Lecture 2: Bogoliubov theory of a dilute Bose gas
Christopher Mudry∗
Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland.
(Dated: March 01, 2010)
Abstract
Second quantization for bosons is reviewed.
Bose-Einstein condensation for noninteracting
bosons is interpreted as an example of spontaneous symmetry breaking. The spectrum of a dilute
Bose gas with hardcore repulsion is calculated within Bogoliubov mean-field theory. It is shown
that a Goldstone mode, an acoustic phonon, emerges in association with spontaneous symmetry
breaking. Landau criterion for superfluidity is presented.
∗
Electronic address: Christopher.Mudry@psi.ch; URL: http://people.web.psi.ch/mudry
1
Contents
3
I. Introduction
3
II. Second quantization for bosons
III. Bose-Einstein condensation: An example of spontaneous symmetry
8
breaking
16
IV. Dilute Bose gas
A. Operator formalism
16
B. Landau criterion for superfluidity
22
C. Path integral formalism
23
1. Noninteracting limit λ = 0
25
2. Random-phase approximation
28
3. Beyond the random-phase approximation
35
A. Some Gaussian integrals
39
B. Bose-Einstein distribution and the residue theorem
40
41
References
2
I.
INTRODUCTION
In this lecture we shall study a dilute Bose gas with a repulsive contact interaction. We
shall see that the phenomenon of superfluidity takes place at sufficiently low temperatures.
Superfluidity is an example of spontaneous symmetry breaking of a continuous symmetry. The continuous symmetry is the global U(1) gauge symmetry which is responsible
for conservation of total particle number. We shall also carefully distinguish Bose-Einstein
condensation from superfluidity. Interactions are necessary for superfluidity to take place.
Interactions are not needed for Bose-Einstein condensation.
From a technical point of view, we begin with the formalism of second quantization for
Bosons. We then interpret Bose-Einstein condensation at zero temperature as an example of
spontaneous symmetry breaking of a continuous symmetry through an explicit construction
of a ground state that breaks the global U(1) gauge symmetry. The emphasis here is on
how the global U(1) gauge symmetry organizes the Hilbert space spanned by eigenstates
of the Hamiltonian. In this construction, the thermodynamic limit plays an essential role.
Next, we treat a repulsive contact interaction through an approximation first proposed by
Bogoliubov. This approximation is called the random-phase approximation. We revisit this
approximation using path integral techniques and see that the random-phase approximation
is nothing but a saddle-point approximation with the inclusion of Gaussian fluctuations
around the saddle-point. We present two effective field theories with different physical
contents. The first one deals with single-particle excitations. The second one deals with
collective excitations.
II.
SECOND QUANTIZATION FOR BOSONS
The terminology “second quantization” is rather unfortunate in that it might be perceived
as implying concepts more difficult to grasp than the passage from classical to quantum mechanics. Quite to the contrary the relation between “second” and “first” quantization1 is
nothing but a matter of convenience. Going from first to second quantization is like going
from a real space representation of Schrödinger equation to a momentum space representation when the Hamiltonian has translation symmetry.
1
By first quantization is meant Schrödinger equation.
3
Second quantization is a formalism that aims at describing a system made of identical
“particles”, bosons or fermions, in which creation and annihilation of particles is easily and
naturally accounted for. Hence, the quantum “particle number” need not be sharp in this
representation very much in the same way as position is not a sharp quantum number for a
momentum eigenstate. Another analogy for the relationship between first quantization, in
which the quantum “particle number” is a sharp quantum number, and second quantization,
in which it need not be, is that between the canonical and grand canonical ensembles of
statistical mechanics. In the canonical ensemble, particle number is given. In the grand
canonical ensemble, particle number fluctuates statistically as it has been traded for a fixed
chemical potential.
The formalism of second quantization can already be introduced at the level of a single harmonic oscillator, but it is for interacting many-body systems that it becomes very
powerful. It is nevertheless instructive to develop the formalism already at the level of a
single-particle Hamiltonian since, too a large extend, many-body physics is glorified perturbative physics about some noninteracting limit.
I will now generalize the construction of a second quantized formalism in terms of creation
and annihilation operators for the one-dimensional harmonic oscillator that I presented in
lecture 1. I will thus consider a finite volume V of d-dimensional space on which the singleparticle Hilbert space H(1) of square integrable and twice differentiable functions is defined.
In turn, the single-particle Hamiltonian is represented by (~ = 1 and ∆ is Laplace’s operator
in d space dimensions)
H =−
∆
+ U(r),
2m
(2.1a)
and possesses the complete, orthogonal, and normalized basis of eigenfunctions
Z
X
Hϕn (r) = εn ϕn (r),
dd r ϕ∗m (r)ϕn (r) = δm,n ,
ϕ∗n (r)ϕn (r ′ ) = δ(r − r ′).
n
V
(2.1b)
The index n belongs to a countable set after appropriate boundary conditions, say periodic, have been imposed at the boundaries of the finite volume V . I assume that the
single-particle potential U(r) is bounded from below, i.e., there exists a single-particle nondegenerate ground-state energy, say ε0 . Hence the energy eigenvalue index runs over the
4
positive integers, n ∈ N. The time evolution of any solution of Schrödinger equation
i∂t Ψ(r, t) = HΨ(r, t),
Ψ(r, t = 0) given,
(2.2a)
can be written as
Ψ(r, t) =
X
−iεn t
An ϕn (r) e
,
An =
n
Z
dd r ϕ∗n (r)Ψ(r, t = 0).
(2.2b)
V
The formalism of second quantization starts with two postulates:
• There exists a set of pairs of adjoint operators ân (annihilation operator) and â†n
(creation operator) labeled by the energy eigenvalue index n and obeying the boson
algebra 2
[âm , â†n ] = δm,n ,
[âm , ân ] = [â†m , â†n ] = 0,
∀m, n.
(2.3)
• There exists a vacuum state |0i that is annihilated by all annihilation operators,
ân |0i = 0,
∀n.
(2.4)
With these postulates in hand, it becomes possible to define the Heisenberg representation
for the operator-valued field (in short, quantum field):
ϕ̂(r, t) :=
X
ân ϕn (r) e−iεn t ,
(2.5)
n
together with its adjoint
ϕ̂† (r, t) :=
X
â†n ϕ∗n (r) e+iεnt .
(2.6)
n
The boson algebra (2.3) endows the quantum fields ϕ̂(r, t) and ϕ̂† (r, t) with the equal-time
algebra3
[ϕ̂(r, t), ϕ̂† (r ′ , t)] = δ(r − r ′ ),
2
3
[ϕ̂(r, t), ϕ̂(r ′ , t)] = [ϕ̂† (r, t), ϕ̂† (r ′ , t)] = 0.
(2.10)
My conventions for the commutator and anticommutator of any two “objects” A and B are [A, B] :=
AB − BA and {A, B} := AB + BA, respectively.
Alternatively, if we start from the classical Lagrangian density
L := (ϕ∗ i∂t ϕ)(r, t) −
1
|∇ϕ|2 (r, t) − |ϕ∗ |2 (r, t)U (r),
2m
5
(2.7)
The quantum fields ϕ̂(r, t) and ϕ̂† (r, t) act on the “big” many-particle space


∞ O
N

M
(1)
F :=
H
.


sym
N =0
N
(1)
Here, each N
sym H is spanned by states of the form
mi
†
Y âi
p
|m0 , · · · , mi−1 , mi , mi+1 , · · · i :=
|0i,
mi ∈ N,
mi !
i
(2.11a)
(2.11b)
with the condition on the positive integers mi that
X
mi = N.
(2.11c)
i
NN
(1)
sym H is the N-th symmetric power of H(1) , i.e., that the state |m0 , · · · , mi−1 , mi , mi+1 , · · · i made of N identical
The algebra obeyed by the â’s and their adjoints ensures that
particles of which mi have energy εi is left unchanged by any permutation of the N particles.
Hence, the “big” many-particle Hilbert space (2.11a) is the sum over the subspaces spanned
by wave functions for N identical particles that are symmetric under any permutation of
the particles labels. This “big” many-particle Hilbert space is called the boson Fock space
in physics.
The rule to change the representation of operators from the Schrödinger picture to the
second quantized language is best illustrated by the following examples:
• The second-quantized representation Ĥ of the single-particle Hamiltonian (2.1a) is
Z
Ĥ :=
dd r ϕ̂† (r, t)H ϕ̂(r, t)
V
=
X
εn â†n ân .
(2.12)
n
As it should be it is explicitly time-independent.
we can elevate the field ϕ(r, t) and its momentum conjugate
π(r, t) :=
δL
= iϕ∗ (r, t)
δ(∂t ϕ)(r, t)
(2.8)
to the status of quantum fields ϕ̂(r, t) and π̂(r, t) = iϕ̂† (r, t) obeying the equal-time bosonic algebra
[ϕ̂(r, t), π̂(r ′ , t)] = iδ(r − r ′ ),
[ϕ̂(r, t), ϕ̂(r ′ , t)] = [π̂(r, t), π̂(r ′ , t)] = 0.
6
(2.9)
• The second-quantized total particle-number operator Q̂ is
Z
Q̂ :=
dd r ϕ̂† (r, t)11ϕ̂(r, t)
V
=
X
â†n ân .
(2.13)
n
It is explicitly time-independent as follows from the continuity equation
0 = (∂t ρ)(r, t) + (∇ · J )(r, t),
(2.14a)
ρ(r, t) := |Ψ(r, t)|2 ,
1
J (r, t) :=
[Ψ∗ (r, t) (∇Ψ) (r, t) − (∇Ψ∗ ) (r, t)Ψ(r, t)] ,
2mi
(2.14b)
(2.14c)
obeyed by Schrödinger equation (2.2a). The number operator Q̂ is the infinitesimal
generator of global gauge transformations by which all states in the boson Fock space
are multiplied by the same operator-valued phase factor. Thus, a global gauge transformation on the Fock space is implemented by the operation
|m0 , · · · , mi−1 , mi , mi+1 , · · · i → e+iqQ̂ |m0 , · · · , mi−1 , mi , mi+1 , · · · i,
∀q ∈ R,
(2.15)
on states, or, equivalently,4
ân → e+iqQ̂ ân e−iqQ̂ = e−iq ân ,
(2.17)
â†n → e+iqQ̂ â†n e−iqQ̂ = e+iq â†n ,
(2.18)
and
for all pairs of annihilation and creation operators. Equation (2.17) tells us that
annihilation operators carry particle number −1. Equation (2.18) tells us that creation
operators carry particle number +1.
4
I made use of
[↠â, â] = ↠ââ − â↠â = ↠ââ − ↠ââ + ↠ââ − â↠â = ↠[â, â] + [↠, â]â = −â,
(2.16a)
and, similarly,
[↠â, ↠] = +↠.
7
(2.16b)
• The second-quantized local particle-number density operator ρ̂ and the particlenumber current density Jˆ are
ρ̂(r, t) := ϕ̂† (r, t)11ϕ̂(r, t),
(2.19a)
and
1 †
Jˆ(r, t) :=
ϕ̂ (r, t) (∇ϕ̂) (r, t) − ∇ϕ̂† (r, t)ϕ̂(r, t) ,
2mi
(2.19b)
respectively. The continuity equation
0 = (∂t ρ̂)(r, t) + (∇ · Jˆ)(r, t)
(2.19c)
that follows from evaluating the commutator between ρ̂ and Ĥ is obeyed as an operator
equation.
The operators Ĥ, Q̂, ρ̂, and Jˆ all act on the Fock space F . They are thus distinct from
their single-particle counterparts H, Q, ρ, and J whose actions are restricted to the Hilbert
N
space H(1) . By construction, the action of Ĥ, Q̂, ρ̂ and Jˆ on the subspace 1sym H(1) of F
coincides with the action of H, Q, ρ, and J on H(1) , respectively.
III.
BOSE-EINSTEIN CONDENSATION: AN EXAMPLE OF SPONTANEOUS
SYMMETRY BREAKING
Given a many-body system made of identical bosons, say atoms carrying an integervalued total angular momentum, how does one construct the ground state? The simplest
answer to this question occurs when bosons are noninteracting. In this case, the ground
state is simply obtained by putting all bosons in the lowest energy single-particle state. If
the number of bosons is taken to be N, then the ground state is |N, 0, · · · i with energy Nε0 .
This straightforward observation underlies the phenomenon of Bose-Einstein condensation:
A finite or macroscopic fraction of bosons occupies the single-particle energy level ε0 below
the Bose-Einstein transition temperature TBE in the thermodynamic limit of infinite volume
V but finite particle density.
From a conceptual point of view, it is more fruitful to associate Bose-Einstein condensation with the phenomenon of spontaneous symmetry breaking of a continuous symmetry
than with macroscopic occupation of a single-particle level. The continuous symmetry in
8
question is the freedom in the choice of the global phase of the many-particle wave functions. This symmetry is responsible for total particle-number conservation. In mathematical
terms, the vanishing commutator
[Ĥ, Q̂] = 0
(3.1)
between the total number operator Q̂ and the single-particle Hamiltonian Ĥ implies a global
U(1) gauge symmetry.
The concept of spontaneous symmetry breaking is subtle. For one thing it can never take
place when the normalized ground state |Φ0 i of the many-particle Hamiltonian (possibly
interacting) is nondegenerate, i.e., unique up to a phase factor. Indeed, the transformation
law of the ground state |Φ0 i under any symmetry of the Hamiltonian must then be multiplication by a phase factor. Correspondingly, the ground state |Φ0 i must transform according
to the trivial representation of the symmetry group, i.e., |Φ0 i transforms as a singlet. In
this case there is no room for the phenomenon of spontaneous symmetry breaking by which
the ground state transforms nontrivially under some symmetry group of the Hamiltonian.
Now, the Perron-Frobenius theorem for finite dimensional matrices with positive entries
[1] or its extension [2] to single-particle Hamiltonians of the form (2.1a) guarantees that
the ground state is nondegenerate for noninteracting N-body Hamiltonians defined on the
N
(1)
Hilbert space N
sym H . Although there is no rigorous proof that the same theorem holds
for interacting N-body Hamiltonians, it is believed that the ground state of interacting
N
(1)
Hamiltonians defined on N
sym H is also unique. As a corollary, it is believed that spontaneous symmetry breaking is always ruled out for interacting Hamiltonian defined on the
N
(1)
Hilbert space N
sym H .
Before evading this no-go theorem by taking advantage of the thermodynamic limit of
infinite volume V but finite particle density, I want to investigate more closely the consequences of having a nondegenerate ground state. I consider the cases of both noninteracting
many-body Hamiltonians such as Ĥ in Eq. (2.12) and interacting many-body Hamiltonians5
that commute with Q̂. The Hilbert space will be the boson Fock space F in Eq. (2.11a)
on which the quantum field operator ϕ̂(r, t) in Eq. (2.5) is defined. We shall see that the
expectation value of ϕ̂(r, t) in the ground state of the many-body system |Φ0 i can be used
5
Interactions are easily introduced through polynomials in creation and annihilation operators of degree
larger than 2.
9
as a signature of spontaneous symmetry breaking of the U(1) symmetry. More generally, we
shall interpret the quantum statistical average of ϕ̂(r, t) as a temperature dependent order
parameter. As follows from Eq. (2.17), the quantum field ϕ̂(r, t) transforms according to
e+iqQ̂ ϕ̂(r, t)e−iqQ̂ = e−iq ϕ̂(r, t),
∀r, t,
(3.2)
under any global gauge transformation. The quantum field ϕ̂(r, t) carries U(1) charge −1
P
as it lowers the boson occupation numbers i mi by one on any state (2.11b) of the boson
Fock space F . By hypothesis, the ground state |Φ0 i of Ĥ is unique. Thus, it transforms like
a singlet under U(1):
∃Q0 ∈ R,
e+iqQ̂ |Φ0 i = e+iqQ0 |Φ0 i,
hΦ0 | e−iqQ̂ = hΦ0 | e−iqQ0 .
(3.3)
What then follows for the expectation value hΦ0 |ϕ̂(r, t)|Φ0 i? It must vanish. Indeed,
+iq Q̂
−iq Q̂
|Φ0 i = e−iq hΦ0 |ϕ̂(r, t)|Φ0 i,
hΦ0 | e
ϕ̂(r, t)e
∀r, t,
(3.4)
∀r, t,
(3.5)
by Eq. (3.2), and
hΦ0 |e+iqQ̂ ϕ̂(r, t) e−iqQ̂ |Φ0 i = hΦ0 |ϕ̂(r, t)|Φ0 i,
by Eq. (3.3), hold simultaneously for any q ∈ R. The vanishing of hΦ0 |ϕ̂(r, t)|Φ0 i – in view
of the fact that ϕ̂(r, t) carries U(1) charge −1 and thus transforms nontrivially under U(1)
– can be traced to the assumption that the ground state |Φ0 i is unique, i.e., that |Φ0 i is an
eigenstate of Q̂. In more intuitive terms, the action of ϕ̂(r, t) on an eigenstate of Q̂ such as
|Φ0 i is to lower the total number of particle by one, thereby producing a state orthogonal to
|Φ0 i. Conversely, a nonvanishing expectation value of ϕ̂(r, t) in some state |φi ∈ F is only
possible if |φi ∈ F is not an eigenstate of Q̂.6 Evading the no-go theorem for spontaneous
symmetry breaking thus requires quantum degeneracy of the ground state with orthogonal
ground states that are related by the action of the U(1) symmetry group. In turn, this can
be achieved by constructing a ground state |φi ∈ F that is an eigenstate of ϕ̂(r, t) and thus
cannot be an eigenstate of Q̂.
A prerequisite to evade the no-go theorem for spontaneous symmetry breaking is that
the thermodynamic limit of infinite volume V but finite particle density be taken. This is
6
It is impossible for ϕ̂(r, t) to acquire an expectation value on
10
NN
(1)
sym H .
certainly true when the bosons making up the many-body system form a weakly interacting
gas of atoms in which the total number of atoms is accessible to experimental control. The
thermodynamic limit is then necessary so as to justify the use of the Fock space F with
unrestricted boson number instead of a Hilbert space with fixed boson number. Indeed, if
the thermodynamic limit is well defined, there is no difference between approaching the thermodynamic limit by working at fixed volume and at fixed particle number with the Hilbert
N
(1)
space N
sym H and taking N, V → ∞ with N/V held fixed, or approaching the thermodynamic limit by working at fixed external pressure and at fixed chemical potential with the
NN
P
(1)
Fock space F = ∞
sym H . The first approach to the thermodynamic limit defines
N =0
the so-called canonical ensemble of quantum statistical mechanics. The second approach to
the thermodynamic limit defines the so-called grand canonical ensemble of quantum statistical mechanics. The thermodynamic limit is also needed to recover spontaneous symmetry
breaking even when7 the Hilbert space of finitely many degrees of freedom is endowed with
the structure of a Fock space.
To underscore the role played by the thermodynamic limit to evade the no-go theorem
for spontaneous symmetry breaking, I now restrict myself to the many-body noninteracting
Hamiltonian
Ĥµ := Ĥ − µQ̂
(3.6a)
with
H=−
∆
2m
(3.6b)
in Eq. (2.1a) so that translation invariance holds at the single-particle level. The real-valued
parameter µ is called the chemical potential. Since Ĥ commutes with Q̂ by hypothesis, an
eigenstate of Ĥ is also an eigenstate of Ĥµ and conversely. Eigenenergies of Ĥ and Ĥµ can
differ, however. For example, the single-particle eigenfunctions ϕn (r) of H in Eq. (2.1a)
are also single-particle eigenfunctions of Ĥµ on H(1) but with the rigidly shifted spectrum
of energy eigenvalues εn − µ. Furthermore, the dimensionality of the eigenspaces of Ĥ
can change dramatically by the addition of −µQ̂. To see this, observe that the choice
7
This occurs when the bosons of the many-body system are collective excitations, say phonons in a solid,
spin waves in an antiferromagnet, or excitons in a semiconductor, i.e., when the finitely many degrees of
freedom are ions, spins, or band electrons, respectively.
11
µ = ε0 insures that the single-particle ground-state energy of Ĥµ vanishes and that the
√
corresponding normalized eigenfunction ϕ(r) = 1/ V .8 This choice also guarantees that
all states
(a† )m0
|0i,
|m0 , 0, · · · i = p0
m0 !
m0 ∈ N,
(3.8)
are orthogonal eigenstates of Ĥµ in F with the same vanishing energy.9 The choice µ = ε0
guarantees that Ĥµ has countably-many orthogonal ground states provided the volume V is
finite.
Any linear combination of states of the form (3.8) is a ground state of Ĥµ with µ = ε0 .
Of all these possible linear combinations, consider the continuous family of normalized10
ground states labeled by the complex-valued parameter φ,
V
2
− V2
|φ|2
|φigs := e− 2 |φ|
= e
V
â0 |0i = 0
[[A, B], A] = [[A, B], B] = 0 =⇒ eA eB = e[A,B]/2 eA+B
2
∞
X
m0 =0
√
V φâ†0
e
= e− 2 |φ| e+
√
= e
√
√
√
|0i
V φâ†0
V (φâ†0 −φ∗ â0 )
m0
Vφ
p
|m0 , 0, · · · i
m0 !
e−
√
V φ∗ â0
|0i
|0i
=: D̂( V φ, 0, · · · )|0i.
(3.10)
√
Here, the unitary operator D̂( V φ, 0, · · · ) rotates the vacuum into the boson coherent state
√
√
†
| V φ, 0, · · · ics := e V φâ0 |0i,
(3.11)
up to the proportionality constant exp(− V2 |φ|2). Bosonic coherent states form an overcomplete set of the Fock space. The overlap between any two coherent states is always
8
A time dependent gauge transformation plays the same role as the chemical potential if one chooses to
work in the canonical instead of the grand canonical statistical ensemble. For example, setting ε0 to 0 in
the single-particle Hilbert space H(1) is achieved with the help of the time-dependent gauge transformation
Ψ(r, t) → eiε0 t Ψ(r, t)
9
10
(3.7)
on the single-particle Schrödinger equation (2.2a).
The same states are also eigenstates of Ĥ in F but with distinct energy eigenvalues m0 ε0 .
Observe that the operator
D(φ1 , φ2 , · · · ) :=
Y
e(φn ân −φn ân ) ,
†
n
is unitary.
12
∗
φ1 , φ2 , · · · ∈ C,
(3.9)
nonvanishing,
cs hα0 , α1 , · · · |β0 , β1 , · · · ics =
cs hα0 , α1 , · · · |
:= h0|
|β0 , β1 , · · · ics :=
The same is true of the overlaps
Y
n
Y
α∗n an
e
Y
∗
eαn βn ,
n
,
αn , βn ∈ C,
n
†
eβn an |0i,
gs hφ|0i
V
αn , βn ∈ C.
2
= e− 2 |φ| ,
′
gs hφ|φ igs
= e−V
αn , βn ∈ C,
(3.12a)
(3.12b)
(3.12c)
(3.13a)
|φ−φ′ |2
2
.
(3.13b)
√
The rational for having scaled the arguments of the unitary operator D̂( V φ, 0, · · · ) by
the square root of the volume V of the system in Eq. (3.10) is to guaranty that all the
rotated vacuum in Eq. (3.10) become orthogonal in the thermodynamic limit. The thermodynamic limit is thus essential in providing an escape to the absence of spontaneous
symmetry breaking in systems of finite sizes. In the thermodynamic limit, we need not
NN
(1)
distinguish Ĥ defined on
sym H from Ĥµ defined on F . It is only in the thermodynamic limit that the ground-state manifold ∼
= C of Ĥµ , µ = ε0 , in Eq. (3.10) becomes the
ground-state manifold ∼
= C of Ĥ. Where does this degeneracy of Ĥ comes from? When V
NN
(1)
and N are finite and Ĥ is restricted to
sym H the ground-state energy is Nε0 . The
NN ±1 (1)
N
(1)
ground-state energy of Ĥ in sym
H differs from that in N
sym H by a term of order
N 0 namely ±ε0 . In the Fock space F , the energy difference between hN, 0, · · · |Ĥ|N, 0, · · · i
and hN ± δN, 0, · · · |Ĥ|N ± δN, 0, · · · i scales like 1/N as the thermodynamic limit N → ∞,
δN/N → 0, and N/V finite is taken. Hence, more and more states have an energy of order
N 0 above the ground-state energy Nε0 as the system size is increased. The surprising result
is that it is not a mere countable infinity of states that become degenerate with the ground
state in the thermodynamic limit but an uncountable infinity.
It remains to verify that each ground state |φigs in Eq. (3.10) is an eigenstate of the
quantum fields ϕ̂(r, t)11 but is not an eigenstate of Q̂:
11
ϕ̂(r, t) |φigs = φ |φigs,
(3.14a)
e−iαQ̂ |φigs = |e−iαφigs .
(3.14b)
√
Remember that the single-particle ground-state wave function ϕ0 (r) is the constant 1/ V . Make then
√
√
√
use of the expansion (2.5) applied to (3.10) whereby √1V â0 | V φics = √1V ( V φ)| V φics must be used.
13
The U(1) “multiplet” structure of the manifold of ground states ∼
= C in Eq. (3.10) is displayed
by Eq. (3.14b). Circles in the complex plane φ ∈ C correspond to U(1) “multiplets”.
√
Normalization of the single-particle eigenfunction ϕ0 (r) = 1/ V and the property that
coherent states are eigenstates of annihilation operators guaranty that the quantum field
ϕ̂(r, t) acquires the expectation value φ ∈ C with the particle density |φ|2 in the groundstate manifold (3.10):
gs hφ|ϕ̂(r, t)|φigs
gs hφ|ϕ̂
†
= φ,
(3.15a)
(r, t)ϕ̂(r, t)|φigs = |φ|2 .
(3.15b)
In an interacting system the noninteracting trick relying on fine tuning of the chemical
potential µ → ε0 to construct explicitly the many-body ground state breaks down. The
chemical potential is chosen instead by demanding that the particle density,
hΦ0 |ϕ̂† (r, t)ϕ̂(r, t)|Φ0 i =
N
,
V
(3.16)
at zero temperature,12 be held fixed to the value N/V as the thermodynamic limit is taken.
At finite temperature the right-hand side is unchanged whereas the left-hand side becomes
a statistical average in the grand canonical ensemble. A degenerate manifold of ground
states satisfying Eqs. (3.15b) is not anymore parametrized by φ ∈ C but by arg(φ) ∈ [0, 2π[
since the modulus |φ|2 = N/V is now given. The U(1) symmetry group parametrized by
exp(iαQ̂), α ∈ [0, 2π[ is said to act transitively on the ground-state manifold. Construction
of the ground-state manifold relies on approximate schemes such as mean-field theory. These
approximations are nonperturbative in the sense that they yield variational wave functions
that cannot be derived from the noninteracting limit to any finite order of the perturbation
theory in the interaction strength.
Spontaneous symmetry breaking is said to occur when the ground state |Φ0 i of a manybody system is no longer a singlet under the action of a symmetry group of the system.
A quantity like hΦ0 |ϕ̂(r, t)|Φ0 i that must vanish when the ground state is a singlet, but
becomes nonzero in a phase with spontaneous symmetry breaking is called an order parameter. An order parameter is a probe to detect spontaneous symmetry breaking. In
12
As before, |Φ0 i denotes the many-body ground state which, in practice, cannot be constructed exactly
when interactions are present. I am implicitly assuming translation invariance. This is the reason why
the right-hand side does not depend on r.
14
condensed matter physics, the order parameter can be directly observed in a static measurement. For example, elastic neutron scattering can show Bragg peaks corresponding to
crystalline or magnetic order. The order parameter can also be indirectly observed in a dynamic measurement. For instance, inelastic neutron scattering can show a gapless branch of
excitations, Goldstone modes, corresponding to phonons or spin waves. Some consequences
of symmetries such as selections rules and degeneracies of the excitation spectrum no longer
hold in their simple form when the phenomenon of spontaneous symmetry breaking occurs.
The mass distributions of mesons, hadrons, photon, W and Z bosons are interpreted as a
manifestation of spontaneous symmetry breaking leading to the standard model of strong,
weak, and electromagnetic interactions.
How does one go about detecting spontaneous symmetry breaking in the canonical ensemble? This question is of relevance to numerical simulations where the dimensionality
of the Hilbert space is necessarily finite. A probe for spontaneous symmetry breaking is
off-diagonal long-range order. Let |ΦN i be the ground state of the many-body system in
NN
(1)
the Hilbert space
sym H . I will denote with ϕ̂(r) the quantum field ϕ̂(r, t = 0) in
the Schrödinger picture. Here, the Schrödinger picture can be implemented numerically
through exact diagonalization of matrices say. I will assume translational invariance, i.e.,
the single-particle potential U(r) = 0 in Eq. (2.1a). Define the one-particle density matrix
by
R(r, r ′) :=
1
hΦN |ϕ̂† (r)ϕ̂(r ′ )|ΦN i.
V
(3.17)
By translation invariance R(r, r ′ ) = R(r − r ′ ) whereby
Z
Z d ′
dd k
d k i(k·r−k′·r′ )
′
R(r, r ) =
e
hΦN |â†k′ âk|ΦN i
d
d
(2π)
(2π)
Z
Z
Z d ′
dd k
d k i[k·(r+y)−k′·(r′ +y)]
1
d
′
′
d y
e
hΦN |â†k′ âk|ΦN i
R(r, r ) = R(r − r )
=
d
V
(2π)
(2π)d
Z
Z d ′
1
dd k
d k
d
′ i(k·r−k′ ·r′ )
=
(2π)
δ(k
−
k
)e
hΦN |â†k′ âk|ΦN i
V
(2π)d
(2π)d
Z
1
dd k ik·(r−r′ )
=
e
hΦN |â†kâk|ΦN i
d
V
(2π)
Z
d
d k ik·(r−r′ )
=:
e
nk.
(3.18)
(2π)d
The ground-state expectation value nk is the number of particles per unit volume with
momentum k. When r − r ′ = 0, the one-particle density matrix R(r − r ′ ) is just the total
15
number of particles per unit volume n0 = N/V . Bose-Einstein condensation means that
nk = n0 (2π)d δ(k) + f (k),
(3.19a)
with f (k) some smooth function that satisfies
Z
dd k
f (k) = 0.
(2π)d
In real space, Bose-Einstein condensation thus amounts to
Z
dd k ik·r
′
′
R(r, r ) = n0 + F (r − r ),
F (r) :=
e f (k),
(2π)d
(3.19b)
lim F (r) = 0. (3.20)
|r|→∞
The nonvanishing of lim|r|→∞ R(r, 0) is another signature of spontaneous symmetry breaking
associated to Bose-Einstein condensation.
I conclude this section with some field theoretical terminology. States |Θi for which
lim
hΘ|Ô1(r1 )Ô2 (r2 )|Θi = hΘ|Ô1 (r1 )|ΘihΘ|Ô2(r2 )|Θi
|r1 −r2 |→∞
(3.21)
holds for any pair of operators Ô1 (r) and Ô2 (r) defined on the Fock space F are said to
satisfy the cluster decomposition property or to be clustering. The ground state |ΦN i in
Eq. (3.20) does not satisfy the clustering property.13 The manifold of states |φ ∈ Cigs in
Eq. (3.10) does satisfy the clustering property by Eq. (3.15b).
IV.
A.
DILUTE BOSE GAS
Operator formalism
Bogoliubov introduced the dilute Bose gas as a model for superfluid 4 He. The model is
not a very good one for superfluid 4 He in that the assumption of pairwise interactions fails
badly, but it may be a more realistic model for alkali metallic vapor Bose condensates. The
model is defined by the second-quantized Hamiltonian
Z
λ † 2
∆
d
†
ϕ̂ ϕ̂ (r, t) .
− µ ϕ̂(r, t) +
Ĥµ,λ = d r ϕ̂ (r, t) −
2m
2
(4.1)
V
13
Choose Ô1 = ϕ̂† and Ô2 = ϕ̂ in which case the left-hand side of Eq. (3.21) is nonvanishing whereas the
right-hand side vanishes.
16
The chemical potential µ determines the number N(µ) of particles in the interacting ground
state |Φgs i from
Z
N(µ) = hΦgs |
V
dd r ϕ̂† ϕ̂ (r, t)|Φgs i.
(4.2)
Conversely, fixing the total particle number to N determines µ(N). The interaction is a
density-density delta function repulsion,
Z
Z
λ
d
Ĥλ :=
d r dd r ′ ρ̂(r, t)δ(r − r ′ )ρ̂(r ′ , t),
2
V
V
ρ̂(r, t) := ϕ̂† ϕ̂ (r, t).
(4.3)
The real parameter λ ≥ 0 measures the strength of the repulsive interaction and carries the
units of energy×volume. Bosons are said to have a hardcore. Periodic boundary conditions
are imposed in the volume V and it is natural to expand the pair of canonical conjugate
quantum fields ϕ̂(r, t) and iϕ̂† (r, t) in the basis of plane waves,
1 X
âk e+i(k·r−εk t) ,
ϕ̂(r, t) = √
V k
1 X † −i(k·r−εk t)
ϕ̂† (r, t) = √
âk e
,
V k
εk =
k2
. (4.4)
2m
Here, the summation over reciprocal space is infinite but countable,
k=
2π
l,
L
l ∈ Zd ,
Ld ≡ V.
(4.5)
The single-particle plane wave with the lowest energy is
1
ϕ0 (r) = √ ,
V
ε0 = 0.
(4.6)
The representation of the Hamiltonian in terms of annihilation and creation operators âk
and â†k is
Ĥµ,λ
X
λ
λ
εk − µ + δ(r = 0) â†kâk +
=
2
2V
k
X
δk1 +k2 ,k3 +k4 â†k1 â†k2 âk3 âk4 . (4.7)
k1 ,k2 ,k3 ,k4
Normal ordering has resulted in the (divergent) shift in the chemical potential − λ2 δ(r = 0).
The strategy that I will use to study the energy spectrum of the dilute Bose gas is to try
a variational Ansatz for the ground state. This variational state is taken to be the ground
state in the noninteracting limit: Define the Bose-condensate wave function |Φ0 i to be state
(3.10) with
√
Vφ=
17
p
N0 .
(4.8)
The variational Ansatz |Φ0 i is the ground state of Eq. (4.1) with µ = λ = 0. It depends on
a single variational parameter, the expectation value N0 of the number operator â†0 â0 in the
state |Φ0 i. The presence of repulsive interactions is encoded by the possibility that N0 be
smaller than N, i.e., causes a depletion of the Bose condensate in the noninteracting limit.
By construction, N0 /V remains finite in the thermodynamic limit whereas the expectation
value of â†kâk in the state |Φ0 i vanishes for all k 6= 0.
In view of the very special role played by the reciprocal vector k = 0, all contributions
to the Hamiltonian that depend on k = 0 are singled out,
λ † †
λ
δ(r = 0) − µ â†0 â0 +
â â â â
Ĥµ,λ =
2
2V 0 0 0 0
X
λ
εk − µ + δ(r = 0) â†kâk
+
2
k6=0
λ X † †
4âkâ0 âkâ0 + â†+kâ†−kâ0 â0 + â†0 â†0 â+kâ−k
+
2V k6=0
λ X † †
+
â0 âk+k′ âkâk′ + â†k+k′ â†0 âkâk′ + â†kâ†k′ â0 âk+k′ + â†kâ†k′ âk+k′ â0
2V
k,k′ 6=0
X
λ
+
δk1 +k2 ,k3 +k4 â†k1 â†k2 âk3 âk4 .
(4.9)
2V k ,k ,k ,k 6=0
1
2
3
4
Interaction terms have been arranged by decreasing number of a0 or a†0 . Momentum conservation prevents terms linear (cubic) in a0 or a†0 that arise from the kinetic energy (interaction). Only the first line contributes to the expectation value of Ĥ in the variational state
|Φ0 i. The new ground-state energy, to first order in λ/V , is thus
p 2
λ
λ p 4
N0 +
N0 .
δ(r = 0) − µ
(4.10)
2
2V
√
It is permissible to replace any â0 or â†0 by N0 on the subspace spanned by acting with
the creation operators â†k, k 6= 0, on the variational Ansatz |Φ0 i. Hence, on this subspace,
λ
λ 2
Ĥµ,λ →
δ(r = 0) − µ N0 +
N
2
2V 0
X †
X
λ
λ
†
†
†
4âkâk + â+kâ−k + â+kâ−k
N
εk − µ + δ(r = 0) âkâk +
+
2
2V 0 k6=0
k6=0
X †
λ p
+ 2
âk+k′ âkâk′ + â†kâ†k′ âk+k′
N0
2V
k,k′ 6=0
X
λ
(4.11)
δ
↠↠â â .
+
2V k ,k ,k ,k 6=0 k1 +k2 ,k3 +k4 k1 k2 k3 k4
1
2
3
4
18
After absorbing the divergent C-number − λ2 δ(r = 0) into a redefinition
λ
µren := µ − δ(r = 0)
2
(4.12)
of the chemical potential µ, Eq. (4.11) suggests the approximation of truncating the rightp
hand side to the first two leading terms in powers of N0 , i.e., the first two lines, provided
the full Fock space F is restricted to the subspace spanned by the tower of states obtained
from acting on |Φ0 i with a†k 6= 0. Hence, the task of solving for the spectrum of Ĥ in the
Fock space F has been replaced by the simpler problem of solving for the spectrum of Ĥmf
in the Fock space Fmf ,
Ĥµ,λ → Ĥmf :=
F → Fmf
X
k6=0
εk − µren â†kâk − µren N0
!
X †
λ
+
,
N N0 +
4âkâk + â†+kâ†−k + â+kâ−k
2V 0
k6=0
)
(
Y † mk
|Φ0 i, mk ∈ N .
:= span
ak
(4.13)
k6=0
This approximation is called a mean-field approximation. It is useful because it can be
solved exactly: Ĥmf is quadratic in creation and annihilation operators. It should be a
good approximation if N0 is very close to N. The self-consistency of this approximation is
verified once the variational parameter N0 has been expressed in terms of the total number
of bosons, or, equivalently, in terms of the chemical potential.
Observe in the mean-field Hamiltonian (4.13) the pure C-number given by
λ 2
N − µren N0 .
2V 0
(4.14)
A first estimate of the variational parameter N0 follows from minimization of this C-number,
µ
N0
= ren .
V
λ
(4.15)
Insertion of N0 = µrenV /λ into the mean-field Hamiltonian then yields
Ĥmf :=
X
k6=0
V
µ X † †
εk − µren â†kâk + ren
â+kâ−k + â+kâ−k − µ2ren .
2 k6=0
2λ
(4.16)
I will discard the last C-number since I am only interested in the dependence on k of the
excitation spectrum of Ĥmf and in the change in the variational wave function |Φ0 i induced
by the interactions within the mean-field approximation.
19
Diagonalization of Eq. (4.16) on the Fock space Fmf is performed with the help of a
canonical transformation (also called a Bogoliubov transformation in this context)
14
â+k = cosh(θ+k) b̂+k + sinh(θ+k) b̂†−k,
â†+k = sinh(θ+k) b̂−k + cosh(θ+k) b̂†+k,
(4.18)
where (see chapter 2 of [3] or chapter 35 of [4])
εk + µren
cosh(2θk) = q
,
2
2
(εk + µren ) − µren
sinh(2θk) = q
−µren
(εk + µren
)2
−
µ2ren
.
(4.19)
This transformation preserves the boson algebra (hence the terminology canonical),
[âk, â†k′ ] = δk,k′ , [âk, âk′ ] = [â†k, â†k′ ] = 0 ⇐⇒ [b̂k, b̂†k′ ] = δk,k′ , [b̂k, b̂k′ ] = [b̂†k, b̂†k′ ] = 0.
(4.20)
Correspondingly, there exists a unitary transformation Û on the mean-field Fock space such
that b̂k = Û âkÛ −1 ,
Û = exp +
X
k6=0
V 2
µren −
Up to the C-number E0 := − 2λ
has become
Ĥmf =
X
P
ξk b̂†k b̂k,
θk â†kâ†−k − â−kâk
1
k6=0 2
!
.
(4.21)
i
h
εk + µren − ξk , the mean-field Hamiltonian
ξk :=
k6=0
q
(εk + µren )2 − µ2ren .
(4.22)
This is the Hamiltonian of a gas of free bosons with dispersion ξk. For small |k|,
s
r
µren
λN0
|k| =
|k| ≡ v0 |k|.
ξk ≈
m
mV
(4.23)
This is the dispersion relation of sound waves in a fluid that propagate with the speed v0 .
For large |k|, the dispersion crosses over into the usual free-particle expression
k2
ξk ≈
.
2m
14
In matrix form the Bogoliubov transformation reads
!
!
!
!
â+k
cosh θk sinh θk
b̂+k
b̂+k
=
⇐⇒
=
â†−k
sinh θk cosh θk
b̂†−k
b̂†−k
where θk = θ−k .
20
(4.24)
cosh θk − sinh θk
− sinh θk cosh θk
!
â+k
â†−k
!
(4.17)
Having found the mean-field excitation spectrum, we must evaluate the change on the
unperturbed ground state |Φ0 i induced by the Bogoliubov transformation Û . The “rotated”
ground state is the one annihilated by all b̂k, k 6= 0. The state annihilated by all b̂k is
|Φmf i := Û |Φ0 i.
(4.25)
With the mean-field ground state at hand, and recalling that the total number of particle
N is the expectation value of the total particle-number operator Q̂ in the ground state, a
better approximation to the relation
N ≈ hΦ0 | a†0 a0 +
= N0
X
k6=0
!
a†kak |Φ0 i
(4.26)
can be found,
N ≈ hΦmf |
a†0 a0
+
= hΦmf | N0 +
X
a†kak
k6=0
X
k6=0
!
|Φmf i
!
bk b†k sinh2 θ−k |Φmf i
!
1
+1
cosh(2θk) − 1
|Φmf i
= hΦmf | N0 +
2
k6=0


X
1
 q εk + µren
= N0 +
− 1 .
2 k6=0
2
(εk + µren )2 − µren
X
b†kbk
(4.27)
The number N of particles present in |Φmf i exceeds by a fraction of order λ the number
N0 in the single-particle condensate |Φ0 i. Hence, the mean-field approximation is indeed
self-consistent for a dilute hardcore Bose gas.
Had we not chosen N0 by minimization, we would have found that ξk2 would not vanish
anymore in the limit k → 0. Hence, for any other value of N0 than the one in Eq. (4.15),
we could lower the trial energy by either removing or adding particles in the condensate,
i.e., varying the parameter N0 of the trial wave function |Φ0 i.
We close the discussion of this mean-field theory with a word of caution. The main
prediction of this mean-field analysis is the existence of a mean-field gapless spectrum. Is
this prediction robust? This prediction is predicated on the minimization (4.15). As such,
it would be robust if and only if this local mimimum is the global one, as shall become clear
21
when we derive the mean-field approximation from the path-integral formalism. In practice,
such a proof can not be achieved and the “validity” of a mean-field approximation rests on
two verifications, namely that it is self-consistent and that it agrees with experiments.
B.
Landau criterion for superfluidity
I shall assume that the mean-field spectrum that was derived for the dilute Bose gas
is exact and I shall assume that the excitations described by the operators b̂k, phonons in
short, are the only ones in the system. Neither assumptions are realistic but the point made
by Landau is that they are sufficient for the phenomenon of superfluidity to occur.
Consider a body of large mass M moving in the dilute Bose gas (the fluid from now on)
at velocity V . By hypothesis, the only way for the body to experience a retarding force or
drag is for it to emit some phonons. In doing so, energy
1
1
δε := MV 2 − M (V − δV )2 = MV · δV + O[(δV )2 ]
2
2
(4.28)
δk := MV − M(V − δV ) = MδV
(4.29)
and momentum
is lost to the phonons with momenta ki and energies εki :
δε = +
X
εki ,
(4.30a)
ki .
(4.30b)
i
δk = +
X
i
By hypothesis phonons in the model have a finite minimum phase velocity
εk
> 0.
v0 = inf k
|k|
(4.31)
The chain of inequalities
|δε| =
X
i
εki ≥ v0
X
i
|ki | ≥ v0 |
X
i
ki | = v0 |δk|
(4.32)
then follows. To leading order in M −1 , we have established that
|δk||V | ≥ |δk · V | = |δε| + O(M −1 ) ≥ v0 |δk| + O(M −1 )
22
(4.33)
for any permitted δk. Such a δk can only exist if
|V | ≥ v0 ,
(4.34)
i.e., the body must exceed a minimum velocity before experiencing any drag. By moving
sufficiently slowly, a heavy body suffers no loss of energy and momentum from the medium.
This property of the medium is called superfluidity. It originates here from the fact that
the mean-field excitation spectrum is bounded from below by a linear dispersion. In turn,
this is a consequence of the interactions conspiring together with spontaneous symmetry
breaking in the existence of Goldstone modes, acoustic phonons. Interactions are essential
to superfluidity. The excitation spectrum remains quadratic in the noninteracting limit and
the velocity threshold below which a moving body does not suffer drag is v0 = 0. BoseEinstein condensation alone (i.e., without Goldstone modes) is not sufficient for superfluidity
to occur.
C.
Path integral formalism
The partition function for the dilute Bose gas at inverse temperature β and chemical
potential µ is
Z := Tr e−β Ĥµ,λ ,
Z
∆
λ † 2
d
†
Ĥ =
d r ϕ̂ (r) −
ϕ̂ ϕ̂ (r) .
− µ ϕ̂(r) +
2m
2
(4.35a)
(4.35b)
V
The total number of bosons N(β, µ) at inverse temperature β and chemical potential µ is
obtained from
N(β, µ) :=
*Z
dd r ϕ̂† ϕ̂ (r)
V
+
≡ β −1 ∂µ ln Z.
Z
We have seen in lecture 1 that the path integral representation
 β

Z
Z
Z
Z
Z =
D[ϕ∗ , ϕ] exp (−SE ) = D[ϕ∗ , ϕ] exp − dτ dd rLE  ,
0
LE
(4.35c)
V
(4.36a)
∆
λ
λ
= ϕ∗ (r, τ ) ∂τ −
− µ + δ(r = 0) ϕ(r, τ ) + [ϕ∗ (r, τ )]2 [ϕ(r, τ )]2 ,
2m
2
2
(4.36b)
23
of this partition function exists. Integration variables are the real and imaginary parts of the
complex-valued function ϕ(r, τ ) or, equivalently, its complex conjugate ϕ∗ (r, τ ) that obey
periodic boundary conditions in imaginary time τ ,
ϕ∗ (r, τ ) = ϕ∗ (r, τ + β).
ϕ(r, τ ) = ϕ(r, τ + β),
(4.36c)
Boundary conditions in space, say periodic ones, are also present. The total number of
bosons N(β, µ) is now represented by
N(β, µ) = β −1
*Zβ
dτ
0
Z
dd r (ϕ∗ ϕ) (r, τ )
V
+
= β −1 ∂µ ln Z.
(4.36d)
Z
The imposition of periodic boundary conditions in space and time suggests to change integration variable in the path integral representation of the partition function by performing
the Fourier transform
1 XX
ϕ(r, τ ) = √
ak,̟l e+i(kr−̟l τ ) ,
βV k l
ak,̟l
1 XX ∗
ak,̟l e−i(kr−̟l τ ) ,
ϕ (r, τ ) = √
βV k l
a∗k,̟l
∗
1
=√
βV
Zβ Z
dτ dd r ϕ(r, τ ) e−i(kr−̟l τ ) ,
0
V
(4.37a)
1
=√
βV
Zβ Z
dτ dd r ϕ∗ (r, τ ) e+i(kr−̟l τ ) .
0
V
(4.37b)
Here,
̟l =
2π
l,
β
l ∈ Z,
k=
2π
l,
L
l ∈ Zd .
(4.37c)
This change of integration variable turns the path integral representation of the partition
function into
Z
Z =
D[a∗ , a] exp (−SE ) ,
SE =
XX
l
k
λ 1
+
2 βV
(4.38a)
2
k
λ
− µ + δ(r = 0) ak,̟l
2m
2
X
X
δk1 +k2 ,k3 +k4 a∗k1 ,̟l a∗k2 ,̟l ak3 ,̟l ak4 ,̟l .
δl1 +l2 ,l3 +l4
a∗k,̟l
l1 ,l2 ,l3 ,l4
−i̟l +
k1 ,k2 ,k3 ,k4
1
2
3
4
(4.38b)
24
The total number of particles N(β, µ) is represented by
+
*
X X
= β −1 ∂µ ln Z.
a∗k,̟l ak,̟l
N(β, µ) = β −1
l
1.
k
(4.38c)
Z
Noninteracting limit λ = 0
In the noninteracting limit λ = 0, we need to solve the quadratic problem
Z
Z =
D[a∗ , a] exp (−SE ) ,
XX
k2
∗
SE =
ak,̟l −i̟l +
− µ ak,̟l .
2m
l
(4.39a)
(4.39b)
k
The path integral is a multidimensional Gaussian integral, one Gaussian integral of the form
Z
Z
d(x − iy)d(x + iy) −(x−iy)K(x+iy)
dz ∗ dz −z ∗ Kz
e
≡
e
2πi
2πi
Z+∞ Z+∞
1
2
2
=
dx
dy e−K(x +y )
π
−∞
−∞
Z+∞
1
2
= (2π)
drre−Kr
π
0
1
1 −r2 0
e = (2π)
π
2K
+∞
1
=
,
K ∈ R+ ,
K
(4.40)
2
k
− µ to K has a positive real
for each pair (a∗k,̟l , ak,̟l ), provided the counterpart −i̟l + 2m
part, i.e.,
k2
> µ.
2m
(4.41)
This is symbolically written as an inverse determinant (see Appendix A)
Z=
1
Det ∂τ −
∆
2m
−µ
≡
YY
l
k
1
,
k2
−µ
−i̟l + 2m
µ < 0.
(4.42)
The total number of bosons N(β, µ) is thus given by the expression
N(β, µ) = β −1 ∂µ ln Z
XX
= β −1
l
k
1
,
k2
−µ
−i̟l + 2m
25
µ<0
(4.43)
in the noninteracting limit. It is shown in appendix B that the imaginary-time summation
can be written as a contour (Γ) integral in the complex z-plane for any given k,
Z
X
1
fBE (z)
dz
= +β
k2
k2
2πi −z + 2m
−i̟l + 2m − µ
−µ
l
Γ
Z
fBE (z)
dz
= −β
k2
2πi z − 2m
+µ
Γ
Z
fBE (z)
dz
,
µ < 0,
≡ −β
2πi z − εk + µ
(4.44a)
Γ
where fBE (z) is the Bose-Einstein distribution function
1
,
eβz − 1
(4.44b)
k2
.
2m
(4.44c)
fBE (z) :=
and εk is the single-particle dispersion,
εk :=
A second application of the residue theorem (see appendix B) turns the z-integral over the
counterclockwise Γ contour into
Z
dz
fBE (z)
−β
= (−)2 βfBE (εk − µ),
2πi z − εk + µ
µ < 0.
(4.45)
Γ
I conclude that the total number of bosons is given by
N(β, µ) =
X
k
fBE (εk − µ) =
X
k
2
−1
k
exp β
−µ −1
,
2m
µ < 0.
(4.46)
When β is fixed, N(β, µ) is a monotonically increasing function of µ. When µ is fixed,
N(β, µ) is a monotonically decreasing function of β. If the temperature dependence of
µ is determined by fixing the left-hand side of Eq. (4.46) to be some constant number,
say the average total particle number N in the grand canonical ensemble, then µ(β) is a
monotonically increasing function of β. However, µ(β) is necessarily bounded from above
by
µc := inf εk = 0,
k
26
(4.47)
for, if it was not, there would be an inverse critical temperature βc above which µ(β) > µc
and the integral over the so-called zero mode 15 (a∗k=0,̟
Z
l
=0 , ak=0,̟l =0 ),
da∗0,0 da0,0
exp +|µ(β)|a∗0,0a0,0 ,
2πi
(4.50)
would diverge in contradiction with the assumption that there exists a well defined vacuum
|0i. The alternative scenario by which µ(β) is pinned to µc above βc is actually what
transpires from a numerical solution of Eq. (4.46) which now reads

P



fBE (εk − µ),
if β < βc .


k


N=



P

−1
2


fBE (εk − µc ), if β ≥ βc .
β |a0,0 | +
(4.51)
k6=0
Equation (4.51) determines the chemical potential as a function of the inverse temperature.
It also determines the macroscopic number of bosons β −1 |a0,0 |2 (β) that occupy the lowest
single-particle energy ε0 above the critical inverse temperature βc . At zero temperature
limβ→∞ fBE (εk − µc ) = 0, k 6= 0, and all N bosons occupy the single-particle ground-state
energy ε0 . Before tackling the interacting case, it is important to realize that Bose-Einstein
condensation did not alter the single-particle dispersion εk. According to the Landau criterion, superfluidity cannot take place in the noninteracting limit. We shall see below how
the excitation spectrum becomes linear at long wave lengths due to a conspiracy between
spontaneous-symmetry breaking and interactions, thus enabling superfluidity.
Before leaving the noninteracting limit, I want to introduce the single-particle Green
function
Gk,̟l := −
15
1
.
k2
−µ
−i̟l + 2m
(4.52)
A zero mode is a configuration ϕ(r, τ ) that does not depend on space or time:
ϕ(r, τ ) = ϕ0 .
(4.48)
The only nonvanishing Fourier component of ϕ0 is
ak=0,̟l =0 =
27
p
βV ϕ0 .
(4.49)
The sign is convention. The Green function (4.52) is, up to a sign, the inverse of the Kernel
in Eq. (4.39b). Furthermore, because of the identities (4.40) and
Z
Z
dz ∗ dz ∗
dz ∗ dz −z ∗ Kz+J ∗ z+Jz ∗ ∂2
−z ∗ Kz
(z z) e
=
e
∗
2πi
∂J ∗ ∂J
2πi
J =J=0
Z
∂2
dz ∗ dz −(z− KJ )∗ K (z− KJ )∗ +J ∗ K1 J =
e
∗
∂J ∗ ∂J
2πi
J =J=0
1
J
2 +J ∗ K
∂ e
Eq. (4.40) = (1/K)
∗
∂J ∂J ∗
J =J=0
(1/K)
,
=
K
K ∈ R+ ,
(4.53)
the Green function (4.52) is the covariance or two-point function
E
D
Gk,̟l := − a∗k,̟l ak,̟l
Z
R
∗
−SE
D[a , a] e
a∗k,̟ ak,̟
l
l
≡ −R
.
−SE
∗
D[a , a] e
2.
(4.54)
Random-phase approximation
The first change relative to the analysis of the noninteracting limit that is brought by
switching a repulsive contact interaction, λ > 0, is the breakdown of the stability argument
that leads to the pinning of the chemical potential. To see this, consider as in Eq. (4.50)
the action of the zero mode
ϕ(r, τ ) := ϕ0 , ∀r, τ =⇒
(0)
SE [ϕ∗0 , ϕ0 ]
λ
4
2
:= βV −µren |ϕ0 | + |ϕ0 | ,
2
(4.55)
where now λ > 0 implies that the renormalized chemical potential
λ
µren := µ − δ(r = 0)
2
(4.56)
can become arbitrarily large as a function of inverse temperature without endangering the
convergence of the contribution
Z0 :=
Z
dϕ∗0 dϕ0 −SE(0) [ϕ∗0 ,ϕ0 ]
e
2πi
(4.57)
from the zero modes to the partition function.
An estimate of Eq. (4.57) that becomes exact in the limit of β, V → ∞ is obtained from
the saddle-point approximation. In the saddle-point approximation, the modulus |ϕ0 (µren)|2
28
is given by the solution [compare with Eq. (4.15)]



0,
if µren < 0,



|ϕ0 (µren )|2 =




 µren , if µ ≥ 0,
ren
λ
(4.58)
to the classical equation of motion
(0)
δSE
,
δ|ϕ0 |2
0=
(4.59)
and
Z0 ≈



1,



if µren < 0.




exp +βV
(4.60)
µ2ren
2λ
, if µren ≥ 0.
In turn, the dependence on β of the renormalized chemical potential is determined by demanding that
N = V |ϕ0 (µren )|2 + β −1
*Zβ
dτ
0
Z
V
d d r (ϕ
e∗ ϕ)
e (r, τ )
+
.
(4.61)
Z/Z0
The tilde over ϕ
e∗ (r, τ ) and ϕ(r,
e τ ) as well as the subscript Z/Z0 are reminders that zero
modes should be removed from the path integral in the second term on the right-hand side
[compare with Eq. (4.36d)], as they would be counted twice when µren ≥ 0 otherwise.
The strategy that I will pursue to go beyond the zero-mode approximation consists in:
• Step 1, assuming that Eq. (4.58) holds with some µren > 0.
• Step 2, choosing a convenient parametrization of the fluctuations ϕ
e∗ (r, τ ) and ϕ(r,
e τ)
about the zero modes.
• Step 3, constructing an effective theory in ϕ
e∗ (r, τ ) and ϕ(r,
e τ ) to the desired accuracy.
• Step 4, solving Eq. (4.61) with the effective theory of step 3 and verify the selfconsistency of step 1 within the accuracy of the approximation made in step 3.
29
This approximate scheme is called the random-phase approximation (RPA) when the effective theory in step 3 is noninteracting. It is nothing but an expansion of the action up to
quadratic order in the fluctuations about the saddle-point or mean-field solution.
The zero-mode approximation in Eq. (4.58) leaves the choice of the phase of the zero mode
ϕ0 arbitrary. This is the classical implementation of the spontaneous-symmetry breaking
of the U(1) symmetry associated with total particle-number conservation. Without loss of
generality, the (linear ) parametrization that I choose is
r
r
µren
µren
∗
∗
ϕ (r, τ ) =
+ϕ
e (r, τ ),
ϕ(r, τ ) =
+ ϕ(r,
e τ ).
λ
λ
(4.62)
Here, I am also assuming that
0 = hϕ
e∗ (r, τ )iZ/Z0 = hϕ(r,
e τ )iZ/Z0 .
(4.63)
This parametrization is the natural one if the approximation to the action is meant to
linearize equations of motion. Correspondingly, the action is expanded up to second order
in the deviations ϕ
e∗ (r, τ ) and ϕ(r,
e τ ) from the saddle point or mean-field Ansatz (4.58)
p
p
(0)
SE [ϕ∗ , ϕ] ≈ SE [ µren /λ, µren /λ]
Zβ
Z
∆
µren ∗
2
d
∗
dτ
d r ϕ
e ∂τ −
+
ϕ
e+
(ϕ
e + ϕ)
e (r, τ )
2m
2
0
V
+ ···
p
p
(0)
= SE [ µren /λ, µren /λ]
Zβ
Z
∆
d
e (r, τ )
+ 2µren (Re ϕ)
dτ
+
d r (Re ϕ)
e −
2m
+
+
0
V
Zβ
Z
dτ
0
V
Zβ
Z
dτ
0
+ ··· .
V
∆
d r (Im ϕ)
e −
2m
d
(Im ϕ)
e (r, τ )
dd r [(Re ϕ)i∂
e τ (Im ϕ)
e − (Im ϕ)i∂
e τ (Re ϕ)]
e (r, τ )
(4.64)
To reach the second equality, partial integrations were performed and all space or time total
derivatives were dropped owing to the periodic boundary conditions in space and time.
A quite remarkable phenomenon is displayed in Eq. (4.64). A purely imaginary term has
appeared in the effective action. Hence, this effective action cannot be interpreted as some
30
classical action. The pure imaginary term is an example of a Berry phase. Upon canonical
be
be
quantization, the commutator between Re ϕ(r)
and Im ϕ(r)
is the same as that between
the position and momentum operators, respectively, in quantum mechanics (see lecture 1).
If we ignore for one instant the Berry phase term, i.e., ignore quantum mechanics, we can
interpret Re ϕ(r,
e τ ) as a massive mode and Im ϕ(r,
e τ ) as a massless mode. The massive
mode Re ϕ(r,
e τ ) originates from the radial motion of a classical particle which, at rest, is
sitting somewhere at the bottom of the circular potential well
(0)
U0 [ϕ∗0 , ϕ0 ]
S [ϕ∗ , ϕ ]
λ
:= E 0 0 = −µren |ϕ0 |2 + |ϕ0 |4 .
βV
2
(4.65)
The massless mode Im ϕ(r,
e τ ) originates from the angular motion of this particle along
the bottom of the circular potential well U0 [ϕ∗0 , ϕ0 ]. At the classical level, the dispersions
above the gap thresholds 2µren and 0 of Re ϕ(r,
e τ ) and Im ϕ(r,
e τ ), respectively, are both
quadratic in the momentum k. Including quantum fluctuations through the Berry phase
dramatically alters this picture. Of course, these “two classical modes” are not independent
since they interact through the Berry phase terms. As we now show, including the Berry
phase couplings allows us to interpret ϕ
e as a mode with a linear (quadratic) dispersion
relation at long (short) wave lengths.
The explicit dispersion can be obtained from Fourier transforming Eq. (4.64) into



k2 + 2µ
X
X
e +k,+̟
ren +̟l  (Re ϕ)
(0)
2m
l 
SE ≈ SE +
(Re ϕ)
e −k,−̟ (Im ϕ)
e −k,−̟ 
2
k
l
l
−̟l
(Im ϕ)
e +k,+̟
l∈Z k∈Zd
2m
l



† 
k2
X X (Re ϕ)
+ 2µren +̟l
(Re ϕ)
e k,̟l
e k,̟l
(0)
2m



 
= SE + 2
2
k
−̟
(Im
ϕ)
e
(Im
ϕ)
e
d
l∈Z k∈N
l
k,̟l
k,̟l
2m



† 
k2
X
X
+
2µ
−̟
(Re
ϕ)
e
(Re
ϕ)
e
ren
l
−k,−̟l 
−k,−̟l   2m
(0)

= SE + 2
.
(4.66)
k2
+̟l
(Im
ϕ)
e
(Im ϕ)
e −k,−̟l
l∈Z k∈Nd
−k,−̟
2m
l
To reach the second line, we have used the fact that the dispersion is even under k → −k,
where the notation k ∈ Zd , or the restriction k ∈ Nd , is a slight abuse of notation, while
the real and imaginary parts of ϕ(r, τ ) are real-valued functions. To reach the third line,
we made the relabeling k → −k and ̟l → −̟l , under which the ̟l -dependence of the
kernel is odd while the k-dependence of the kernel is even.
31
Define the 2 × 2 matrix-valued Green function by its matrix elements in the k-̟l basis,
†
*
+
(Re ϕ)
e −k,−̟
l 
G−k,−̟l := − 
.
(4.67)
(Re ϕ)
e −k,−̟ (Im ϕ)
e −k,−̟
l
l
(Im ϕ)
e −k,−̟l
Z/Z0
The − sign is convention. Evaluation of the Green function (4.67) is an exercise in Gaussian
integrations that is summarized in appendix A. Hence,

−1
k2
+ 2µren −̟l
1

G−k,−̟l = −  2m
k2
2
+̟l
2m


k2
+̟
1
1
l
,
 2m
= − k2 k2
2
2
k
2 2m 2m + 2µren + (̟l )
+ 2µren
−̟l
2m
2π
k ∈ Nd ,
L
l ∈ Z.
(4.68)
The factor 1/2 comes from the fact that only k ∈ Nd are to be counted as independent
momenta, for (Re ϕ)(r,
e
τ ) and (Im ϕ)(r,
e
τ ) are real valued.
The Green function (4.68) has first-order poles whenever
s
k2 k2
i̟l = ±
+ 2µren
2m 2m
|k| p 2
k + 4mµren
= ±
2m
|k| p 2
k + (k0 )2
≡ ±
2m
≡ ±ξk
(4.69)
and we recover with ξk the dispersion in Eq. (4.22). For long wave lengths, the dispersion
is linear
ξk =
k0
|k| + O
2m
"
|k|
k0
2 #
,
(4.70a)
with the sound velocity
k0
=
2m
r
µren
.
m
For short wave lengths, the free particle dispersion relation emerges,
" #
2
k2
k0
ξk =
.
+O
2m
|k|
32
(4.70b)
(4.71)
It is time to verify the selfconsistency of the assumptions encoded by Eqs. (4.58), (4.62),
and (4.64). This we do by solving Eq. (4.61) in the Gaussian approximation. Fourier
transform of Eq. (4.61) yields
N = V (ϕ0 )2 + δN,
(4.72a)
where
δN := β −1
*Zβ
Z
dτ
0
= β −1
V
d d r (ϕ
e∗ ϕ)
e (r, τ )
X X D
l∈Z k∈Zd
+
Z/Z
0
E
Im ϕ
e−k,−̟l
e+k,+̟l
Re ϕ
e−k,−̟l + Im ϕ
Re ϕ
e+k,+̟l
Z/Z0
E
X X D
Re ϕ
e−k,−̟l
e+k,+̟l
Im ϕ
e−k,−̟l − Im ϕ
+iβ −1
Re ϕ
e+k,+̟l
l∈Z k∈Zd
Z/Z0
.
(4.72b)
The Gaussian approximation estimates
δN ≈ β
−1
X X ei0+ ̟l k2
+ µren + i̟l ,
2
2
̟
+
ξ
2m
l
k
d
l∈Z
(4.73)
k∈Z
as can be read from the Green function (4.68). A convergence factor exp(i0+ ̟l ) was introduced to regulate the poles of the Green function (4.68). One verifies that the summand
with k and l fixed, while µ = 0 in Eq. (4.43) is recovered in the noninteracting limit µren = 0.
+∞
R
P
At zero temperature, the summation over l turns into an integral l∈Z → β
d̟/(2π).
−∞
The integrand is nothing but nk. As a function of ̟ ∈ C, it has two first-order poles
2
k
+ µren ∓ ξk . The convergence
along the imaginary axis at ±iξk with residues ± i2ξ1 2m
k
+
factor exp(i0 ̟) entitles us to close the real-line integral by a very large circle in the upper
complex plane ̟ ∈ C and application of the residue theorems yields
2
1
k
nk ≈
+ µren − ξk
2ξk 2m
!
k2 + (k0 )2 /2
1
p
−1 .
=
2 |k| k2 + (k0 )2
(4.74)
In the thermodynamic limit,
V
−1
X
k6=0
d
nk ≈ γd (k0 ) ,
Ωd
γd :=
2(2π)d
Z∞
0
33
d−2
dxx
x2 + 1/2
√
−x .
x2 + 1
(4.75)
√
Here, Ωd is the area of the unit sphere in d-dimensions. Since k0 = 2 mλϕ0 , I conclude
that ϕ0 is determined by
N
V
= (ϕ0 )2 + 2d γd (mλ)d/2 (ϕ0 )d
= (ϕ0 )2 1 + 2d γd (mλ)d/2 (ϕ0 )d−2 .
n :=
(4.76)
The Gaussian approximation is selfconsistent if the quantum correction 2d γd (mλ)d/2 (ϕ0 )d
is smaller than the semiclassical result (ϕ0 )2 , i.e., if
ϕ0 ∼
√
n
−1/(d−2)
≪ 2d γd (mλ)d/2
.
(4.77)
The constant γd is finite if and only if 1 < d < 4. For d = 1, γ1 has an infrared logarithmic
divergence. For d = 4, γ4 has an ultraviolet logarithmic divergence. When Eq. (4.77) is
satisfied,
(ϕ0 )2 ≈ n − 2d γd (mλ)d/2 nd/2 .
(4.78)
The RPA (Gaussian approximation) in d = 3 is thus appropriate in the dilute limit or
when the interacting coupling constant λ is small. For d = 2, Eq. (4.78) indicates that
quantum corrections to the semi-classical result scale in the same way as a function of the
particle density n, but with the opposite sign. This is an indication that fluctuations are
very important in two-space dimensions and that the RPA might then break down.
Two dimensions is indeed very special. To see this one can estimate the size of the
fluctuations about the semiclassical value of the order parameter by calculating the rootmean-square deviation
s
h|ϕ0 |2 − ϕ∗ (r, τ )ϕ(r, τ )iZ
|ϕ0 |2
(4.79)
within the RPA. The root-mean-square deviation should be smaller than one in the thermodynamic limit if there is true long-range order, i.e., below the transition temperature. It
should diverge upon approaching the transition temperature from below. To evaluate
hϕ∗ (r, τ )ϕ(r, τ )iZ
34
(4.80)
at inverse temperature β one must perform a Fourier integral over the entries of the Green
function (4.68). These integrals are dominated at long wave lengths by the contribution
Z d
d k
−1
β
(4.81)
k2
coming from the acoustic mode. This contribution is logarithmically (linearly) divergent in
d = 2 (d = 1) whenever β < ∞. Hence, the root-mean-square deviation diverges in the thermodynamic limit and within the RPA, signaling the breakdown of spontaneous-symmetry
breaking and off-diagonal long-range order at any finite temperature when d ≤ 2. It is
said that d = 2 is the lower-critical dimension at which and below which the U(1) continuous symmetry cannot be spontaneously broken at finite temperatures within the RPA.
Absence of spontaneous-symmetry breaking of the U(1) symmetry in the dilute Bose gas
within the RPA is an example of the Mermin-Wagner-Hohenberg-Coleman theorem. It can
be shown that thermal fluctuations due to acoustic modes downgrade the long-range order of the ground state to quasi long-range-order within the RPA. Quasi long-range-order
is the property that the one-particle density matrix in Eq. (3.17) decays algebraically fast
with |r − r ′ | at long separations. Quasi long-range order cannot maintain itself at arbitrary
high temperatures. The mechanism by which quasi long-range order is traded for exponentially fast decaying spatial correlations is called the Kosterlitz-Thouless transition. The
Kosterlitz-Thouless transition cannot be accounted for within the RPA since it is intrinsically
a nonlinear phenomenon. I will devote a lecture to the Kosterlitz-Thouless transition.
3.
Beyond the random-phase approximation
I would like to close the discussion of a repulsive dilute Bose gas by sketching how one
can go beyond the RPA defined by Eq. (4.64). The key to capturing physics beyond the
RPA (4.64) is to choose the parametrization
ϕ∗ (r, τ ) =
p
ρ(r, τ ) e−iθ(r,τ ) ,
ϕ(r, τ ) =
p
ρ(r, τ ) e+iθ(r,τ ) ,
(4.82)
of the fields entering the path integral (4.36). This parametrization is nonlinear. It reduces to the linear parametrization (4.62) if one works to linear order in θ and makes the
identifications
ρ(r, τ ) →
µren
,
λ
iρ(r, τ )θ(r, τ ) → ϕ(r,
e τ ).
35
(4.83)
If we were able to solve the repulsive dilute Bose gas model exactly, the choice of parametrization of the fields in the path integral (4.36) would not matter. However, performing approximations, say linearization of the equations of motion, can lead to very different physics
depending on the initial choice of parametrization of the fields in the path integral. For
example, the RPA on the linear parametrization (4.62) breaks down in d = 1, 2 due to the
dominant role played by infrared fluctuations of the Goldstone mode. On the other hand, I
shall argue that the nonlinear parametrization (4.82) can account for these strong fluctuations. The physical content of Eq. (4.82) is to parametrize the fields in the path integral of
Eq. (4.36) in terms of the density
ρ(r, τ ) := ϕ∗ (r, τ )ϕ(r, τ )
(4.84)
and the currents
1
[ϕ∗ (r, τ ) (∇ϕ) (r, τ ) − (∇ϕ∗ ) (r, τ )ϕ(r, τ )]
2mi
1
ρ(r, τ )(∇θ)(r, τ )
=
m
J (r, τ ) :=
(4.85)
associated to the global U(1) gauge invariance of the theory.
I begin by inserting Eq. (4.82) into the Lagrangian in Eq. (4.36)
1
1
λ
2
2
LE = iρ∂τ θ +
(∇ρ) + ρ(∇θ) − µren ρ + ρ2
2m 4ρ
2
1
1 2
+ (∂τ ρ) − (∇ ρ) − i∇ [ρ(∇θ)] .
2
2
(4.86)
The second line does not contribute to the action since fields obey periodic boundary conditions in imaginary time and in space.
Next, I expand the first line to quadratic order in powers of δρ, to zero-th order in powers
of (∇δρ)2 , and to zero-th order in powers of (δρ)(∇θ)2 , where
ρ(r, τ ) = ρ0 + δρ(r, τ ),
µ
ρ0 := ren ,
λ
Zβ
0
dτ
Z
dd r δρ(r, τ ) = 0.
(4.87)
V
This expansion is a good one at low temperatures when the renormalized chemical potential
is strictly positive. In particular, note that, to the contrary of the RPA (4.64), we are not
36
assuming that θ is small (only ∇θ is taken small). I find the quadratic action
SE
ρ0
λ
2
2
=
+ dτ d r i(δρ)∂τ θ +
(∇θ) + (δρ) − (δµ)(δρ) + · · ·
2m
2
0
V

† 


̟l
λ
X
X
δρ
+ 2
δρ
(0)
 +k,+̟l   2
  +k,+̟l 
= SE + 2
̟l
ρ0 2
θ+k,+̟
− 2 2m k
θ+k,+̟
l k∈N
l
l
XX
δµ−k,−̟l δρ+k,+̟l + · · · ,
− 2
(0)
SE
Zβ
l
k∈N
Z
d
(4.88)
and the currents
J (r, τ ) =
ρ0
(∇θ)(r, τ ) + · · · .
m
(4.89)
Here, I have also substituted µren by µren + δµ(r, τ ).
The source term δµ(r, τ ),
Rβ R d
dτ d r (δµ)(r, τ ) = 0, is a mathematical device to probe the response to an external
0
V
“scalar” potential.
The first term on the right-hand side of the first line of Eq. (4.88) is the Berry phase
that converts the radial (δρ) and angular (θ) semi-classical modes into a quantum harmonic
oscillator. It implies that δρ and θ are coupled through the classical equations of motion
0 = +i∂τ θ + λδρ − δµ ≡ +i∂τ θ − δµeff ,
ρ
0 = −i∂τ δρ − 0 ∆θ ≡ −i∂τ δρ + ∇ · J ,
m
(4.90)
in imaginary time. These equations are called the Josephson equations. We will devote two
lectures to study them in the context of superconductivity.
The second term on the right-hand side of the first line of Eq. (4.88) is only present
below the U(1) symmetry-breaking transition temperature. It endows the angular degree
of freedom θ with a rigidity since, classically,
ρ0
(∇θ)2
2m
is the penalty in elastic energy paid
by a gradient of the phase. Alternatively, this term corresponds to the kinetic energy of a
“point-like particle” of mass m/ρ0 .
The third term on the right-hand side of the first line of Eq. (4.88) represents the potentialenergy cost induced by the curvature of the semi-classical potential well (4.65) if the “pointlike particle” moves by the amount δρ away from the bottom of the well.
Poles in the counterpart
Gcol
k,̟l = −
1/2
1 λρ0 2
k
4 m
+
37
̟l
2
2

ρ0 2
k
 2m
+̟l
−̟l
λ
2


(4.91)
for δρ and θ to the Green function (4.68) give the linear dispersion
r
r
λρ0
µren
col
ξk =
|k| =
|k|.
m
m
(4.92)
There is a one-to-one correspondence between the existence of a linear dispersion relation
and the existence of a rigidity ∝ ρ0 in our effective model. If we interpret the rigidity
of the phase θ as superfluidity, we have establish the one-to-one correspondence between
superfluidity and the existence of a Goldstone mode associated with spontaneous symmetry
breaking. In turn, this Goldstone mode can only exist when λ > 0, i.e., Bose-Einstein
condensation at λ = 0 cannot produce superfluidity.
The Green functions (4.68) and (4.91), although very similar, have a very different physical content. In the former case, the Green function describes single-particle properties. In
the latter case, the Green function describes collective excitations (δρ and θ). For example,
the equal-time density-density correlation function Sk is given by
Sk :=
=
m
(ξkcol )2 + (̟l )2
Nβ
l
ρ0
|k|2
2m
ξkcol
2
V
+ O(β −1 )
N
|k|
≈
+ O(β −1 ).
2mξkcol
=
ρ0 ≈ N
V
V 1 X
δρ+l,+kδρ−l,−k
Nβ l
ρ0
|k|2
V 1X
(4.93)
The so-called Feynman relation
ξkcol
|k|2
,
≈
2mSk
(4.94)
which is valid at zero temperature, implies that the long-range correlation
Sk ∝ |k|
(4.95)
is equivalent to an acoustic wave dispersion for density fluctuations. Feynman relation can
be used to establish the existence of superfluidity in d = 2 when the criterion built on
the RPA (4.64) fails. However, effective theory (4.88) is still too crude for a description of
superfluidity in d = 2.
38
APPENDIX A: SOME GAUSSIAN INTEGRALS
Path integrals are generalizations of multidimensional Riemann integrals. Integrands of
path integrals for noninteracting bosons are exponential of quadratic forms. Hence, their
evaluations require generalizations of the Gaussian integrals
Z
dz ∗ dz −z ∗ az+j ∗ z+jz ∗
∗
Za (j , j) :=
e
2πi
Z
dz ∗ dz −z ∗ az+j ∗ z+jz ∗−j ∗ a−1 j+j ∗a−1 j
=
e
2πi
Z
dz ∗ dz −(z−a−1 j)∗ a(z−a−1 j)
+j ∗ a−1 j
= e
e
2πi
Z
dz ∗ dz −z ∗ az
+j ∗ a−1 j
= e
e
2πi
∗ −1 Z
dz ∗ dz −z ∗ z
e+j a j
e
=
a
2πi
∞
∗ −1 Z
2π −r2
e+j a j
drr
e
=
a
π
0
+j ∗ a−1 j
=
hz ∗ zia :=
e
a
Z
,
dz ∗ dz ∗ −z ∗ az
z ze
2πi
,
Za (j ∗ , j)|j∗=j=0
1
∂ 2 Za (j ∗ , j) =
Za (j ∗ , j) ∂j∂j ∗ j∗=j=0
1
∀a ∈ C, Re a > 0,
= ,
a
(A1)
to path integrals. The function Za (j ∗ , j) is called a generating function. From it all moments
of the form
1
∂ 2n Za (j ∗ , j) h(z z) ia :=
,
Za (j ∗ , j) ∂j n ∂j ∗n j∗=j=0
∗
n
n ∈ N,
(A2)
can be calculated. Generalization of Eqs. (A1) and (A2) to N-dimensional Riemann integrals
is straightforward. Replace the complex conjugate pair z ∗ and z by N-dimensional vectors
z † and z, respectively. Replace the complex number a with a strictly positive real part by
the N × N positive definite Hermitean matrix A. Define the generating functional
Z N † N
d z d z −z† Az+j† ·z+z† ·j
†
ZA (j , j) :=
e
,
(2πi)N
39
(A3)
from which all moments
n
2
†
Y
1
∂
Z
(j
,
j)
A
∗
h
zm
zm iA :=
† , j)
∗
Z
(j
∂j
∂j
m
m A
m=1
m=1
n
Y
,
j∗=j=0
n ∈ N,
(A4)
can be calculated. Since the measure of the generating functional is invariant under any
unitary transformation of CN , we can choose a basis of CN that diagonalizes the positive
definite Hermitean matrix A, in which case Eq. (A1) can be used for each independent
integration over the N normal modes. Thus,
†
ZA (j † , j) =
−1
e+j A j
,
det A
∗
hzm
zn iA = A−1
mn
,
m, n = 1, · · · , N.
(A5)
Imposing periodic boundary conditions for continuous systems results in having a countable
infinity of normal modes. In this case Eq. (A5) is generalized by replacing A, whose determinant is made of a finite product of eigenvalues, by a kernel, whose determinant is made of a
countable product of eigenvalues. For infinite dimensional vector spaces, I use the notation
Det · · · for the determinant of the kernel · · · . After taking the thermodynamic limit, the
number of normal modes is uncountable. The logarithm of the determinant of the kernel
becomes an integral instead of a sum.
APPENDIX B: BOSE-EINSTEIN DISTRIBUTION AND THE RESIDUE THEOREM
The Bose-Einstein distribution
fBE (z) :=
eβz
1
−1
(B1)
is analytic in the complex plane except for the equidistant first-order poles
zl =
2πi
l,
β
l ∈ Z,
(B2)
on the imaginary axis. Each pole zl = (2πi/β)l of fBE (z) has the residue 1/β since
exp [β(zl + z)] − 1 = βz + O(z 2 ),
Let g(z) be a complex function such that:
40
∀l ∈ Z.
(B3)
• g(z) decreases sufficiently fast at infinity,
lim |z|fBE (z)g(z) = 0.
|z|→∞
(B4)
• g(z) is analytic everywhere in the complex planes except for two poles on the real axis
away from the origin, say at z = ±x 6= 0.
Let Γ be a closed path infinitesimally close to the imaginary axis and running antiparallel
(parallel) to the imaginary axis when Re z < 0 (Re z > 0). Let ∂U±x be circular paths
running clockwise and centered about ±x. Then, path Γ can be deformed into path ∂U−x ∪
∂U+x by Cauchy theorem, and the residue theorem yields
Z
X
dz
g(zl ) = +β
fBE (z)g(z)
2πi
l
Γ


Z
Z

 dz
= +β 
+
f (z)g(z)

2πi BE
∂U−x
∂U+x
= −β [Res (fBE g)(−x) + Res (fBE g)(+x)] .
(B5)
[1] See chapter 5 from “Matrices: Theory and Applications,” by D. Serre, Springer (New-York
2002). See also chapter A4.1 from “Quantum Field Theory and Critical Phenomena,” by
J. Zinn-Justin, (Oxford University Press, New-York, 1996).
[2] J. Glimm and A. Jaffe, Quantum physics, (Springer Verlag, New York, 1987).
[3] C. Kittel, Quatum theory of solids, (John Wiley, New York, 1963).
[4] A. L. Fetter and J.D. Walecka, Quatum theory of many-particle systems, (mcGraw-Hill, New
York, 1971).
41