Quantum Field Theory in Condensed Matter Physics Luca DellAnna and - - PowerPoint PPT Presentation

Quantum Field Theory in Condensed Matter Physics

Luca Dell’Anna and Luca Salasnich

Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universit` a di Padova

PhD School of Physics, February 2016 Description: an introduction to quantum field theory and quantum entanglement for macroscopic quantum phenomena: superfluidity in ultracold atomic gases and liquid helium and superconductivity in metals. Theoretical framework: finite-temperature functional integration of bosonic and fermionic fields with broken symmetry.

Program of the course (I)

First part (Salasnich, 12 hours) Partition function of bosons. Bosonic fields. Bosonic coherent

• states. Functional integration of bosonic fields.

Broken symmetry and ideal Bose gas. Interacting Bose gas. Saddle-point approximation and gaussian fluctuations. Bogoliubov spectrum and dimensional regularization. Gross-Pitaevskii equation of the Bose-Einstein condensate. Solitons and quantized vortices. Critical velocity and Landau criterion for ultracold atoms and liquid helium. Von Neuman entropy and entanglement for bipartite systems. Interacting bosons in a double-well potential: coherence visibility, Fisher information and entanglement entropy.

Program of the course (II)

Second part (Dell’Anna, 12 hours) Partition function of fermions. Fermionic coherent states. Gaussian integrals with Grassman variables. Functional integration of fermionic fields. BCS theory of metals by functional integration. Hubbard-Stratonovich transformation and the bosonic field of pairing. Saddle point approximation: gap equation and critical temperature. Ginzburg-Landau theory from the BCS effective action. Gaussian fluctuations: Goldstone mode, Meissner effect and Higgs mechanism.

Suggested Books

• N. Nagaosa, Quantum Field Theory in Condensed Matter Physics

(Springer, 1999).

• A. Altland and B. Simons, Condensed Matter Field Theory

(Cambridge Univ. Press, 2006).

• M. Le Bellac, A Short Introduction to Quantum Information and

Quantum Computation (Cambridge Univ. Press, 2006). H.T.C. Stoof, K.B. Gubbels, and D.B.M. Dickerscheid, Ultracold Quantum Fields (Springer, Dordrecht, 2009).

1.1 Partition function of bosons (I)

Let us consider a gas of non-interacting bosons in thermal equilibrium with a bath at the temperature T. The relevant quantity to calculate all thermodynamical properties of the system is the grand-canonical partition function Z, given by Z = Tr[e−β( ˆ

H−µˆ N)]

(1) where β = 1/(kBT) with kB the Boltzmann constant, ˆ H =

• α

ǫα ˆ Nα , (2) is the quantum Hamiltonian, ˆ N =

• α

ˆ Nα (3) is total number operator, and µ is the chemical potential, fixed by the conservation of the average particle number. Here α represents the set of single-particle quantum numbers.

1.1 Partition function of bosons (II)

Figure: Ground-state of non-interacting bosons (a) and fermions (b) is a harmonic trap. Horizontal lines describe single-particle energies ǫα.

The single-particle-state number operator ˆ Nα is given by ˆ Nα = ˆ c+

α ˆ

cα , (4) where the operators ˆ cα and ˆ c+

α act in the Fock space of the identical

• bosons. A generic state of this Fock space is given by

|{nα} = | ... nα ... nβ ... , (5) meaning that there are nα bosons in the single-particle state |α, nβ bosons in the single-particle state |β, et cetera.

1.1 Partition function of bosons (III)

The operators ˆ cα and ˆ c+

α are called annihilation and creation operators

because they respectively destroy and create one boson in the single-particle state |α, namely ˆ cα| ... nα ... = √nα | ... nα − 1 ... , (6) ˆ c+

α | ... nα ...

= √ nα + 1 | ... nα + 1 ... . (7) Note that these properties follow directly from the commutation relations [ˆ cα, ˆ c+

β ] = δα,β ,

[ˆ cα, ˆ cβ] = [ˆ c+

α , ˆ

c+

β ] = 0 ,

(8) where [ˆ A, ˆ B] = ˆ Aˆ B − ˆ B ˆ A.

1.1 Partition function of bosons (IV)

The vacuum state, where there are no particles, can be written as |0 = |{0} = | ... 0 ... 0 ... , (9) and ˆ cα|0 = 0 , ˆ c+

α |0 = |1α = |α ,

(10) where |α is such that r|α = φα(r) . (11) From Eqs. (6) and (7) it follows immediately that ˆ Nα counts the number

• f bosons in the single-particle state |α, i.e.

ˆ Nα| ... nα ... = nα | ... nα ... . (12)

1.1 Partition function of bosons (V)

Taking into account the properties of Fock state, from the definition Z = Tr[e−β( ˆ

H−µˆ N)]

(13)

• f the grand-canonical partition function we find

Z =

• {nα}

{nα}|e−β( ˆ

H−µˆ N)|{nα} =

• {nα}

{nα}|e−β P

α(ǫα−µ) ˆ

Nα|{nα}

=

• {nα}

e−β P

α(ǫα−µ)nα =

• {nα}
• α

e−β(ǫα−µ)nα =

• α
• nα

e−β(ǫα−µ)nα =

• α

∞

• n=0
• e−β(ǫα−µ)n

(14) and finally Z =

• α

1 1 − e−β(ǫα−µ) . (15)

1.1 Partition function of bosons (VI)

Quantum statistical mechanics dictates that the thermal average of any

• perator ˆ

A is obtained as ˆ A = 1 Z Tr[ˆ A e−β( ˆ

H−µˆ N)] .

(16) It is then quite easy to show that ˆ H′ = 1 Z Tr[(ˆ H−µˆ N) e−β( ˆ

H−µˆ N)] = − ∂

∂β ln

• Tr[e−β( ˆ

H−µˆ N]

• = − ∂

∂β ln(Z) . (17) By using Eq. (15) we immediately obtain ln(Z) =

• α

ln

• 1 − e−β(ǫα−µ)

, (18) and finally from Eq. (17) we get ˆ H =

• α

ǫα eβ(ǫα−µ) − 1 and also ˆ N =

• α

1 eβ(ǫα−µ) − 1 . (19)

1.2 Bosonic fields (I)

Introducting the bosonic field operator ˆ ψ(r) =

• α

ˆ cα φα(r) , (20)

• ne immediately finds

ˆ H =

• d3r ˆ

ψ+(r)

• − 2

2m∇2 + U(r)

• ˆ

ψ(r) =

• α

ǫα ˆ c+

α ˆ

cα , (21) where U(r) is the external trapping potential of bosons and φα(r) are the

• rthonormalized single-particle eigenfunctions of the single-particle

stationary Schr¨

• dinger equation
• − 2

2m∇2 + U(r)

• φα(r) = ǫαφα(r)

(22) with single-particle eigenvalues ǫα. Again α represents the set of single-particle quantum numbers.

1.2 Bosonic fields (II)

A remarkable property of the adjoint field operator ˆ ψ+(r) =

• α

ˆ c+

α φ∗ α(r) ,

(23) is the following: ˆ ψ+(r)|0 = |r . (24) That is the operator ˆ ψ+(r) creates a particle in the state |r from the vacuum state |0. In fact, ˆ ψ+(r)|0 =

• α

ˆ c+

α φ∗ α(r)|0 =

• α

ˆ c+

α α|r|0

=

• α

ˆ c+

α |0α|r =

• α

|αα|r = |r , because of the completeness relation (closure)

• α

|αα| = 1 . (25)

1.2 Bosonic fields (III)

It is straightforward to show that the bosonic field operators satisfies the following commutation rules [ ˆ ψ(r), ˆ ψ+(r′)] = δ(r − r′) . (26) and also [ ˆ ψ(r), ˆ ψ(r′)] = [ ˆ ψ+(r), ˆ ψ+(r′)] = 0 . (27) Let us prove Eq. (26). By using the expansion of the field operators one finds [ ˆ ψ(r), ˆ ψ+(r′)] =

• α,β

φα(r) φ∗

β(r′) [ˆ

cα, ˆ c+

β ]

=

• α,β

φα(r) φ∗

β(r′) δα,β =

• α

φα(r) φ∗

α(r′)

=

• α

r|α α|r′ = r|

• α

|α α| r′ = r|r′ = δ(r − r′) .

1.3 Bosonic coherent states (I)

It is interesting to observe that nα|ˆ cα|nα = nα|ˆ c+

α |nα = 0 .

(28) This result of is due to the fact that the expectation value is performed with the Fock state |nα, which simply means that the number of bosons in the single-particle state α is fixed because ˆ Nα|nα = nα|nα , (29) with ˆ Nα = ˆ c+

α ˆ

cα. In some case the number of bosons is not fixed, in

• ther words the system is not in a pure Fock state. For example, the

radiation field of a well-stabilized laser device operating in a single mode |α is described by a coherent state |cα, such that ˆ cα|cα = cα|cα . (30) The coherent state |cα, introduced in 1963 by Roy Glauber, is thus the eigenstate of the annihilation operator ˆ cα with complex eigenvalue cα = |cα|eiθα.

1.3 Bosonic coherent states (II)

The coherent state |cα does not have a fixed number of photons, i.e. it is not an eigenstate of the number operator ˆ Nα, and it is not difficult to show that |cα can be expanded in terms of number (Fock) states |nα as follows |cα = e−|cα|2/2

∞

• nα=0

cnα

α

√nα!|nα = e−|cα|2/2 ecαˆ

c+

α|0 ,

(31) taking into account the normalization to one, i.e. cα|cα = 1 . (32) From Eq. (30) one immediately finds ¯ Nα = cα|ˆ Nα|cα = |cα|2 , (33) and it is natural to set cα =

• ¯

Nα eiθα , (34) where ¯ Nα is the average number of photons in the coherent state, while θα is the phase of the coherent state.

1.3 Bosonic coherent states (III)

We observe that cα|ˆ N2

α|cα = |cα|2 + |cα|4 = ¯

Nα + ¯ N2

α

(35) and consequently cα|ˆ N2

α|cα − cα|ˆ

Nα|cα2 = ¯ Nα , (36) while for the Fock states nα|ˆ N2

α|nα = n2 α and consequently nα|ˆ

N2

α|nα − nα|ˆ

Nα|nα2 = 0 . (37)

1.3 Bosonic coherent states (IV)

The coherent state |cα satisfies the remarkable completeness relation dc∗

αdcα

2πi |cαcα| = 1 . (38) Let us suppose that |cα is the coherent state of the single-particle state |α and |cβ is the coherent state of the single-particle state |β. It is quite easy to prove that they are not orthogonal: cα|cβ = e− 1

2 [|cα|2+|cβ|2−2c∗ αcβ] .

(39) In fact, cα|cβ = cα|

∞

• nβ=0

e−|cβ|2/2 cnβ

β

• nβ!|nβ

=

∞

• nα=0

∞

• nβ=0

e−|cα|2/2e−|cβ|2/2 (c∗

α)nα

√nα! cnβ

β

• nβ!nα|nβ

= e−(|cβ|2+|cα|2)/2

∞

• nα=0

(c∗

αcβ)nα

nα! = e−(|cβ|2+|cα|2)/2 ec∗

αcβ .

1.3 Bosonic coherent states (V)

We can now introduce the general coherent state |ψ =

• α

|cα = e− P

α |cα|2/2 e

P

α cαˆ

c+

α|0 ,

(40) such that ˆ cα|ψ = cα|ψ for any α . (41) Notice that |ψ is the tensor product of all coherent states |cα taking into account their normalization to one. As a consequence, we get ψ|ψ = 1 . (42) The overlap between two general coherent states |ψ and | ˜ ψ is instead given by ψ| ˜ ψ = e− 1

2

P

α[c∗ α(cα−˜

cα)−(c∗

α−˜

c∗

α)˜

cα] .

(43)

1.3 Bosonic coherent states (VI)

Moreover, it is possible to prove that the set of coherent states {|cα}

• btained by varying α is (over)complete and the generalized coherent

state |ψ satisfies the completeness relation

• d[ψ∗, ψ] |ψψ| = 1 ,

(44) where the integration measure is defined by

• d[ψ∗, ψ] =
• α

dc∗

αdcα

2πi . (45)

1.3 Bosonic coherent states (VII)

Given the field operator ˆ ψ(r) =

• α

ˆ cα φα(r) (46)

• ne immediately finds that the generalized coherent state |ψ is

eigenstate of the field operator ˆ ψ(r), namely ˆ ψ(r)|ψ = ψ(r)|ψ , (47) where ψ(r) =

• α

cα φα(r) (48) is the “classical” complex field associated to the field operator ˆ ψ(r). Notice that cα and ψ(r) represent the same complex field in reciprocal spaces, and the integration measure can also be written as

• d[ψ∗, ψ] =
• r

dψ∗(r)dψ(r) 2πi . (49)

1.4 Functional integration of bosonic fields (I)

The partition function Z = Tr[e−β( ˆ

H−µˆ N)]

(50)

• f interacting bosons cannot be calculated exactly. In general, the

quantum-field-theory Hamiltonian is given by ˆ H =

• d3r ˆ

ψ+(r)

• − 2

2m∇2 + U(r)

• ˆ

ψ(r) + 1 2

• d3r d3r′ ˆ

ψ+(r) ˆ ψ+(r′) V (r, r′) ˆ ψ(r′) ˆ ψ(r) , (51) where V (r, r′) is the inter-particle interaction, while the number operator reads ˆ N =

• d3r ˆ

ψ+(r) ˆ ψ(r) . (52)

1.4 Functional integration of bosonic fields (II)

By expanding the field operator ˆ ψ(r) as ˆ ψ(r) =

• α

ˆ cα φα(r) (53) where φα(r) are the orthonormal eigenfunctions of the non-interacting problem with single-particle enegies ǫα, one finds ˆ H =

• α

ǫα ˆ c+

α ˆ

cα +

• αβγδ

Vαβγδ ˆ c+

α ˆ

c+

β ˆ

cδˆ cγ , (54) where Vαβδγ =

• d3r d3r′ φ∗

α(r) φ∗ β(r′) V (r, r′) φδ(r′) φγ(r) ,

(55) and also ˆ N =

• α

ˆ c+

α ˆ

cα . (56)

1.4 Functional integration of bosonic fields (III)

The partition function Z can be expressed as a functional integral taking into account that the trace of an operator can be performed by using any complete set of basis states, and in particular the basis of generalized coherent states. Thus, we write Z = Tr[e−β( ˆ

H−µˆ N)] =

• d[ψ∗, ψ] ψ|e−β( ˆ

H−µˆ N)|ψ ,

(57) where |ψ is the generalized coherent state and

• d[ψ∗, ψ] =
• α

dc∗

αdcα

2πi =

• r

dψ∗(r)dψ(r) 2πi . (58) Moreover, we observe that ψ| ˜ ψ = e− 1

2

P

α[c∗ α(cα−˜

cα)−(c∗

α−˜

c∗

α)˜

cα]

= e− 1

2

R d3r[ψ∗(r)(ψ(r)− ˜ ψ(r))−(ψ∗(r)− ˜ ψ∗(r)) ˜ ψ(r)] .

(59)

1.4 Functional integration of bosonic fields (IV)

The main problem in the functional representation of the partition function Z is the calculation of ψ|e−β( ˆ

H−µˆ N)|ψ .

(60) The simple semiclassical approximation ψ|e−β( ˆ

H−µˆ N)|ψ ≃ e−βψ|( ˆ H−µˆ N)|ψ

(61) gives Z =

• d[ψ∗, ψ] e−β(E[ψ∗,ψ]−µN[ψ∗,ψ])

(62) where E[ψ∗, ψ] =

• d3r ψ∗(r)
• − 2

2m∇2 + U(r)

• ψ(r)

+ 1 2

• d3r d3r′ |ψ(r)|2 V (r, r′) |ψ(r′)|2 ,

(63) and N[ψ∗, ψ] =

• d3r ψ∗(r)ψ(r) .

(64)

1.4 Functional integration of bosonic fields (V)

To better treat the expectation value ψ|e−β( ˆ

H−µˆ N)|ψ

(65) we write |ψ0 instead of |ψ and also (with M → ∞) ∆τ = β M . (66) In this way ψ0|e−β( ˆ

H−µˆ N)|ψ0

= ψ0|e−M∆τ( ˆ

H−µˆ N)/|ψ0

= ψ0|

• e−∆τ( ˆ

H−µˆ N)/M

|ψ0 (67) = ψ0|e−∆τ( ˆ

H−µˆ N)/ ... e−∆τ( ˆ H−µˆ N)/|ψ0 .

Then we insert M − 1 completeness relations for the general coherent state |ψj (j = 1, ..., M − 1)

• d[ψ∗

j , ψj] |ψjψj| = 1

(68)

1.4 Functional integration of bosonic fields (VI)

and obtain ψ0|e−β( ˆ

H−µˆ N)|ψ0 =

 

M−1

• j=1

d[ψ∗

j , ψj]

 

M

• j=1

ψj|e−∆τ( ˆ

H−µˆ N)/|ψj−1

(69) imposing that |ψ0 = |ψM. In the limit M → ∞ one has ∆τ → 0 and also ψj|e−∆τ( ˆ

H−µˆ N)/|ψj−1 = ψj|ψj−1 e−∆τ(E[ψ∗

j ,ψj]−µN[ψ∗ j ,ψj])/

(70) Moreover, the overlap between two general coherent states reads ψj|ψj−1 = e− ∆τ

2

R d3r

• ψ∗

j (r) [ψj (r)−ψj−1(r)] ∆τ

−ψj(r)

[ψ∗ j (r)−ψ∗ j−1(r)] ∆τ

• .

(71) Finally, setting τ = j∆τ, ψ(r, τ) = ψj(r) and ∂ ∂τ ψ(r, τ) = [ψj(r) − ψj−1(r)] ∆τ (72)

1.4 Functional integration of bosonic fields (VII)

we obtain for the partition function Z the elegant result Z =

• D[ψ, ψ∗] exp
• −S[ψ, ψ∗]
• ,

(73) where S[ψ, ψ∗] = β dτ

• d3r L(ψ∗, ψ)

(74) is the Euclidean action of Lagrangian density L(ψ∗, ψ) =

• 2
• ψ∗(r, τ) ∂

∂τ ψ(r, τ) − ψ(r, τ) ∂ ∂τ ψ∗(r, τ)

• +

ψ∗(r, τ)

• − 2

2m∇2 + U(r) − µ

• ψ(r, τ)

(75) + 1 2

• d3r′ |ψ(r′, τ)|2 V (r, r′) |ψ(r, τ)|2

with ψ(r, 0) = ψ(r, β) and

• D[ψ, ψ∗] =
• (r,τ)

dψ∗(r, τ) dψ(r, τ) 2πi . (76)

2.1 Broken symmetry and ideal Bose gas (I)

We consider a D-dimensional (D = 1, 2, 3) Bose gas of ultracold and dilute neutral atoms either noninteracting or with a contact interaction, namely V (r − r′) = g δ(r − r′) , (77) where δ(x) is the Dirac delta function and g the strengh of the

• intraction. We adopt the path integral formalism, where the atomic

bosons are described by the complex field ψ(r, τ). The Euclidean Lagrangian density of the free system in a D-dimensional box of volume LD and with chemical potential µ is given by L = ψ∗

• ∂τ − 2

2m∇2 − µ

• ψ + 1

2 g |ψ|4 , (78) where g is the strength of the contact inter-atomic coupling.

2.1 Broken symmetry and ideal Bose gas (II)

We have seen that the partition function Z of the system at temperature T can then be written as Z =

• D[ψ, ψ∗] exp
• −S[ψ, ψ∗]
• ,

(79) where S[ψ, ψ∗] = β dτ

• LD dDr L(ψ, ψ∗)

(80) is the Euclidean action and β ≡ 1/(kBT) with kB being Boltzmann’s constant. The grand potential Ω of the system, which is a function of µ, T and g, is then obtained as Ω = − 1 β ln Z . (81)

2.1 Broken symmetry and ideal Bose gas (III)

We work in the superfluid phase where the global U(1) gauge symmetry

• f the system is spontaneously broken. For this reason we set

ψ(r, τ) = ψ0 + η(r, τ) , (82) where η(r, τ) is the complex field of bosonic fluctuations around the

• rder parameter ψ0 (condensate in 3D or quasi-condensate in 1D and

2D) of the system. We suppose that ψ0 is constant in time, uniform in space and real. First we analyze the case with g = 0, where exact analytical results can be obtained in any spatial dimension D. In this case the Lagrangian density reads L = −µ ψ2

0 + η∗(r, τ)

• ∂τ − 2

2m∇2 − µ

• η(r, τ) .

(83)

2.1 Broken symmetry and ideal Bose gas (IV)

The Euclidean action of the ideal Bose gas can be written in a diagonal form as S[ψ, ψ∗] = −µ ψ2

0 β LD +

• Q

λQ η∗

QηQ

(84) where Q = (q, iωn) is the D + 1 vector denoting the momenta q and bosonic Matsubara frequencies ωn = 2πn/(β), and λQ = β (−iωn + 2q2 2m − µ) . (85) Here ηQ = ηq,iωn = ηq,n is the Fourier transform of η(r, τ) and the quantization of frequencies ωn is a consequence of the periodicity η(r, τ + β) = η(r, τ).

2.1 Broken symmetry and ideal Bose gas (V)

Taking into account that

• D[η, η∗] exp
• −
• Q

λQη∗

QηQ

• =

1

• Q λQ

(86) and

+∞

• n=−∞

ln (−i n + a) = 1 2

+∞

• n=−∞

ln (n2 + a2) , (87)

• ne finds the grand potential

Ω = −µ ψ2

0LD + 1

β

• Q

ln (λQ) = −µ ψ2

0LD + 1

2β

• q

+∞

• n=−∞

ln [β2(2ω2

n + ξ2 q)] ,

(88) where ξq is the shifted free-particle spectrum, i.e. ξq = 2q2 2m − µ . (89)

2.1 Broken symmetry and ideal Bose gas (VI)

The sum over bosonic Matsubara frequencies gives 1 2β

+∞

• n=−∞

ln [β2(2ω2

n + ξ2 q)] = ξq

2 + 1 β ln (1 − e−β ξq) . (90) In fact, the derivative with respect to ξq of Eq. (90) reads 1 β

+∞

• n=−∞

ξq ω2

n + ξ2 q

= 1 2 + 1 eβξq − 1 = 1 2 coth (βξq 2 ) , (91) and this formula is a straightforward consequence of the exact result

+∞

• n=−∞

1 n2 + a2 = π a coth (π a) (92) remembering that the Matsubara frequencies are given by ωn = 2πn/β.

2.1 Broken symmetry and ideal Bose gas (VII)

The grand potential finally reads Ω = Ω0 + Ω(0) + Ω(T) , (93) where Ω0 = −µ ψ2

0LD

(94) is the grand potential of the order parameter, Ω(0) = 1 2

• q

ξq (95) is the zero-point energy of bosonic single-particle excitations, i.e. the zero-temperature contribution of quantum fluctuations, and Ω(T) = 1 β

• q

ln

• 1 − e−βξq

(96) takes into account thermal fluctuations.

2.1 Broken symmetry and ideal Bose gas (VIII)

In the continuum limit, where

q → LD

dDq/(2π)D, the zero-point energy Ω(0) LD = 1 2 SD (2π)D +∞ dq qD−1 2q2 2m − µ

• (97)
• f the ideal Bose gas is clearly ultraviolet divergent at any integer

dimension D, i.e for D = 1, 2, 3. Here SD = 2πD/2/Γ(D/2) is the solid angle in D dimensions with Γ(x) the Euler gamma function. We shall show that this divergent zero-point energy of the ideal Bose gas is completely eliminated by dimensional regularization. Consequently, the exact grand potential of the ideal Bose gas is given by Ω LD = −µ ψ2

0 +

1 βLD

• q

ln

• 1 − e−βξq

. (98)

2.1 Broken symmetry and ideal Bose gas (IX)

We notice that ψ0 is not a free parameter but must be determined by minimizing Ω0, namely ∂Ω0 ∂ψ0

• µ,T,LD = 0 ,

(99) from which one finds that ψ0 = if µ = 0 any value if µ = 0 (100) The number density n = N/LD is obtained from the thermodynamic relation n = − 1 LD ∂Ω ∂µ

• T,LD,ψ0

, (101) which gives: n = ψ2

0 + 1

LD

• q

1 eβξq − 1 . (102)

2.1 Broken symmetry and ideal Bose gas (X)

In the continuum limit

q → LD

dDq/(2π)D, by setting µ = 0 (condensed phase) from Eq. (102) one gets n = n0 +

• dDq

(2π)D 1 e

2q2 2mkB T − 1

, (103) which gives the condensate density n0 = ψ2

0 as a function of the

temperature T. The critical temperature Tc is obtained setting ψ0 = 0 in the previous

• equation. In this way one finds

kB Tc =    no solution for D = 1 for D = 2

1 2πζ(3/2)2/3 2 m n2/3

for D = 3 (104) where ζ(x) is the Riemann zeta function.

2.1 Broken symmetry and ideal Bose gas (XI)

In the three-dimensional case (D = 3) from the equation n = n0 +

• d3q

(2π)3 1 e

2q2 2mkB T − 1

, (105) we find n = n0 + ζ(3/2) mkBT 2π2 3/2 . (106) It follows that n0 n = 1 − ζ(3/2) mkB

2π2

3/2 T 3/2 n = 1 − ζ(3/2) mkB

2π2

3/2 T 3/2 ζ(3/2) mkB

2π2

3/2 T 3/2

c

. (107) Thus, the condensate fraction reads n0 n = 1 − T Tc 3/2 . (108)

2.2 Interacting Bose gas (I)

Let us now consider a system of bosons with a repulsive contact interaction, i.e. let us set g > 0 in L = ψ∗

• ∂τ − 2

2m∇2 − µ

• ψ + 1

2 g |ψ|4 . (109) In this case one finds immediately the partition function of the uniform and constant order parameter ψ0 as Z0 = exp

• −S0
• = exp {−β Ω0} ,

(110) where S0 = S[ψ0] and the grand potential Ω0 reads Ω0 LD = −µ ψ2

0 + 1

2 g ψ4

0 .

(111) See the figure in the next slide.

2.2 Interacting Bose gas (II)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

ψ0

• 1

1 2 3

Ω0 / L

D

µ < 0 µ > 0

Figure: Mean-field grand potential Ω0 as a function of the real order parameter ψ0 for an interacting Bose gas, see Eq. (111).

2.2 Interacting Bose gas (III)

Again, the constant, uniform and real order parameter ψ0 is obtained by minimizing Ω0 as ∂Ω0 ∂ψ0

• µ,T,LD = 0 ,

(112) from which one finds the relation between order parameter and chemical potential µ = g ψ2

0 .

(113) showing that in the superfluid broken phase the chemical potential is positive and ψ0 = µ g , (114) Inserting this relation into Eq. (111) we find Ω0 LD = − µ2 2 g . (115) Clearly, this equation of state is lacking important informations encoded in quantum and thermal fluctuations.

2.2 Interacting Bose gas (IV)

Thus we consider again ψ(r, τ) = ψ0 + η(r, τ) (116) and expand the action S[ψ, ψ∗] around ψ0 up to quadratic (Gaussian)

• rder in η(r, τ) and ¯

η(r, τ). We find Z = Z0

• D[η, η∗] exp
• −Sg[η, η∗]
• ,

(117) where Sg[η, η∗] = 1 2

• Q

(η∗

Q, η−Q) MQ

• ηQ

η∗

−Q

• (118)

is the Gaussian action of fluctuations in reciprocal space with Q = (q, iωn) the D + 1 vector denoting the momenta q and bosonic Matsubara frequencies ωn = 2πn/(β), and

2.2 Interacting Bose gas (V)

MQ = β

• −iωn + 2q2

2m − µ + 2gψ2

−gψ2 −gψ2 iωn + 2q2

2m − µ + 2gψ2

• (119)

is the inverse fluctuation propagator. Integrating over the bosonic fields η(Q) and ¯ η(Q) in Eq. (117) one finds the Gaussian grand potential Ωg = 1 2β

• Q

ln Det(MQ) = 1 2β

• q

+∞

• n=−∞

ln [β2(2ω2

n + E 2 q )] ,

(120) where Eq is given by Eq = 2q2 2m − µ + 2gψ2 2 − g 2ψ4

0 .

(121)

2.2 Interacting Bose gas (VI)

By using ψ0 =

• µ/g the spectrum becomes

Eq =

• 2q2

2m 2q2 2m + 2µ

• ≃ cB q

for small q , (122) which is the familiar Bogoliubov spectrum, with cB =

• µ/m.

0.5 1 1.5 2 2.5

q

1 2 3 4

Eq

Bogoliubov spectrum Phonon spectrum

Figure: Bogoliubov spectrum, given by Eq. (122), and its low-momentum phonon spectrum Eq = cB q, where cB = p µ/m is the sound velocity. Energy Eq in units of µ and momentum q in units of p mµ/2.

2.2 Interacting Bose gas (VII)

Again, the sum over bosonic Matsubara frequencies gives 1 2β

+∞

• n=−∞

ln [β2(2ω2

n + E 2 q)] = Eq

2 + 1 β ln (1 − e−βEq) . (123) The total grand potential may then be written as Ω = Ω0 + Ω(0)

g

+ Ω(T)

g

, (124) where Ω0 is given by Eq. (115). Ω(0)

g

= 1 2

• q

Eq (125) is the zero-point energy of bosonic collective excitations, i.e. the zero-temperature contribution of quantum Gaussian fluctuations, while Ω(T)

g

= 1 β

• q

ln

• 1 − e−βEq

(126) takes into account thermal Gaussian fluctuations.

2.2 Interacting Bose gas (VIII)

We notice that the continuum limit of the zero-point energy for the interacting Bose gas Ω(0)

g

LD = 1 2 SD (2π)D +∞ dq qD−1

• 2q2

2m 2q2 2m + 2µ

• (127)

is ultraviolet divergent at any integer dimension D. We may rewrite Eq. (127) for the zero point energy of a repulsive Bose gas in dimension D as: Ω(0)

g

LD = SD(2µ)

D 2 +1

4(2π)D 2m 2 D

2

B D + 1 2 , −D + 2 2

• ,

(128) where B(x, y) is the Euler Beta function B(x, y) = +∞ dt tx−1 (1 + t)x+y , Re(x), Re(y) > 0 (129) which may be continued to complex values of x and y as B(x, y) = Γ(x) Γ(y) Γ(x + y) . (130)

2.2 Interacting Bose gas (IX)

We rewrite Eq. (127) using Eq. (130) to get Ω(0)

g

LD = SD(2µ)

D 2 +1

4(2π)D 2m 2 D

2 Γ( D+1

2 ) Γ(− D+2 2 )

Γ(− 1

2)

. (131) This expression is now finite when D = 1 or D = 3, while it is still divergent if D = 2 since Γ(p) diverges for integers p ≤ 0. In particular, setting D = 3 in Eq. (131) we get Ω(0)

g

L3 = 8 15π2 ( m 2 )3/2µ5/2 . (132) In conclusion, the total grand potential of the three-dimensional Bose gas is then given by Ω L3 = − µ2 2 g + 8 15π2 ( m 2 )3/2µ5/2 + 1 βL3

• q

ln

• 1 − e−βEq

. (133)

3.1 Gross-Pitaevskii equation (I)

Let us consider again a system of bosons with a repulsive contact interaction, i.e. let us set g > 0 in L = ψ∗

• ∂τ − 2

2m∇2 − µ + U(r)

• ψ + 1

2 g |ψ|4 , (134) where U(r) is the external trapping potential. In this inhomogeneous case the partition function of the mean-field (saddle-point) order parameter ψ0(r, τ) reads Z0 = exp

• −S0
• = exp {−β Ω0} ,

(135) where S0 = S[ψ0(r, τ), ψ∗

0(r, τ)] =

• dt
• d3r L(ψ0, ψ∗

0)

(136) and ψ0(r, τ) is the field ψ(r, τ) which minimizes S[ψ(r, τ)], namely δS[ψ(r, τ), ψ∗(r, τ)] = 0 . (137)

3.1 Gross-Pitaevskii equation (II)

The minimization of the action S gives the Euler-Lagrange equation ∂L ∂ψ∗ − ∂τ ∂L ∂(∂τψ∗) − ∇ ∂L ∂(∇ψ∗) + ∇2 ∂L ∂(∇2ψ∗) = 0 (138) from which we get ∂τψ0(r, τ) =

• − 2

2m∇2 − µ + U(r) + g |ψ0(r, τ)|2

• ψ0(r, τ) .

(139) This is the Gross-Pitaevskii equation with imaginary time τ. Moreover, at the mean-field level, the number of bosons reads N = −∂Ω0 ∂µ =

• d3r |ψ0(r, τ)|2 .

(140)

3.1 Gross-Pitaevskii equation (III)

Going back to real time t = iτ we obtain i∂tψ0(r, t) =

• − 2

2m∇2 − µ + U(r) + g |ψ0(r, t)|2

• ψ0(r, t) ,

(141) that is the real-time (time-dependent) Gross-Pitaevskii equation, with the normalization condition N =

• d3r |ψ0(r, t)|2 .

(142) Immediately one deduces the corresponding stationary Gross-Pitaevskii equation

• − 2

2m∇2 + U(r) + g |ψ0(r)|2

• ψ0(r) = µ ψ0(r) .

(143)