behavior of the anomalous correlation function in a uniform two-dimensional bose gas
TRANSCRIPT
PHYSICAL REVIEW A 86, 043608 (2012)
Behavior of the anomalous correlation function in a uniform two-dimensional Bose gas
Abdelaali Boudjemaa*
Department of Physics, Faculty of Sciences, Hassiba Benbouali University of Chlef P.O. Box 151, 02000 Chlef, Algeria(Received 8 July 2012; published 9 October 2012)
We investigate the behavior of the anomalous correlation function in two-dimensional Bose gas. In the localcase, we find that this quantity has a finite value in the limit of weak interactions at zero temperature. The effectsof the anomalous density on some thermodynamic quantities are also considered. These effects can modify,in particular, the chemical potential, the ground-state energy, the depletion, and the superfluid fraction. Ourpredictions are in good agreement with recent analytical and numerical calculations. We show also that theanomalous density presents a significant importance compared to the noncondensed one at zero temperature. Thesingle-particle anomalous correlation function is expressed in two-dimensional homogenous Bose gases by usingthe density-phase fluctuation. We then confirm that the anomalous average accompanies in analogous mannerthe true condensate at zero temperature, while it does not exist at finite temperature.
DOI: 10.1103/PhysRevA.86.043608 PACS number(s): 03.75.Hh, 05.30.Jp
I. INTRODUCTION
The experimental progress of the ultracold gases in twodimensions (2D) [1–8] has recently attracted great attention.The properties of these fluids are radically different fromthose in 3D. The famous Mermin-Wagner-Hohenberg theorem[9,10] states that long-wavelength thermal fluctuations destroylong-range order in a homogeneous 1D Bose gas at alltemperatures and in a homogeneous 2D Bose gas at anynonzero temperature, preventing formation of condensate.
Since the earlier works of Schick [11] and Popov [12],several theoretical studies of fluctuations, scattering properties,and the appropriate thermodynamics have been performedin [13–16]. In fact, in most of the previous references, theanomalous density is neglected under the claim that it is adivergent and unmeasured quantity, as well as its contributionbeing very small compared to the other terms. Otherwise,the importance of the anomalous density in 3D Bose gashas been shown in our recent theoretical results [17,18] andalso by several authors [19–24] using different approaches.Theoretically, the anomalous average arises of the symmetry-breaking assumption [17,19,24]. It quantifies the correlationsof pairs of noncondensate atoms with pairs of condensateatoms due to the Bogoliubov pair promotion process in whichtwo condensate atoms scatter each other out of the condensatewhich is responsible for the well-known Bogoliubov particle-hole structure of excitations in the system [24]. The anomalousdensity can also be interpreted as a measure of the squeezingof the noncondensate field fluctuations [25]. Certainly, thepresence of this quantity adds new features to the well-knownproblems and attracts our attention to the 2D systems. Anumber of questions arise naturally in this paper. Does theanomalous density exist even at finite temperature in 2D Bosegas? How does its behavior compare with the normal densityat zero temperature? What are the effects of this quantity onthe thermodynamics of the system?
Due to the complexity and the particularity of dilute2D Bose gases, many analytical investigations have beenperformed recently to find corrections beyond mean field
at zero temperature. One should cite at this stage thatPricoupenko [26] employs the pseudopotential with a Gaussianvariational approach. Mora and Castin [27], on the other hand,used their lattice model, which is a sort of regularizationscheme to treat ultraviolet divergences. Cherny et al. [28] useda reduced-density matrix of second order and a variationalprocedure to derive results identical to those of Refs. [26,27]for equation of state (EoS) and ground-state energy. Theabove analytical results have been checked using MonteCarlo calculations to find numerical agreement with beyond-mean-field terms in 2D [29,30]. Recently, Mora and Castin[31] have been also extended their approach [27] one stepbeyond Bogoliubov theory, which gives good accuracy withthe simulations of [30]. Another kind of extension has beendeveloped recently by Sinner et al. [32] which is based onusing the functional renormalization group to study dynamicalproperties of the 2D Bose gas at T = 0. The approach is freefrom infrared divergences and satisfies both the Hungeholtz-Pines (HP) [33] relation and the Nepomnyashchy identity[34], which states that the anomalous self-energy vanishesat zero frequency and momentum. The spectrum energy thussatisfies a Bogoliubov-type expression with a renormalizedsound velocity. Although the above approaches provide goodpredictions for the thermodynamic of 2D Bose gas in theuniversal regime, they are limited only at zero temperature.
The present paper deals with extending our variational time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory to thecase of 2D Bose systems. The theory was previously presentedfor 3D systems in [17,18]. In fact, the main difference betweenour approach and the earlier variational HFB treatments isthat in our variational theory we do not minimize only theexpectation values of a single operator like the free energy inthe standard HFB approximation. Conversely, our variationaltheory is based on the minimization of an action also with aGaussian variational ansatz. The action to minimize involvestwo types of variational objects: one related to the observablesof interest and the other akin to a density matrix [35,36].
The paper is organized as follows. In Sec. II, we brieflyreview the derivation of the TDHFB formalism and givethe different quantities which we study in 2D homogeneoussystem. In Sec. III, we restrict ourselves to the behavior ofthe anomalous density and its effects on the depletion, the
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ABDELAALI BOUDJEMAA PHYSICAL REVIEW A 86, 043608 (2012)
chemical potential, and the ground-state energy. We thereforecompare our results with recent Monte Carlo simulationsand analytic predictions. The validity of the HP theoremand Nepomnyashchy identity are also discussed within thepresent formalism. In Sec. IV, we extend our results at finitetemperature where we calculate, in particular, the one-bodyanomalous correlation function. In Sec. V, we apply ourformalism to analyze the behavior of the superfluid fraction.We then emphasize the importance of the anomalous densityfor the occurrence of superfluid transition and sound velocity.Our conclusion and perspectives are drawn in Sec. VI.
II. FORMALISM
Our starting point is the TDHFB equations which describethe dynamics of d-dimensional interacting trapped Bosesystems. For a short-range interaction potential and sufficientlydilute gas, the TDHFB equations read
ih�(r) = [hsp + gnc(r) + 2gdn(r)]�(r) + gdm(r)�∗(r),
(1a)
ihdρ
dt= �ρ − ρ�+, (1b)
where hsp = − h2
2m� + Vext(�r) − μ is the single-particle
Hamiltonian, Vext(�r) is the external trapping potential, μ isthe chemical potential, and gd is the interaction parameter ind dimensions.
Here we have defined the 2 × 2 matrices
�(r,r ′) =(
h(r,r ′) �(r ′,r ′)−�∗(r,r) −h∗(r,r ′)
)and
ρ(r,r ′) =(
n(r,r ′) m(r,r ′)m∗(r,r ′) n∗(r,r ′) + δ(r,r ′)
),
where h(r,r ′) = hsp(r)+2gd [n(�r ′,�r ′) + �∗(�r ′)�(�r ′)],�(r,r) =gd [m(�r,�r) + �(�r)�(�r)], and
n(�r,�r ′) ≡ n∗(�r,�r ′) = 〈�+(�r)�(�r ′)〉 − �∗(�r)�(�r ′),(2)
m(�r,�r ′) ≡ m(�r ′,�r) = 〈�(�r)�(�r ′)〉 − �(�r)�(�r ′),
are, respectively, the normal and anomalous single-particlecorrelation functions. In the local case they play the role ofthe noncondensed and anomalous densities. Moreover, ourformalism provides a direct link between these two laterquantities as
I (�r,�r ′)=∫
d�r ′′[ρ11(�r,�r ′′)ρ22(�r ′′,�r ′) − ρ12(�r,�r ′′)ρ21(�r ′′,�r ′)].
(3)
Notice that Eq. (3) is often known as the Heisenberginvariant; it is a direct consequence of the conservation ofthe von Neumann entropy S = −TrD lnD. For the pure stateEq. (3) takes the form I (�r,�r ′) = δd (�r − �r ′) [37].
Among the advantages of the TDHFB equations is thatthe three densities are coupled in a consistent and closedway. Second, they should in principle yield the general time,space, and temperature dependence of the various densities.Furthermore, they satisfy the energy and number conservinglaws. In addition, the most important feature of the TDHFB
equations is that they are valid for any Hamiltonian H andfor any density matrix operator. Interestingly, our TDHFBequations can be extended to provide self-consistent equationsof motion for the triplet correlation function by using thepost-Gaussian ansatz.
In the uniform case [Vext(r) = 0] and for a thermal distri-bution at equilibrium, by working in the momentum space,
ρij (�r,�r′) =∫
ddk
(2π )dei�k·(�r−�r′)ρij (k), (4)
where ρij (k) is the Fourier transform of ρij (�r,�r ′). We can theneasily rewrite Eq. (3) as
Ik = nk(nk + 1) − |mk|2 = 1
4 sinh2(εk/2T ), (5)
where εk is the Bogoliubov energy spectrum given below.The physical meaning of Eq. (5) is that it allows us
to calculate in a very useful way the dissipated heat ford-dimensional Bose gas as
Qd = 1
n
∫EkIk
ddk
(2π )d, (6)
where Ek = h2k2/2m is the energy of a free particle.Furthermore, at zero temperature, Eq. (5) reduces to
|mk|2 = nk(nk + 1), which constitutes an explicit relationshipbetween the normal and the anomalous densities at zerotemperature and indicates that these two quantities are of thesame order of magnitude at low temperatures which leads tothe fact that neglecting m while maintaining n is a quite unsafeapproximation.
The excitation energy εk is determined in our formalismvia the random-phase approximation (RPA) [36], which canbe found by expanding all quantities around their equilibriumsolution. The RPA appears as a direct application of the generalBalian-Veneroni formalism to the Lie algebra of single bosonoperators [36,37].
Thus, we write
�(k,t) = �(k) + δ�(k,t), n(k,t) = n(k) + δn(k,t),(7)
m(k,t) = m(k) + δm(k,t).
Then, we have written these quantities on a diagonal basis(RPA matrix), which derived from the set (1), and kept only thefirst-order terms. After a long, but straightforward, calculation,we arrive at the gapless expression of the Bogoliubov spectrum[18,38]
εk =√
(Ek−μ + 11)2 − 212, (8)
where 11 = 2gdn and 12 = gd(nc + m) are, respectively,the first-order normal and anomalous self-energies, where n =nc + n is the total density.
A detailed derivation of the Bogoliubov spectrum with theRPA method will be given elsewhere.
Note that Eq. (8) can be also obtained using the Green’sfunctions (see, e.g., [38]). It provides a useful finite-temperature version of the healing length and the sound
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BEHAVIOR OF THE ANOMALOUS CORRELATION . . . PHYSICAL REVIEW A 86, 043608 (2012)
velocity csas
ξ = h/√
m12 = h
/√mncgd
(1 + m
nc
)= h/mcs. (9)
In order to get explicit formulas for the noncondensed and theanomalous averages in d-dimensions we may use Eq. (5). Asimple calculation yields
n = 1
2
∫ddk
(2πL−1)d
[Ek + gd (nc + m)
εk
√Ik − 1
], (10a)
m = −1
2
∫ddk
(2πL−1)d
[gd (nc + m)
εk
√Ik
]. (10b)
It is worth noting that Eqs. (1a) and (10) together form thegeneralized HFB equations. This shows that, in the static case,our formalism recovers easily the full HFB equations at bothfinite and zero temperatures.
III. ANOMALOUS DENSITY AT ZERO TEMPERATURE
Let us now discuss the behavior of the normal andanomalous densities in homogeneous 2D Bose gas at bothzero and finite temperatures. From this point we consider theregime in weakly repulsive interaction at zero temperature,where
√Ik = 1. In 2D Bose gas, the interaction parameter
(gd = g2) depends logarithmically on the chemical potential as
g2 =[
4πh2
m
1
ln(2h2/mμa2)
], (11)
where a is the 2D scattering length among the particles andg2 is the two-body T matrix (see, e.g., [15,26,27]).
The calculation of the integral in Eq. (10a) leads us to thefollowing expression of the depletion:
n
n= 1
4πnξ 2. (12)
This equation is in good agreement with that obtained in [26].The integral in Eq. (10b) has an ultraviolet divergence in
both two and three dimensions. This divergence is well knownand arises due to the use of the contact potential. To regulatethe ultraviolet divergences, we may use the dimensionalregularization [39,40]. In such a technique one calculatesthe loop integrals in d = 2 − 2η dimensions for values of η
where the integrals converge. One then analytically continuesback to d = 2 dimensions. With dimensional regularization,an arbitrary renormalization scale M is introduced. Thisscale can be identified with the simple momentum cutoff.An advantage of dimensional regularization is that in 2Dsystems it automatically sets power divergences to zero, whilelogarithmic divergences show up as poles in η [40]. Using thistechnique one gets for the anomalous density
m�T =0 = −�2
4J0,1, (13)
where � is the regularized part, which is related to the sizeof particles and interactions as � = 2/ξ . This parameter issimilar to that used in [26,32,40].
And J0,1 = 1
4π
[1
η− L + O(η)
]with L = ln(�2/4M2).
Thus, the convergent part of the anomalous density provides
m�T =0 = �2
16πln(�2/4M2). (14)
A useful remark at this level, the noncondensed density ofEq. (12), can be rewritten also in terms of �as n = �2/16π .
At T = 0, the condensed density has a significant value andhence constitutes the dominant quantity in the system, whileboth the noncondensed and the anomalous densities vanishfor � → 0, which ensures that �=0
11 = 2μ and �=012 =
μ, in good accordance with the Hugenholtz-Pines theorem�=0
11 − �=012 = μ [33]. On the other hand, once we find
�=012 �= 0, this means that the actual version of our extended
variational TDHFB including a dimensional regularizationdoes not satisfy the Nepomnyashchy identity [32] as it shouldfor any limited approximation order (see, for example, Griffinand Shi [38], Yukalov and Graham [19], Pricoupenko [26], andAndersen and Haugerud [40]). Indeed, in a Bose-condensedsystem, the anomalous self-energy must be nonzero in orderto define a meaningful nonzero sound velocity and healinglength. In addition, a zero sound velocity leads evidently to anunstable system [see Eq. (9)].
Indeed, our approach can be an effective way to verifythe Nepomnyashchy identity, but on the condition that wesum over all terms of perturbation theory for the self-energywith renormalization of the sound velocity to ensure thestabilization of the system as it has been demonstrated in [32].
Conversely, at finite-temperature 3D Bose gas, the chemicalpotential satisfies the generalized version of HP theorem givenby Hohenberg and Martin [41] �
11 − �12 = μ. In such a sit-
uation, when T � Tc, both the condensate and the anomalousdensities vanish [17–19,24,25] whatever the value of�, whichimplies directly that �
12 = 0 and �11 = 2gn ≈ 2μ. Conse-
quently, a vanishing anomalous self-energy is further guaran-teed at high temperature and momentum in 3D systems. Phys-ically, this result is reasonable because the gas becomes com-pletely thermalized and therefore there is neither superfluid noracoustic waves when the temperature reaches its critical value.
On the other side, the dimensional regularization gives anasymptotically exact result at weak interactions (g2 → 0). Theextrapolation to finite interactions requires that the limitingcondition m/nc 1 be verified. In real systems, however,the interactions have a finite range a and so 1/a provides anatural ultraviolet cutoff M [40]. Hence, using this techniquepresupposes that value of the healing length in Eqs. (12)and (13) takes the form ξ = h/
√mncg2. In the case where
nc ≈ n, we recover easily the well-known result of Schick [11]for the depletion, while Eq. (14) has no analog in the literature.Under these conditions, the anomalous average turns out to begiven as
mT =0 = mμ0
4πh2 ln
(mμ0a
2
h2
), (15)
where μ0 = g2dn.Figure 1 shows that the noncondensed and the anomalous
densities, as function of the dimensionless parameter x =mμ0a
2/h2, are competitive contributions at zero temperature.For small values ofx, we observe that the noncondensed
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ABDELAALI BOUDJEMAA PHYSICAL REVIEW A 86, 043608 (2012)
FIG. 1. Noncondensed (dashed line) and anomalous (solid line)densities as function of x = mμ0a
2/h2.
density is greater than the anomalous one while this laterbecomes the dominant quantity for the whole range of x(x >
10). n and m are comparable only for x ≈ 10. Therefore, wededuce that omitting the anomalous density, while keeping thenormal one, is physically and mathematically inappropriate. Itis noticed that this behavior holds also in 3D Bose gas [17–25].
It is important now to discuss how the anomalous densitycan modify the chemical potential and hence the otherthermodynamic quantities of dilute Bose gas. The first-orderquantum corrections to the chemical potential are given byδμ = gd (n + m) [42]. Therefore, using the results obtained inEqs. (12) and (15) with the assumption n ≈ nc at T = 0. Oneobtains for the chemical potential
μ = μ0
[1 + mμ0
4πh2nln
(mμ0a
2e
h2
)]. (16)
The ground-state energy is obtained through
E/N =∫ n
0μdn =μ0
2
[1 + mg2
8πh2 ln
(mμ0a
2√e
h2
)]. (17)
The leading term in Eq. (16) was first obtained by Schick[11], while the second represents our correction to the chemicalpotential. Clearly, this correction is universal, depending onlyon the interactions and scattering length. It is worth mentioningthat the additional logarithm term in Eq. (16) is analogous to
0.00 0.05 0.10 0.15 0.20
5�10�5
1�10�4
5�10�4
0.001
0.005
0.010
x
na2
FIG. 2. Equation of state of a 2D homogeneous Bose gas. Solidline, our extended variational approach; dashed line, EoS predictedin [26,27]; dotted line, first correction beyond Bogoliubov theory [31].
FIG. 3. (Color online) Ground-state energy of a 2D Bose gas, as afunction of the gas density, in units of the mean-field prediction EMF.Brown solid line, our calculations; dark solid line, energy obtainedfrom the beyond Bogoliubov [31]; dashed line, analytical predictionof [28]; dotted line, analytical calculations of [26]. Plotting symbolswith error bars, numerical results of [29], for interactions given byhard disks (crosses) and by soft disks (circles); numerical resultsof [30], for dipolar interactions (diamonds).
that found recently in [31]. Moreover, what is interesting inEq. (16) is that if we invert it and take the limit of vanishingdensity, we thereby recover the well-known Popov’s EoS [12].
Before plotting Figs. 2 and 3, we use the dimensionlessrelation na2 = x2[ 1
2 − ln( xeγ
2 )], where γ is the Euler’s constant[26,31].
One can see from Fig. 2 that for the value of the gasparameter around 5 × 10−5 there is a difference of 10%between diverse EoS, while for na2 larger than 5 × 10−3 allEoS are practically identical.
Figure 3 shows that our expression of the ground-sateenergy [Eq. (17)] gives also an estimate compatibility withinerror bars of Monte Carlo simulations [29,30] and analyticresults of [26–28,31]. Furthermore, it is clearly seen fromFigs. 2 and 3 that there is an upward shift of our curve relativeto that of Ref. [31], which is indeed due to a prefactor whichappears in the expansion of the later reference.
Another important feature revealed in Fig. 3 is that there isno difference between dipolar and short-range interaction fordensities lower than 10−10 [30].
IV. ANOMALOUS DENSITY AT FINITE TEMPERATURE
Now we turn to analyze the finite-temperature case. Aswe mentioned in the Introduction, the finite-temperatureuniform 2D Bose gas is characterized by the absence of atrue Bose-Einstein condensate and long-range order [9,10].So the physics of 2D Bose gas at finite T can be understood inthe context of the density-phase representation. Accordingly,the single-particle anomalous correlation function is found byusing the field operator in the form � = √
neiφ . Followingthe hydrodynamic approach described in [14,15,43] with theassumptions m/nc 1 and nc ≈ n for T → 0. Then onthe basis of Eq. (2), we obtain for the single-particle anomalous
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BEHAVIOR OF THE ANOMALOUS CORRELATION . . . PHYSICAL REVIEW A 86, 043608 (2012)
correlation function
m(�r,0) = n exp
[− 1
2n
∫d�k εk
Ek
coth(εk/2T ) cos2
( �k · �r2
)].
(18)
At low temperatures (T μ) the main contribution to theintegral of Eq. (18) comes from the region of small momen-tum; then the single-particle anomalous correlation functionundergoes a slow law decay at large distances:
m (r) = n
(ξ
r
)T/2Td
, (19)
where Td = 2πh2nm
is the temperature of quantum degeneracy.We can infer from these results that the anomalous average
does not exist at finite temperature. This is strictly confirmedby Eq. (19), where one finds that m(r) vanishes for r → ∞.Similarly to the 2D situation at finite temperature regardingthe normal correlator n(�r,�r ′) = 〈�+(�r)�(�r ′)〉 − �∗(�r)�(�r ′),which tends to zero asr → ∞, confirming that there is notrue condensate, but one identifies instead the existence of aquasicondensate. However, this result also implies that thereis no symmetry breaking, and consequently the anomalousaverage should not exist at any nonzero temperature. The butterof this result is that the anomalous density accompanies ina manner analogous the true condensate in a system of 2Dhomogeneous Bose gas.
V. SUPERFLUID FRACTION
Usually, Bose-Einstein condensation (BEC) is accompa-nied by superfluidity. However, in a 2D system at finitetemperature, there is no BEC, but there still exists superfluidity.The relation between them depends on the Bogoliubov-typenature of the spectrum Eq. (7) [13]. Also, what is importantis that our formalism provides a useful relation between thesuperfluid fraction and the dissipated heat, which is equivalentto that obtained in Refs. [13,19],
fs = ns
n= 1 − h2
2mnT
∫d2k
(2π )2k2 eεk/T
(eεk/T − 1)2
= 1 − Qd=2
T, (20)
where ns is the superfluid density.It is very important to mention that the superfluid fraction fs
will be a divergent quantity and thus the superfluid transitiondoes not occur when the anomalous average is omitted inEq. (20).
At low temperature and weak interaction, we get
fs = 1 − 3ς (3)
2πh2mnc4s
T 3, (21)
where ς (3) is a Riemann ζ function and the sound velocityturns out to be given
cs = c(0)s
√(1 − n
n+ m
n
), (22)
where c(0)s = √
μ0/m is the zero-order sound velocity.Upon neglecting the normal and the anomalous fractions
we recover straightforwardly the superfluid fraction obtainedearlier by Popov [12] and by Fisher and Hohenberg [13].
VI. CONCLUSION
We have studied in this paper the behavior of the anomalousdensity in 2D homogeneous Bose gases. We find that thisquantity has a finite value in the limit of weak interactions.We have discussed also the effects of the anomalous averageon some thermodynamic quantities. As an example, we havegiven formulas for the chemical potential, ground-state energy,the depletion, and superfluid fraction. The later does not occurif the anomalous density is neglected. In the ultradilute limit,the known results are reproduced. This feature makes ourpredictions in accordance with Monte Carlo simulations andanalytical calculations. Also, we have shown that our approachwell satisfies the HP theorem at zero temperature, while itdoes not verify the Nepomnyashchy identity as it should befor any limited approximation. Moreover, the importance ofthe anomalous density compared to the normal one at lowtemperature has been also highlighted. In addition, by usingthe density phase fluctuation we found that the single-particleanomalous correlation function undergoes a slow law decayat large distances. Such a result implies that the anomalousaverage does not exist at finite temperature.
Finally, an interesting question to ask is whether somequantity can exist in this system which accompanies thequasicondensate density in a manner analogous to that in whichthe anomalous density accompanies a true condensate?
The goal of our next work is to use our approach toanswer this important question. On the other hand, we willtry to extract something useful about superfluidity in 2D Bosesystem. The idea is to relate our predictions with appropriatenumerical simulations for some realistic experiments in trapsto study, for example, the vortex stability without rotatingfluid.
ACKNOWLEDGMENTS
We acknowledge Ludovic Pricoupenko, Gora Shlyapnikov,Jean Dalibard, and Usama Al-Khawaja for many usefulcomments about this work. We are grateful to J. Andersenand V. Yukalov for helpful discussions. We are indebted toYvan Castin for giving us the numerical data.
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