SEMI-ANALYTIC CALCULATION OFTHE SHIFT IN THE CRITICAL

TEMPERATURE FOR BOSE-EINSTEINCONDENSATION

Dr. Eugeniu Radescu

1

OUTLINE

• General Notions

• The problem of Tc

• Method

• Results

• Conclusions

2

1 Introduction

• According to Louis de Broglie, very cold particles behave like waveswhose wavelengths increase as their velocity drops. The particle isdelocalized over a distance corresponding to the de Broglie wave-length

λdB = h/mv,

where h = h/2π is Planck’s constant.

• When a gas is cooled down to very low temperatures, the individualatomic de Broglie waves become very long and eventually overlap.

• If the gas consists of bosonic particles all being in the same quan-tum state, the de Broglie waves of the individual particles con-structively interfere and build up a large coherent matter-wave.

3

• The transition from a gas of individual atoms to the macroscopicquantum state occurs as a phase transition and is named Bose-

Einstein condensation (BEC) after Shandrasekhar Bose and AlbertEinstein.

• Bose-Einstein condensation occurs as a phase transition from a gasto a new state of matter that is in many respects dissimilar to allusual macroscopical states such as solids, liquids or gases.

• The experimental verification of Bose-Einstein condensation hasbeen achieved in 1995 (Anderson et al., Davis et al., Bradley etal.). The observation of Bose-Einstein condensation has now beenconfirmed by more than twenty groups worldwide and triggered anenormous amount of theoretical and experimental work on Bose-condensed gases.

4

2 Basic Notions

• For a gas composed of particles of mass m at temperature T , thevelocity distribution is given by the Maxwell-Boltzmann law:

g(v) =

(

m√2πkBT

)3

exp

(

− mv2

2kBT

)

,

where kB is the Boltzmann constant.

• This law describes well the behavior of atoms at low density andhigh temperatures. Deviations from it are insignificant until quan-tum mechanical effects assert themselves, and this does not occuruntil the temperature becomes so low that the atomic de Brogliewavelength λdB becomes comparable to the mean distance betweenparticles.

5

• Since the typical momentum mv is of the order (mkBT )1/2, thede Broglie wavelength of a typical particle is of order the thermalwavelength defined by

λT =

√

2πh2

mkBT,

• For a general system with density n, the mean distance betweenparticles is n−1/3. Quantum effects are expected to show up forn−1/3 ∼ λT .

Example: an atomic gas at room temperature (T ∼ 300K) and withthe density of air at sea level (n ∼ 3 × 1019 cm−3) is safely within theclassical regime, since n−1/3 ∼ 3 × 10−7 cm � λT = 1 × 10−10 cm.

• To witness quantum effects, one needs atoms at low temperatureand relatively high density.

6

3 Bose-Einstein Condensation of Ideal Gasof Bosons

Ideal gas is a gas of point-like noninteracting particles.

Bosons follow a quantum statistical distribution called Bose-Einstein

distribution. The basic difference between Maxwell-Boltzmannstatistics and Bose-Einstein statistics is that the former applies toparticles with the same mass that nevertheless are distinguishablefrom one another, while the latter describes identical indistinguish-able particles.

• Bosons, in contrast to fermions, enjoy sharing a quantum state andeven encourage other bosons to join them.

7

Bose-Einstein distribution for a non-degenerate quantum state withenergy ε when the system is held at temperature T is:

f(ε) =1

eβ(ε−µ) − 1,

where β ≡ 1/kBT and µ ≤ 0 is chemical potential.

• The expression for the number density of particles is

n =2π

√2m

3

h3

∫

∞

0

ε1/2dε

eβ(ε−µ) − 1.

• Since µ ≤ 0, this expression suggests that for fixed T , there is amaximum number density n = N/V :

n ≤ 2π√

2m3

h3

∫

∞

0

ε1/2dε

eβε − 1= ζ(3/2)

(

m

2πh2

)3/2

(kBT )3/2 .

8

• Equivalently, for fixed number density, there is a minimum tem-perature Tc to which the bosons can be cooled.

kBTc =2π

ζ( 32)2/3

h2n2/3

m.

• Einstein pointed out that there is no minimum temperature. In-stead, for T < Tc there is a macroscopic occupancy of the groundstate. The number N0 of atoms in the ε = 0 state is

N0 = N

[

1 −(

T

Tc

)3/2]

.

9

4 Atomic Interactions in NonidealHomogeneous Bose-Gas

• Real systems are always affected by particle interactions.

• The formula for Tc implies that the interparticle spacing n−1/3 andthe thermal wavelength λT = (2πh2/mkBT )1/2 are comparable.

• We assume that they are both large compared to the range R ofthe two-body potential: n−1/3, λT � R.

• In this case, the interaction between the bosons can be character-ized entirely by the s-wave scattering length a.

• We assume a to have magnitude of order R. Thus we require

a� n−1/3, λT .

10

5 Shift in Tc

• If a is the only parameter, then an1/3 is dimensionless and thetransition temperature must take the form

Tc(a) = Tc(a = 0) f(an1/3)

• The leading order shift in Tc is dominated by the infrared regionof momentum space (the typical momenta involved are of ordera/λ2

T � 1/λT ) and is insensitive to the ultraviolet region (Baym,1999)

• Effective field theory reveals that as an1/3 → 0, the leading correc-tion is linear in an1/3:

Tc(a) = Tc(a = 0){1 + c a n1/3 +O(a2n2/3)},

where c is a numerical constant.

11

• Several theoretical predictions for the value of the coefficient c canbe found in literature. The values span a range from −1 to 5.

• Recently (Kashurnikov et al. and Arnold et al., 2001), the latticeMonte Carlo simulations gave a definitive answer to the problemof finding the leading order correction to the shift in Tc, giving forthe coefficient c the value

∆Tc

Tc= c an1/3,

c = 1.32 ± 0.02

• Is there a simpler way to derive this quantity?

12

−1

01

23

45

Gruter et al. [3]

Holzmann et al. [4]

Arnold, Tomasik (NLO 1/N) [5]

Baym et al. (LO 1/N) [7]

Baym et al. [1]

Stoof [9]

Holzmann, Krauth [6]

de Souza Cruz et al. [8]

Wilkens et al. [2]

Figu

re1:

13

6 Method

• This system can be described by a second-quantized Schrodingerequation. The corresponding imaginary-time Lagrangian is

L = ψ∗∂

∂τψ − 1

2mψ∗∇2ψ − µψ∗ψ +

2πa

m(ψ∗ψ)2,

For simplicity we use units such that h = kB = 1.

• The field ψ(x, t) can be decomposed into Fourier modes ψn (x) withfrequencies ωn = 2πnT. At sufficiently large distance scales (� λT )and small chemical potential (|µ| � T ), the n 6= 0 frequency modesdecouple from the dynamics leaving an effective theory of only thezero modes (Baym, 1999).

14

• The effective lagrangian density for the dimensionally reduced the-ory can then be written as

Leff = −1

2~φ · ∇2~φ+

1

2r~φ 2 +

1

24u

(

~φ 2)2

,

where we replace the complex field ψ0 by the 2-component realfield ~φ = (φ1, φ2) defined by

ψ0(x) = (mT )1/2 (φ1(x) + iφ2(x)) ,

and the new parameters are

r = −2mµ,

u = 48πamT.

15

• The leading order shift ∆Tc in the critical temperature at fixednumber density is related to the leading order shift ∆nc in thecritical density at fixed temperature by

∆Tc

Tc= −2

3

∆nc

nc= −2

3

mT

nc∆,

where ∆ has a diagramatic expansion within the framework of theeffective theory.

• The number density to the accuracy required to calculate the shiftin Tc to leading order in an1/3, is

n = n0 +mT 〈~φ 2〉,

where n0 is the short-distance contribution (coming from momen-tum scales ∼ 1/λT ).

16

P

Figure 2: Diagram for 〈~φ 2〉 at the critical point. The black blob rep-

resents the ~φ 2 vertex, while the shaded blob represents the completepropagator with subtracted self-energy Σ(p) − Σ(0).

17

• 〈~φ 2〉 can be expressed as

〈~φ 2〉 = 2

∫

p

[

p2 + r + Σ(p)]

−1,

where Σ(p) is the self-energy of the ~φ field.

• ∆ can be calculated within the effective 3-dimensional field theory:

∆ = 2

∫

p

[

[

p2 + Σ(p) − Σ(0)]

−1 −(

p2)

−1]

,

We made use of the condition that the correlation length at thecritical point is infinite:

r + Σ(0) = 0.

• At the phase transition, there is only one relevant length scale andit is set by the parameter u ∼ a. Since ∆ has the same dimensionas u, it must be proportional to u by simple dimensional analysis,and therefore ∆Tc is linear in a.

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• Although ∆ has a well-defined diagramatic expansion, any attemptto calculate ∆Tc using ordinary perturbation theory in u is doomedto failure.

• Nonperturbative methods that have been applied to this prob-lem include numerical simulations, large-N techniques, variationalmethods and self-consistent schemes.

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7 Linear δ-expansion

The linear δ-expansion (LDE) is a particularly simple variationalmethod for obtaining nonperturbative results using perturbative tech-niques. The convergence properties of the LDE have been studied exten-sively for the quantum mechanics problem of the anharmonic oscillator.

• It can be defined by a lagrangian whose coefficients are linear in aformal expansion parameter δ.

• If L is the lagrangian for the system of interest, the lagrangian thatgenerates the LDE has the form

Lδ = (1 − δ)L0 + δL,

where L0 is the lagrangian for an exactly solvable theory.

20

• In our case the lagrangian Lδ can be written L = L0 +Lint, where

L0 = −1

2~φ · ∇2~φ+

1

2m2~φ 2,

Lint =δ

2

(

r −m2)

~φ 2 +δ u

24

(

~φ 2)2

.

• Calculations are carried out by using δ as a formal expansion pa-rameter, expanding to a given order in δ, and then setting δ = 1.

• The lagrangian L0 for the exactly solvable theory involves an ar-bitrary dummy parameter m which is treated as a variational pa-rameter.

• A prescription for m is required to obtain a definite prediction.A simple prescription for m is the principle of minimal sensitivity

(PMS) that the derivative with respect tom should vanish (Steven-son,1981).

• The LDE has been proven to converge in the Anharmonic Oscil-lator calculations for appropriate order-dependent choices of thevariable m that include the PMS criterion as a special case.

21

• A previous application of the LDE to our problem gave a resultwhich scales incorrectly with the number of the scalar fields in thetheory (de Souza Cruz et al., 2001).

• To circumvent this problem, we apply LDE only to the self-energypart Σ(p) − Σ(0) of the propagator.

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8 Large N limit

The large-N limit is defined byN → ∞, u→ 0 withNu fixed, whereN is the number of scalar fields in the Lagrangian (we set N = 2 at theend of the computation, making the contact with the initial Lagrangian.)

Figure 3: The 4th in the series of diagrams for Σ(p) that survive in thelarge-N limit.

• In the large-N limit, the shift in Tc can be computed analytically.

23

• The series in δ in the large-N limit involves a particular class ofself-energy diagrams which can be computed numerically to a veryhigh order in expansion. In this way we are able to see the rate ofthe convergence to the analytic result.

• It is easy to show that if the LDE converges, it converges to thecorrect analytic result.

• The LDE seems to converge for all µ above some critical value near0.7.

• The PMS prescription improves the convergence rate by allowingthe variational parameter µ to vary with the order in δ.

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0.6 0.8 1 1.2 1.4

µ

0

0.5

1

1.5

∆ /(-

Nu/

96π2 )

n=4

n=35

n=3n=5

Figure 4: ∆/(−Nu/96π2) in the large-N limit as a function of µ at nth

order in the LDE for n = 3, 4, 5, 7, 11, 19, 35. The curves for n = 7, 11,and 19 appear in order between those labelled n = 5 and 35.

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n ∆

3 0.48145 0.6117 0.675

11 0.74219 0.80135 0.850

Table 1: The values of ∆/(−Nu/96π2) in the large-N limit at nth orderin the δ. The analytic result is equal to 1.

26

9 Full O(N) (N = 2) effective theory

After the results (convergence, but a very slow convergence) for theparticular class of large-N diagrams, we are ready to compute severalorders of the full O(N) theory. The 1PI self-energy diagrams needed areshown in the following figures.

27

a b c

Figure 5: The diagrams that contribute to Σ(p) − Σ(0) at order δ3.

a b c d e

Figure 6: Four-loop diagrams that contribute to Σ(p)−Σ(0) at order δ4.

28

� � � �� �

�� � �� �

�� �

�� �

�� � � �

� � �� �

�� � � �

Figure 7: Five-loop diagrams that contribute to Σ(p)−Σ(0) at order δ5.

29

• For N = 2, the 3rd, 4th and 5th order approximations differ fromthe lattice Monte Carlo result (0.57) by about 66%, 63% and 59%respectively. These predictions are not as accurate as in the largeNlimit, where the errors in the 3rd, 4th and 5th order approximationsare about 52%, 44% and 40% respectively.

• The LDE seems to be approaching the correct result for N = 2,albeit very slowly.

30

n ∆/(−u/48π2)

2

3 0.192

4 0.2088

5 0.2373

Table 2: The values of ∆/(−u/48π2) for N = 2 at nth order in the LDE.The lattice result is 0.57 ± 0.1

31

10 Variational Perturbation Theory (VPT)applied to the problem of the shift in Tc

The approach to a second order phase transition is characterizedby nontrivial critical exponents that differ from the values predicted bydimensional analysis. For example, the scaling dimension of the scalarfield φ changes from the naive value d/2 − 1 (1/2 in our case) to d/2 −1 + η/2, where η ≈ 0.033 is a critical exponent.

• This information can be incorporated into a variational procedurewhich accelerates the convergence compared to the application ofthe LDE.

• The method introduces a new variational parameter q which gov-erns the behavior of the quantity ∆ in the strong coupling (mass-less) limit.

32

• The method uses the fact that the quantity ∆/u approach a con-stant as the weak-coupling expansion parameter u/m goes to in-finity.

∆/u = f(u/m) → f(∞). (1)

• We will use the PMS prescription to fix the parameters s and q ateach order in the expansion.

• For q = 1 the VPT method reduces to the LDE method.

• VPT method increases the rate of the convergence, by having q asa variational parameter allowing m to approach 0 with a nontrivialexponent.

• We know only the truncated weak-coupling expansion

fn(u/m) =

N∑

n=1

an(u/m)n (2)

• Construct function Fn(u/m, q) bya) setting u→ δu,m→ m(1 − δ)q

33

b) truncate after nth order in δc) setting δ = 1

Fn(u/m, q) =N

∑

n=1

An(q)(u/m)n (3)

• Modulo interchange of limits

Fn(u/m, q) → f(∞). (4)

34

4 6 8 10 12n

-1

-0.9

-0.8

-0.7

-0.6

-0.5<Fi^2>

Figure 8: ∆/(Nu/96π2) in the large-N limit for n from 4 to 12. q = 1(LDE) results are shown with empty triangles. Empty squares are forfixed value q = 2. Results for variationally determined values of qn areshown with black squares. The analytic result −1 is shown with dashedline. 35

4 5 6 7 8n

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

<Fi^

2>

Figure 9: ∆/(Nu/96π2) for the O(2) theory for F opt2 , F opt

3 , F opt4 . q = 1

(LDE) results are shown with empty triangles. Empty squares are forfixed value q = 2. Results for variationally determined values of qn areshown with black squares. The lattice result is shown with two linesdenoting the maximal and the minimal value separated by 2 error bars.

36

11 Conclusions

• This work is among the few applications of the LDE to a nonper-turbative field-theoretical problem which gives unequivocal results.

• It shows the importance of the correct understanding of the energyscales involved in the problem. The power of an1/3, the sign ofthe coefficient and even its order of magnitude can be deducedwithout calculation just by applying correctly the effective fieldtheory method.

• The LDE shows itself as a systematically improvable scheme tocompute quantitatively nonperturbative quantity. Applied to theproblem of Tc, it seems to converge very slowly towards the correctresult.

• By incorporating the knowledge about the behavior of the strong-coupling limit, VPT shows itself as a strong tool which competesvery well with lattice Monte Carlo simulations.

37