Introduction to superfluidity and superconductivity

 
 

Министерство образования и науки Российской Федерации 
Федеральное государственное бюджетное образовательное 

учреждение высшего образования 

«Московский педагогический государственный университет» 
 

 
 
 
 
С. А. Рябчун 
 
 
INTRODUCTION TO SUPERFLUIDITY 
AND SUPERCONDUCTIVITY 
 
 
Учебное пособие 
 
 
 
 
 
 
 
 
 
 
 
 
 
МПГУ 
Москва • 2017 
 

УДК 538.945 
ББК 22.268.3 

Р985 
Рецензенты: 
Г. М. Чулкова, доктор физико-математических наук, профессор 
кафедры общей и экспериментальной физики факультета физики 
и информационных технологий МПГУ 
В. А. Ильин, доктор физико-математических наук, профессор кафедры 
общей и экспериментальной физики факультета физики 
и информационных технологий МПГУ 

Рябчун, Сергей Александрович. 
Р985  Introduction to superfluidity and superconductivity : учебное 
пособие / С. А. Рябчун. – Москва : МПГУ, 2017. – 72 с.  

ISBN 978-5-4263-0572-4 
These notes have appeared as a result of a one-term course in 
superfluidity and superconductivity given by the author to fourth-year 
undergraduate students and first-year graduate students of the Department of 
Physics, Moscow State University of Education. The goal was not to give a 
detailed picture of these two macroscopic quantum phenomena with an 
extensive coverage of the experimental background and all the modern 
developments, but rather to show how the knowledge of undergraduate 
quantum mechanics and statistical physics could be used to discuss the basic 
concepts and simple problems, and draw parallels between superconductivity 
and superfluidity. 

УДК 538.945 
ББК 22.268.3  

ISBN 978-5-4263-0572-4 
© МПГУ, 2017  
© Рябчун С. А., 2017 

CONTENTS 
 
Preface ................................................................................................... 5 
1. Second quantisation ........................................................................... 7 

1.1. General remarks .......................................................................... 7 
1.2. Second quantisation for fermions ............................................... 8 
1.3. Second quantisation for bosons ................................................ 10 
1.4. Operators in second-quantised form ......................................... 11 
Exercises ........................................................................................... 16 

2. The uniform weakly interacting Bose gas ....................................... 17 

2.1. The ideal Bose gas .................................................................... 17 
2.2. Elements of scattering theory.................................................... 21 
2.3. Costructing the Hamiltonian ..................................................... 24 
2.4. Bogoliubov transformations...................................................... 25 
2.5. The two-fluid picture ................................................................ 27 
Exercises ........................................................................................... 29 

3. The non-uniform weakly interacting Bose gas ............................... 31 

3.1. The time-independent Gross-Pitaevskii equation ..................... 31 
3.2. The coherence length ................................................................ 33 
3.3. The time-dependent Gross-Pitaevskii equation ........................ 33 
3.4. Bogoliubov equations ............................................................... 35 
3.5. Vortex states .............................................................................. 36 
Exercises ........................................................................................... 40 

4. The uniform superconducting state ................................................. 42 

4.1. The ideal Fermi gas ................................................................... 42 
4.2. Constructing the Hamiltonian ................................................... 43 
4.3. Excitations and the energy gap ................................................. 47 
4.4. The critical temperature ............................................................ 49 
4.5. The discontinuity of the electronic heat capacity ..................... 50 
4.6. The Meissner effect ................................................................... 51 
4.7. Tunnelling ................................................................................. 54 

4.7.1. A simple problem ................................................................ 54 
4.7.2. NN-contact.......................................................................... 55 
4.7.3. NS-contact .......................................................................... 55 

Exercises ........................................................................................... 57 

5. The non-uniform superconducting state near TC ............................ 58 

5.1. Thermodynamics of superconductors ....................................... 58 
5.2. The Ginzburg-Landau equations ............................................... 59 
5.3. Characteristic lengths of the Ginzburg-Landau theory............. 61 
5.4. Flux quantisation ....................................................................... 62 
5.5. Energy of the SN interface ........................................................ 63 
5.6. Critical field of a thin film ........................................................ 65 
5.7. Critical current of a thin film .................................................... 66 
5.8. Vortex solutions ......................................................................... 67 
Exercises ........................................................................................... 71

PREFACE 

These notes have appeared as a result of a one-term course in superfluidity 

and superconductivity given by the author to fourth-year undergraduate students 
and first-year graduate students of the Department of Physics, Moscow State 
University of Education. The goal was not to give a detailed picture of these two 
macroscopic quantum phenomena with an extensive coverage of the experimental 
background and all the modern developments, but rather to show how the 
knowledge of undergraduate quantum mechanics and statistical physics could be 
used to discuss the basic concepts and simple problems, and draw parallels 
between superconductivity and superfluidity. 

Superconductivity and superfluidity are two phenomena where quantum 

mechanics, typically constrained to the microscopic realm, shows itself on the 
macroscopic level. Conceptually and mathematically, these phenomena are 
related very closely, and some results obtained for one can, with a few 
modifications, be immediately carried over to the other. However, the student of 
these notes should be aware of important differences between superconductivity 
and superfluidity that stem mainly from two facts: (1) electrons in a 
superconductor carry a charge, therefore one has to take into account interaction 
with electromagnetic radiation; (2) electrons move in a lattice, therefore phonons 
play a role not only a mediators of attractive interaction between pairs of 
electrons, but also as scatterers of charge carriers. 

Although these are notes on superfluidity and superconductivity, and there 

are a few cross-references, the two subjects can be studied independently with, 
perhaps, a little extra work by the student to fill in the gaps resulting from such 
study. The material of Chapter 1 introduces the method of second quantisation  
that is commonly used to discuss systems with many interacting particles. It is 
then applied in Chaper 2 to treat the uniform weakly interacting Bose gas within 
the approach by N. Bogoliubov, and in Chapter 4 to formulate the theory of the 
uniform superconducting state put forth by J. Bardeen, L. Cooper and R. 
Schrieffer. Chapter 3 presents the theory proposed independently by E. Gross and 
L. Pitaevskii of a non-uniform weakly interacting Bose gas, with a discussion of 
vortices, rotation of the condensate, and the Bogoliubov equations. In Chapter 5 
we discuss the Ginzburd-Landau theory of a non-uniform superconductor near the 
critical temperature and apply it to a few simple problems such as the surface 
energy of the boundary between a normal metal and a superconductor, critical 
current and critical magnetic field, and vortices. 

A lovely introduction to both subjects, without formidable mathematics and 

with a good coverage of experiment is presented in the book Superconductivity 
and superfluidity by D.R. Tilley and J. Tilley, Institute of Physics Publishing, 3rd 
edition, 1990. 

A good presentation of second quantisation with applications to many 

topics of condensed matter can be found in the book by P. L. Taylor and O. 
Heinonen, A Quantum Approach to Condensed Matter Physics, Cambridge 
University Press, 2002. 

The student wishing to focus of superfluidity can study Chapters 1 – 3 of 

these notes and then consult the following books: 

• 
C. J. Pethick and Harry Smith, Bose-Einstein Condensation in 

Dilute Gases, 2nd edition, Cambridge University Press, 2008. 

• 
P. Nozieres, D. Pines, Theory Of Quantum Liquids: Superfluid 

Bose Liquids, Advanced Books Classics, 1994. 

• 
I.M. Khalatnikov, An introduction to the theory of 

superfluidity, Advanced Books Classics, 2000. 
 
If, however, the student is more inclined to take up the subject of 

superconductivity, then, after studying Caps. 1, 4 and 5 of the notes, he may find 
useful the following sources: 

• 
J. B. Ketterson and S. N. Song, Superconductivity, Cambridge 

University Press, 1999. 

• 
M. Tinkham, Introduction to Superconductivity, Dover 

Publications, 2nd edition, 2004. 

• 
P.G. de Gennes, Superconductivity of metals and alloys, 

Advanced Books Classics, 1999. 
This list of books is by no means extensive and merely reflects the tastes of 

the author of these notes. 

 
This course was taught within the project No. 14.B25.31.0007 funded 

by the Ministry of Education and Science of the Russian Federation. 
. 

1. SECOND QUANTISATION 

1.1. General remarks

When we deal with a system of many identical particles in quantum 

mechanics, we should make sure that the wave function is properly normalised to 
take account of the fact that identical particles are indistinguishable. Take for 
example two non-interacting particles. The properly symmetrised wave function 
is 

𝛹(x1,x2) = 1

√2

[𝜓1(x1)𝜓2(x2) ± 𝜓1(x2)𝜓2(x1)], 
(1) 

with the plus sign for bosons and the minus sign for fermions. In the 

language of state vectors 

∣𝛹〉 = 1

√2

[∣∣𝜓1〉∣∣𝜓2〉 ± ∣∣𝜓2〉∣∣𝜓1〉]. 
(2) 

 
If the particles interact the state vector can no longer be represented 

by (2) and is a linear combination of a complete set of properly symmetrised 
vectors corresponding to non-interacting particles. We define the following basis 
kets 

∣0〉 
 the vacuum state, i.e. a state with no particles; 

{∣𝛼〉} 
a complete set of one-particle states; 

{∣∣𝛼𝛽〉} 
a complete set of properly symmetrised two
particle states; 

   . . . . . . . . . . . . . . . . 

{∣∣𝛼𝛽. . . 𝜔〉

⏟
𝑛

} 
a complete set of n-particle states. 

In these definitions the letters α, β etc. stand for quantum numbers or a 

collection of quantum numbers that specify the state occupied by a particle. 

 
Now define the permutation operator, which interchanges two 

particles: 

𝑃∣∣𝛼𝛽〉 ≡ ∣∣𝛽𝛼〉. 
(3) 

A permutation of two particles cannot produce to a new physical state, so 

the state vector∣∣𝛽𝛼〉can only differ from the state vector∣∣𝛼𝛽〉by at most a phase 
factor with a unit modulus: 

∣∣𝛽𝛼〉 = 𝑒𝑖𝜙∣∣𝛼𝛽〉, 
(4) 

which means that the original state is an eigenstate of the permutation 

operator. Applying the permutation operator to the state∣∣𝛽𝛼〉 should, on the one 
hand, return us to the original state. One the other hand, in view of (4), the state 
vector will have acquired another phase factor : 

∣∣𝛼𝛽〉 = 𝑃2∣∣𝛼𝛽〉 = 𝑃(𝑒𝑖𝜙∣∣𝛽𝛼〉) = 𝑒2𝑖𝜙∣∣𝛼𝛽〉. 
(5) 

Therefore there are two possibilities for the phase factor, or two eigenvalues 

of the permutation operator: 

𝑒𝑖𝜙 = ±1 ⇒ ∣∣𝛽𝛼〉 = {
∣∣𝛼𝛽〉,bosons;

−∣∣𝛼𝛽〉,fermions. 
(6) 

As one can see from the above brief discussion, working with properly 

symmetrised wave functions (or, in general, state vectors) is not very convenient, 
and a language is needed that will automatically take the symmetrisation 
requirement into account. This language is developed in the subsequent sections. 

1.2. Second quantisation for fermions

 
Consider fermions first. Define creation operators as follows 

𝑐𝛼

†∣0〉 ≡ ∣𝛼〉,

𝑐𝛽

†∣𝛼〉 = ∣∣𝛽𝛼〉 = −∣∣𝛼𝛽〉. 
(7) 

For convenience, let's agree on the following convention: the symbol 

representing the newly created state is placed immediately after the vertical line. 
The operators only act on the state represented by the symbol immediately after 
the vertical line.  We have two corollaries of (7): 

𝑐𝛼

†𝑐𝛽

† = −𝑐𝛽

†𝑐𝛼

†,

(𝑐𝛼

†)

𝑛 = 0, for𝑛 > 1.

 
(8) 

The creation operators thus anticommute. We see that the creation operators 

automatically take into account the symmetry of a state with two or more 
fermions, and ensure that a given state can only be occupied by one fermion (the 
Pauli principle). 

 
To proceed further we need more elaborate notation: 

𝑐𝛼

†∣… 𝛼̃〉 = ∣𝛼 … 〉, 
(9) 

with the tilde indicating that the state α had not been occupied before the 

creation operator acted, and the dots referring to other states that differ from α 
(occupied or not as the case may be). With this, we can define an operator 
conjugate to the creation operator: 

𝑐𝛼

†∣… 𝛼̃〉 = ∣𝛼 … 〉 ⇒ 〈… 𝛼∣ = 〈𝛼̃ … ∣𝑐𝛼. 
(10) 

Although we do not know the meaning of the Hermitian-conjugate operator 

yet, we can nevertheless write 

𝑐𝛼

†∣𝛼 … 〉 = 0 ⇔ 〈… 𝛼∣𝑐𝛼 = 0, 
(11) 

where the first identity is a manifestation of the Pauli principle that no state 

can be occupied by more that one fermion. With what we have established so far, 
we can write down the following identities: 

a)〈… 𝛼∣𝑐𝛼

†∣… 𝛼̃〉 = 1;

b)〈𝛼̃ … ∣𝑐𝛼∣𝛼 … 〉 = 1;
c)〈… 𝛼∣𝑐𝛼∣∣𝜓〉 = 0∀∣∣𝜓〉;

d)〈𝜓∣∣𝑐𝛼

†∣… 𝛼̃〉 = 0if〈𝜓 ∣ 𝛼 … 〉 = 0;

e)〈𝛼̃ … ∣𝑐𝛼∣∣𝜓〉 = 0if〈𝜓 ∣ 𝛼 … 〉 = 0.

 
(12) 

Putting∣∣𝜓〉 = ∣0〉in (12) c) and e), we shall have that the ket 𝑐𝛼∣0〉is orthogonal 

to any bra regardless of whether the state α is occupied or not, i.e. it is orthogonal 
to any bra. This can only be if 𝑐𝛼∣0〉 ≡ 0.if we now choose ∣∣𝜓〉 = ∣… 𝛼̃〉,then from (12) 
c) and e) we shall conclude that 𝑐𝛼∣… 𝛼̃〉is too orthogonal to any bra regardless of 
whether the state α is occupied or not, i.e. it is too orthogonal to any bra. Thus we 
may conclude 

𝑐𝛼∣… 𝛼̃〉 = 0;

𝑐𝛼∣𝛼 … 〉 = ∣… 𝛼̃〉. 
(13) 

The operator𝑐𝛼is called the annihilation operator. Taking the Hermitian 

conjugate of the first line of (8), we write 

𝑐𝛼𝑐𝛽 = −𝑐𝛽𝑐𝛼. 
(14) 

 
We now build two operators𝑐𝛼𝑐𝛽

†and 𝑐𝛽

†𝑐𝛼,and let them act in turn on a 

ket. This ket must  represent a situation with the state α occupied and the state β 
free, otherwise we shall get zero: 

𝑐𝛼𝑐𝛽

†∣∣𝛼 … 𝛽̃〉 = 𝑐𝛼∣∣𝛽𝛼 … 〉 = −𝑐𝛼∣∣𝛼𝛽 … 〉 = −∣∣𝛽 … 𝛼̃〉;

𝑐𝛽

†𝑐𝛼∣∣𝛼 … 𝛽̃〉 = ∣∣𝛽 … 𝛼̃〉.

 
(15) 

 
The number-of-particles operator 

𝑁𝛼 ≡ 𝑐𝛼

†𝑐𝛼 
(16) 

acting on a ket with the state α occupied reproduces that ket: 

𝑁𝛼∣𝛼 … 〉 = 𝑐𝛼

†𝑐𝛼∣𝛼 … 〉 = 𝑐𝛼

†∣… 𝛼̃〉 = ∣𝛼 … 〉. 
(17) 

 
If we introduce the anticommutator of two operators 

{𝐴,𝐵} ≡ 𝐴𝐵 + 𝐵𝐴, 
(18) 

We can neatly summarise our main achievements: 

{𝑐𝛼

†,𝑐𝛽

†} = {𝑐𝛼,𝑐𝛽} = 0.

{𝑐𝛼

†,𝑐𝛽} = 𝛿𝛼𝛽,

 
(19) 

where δαβ is the Kronecker delta. 

1.3. Second quantisation for bosons

Since any number of bosons can occupy a given state, we shall be using 

different notation for a ket-vector of a many-boson system: 

∣∣𝜓〉 = ∣∣𝑛1,𝑛2, … ,𝑛𝛼, … 〉, 
(20) 

where the occupation numbers nα show how many particles there are in the 

state α. A creation operator acting on such a vector will add one particle to the 
appropriate state, and when acting on the vacuum ket it will create one particle in 
the appropriate state: 

𝑏𝛼

†∣0〉 = ∣∣… 𝑛𝛼 = 1 … 〉,

𝑏𝛼

†∣∣… 𝑛𝛼 … 〉 ∝ ∣∣… 𝑛𝛼 + 1 … 〉,

 
(21) 

where in the second line we have allowed for the possibility of a numerical 

factor. Similarly, a destruction operator can be defined: 

𝑏𝛼∣0〉 = 0,

𝑏𝛼∣∣… 𝑛𝛼 … 〉 ∝ ∣∣… 𝑛𝛼 − 1 … 〉. 
(22) 

 
To fix the numerical factors we define the number-of-particles 

operator as the operator whose eigenvectors are kets (20) corresponding to 
eigenvalues that are numbers of particles in a certain state: 

𝑁𝛼∣∣… 𝑛𝛼 … 〉 ≡ 𝑏𝛼

†𝑏𝛼∣∣… 𝑛𝛼 … 〉 ≡ 𝑛𝛼∣∣… 𝑛𝛼 … 〉. 
(23) 

This definition gives 

∥ 𝑏𝛼∣∣… 𝑛𝛼 … 〉 ∥2= 〈… 𝑛𝛼 … ∣∣𝑏𝛼

†𝑏𝛼∣∣… 𝑛𝛼 … 〉 = 𝑛𝛼〈… 𝑛𝛼 … ∣ ⋯ 𝑛𝛼 … 〉 = 𝑛𝛼, 
(24) 

which fixes the numerical factor in the definition of the annihilation 

operator: 

𝑏𝛼∣∣… 𝑛𝛼 … 〉 = √𝑛𝛼∣∣… 𝑛𝛼 − 1 … 〉. 
(25) 

 
For the creation operator write 

𝑏𝛼

†∣∣… 𝑛𝛼 … 〉 = 𝑥∣∣… 𝑛𝛼 + 1 … 〉, 
(26) 

where x is a yet unknown numerical factor. Consider the following chain: 

𝑏𝛼

†𝑏𝛼𝑏𝛼

†∣∣… 𝑛𝛼 … 〉
=
𝑏𝛼

†𝑥√𝑛𝛼 + 1∣∣… 𝑛𝛼 … 〉 = 𝑥2√𝑛𝛼 + 1∣∣… 𝑛𝛼 + 1 … 〉

=
𝑏𝛼

†𝑏𝛼𝑥∣∣… 𝑛𝛼 + 1 … 〉 = 𝑥(𝑛𝛼 + 1)∣∣… 𝑛𝛼 + 1 … 〉.

 
(27) 

Comparing the two lines we write 

𝑏𝛼

†∣∣… 𝑛𝛼 … 〉 = √𝑛𝛼 + 1∣∣… 𝑛𝛼 + 1 … 〉. 
(28) 

 
The following commutation relations follow immediately from the 

definitions of the boson creation and annihilation operators: 

[𝑏𝛼

†,𝑏𝛽

†] = [𝑏𝛼,𝑏𝛽] = 0;

[𝑏𝛼,𝑏𝛽

†] = 𝛿𝛼𝛽.

 
(29) 

1.4. Operators in second-quantised form

As one can see, the procedure of second quantisation established in the 

previous two sections consists in the replacement of the state vector with an 
operator. It is in this sense that the term “second quantisation” is used (when we 
perform the first quantisation, we assign a vector to each state and replace the 
classical variables with operators that act on these vectors). We now need to 
rewrite the familiar operators in the language of second quantisation. 

The simplest operator is the number of particles. Indeed, we have already 

seen that regardless of the statistics of the particles the eigenvalue of the operator 

𝑁𝛼
̂ = 𝑐̂𝛼

†𝑐̂𝛼is the number of particles in the state α. Therefore the operator 

𝑁̂ = ∑

𝛼

𝑐̂𝛼

†𝑐̂𝛼 
(30) 

is the total number of particles in the system†. From the above discussion 

we can easily see that a ket representing a many-particle state is an eigenket of the 
number-of-particles operator corresponding to the eigenvalue that is the total 
number of particles in the system. Indeed, 

𝑁̂∣∣𝜓〉 = ∑

𝛼

𝑐̂𝛼

†𝑐̂𝛼∣∣𝑛1,𝑛2, … 〉 = (∑

𝛼

𝑛𝛼) ∣∣𝑛1,𝑛2, … 〉 = 𝑁∣∣𝜓〉. 
(31) 

 
Since the total number of particles in the system cannot depend on 

our choice of the basis kets that are used to represent the state ket, the operator 
(30) must be invariant under the transformation of the basis. To check if this is 
indeed the case, let's go over to a different set of basis vectors and the 
corresponding set of operators: 

∣𝑛〉 ≡ 𝑏̂𝑛

†∣0〉. 
(32) 

Since the basis kets form a complete set, we have 

𝑐̂𝛼

†∣0〉 = ∣𝛼〉 = ∑

𝑛

〈𝑛 ∣ 𝛼〉∣𝑛〉 = ∑

𝑛

〈𝑛 ∣ 𝛼〉𝑏̂𝑛

†∣0〉, 
(33) 

which provides us with the law of transformation between the two sets of 

operators: 

𝑐̂𝛼

† = ∑

𝑛

〈𝑛 ∣ 𝛼〉𝑏̂𝑛

†,

𝑐̂𝛼 = ∑

𝑛

〈𝛼 ∣ 𝑛〉𝑏̂𝑛.

 
(34) 

So the structure of the number of particles is indeed invariant: 

𝑁̂ = ∑

𝛼𝑛𝑛′

〈𝑛 ∣ 𝛼〉〈𝛼 ∣ 𝑛′〉𝑏̂𝑛

†𝑏̂𝑛′ = ∑

𝑛

𝑏̂𝑛

†𝑏̂𝑛, 
(35) 

                                                 
†In this sectrion we shall be putting hats on operators in order to avoid confusing them with c-numbers. 

where we have made use of the completeness and orthonormality of the 

bases. This shows that the definition of the number of particles adopted at the 
beginning is reasonable. 

 
Next we define operators that create of destroy a particle at a 

particular position. They are called field operators: 

∣x〉 ≡ 𝜓̂†(x)∣0〉. 
(36) 

It is left as an exercise to the reader to prove the transformation formulas 

𝑐̂𝛼

† = ∫ 𝑑3x〈x ∣ 𝛼〉𝜓̂†(x),

𝜓̂†(x) = ∑

𝛼

〈𝛼 ∣ x〉𝑐̂𝛼

†. 
(37) 

In particular, transformation from the momentum basis to the coordinate 

basis will be 

𝑐̂k

† = ∫ 𝑑3x〈x ∣ k〉𝜓̂†(x) = ∫ 𝑑3x𝜓̂†(x) 𝑒𝑖k⋅x

√𝛺

,

𝜓̂†(x) = ∑

k

〈k ∣ x〉𝑐̂k

† = ∑

k

𝑐̂k

† 𝑒−𝑖k⋅x

√𝛺

,

 
(38) 

where the momentum eigenfunctions are normalised so that the particle can 

be found with certainty in the box of volume Ω. 

 
Using the transformation formulas (37) and the commutation 

relations for fermion and boson operators one can prove the following the 
commutation relations for the field operators: 

𝜓̂†(x)𝜓̂†(x′) ± 𝜓̂†(x′)𝜓̂†(x) = 0,

𝜓̂(x)𝜓̂(x′) ± 𝜓̂(x′)𝜓̂(x) = 0,

𝜓̂(x)𝜓̂†(x′) ± 𝜓̂†(x′)𝜓̂(x) = 𝛿(x − x′),

 
(39) 

where the upper sign is for fermions, and the lower for bosons. 
 
We can also express the number operator in terms of the field 

operators as follows: 

𝑁̂
=
∑

𝛼

(∫ 𝑑3x〈x ∣ 𝛼〉𝜓̂†(x))(∫ 𝑑3x′〈𝛼 ∣ x′〉𝜓̂(x′))

=
∫ 𝑑3x∫ 𝑑3x′𝛿(x − x′)𝜓̂†(x)𝜓̂(x′) = ∫ 𝑑3x′𝜓̂†(x)𝜓̂(x) ≡ ∫ 𝑑3x′𝜌̂(x),

 
(40) 

where the number-density operator has been introduced: 

𝜌̂(x) = 𝜓̂†(x)𝜓̂(x). 
(41) 

This operator is Hermitian, as it should be. If we express the field operators 

in terms of operators in the momentum representation, we shall have 

𝜌̂(x) = 1

𝛺 ∑

k, k'

𝑒𝑖(k'−k)⋅x𝑐̂k

†𝑐̂k' ≡ 1

𝛺 ∑

k, q

𝑒−𝑖q⋅x𝑐̂k + q

†
𝑐̂k ≡ ∑

q

𝑒−𝑖q⋅x𝜌̂q, 
(42) 

where we have introduced the number-density operator in the momentum 

representation: