the densities for various temperatures are shown in fig .  1 for @xmath29 and @xmath30 atoms . figure 1 illustrates that even though we are not in the thermodynamic limit , the single - particle density distribution exhibits an enormous accumulation of atoms near the center of the trap for @xmath31 . this type of behaviour is consistent with the confined gas having undergone a phase transition to the bec state . the usual approach for confirming that this phase transition is indeed bec is to examine the temperature dependence of the fractional occupancy of the lowest oscillator level , @xmath32 . the inset to fig . 1 shows that below @xmath33 , the ground state of the system becomes macroscopically occupied , with already @xmath34 of the atoms in the lowest level at @xmath35 . remarkably , even at @xmath30 atoms , the fractional occupancy closely follows the quadratic temperature dependence obtained strictly in the thermodynamic limit ( dashed line in figure inset ) , viz . , = 1 - ( ) ^2  . although @xmath32 gives us a strong indication that the finite system has a bec transition at @xmath33 , it does not provide us with a definitive test for the presence of odlro in the system . it is well known since the early work of penrose and onsager @xcite that for a _ homogeneous _ system , the phenomenon of bec is intimately related to the presence of odlro in the off - diagonal one - particle density matrix , namely , _ |-|^(0)(, ; ) 0  . in fact , the low temperature behaviour of @xmath36 is directly related to the phase fluctuations of the condensate @xcite and the non - zero limiting value of eq . ( 8) yields the value of the order parameter ( i.e. , condensate fraction ) in the system . for finite inhomogeneous systems however , the above criterion has to be modified to account for the fact that odlro can not extend beyond the physical size of the system . this is accomplished by defining a normalized correlation function ( sometimes called the first - order coherence function ) by g^(0)(, ; ) =  , such that _ |-|g^(0)(,;)0  . the above correlation function describes a local measure of coherence in the system . for example , the ability of two bec s , intially separated by @xmath37 , to form an interference pattern is quantified by @xmath38 @xcite . in this sense , the correlation function also characterizes local fluctuations of the phase of the condensate order - parameter . the quantity @xmath39 is the large - distance length - scale over which the odlro of the finite system is manifested . at zero temperature , @xmath40 , whereas for @xmath41 , @xmath39 is taken to be the spatial extent of the single - particle density @xmath28 . to simplify our calculations , we fix one of the coordinate , @xmath42 to be at the center of the trap . note that in eq . ( 9 ) , dividing the off - diagonal density matrix by the square root of the single - particle densities removes any effects of the local - density of particles on the correlation function , and gives @xmath43 in the case of perfect coherence .    the solid curves in fig . 2 depict the correlation function @xmath44 for @xmath29 and @xmath30 at various temperatures . from left to right , we have @xmath45 . above the critical temperature , the atoms remain in the uncondensed phase and the system shows no odlro . however , as the gas is cooled below the transition temperature , we see an abrupt jump in the large - distance decay of the spatial correlations . a direct application of eq .  ( 10 ) yields condensate fractions of @xmath46 , for temperatures @xmath47 , respectively . thus , below @xmath17 , the finite noninteracting 2d bose gas contains a non - zero order - parameter ( i.e. , a bec ) . it also proves very useful to compare the spatial extent of the single - particle density , @xmath28 , to the spatial decay of the correlation function @xmath44 . this information is also presented in fig .  2 where the exact quantum mechanical densities are represented by dashed lines . these densities have been scaled by their value at @xmath48 to allow for a comparison of the length scales between @xmath28 and @xmath44 . from right to left , the dashed lines correspond to densities at @xmath49 . densities at lower temperatures are almost indistinguishable on the scale of the figure . we note that at low temperatures , the correlation function is practically a constant @xmath50 over the entire region for which the density ( i.e. , physical size of the system ) of the gas is nonzero . this means that phase fluctuations are very small , and the density distribution corresponds to that of a `` true condensate '' . at higher temperatures ( i.e. , @xmath51 ) the single - paricle density can become comparable to that of @xmath44 , and the gas can be said to be a `` quasi - condensate '' in the sense that the phase correlations do not extend to all of the atoms in the trap . for @xmath52 , the spatial extent of the single - particle density greatly exceeds characteristic length of the correlations , and system is in the normal state .    having obtained the exact expression for @xmath53 , we now wish to examine its semiclassical approximation , namely , its behaviour when @xmath54 . it is easy to show that eq  ( 5 ) reduces to @xmath55/2\right]^j\exp\left[-|\bfr-\bfr'|^2)/2\beta\right]^{1/j}\\ & = & \frac{1}{(2\pi\beta)^{d/2}}g_{d/2}(\exp\left [ \beta(\mu- [ v(\bfr)+v(\bfr')]/2)\right],\exp\left[-|\bfr-\bfr'|^2)/2\beta\right])~,\end{aligned}\ ] ] where @xmath56 , and as in ref . @xcite we have introduced the generalized bose function g_(x , y ) = _ j=1^  . for @xmath57 and @xmath29 , eq . ( 12 ) becomes ^(0 ) ( ) =  , with the normalization condition n =  ^(0 ) ( )  d^2  . if one attempts to solve eqs . ( 14,15 ) for all @xmath58 , it is readily shown that there exists a temperature @xmath59 below which the equations can no longer be satisfied  @xcite . in fact , the temperature @xmath59 is identical to the the critical temperature @xmath3 obtained by bagnato and kleppner  @xcite . thus , in order to obtain solutions below @xmath3 , a condensate order parameter is required . we introduce the macroscopic occupation of the ground state in the one - particle density matrix according to the _ ansatz _ ^(0)(, ) = n_0 ^(0)_0 ( ) ^(0)_0( ) + g_d/2 ( , )  , where @xmath60 is the zero temperature ground - state eigenfunction of the ho and n_0 =  . the 2d density is then given by eq . ( 16 ) with @xmath57 , ^(0 ) ( ) = n_0|^(0)_0()|^2 +  . the semiclassical densities obtained from eq .  ( 18 ) are indistinguishable from the exact densities obtained from eq . the semiclassical correlation function , eq . ( 16 ) , at various temperatures is represented in fig .  2 as filled circles . the superb agreement between the exact ( solid curves ) and semiclassical results shows that the semiclassical approximation is valid well below the critical temperature provided one takes into account the macroscopic occupation of the ground state separately . if the atoms had remained in the uncondensed phase , eqs . ( 14,15 ) would have been sufficient to describe the gas for all @xmath61 . the thermodynamic properties of the interacting bose gas are most commonly described by the self - consistent mean - field hfb equations . in most calculations ( see ref . @xcite and references therein for details ) , the semiclassical approximation is applied to the discrete set of coupled bogliubov equations , leading to the considerably simplified semiclassical hfb theory . the semiclassical hfb approximation is well - known to be an excellent tool for describing the interacting 3d bose gas @xcite . the 2d version of the semiclassical hfb theory is formally identical to the 3d version and has already been examined by mullin @xcite . we present only the essentials of the model here for completeness .    in the semiclassical hfb approach , the condensed and uncondensed atoms are described by a set of coupled , self - consistent equations . the macroscopic condensate wavefunction , @xmath62 , is characterized by the finite - temperature gross - pitaevskii equation which is given by _ 0 ( ) = 0  , where @xmath63 is the coupling constant associated with the assumed repulsive zero - range two - body pseudo - potential . the total density , @xmath64 , is made up of a condensate @xmath65 , and a noncondensate density , _ t ( ) =  \ { f ( , ) + v^2 ( , ) }  d^2  , where f ( , ) =  , u^2 ( , ) =  , v^2 ( , ) =  , = + r^2 - + 2 ( )  , and ( , ) =  . the total number of particles satisfies n =  [ _ 0 ( ) + _ t ( ) ]  d^2 . in the absence of a condensate , it is easy to show that the above set of hfb equations reduce to _ t ( ) =  , with n =  _ t ( )  d^2 .    equations ( 27,28 ) are equivalent to the so - called self - consistent thomas - fermi approximation used in refs .  @xcite . putting @xmath66 in eq . ( 27 ) yields an expression that is identical to the semiclassical approximation of the exact diagonal density matrix , viz . , eq . ( 14 ) , found in the previous section ; recall from sec . [ noninteracting ] that below @xmath17 , eqs . ( 27,29 ) can not be solved self - consistently . however , if one puts any finite @xmath67 in eq . ( 27 ) , self - consistent solutions to the hfb equations can be obtained all the way down to @xmath68  @xcite . to illustrate this , we show in fig . 3 the density and chemical potential ( figure inset ) of the gas for @xmath69 ( long - dash - short - dashed line ) , @xmath70 ( solid line ) , and @xmath71 ( dashed line ) . in contrast to the noninteracting 2d gas ( where a condensate order - parameter has to be introduced below @xmath17 ) , the interacting system remains in the uncondensed phase for all @xmath61 . although no condensation occurs , it is interesting to compare the densities of the noninteracting ( fig .  1 ) and interacting gases ( fig . 3 ) just above and below @xmath17 . the essential point to be made from this comparison is that in both cases , there is an increase in the density of atoms near the center of the trap for @xmath72 . while the noninteracting density shows a much larger enhancement at @xmath48 , the interacting gas also displays a similar ( albeit diminished ) behaviour . since an increase in the density of atoms near the center of the trap is associated with odlro in the noninteracting gas , it is worthwhile investigating the off - diagonal density matrix for the interacting 2d gas within the semiclassical hfb approximation . the 2d semiclassical hfb off - diagonal density matrix can be obtained from the leading order term in the wigner - kirkwood semiclassical expansion of the quantum mechanical off - diagnonal density matrix @xcite . the noncondensed particles are treated as bosons in an effective potential , and we obtain ( ,  ; ) = g_1 ( , )  , where v_eff ( ) = r^2 + 2 ( )  . since there is no condensate for @xmath73 , we _ have not _ included the first term in eq . of course , for @xmath74 , eq .  ( 29 ) reduces to eq . ( 27 ) .    as in sec . [ noninteracting ] , the normalized correlation function for the finite sized system is given by g(,0 ; ) =  , where again , we have fixed @xmath75 . the temperature dependence of @xmath76 is shown in figs .  4 , 5 for coupling strengths @xmath77 , respectivley .    above @xmath17 , the interacting and noninteracting ( see fig . 2 ) systems have similar correlations . however , as we cool the gas to temperatures just below @xmath17 , there is an drastic change in the behaviour of the interacting off - diagonal density matrix , which to the best of our knowledge , has never been noticed before . in particluar , we observe that the decay of the correlations in the interacting system are qualitatively different from the noninteracting gas . for the interacting gas , values of @xmath78 lead to the tail of the correlation function decreasing slowler than the noninteracting case . in contrast , for @xmath79 , the interacting correlation function has a much faster spatial decay than the noninteracting system . we have already discussed the fact that the near constancy of the noninteracting correlation function , @xmath38 , over the size of the sample ( @xmath80 ) is indicative of negligible phase fluctuations in the condensate @xcite . thus , the re - appearance of odlro in @xmath81 below @xmath17 suggests that there may be larger fluctuations in the phase as compared to the noninteracting gas . if we apply eq . ( 10 ) to @xmath81 ( with @xmath70 ) , we obtain @xmath82 at @xmath83 respectively , for the asymptotic behaviour of the correlation fucntion . even though we may not be able to interpret these numbers as condensate fractions , it is tempting to conjecture that they represent a nonzero value of an order - parameter in the system .    we can be more quantitative about the nature of the correlations displayed by @xmath81 below @xmath17 as follows . in the interacting _ homogeneous _ 2d bose gas , popov @xcite has already calculated the low temperature correlation function : g(, ; ) r^-  , where r ,   =  , and @xmath84 is the so - called superfluid density . algebraic _ odlro exhibited by @xmath81 implies that at low temperature , correlations can extend over macroscopic ( i.e. , size of the system ) distances , although phase fluctutations will be nonzero . this is reminiscent of the quasi - condensate described in ref . @xcite , and it is only identically at @xmath11 that the system will attain a `` true condensate '' .    if the system is finite ( i.e. , in a 2dho potential ) , then bogliubov _ et al . _ @xcite , have shown that r ,   =  , where @xmath85 . ( 34 ) is formally identical to ( 33 ) except for the slowly varying factor @xmath86 which is the inhomogeneous boson density . this additional spatial dependence results in a deviation from the strict power law dependence of the uniform gas . we have found that at low temperatures , the power law dependence in eq .  ( 34 ) agrees well with the observed numerical behaviour of @xmath87 . this suggests that at low temperatures , the interacting system exhibits algebraic odlro similar to that of the superfluid state in the uniform bose gas . at exactly @xmath11 , the gas will have a true condensate . we have investigated the finite - temperature correlations in both the ideal and interacting trapped , 2d bose gas . our study of the noninteracting gas provided an exact framework from which we could compare the validity of the semiclassical approximation over a wide temperature range . our results indicate that the semiclassical approach works extremetly well , even below the critical temperature , provided the macroscopic occupations of the ground state is treated separately . a careful examination of the correlation function reveals that , as in ref . @xcite , there are two bec regimes for the noninteracting gas . namely , for temperatures @xmath51 , the system is a quasi - condensate in the sense that the phase correlations are comparable to the spatial extent of the single - particle density . on the other hand , when the temperature is lowered to @xmath88 , phase fluctuations become negligible , and the gas is a true condensate . it is straightforward to apply our results to study the finite @xmath27 dependence on the coherence properties of trapped gases in dimensions @xmath89 . ( see e.g. , ref . @xcite for work relating to 3d ) .    in the interacting gas , we found that even though the system does not contain a condensate , there is a revival of odlro for @xmath90 . the low temperature behaviour of the interacting correlation function was found to be consistent with an algebraic ( i.e. , power law ) decay in the phase correlations , similar to what is found in the uniform 2d superfluid bose gas . whether or not this re - entrant odlro is to the superfluid state is presently being investigated .    finally , it is interesting to note that margo and ceperely @xcite have found conclusions analgous to ours for the uniform 2d charged bose fluid ( cbf ) with @xmath91 interactions . specifically , they found that even though the 2d cbf _ does not _ undergo a condensation , the noncondensed fluid exhibits a power - law decay of the one - particle density matrix similar to what is found in the present work . it is not immediately clear what relevance their findings may have on our results for the interacting , inhomogeneous 2d bose gas . it is a pleasure to thank drs . wonkee kim , rajat bhaduri and j. p. carbotte for illuminating discussions . this work was supported in part by the national sciences and the engineering research council of canada ( nserc ) and by the canadian institute for advanced research ( ciar ) .
we investigate the presence of off - diagonal long - range order in a harmonically confined two - dimensional bose gas . in the noninteracting case , an analytical calculation of the the finite - temperature one - particle density martix 
 provides an exact description of the spatial correlations known to be associated with the existence of a bose - einstein condensate below the transition temperature @xmath0 . 
 we treat the effects of repulsive interactions within the semiclassical hartree - fock - bogliubov approximation and find that even though the system remains in the same _ uncondensed phase _ for all @xmath1 , there appears to be a revival of off - diagonal long - range order for temperatures @xmath2 . 
 we suggest that this reentrant order is related to a phase transition in the system which _ is not _ the bec state .