in-medium wave functionsin-medium pair-wave functions and spatial pair particle correlationsmomentum condensation and ODLRO (off-diagonal long range order)U(1) symmetry breaking and anomalous averages, the superconducting order parameter as a wave function of a Cooper pairfrom the Hartree and the Hartree-Fock to the Hartree-Fock-BogoliubovapproximationsBogoliubov mean-field theory and effective Hamiltonian

(by A. A. Shanenko)

The initial point of most textbooks on the microscopic superconductivity is the BCS ansatzfor the N-particle ground-state wave function of the superconducting electrons. This often causes serious troubles with understanding the main principles of the Bogoliubov mean-field theory, the corner-stone approach for the Bogoliubov-de Gennes equations and present-day reformulation of the Gor'kov formalism.

The point is that the links between the BCS ground-state ansatz and the Bogoliubov theory are not simple and transparent. The most problems are related to anomalous averages of the field operators. We can say that people can often calculate but their understanding ofthe problem appear to be poor.

Keeping this in mind, the Bogoliubov theory is chosen below as the initial point for the introduction to the microscopic (nanoscale) superconductivity. The basic point of the Bogoliubov mean-field theory is the anomalous averages and their relation to the Cooper-pair wave function. This is why we are going to discuss the following important links:

in-medium pair wave functions

the Cooper pair wave function

the superconducting order parameter

anomalous averages of the field operators

spontaneous breakdown of U(1) symmetry

All the physical information about the system properties is contained in the N-particledensity matrix

However, the N-particle density matrix contains too much information, which significantlycomplicates any investigation (this is why constructing an ansatz for the N-particle wavefunction is not always the most efficient way). A more elegant procedure involves theso-called reduced density matrices. For instance, the reduced density matrix of them-order (the m-matrix) is defined as

The concept of in-medium wave functions.Generally, two particles in medium have no wave function. So, the concept of anin-medium wave functions looks controversial at first sight (the same concerns theCooper pair wave function). However, such a concept is not a fiction but a veryuseful tool introduced and developed by several theorists. The most importantcontribution is due to Bogoliubov (1958). To go in more detail, let us start with agas of superconducting (superfluid) fermions being in the ground-state:

(1)

So, we keep only information about m-particle “cluster”, say, embedded into a systemN-m other particles. It turns out that working with the reduced density matrices allowsone to elegantly introduce the concept of the in-medium wave functions. The m-matrixis hermitian, and we can write

Then, the m-matrix can be represented as

where stands for the eigenfunction of the m-matrix and is theeigenvalue. The orthogonality condition for the eigenfunctions reads

(2)

and using this condition together with

results in

(3)

These eigenfunctions are a natural generalization of a wave function for a complexof m particles out of the medium, and the eigenvalues of the m-matrix control the probabilityto find an in-medium complex of m particles in the state . The most interesting situationsfor the many-body problem are m=1,2. Working with the Bose-Einstein condensation,

ν

),..,( 1)(

mm xxνξ )(mwν

the BEC condensate is considered in the context of the 1-matrix. When investigating thesuperconducting (superfluid) fermions, we should study the 2-matrix and the in-mediumpair wave functions:

For illustrative purposes, let us assume that a 2-particle in-medium complex has no anycorrelations (even quantum) with the other N-2 particles in the system. In such anextreme case the N-body wave function is factorized into the two parts

This immediately results in the following factorization of the 2-matrix

As seen, working with the 2-particle complex in this case, we get the unique wave functionrather than an ensemble of wave functions, which is quite expected. Thus, we get a nice(and helpful) generalization of the concept of the “bare” wave function to the in-mediumsituation. In particular, the 2-matrix and its eigenfunctions are a promising way to treatthe in-medium bound states, e.g., the Cooper pairs. Recall that our initial point was theground-state wave function for the system of N superconducting (superfluid) fermions.The finite temperature generalization is straightforward. For finite temperatures one shouldconstruct the N-particle density matrix with the help of the Gibbs statistical operator (thestatistical ensemble of N-particle wave functions). Then, one can invoke the same procedureas above.

Now, let us turn to the problem of the Cooper pairing and in-medium pair waves.

In-medium pair waves.

In most cases we can ignore any correlations between the spin and spatial coordinates,and, so, we get the product

(4)

(5)

The spin part is given by

zS

It is instructive to study the 2-matrix in the context of superconducting (superfluid) fermions .The eigenfunctions of the 2-matrix are called in-medium pair wave functions. Let us firstconsider such eigenfuctions in bulk (the Bogoliubov-de Gennes equations being helpfulin the presence of quantum confinement are discussed in the next lecture). In thehomogeneous case the total system momentum, the total system spin and its z-projectionare the conserved quantities. The pair momentum , the pair spin and its z-projection

commute with the total system momentum, the total system spin and its z-projection,respectively. Thus, we can expect that , with the set of other,additional quantum numbers. Therefore, the in-medium pair wave function can bewritten in the form [with the center-of-mass (Wigner) coordinates]

Qh S

λ},,,{ SmSQr

λν =

For the spin-singlet states we get

and for the spin-triplet states we obtain

and(6)

and(7)

Another important thing about the in-medium pair wave functions concerns the additionalset of quantum numbers . We have two options: the scattering pair states and boundpair states. For in-medium scattering (dissociated) states one can write

For in-medium bound pair states we have

Note that we have only two possibilities:(i) all in-medium pair states are dissociated;(ii) some of the pair states are bound and

others are scattering.There is no possibility that all the eigenstates of the2-matrix are in-medium bound states!

λ

Thus, starting from the general expansion of the 2-matrix in terms of the pair wavefunctions

one can find [with Eqs. (4)-(7)] the following expression

Don’t be confused, even when all the present particles create in-medium bound pairs,there exists the sector of the scattering (dissociated) states (two fermions, the firstis from one bound pair and the second is from another, are in a scattering pair state).

Spatial particle correlations.

Now the question arises how the in-medium pair-wave functions can be related to thebasic thermodynamic quantities. A good avenue for such a relation is through thepair-correlation function

(9)

(8)

that is connected with the 2-matrix as

(10)

Here the creation and annihilation field operators obey the fermionic permutation relations

with . The diagonal pair-correlation function determines the meaninteraction energy

To prove Eq. (10), one should first derive a similar relation between the 1-matrix andone-particle correlation function

(11)

where

Below we give a proof of Eq. (11), and this proof can easily be generalized to Eq. (10).

Proof of Eq. (11). For the N-particle ground state we have the representation in terms of thefield operators

ABBABA ˆˆˆˆ}ˆ,ˆ{ +=

and

So, for the one-particle correlation function we have

This is the product of the bra-vector and ket-vector . Let us considerthe ket-vector and rearrange it as follows:

)(yN +ψ Ny )'(ψ

N steps

Hence, we get

and

These expressions makes it possible to find that

Now, taking account of

one can find that

This is nothing more but Eq. (11). The proof of Eq. (10) is very similar but here twofield operators should move as a single file towards the vacuum vector. Note thatthe proof is readily generalized to finite temperatures when we have a statisticalensemble of N-particle wave functions rather than the ground-state one.

Then, the pair correlation function can be written as

Here double number of the in-medium boundpairs in the state .

double number of pairs in the scattering(dissociated) state .

In the system of N fermions the total number of pairs is N(N -1)/2. In the thermodynamiclimit , we can rewrite Eq. (12) as ∞→VN ,

SmSQn ,,,r

SmSQq ,,,rr

(12)

(13)

where stands for the number of the in-medium bound pairs with the quantum numbers situated in around the point ; and is the number of dissociated pairs with the spin coordinates situated in around the point .

QdQVSmSn

3,, )(

rρ

SmSn ,, Qr

Qd 3 QqddQqVSmS

33,

2 ),(rrρ

SmS , qdQd 33 qQrr

,

Correlation-weakening principle and ODLRO (Off-Diagonal Long Range Order).

To proceed further, we need to employ a basic principle of the quantum statisticalmechanics, i.e., the correlation-weakening principle

(14)

This principle is the corner-stone hypothesis checked with exactly solvable models. A more complex form of (14) can be written as

Based on Eq. (14), one can expect

NENHNNNN N== ˆ ,ˆ

The system Hamiltonian commutes with the particle-number operator . It means that the eigenstates of the Hamiltonian are at the same time the eigenstates of the particle number operator . Thus, one can expect that the anomalous averages appearing in Eq. (15) should be equal to zero:

0]ˆ,ˆ[ =NH

!!!0=++ NN ψψ

(15)

This results in

In other words, there is no off-diagonal long range order for the 2-matrix.

Let us check this by means of the expansion of the density matrix in terms of the pair waves (13):

and we can immediately conclude that above limit follows from this expansion by virtue of the Riemann theorem (when we have reasonable behavior of and ). However, we should not forget that Eq. (13) is written in the thermodynamic limit and such an integral form with smooth densities does not cover all the physical situations. The most important exclusion is the center-of-mass momentum condensation (an analogue of the Bose-Einstein condensation in the single-particle density matrix). Let us consider an example of such a condensation:

)(,, QSmSn

rρ )(,, Q

SmSn

rρ

(16)

with . Inserting Eq. (16) into Eq. (13), we can getjiji ,)( δ=−∆

and this is the off-diagonal long range order (ODLRO) first discussed by Yang (1962). According to ODLRO one gets

(17)

which is not zero!!! What about the conservation of the particle number?

Solution of the paradox: spontaneous U(1) symmetry breaking.

The concept of the spontaneous symmetry breaking was introduced by Bogoliubov (1961). This concept is not only related to superconducting (superfluid) fermions but a rather general one. In our case, the solution of the above paradox is the spontaneous U(1) symmetry breaking, the same as for the phenomenon of the Bose-Einstein condensation. Let us consider the unitary transformation given by

where, recall, stands for the particle-number operator. The system Hamiltonian under investigation is of the form

N

(19)

where

(18)

Here , however . On the other side,

We can show that such a Hamiltonian is invariant under the unitary transformation (18). First, let us show (for an arbitrary ket-vector ) that

~

NN ≠ NNNNNNNN~~ˆ ,ˆ ==

χ

So, we get

(20)

Now, let us consider the transformed Hamiltonian

Using Eq. (20), one can rearrange the above expression as follows:

This allows one to write (for the unitary transformation given by Eq. (18))

(21)

which is a reflection of the fact that the Hamiltonian given by Eq. (19) is permutable with the particle-number operator. From (21) we can immediately conclude that when our Hamiltonian is invariant under the unitary transformation (18), all the anomalous averages are equal to zero. Indeed, we have

and, in turn,

As seen, subtracting the second result from the first one, we obtain

Fluctuations. Now the question arises why this breaking is called spontaneous? And whatis a reason for this? How general is such a phenomenon? This symmetry breaking is called spontaneous because FLUCTUATIONS are the main factor responsible for violating the system symmetry. In more detail, there are a lot of fluctuating physical fields of differentnature which can contribute to the Hamiltonian. At first sight, these extremely smallfluctuating terms (they should be small if we are based on a reasonable approximation forthe Hamiltonian!) can not produce any effect,

Thus, we have

However, they can reduce the Hamiltonian symmetry. In most cases such a reduction does not result in any serious consequence: thermodynamically, the broken-symmetry state is nearly the same as the unbroken one. Yet, sometimes, especially at low temperatures, this is not true. When the broken-symmetry state becomes more advantageous from the thermodynamic point of view, we get the phase transition associated with the corresponding symmetry breaking. For example, in the Heisenberg model, the averaged value of the total spin z-projection is, according to the Hamiltonian symmetry, zero: all the space directions are equivalent. However, at sufficiently low temperatures the system undergoes the phase transition into a state where the average z-projection of the total spin is nonzero. How is it possible when all the space directions are equivalent? This is due to the presence of an infinitesimal magnetic field. It fixes the magnetization direction.

This is an example of very general and important situation: the symmetry of the Hamiltonian is higher than the symmetry of the ground state. That is, the symmetry is spontaneously broken(this remove the degeneracy of the ground state).

Buridan's donkey is a figurative description of a man of indecision. It refers to a paradoxical situation wherein a donkey, placed exactly in the middle between two stacks of hay of equal size and quality, will die since it cannot make any rational decision to start eating one rather than the other (the symmetry is not broken). The paradox is named after the 14th century French philosopher Jean Buridan.

Thus, according to Bogoliubov, to take into account infinitesimal external fields, one should always keep in mind an additional symmetry-reducing term in the Hamiltonian. The coupling constant controlling interaction with such external fields should be set as coming to zero. In particular, for a superconducting (superfluid) fermions, this can be done by formally introducing the sources of the Cooper pairs:

Then, we can invoke the mean-field approximation for the reduced-symmetry Hamiltonian. This is often called the Hartree-Fock-Bogoliubov approximation. One of the important features is that in the presence of the broken U(1) symmetry associated with the particle-number operator, we have to work in the grand canonical formulation to fix the average number of fermions. This is why the chemical potential appears in the Bogoliubov theoryof superconductivity.

Thus, in the presence of the pair condensate, U(1) symmetry related to the conservation of the number of particles should be broken (notice that the same is for the Bose-Einstein condensation). Notice that the breakdown of the symmetry associated with the particle-number conservation is not the only possibility in superconducting systems. There is one more example: the formation of the Fulde-Ferrel-Larkin-Ovchinnikov pairs, which can be treated as the condensation of in-medium bound pairs with a nonzero center-of-mass momentum. In this example we get also the breakdown of the symmetry associated with the total-momentum conservation.

Order parameter.In the presence of the superconducting (superfluid) phase transition associated with the pair-momentum condensation in the bound sector of the 2-matrix, it is convenient to introduce the corresponding order parameter. The best candidate is the anomalous average given by Eq.(17). Inserting

into Eq. (17), we get

which can be rewritten as

(22)

where, recall, is the wave function for the in-medium condensed pair of fermions with the quantum number and in the spin-singlet state. For the s-wave pairing, it is of convenience to use the diagonal order parameter (it is not spatially dependent in bulk):

(23)

)( 210,0,0rrnrr

−ϕ0n

Thus, the off-diagonal superconducting order parameter reads (g the coupling constant)

For the d-wave pairing the diagonal order parameter is exactly zero and, so, only the off-diagonal superconducting order parameter is possible:

The most interesting case is realized in nanoscale superconductors: for the spin-singlet pairing all-even-parity-waves can be in play!

Hartree-Fock-Bogoliubov approximation.

1. Hartree approximation:

It is interesting to express in terms of the pair-wave functions. )',';,( 2121)(

2 xxxxF H

Now, let us introduce the Wigner coordinates

(the center-of-mass andrelative wave vectors)

momentum and spin conservation

Taking into account this rearrangement, we can write

which, with the help of the completeness relation

can be represented in the following form

In the thermodynamic limit, this expression reduces to

Compare it to Eq. (13), the general form of the pair correlation function,

(24)

(25)

Comparing Eq. (25) to Eq. (13), we can conclude that in the Hartree approximation, the pair-correlation function (or the 2-matrix) does not include the sector of bound pair states. The internal pair-wave functions (dependent on relative coordinates) are usual plane waves:

),(, QqSmS

rrρ

(27)

where, recall, stands for the number of the scattering states with the spin specifications and situated in around . As there are no bound pairs, one can expect

QqddQqVSmS

33,

2 ),(rrρ

SmS , qdQd 33 qQrr

,

As follows from Eq. (27), this relation is fulfilled (in the thermodynamic limit).

2. Hartree-Fock approximation:

Let us now check what changes in the pair-wave functions take place in this approximation as compared to the Hartree approach. Following the same procedure as before (expressing the averages of the field operators in terms of the momentum distribution), we arrive at

no quantum-statistical effects (symmetrization or antisymmetrization); no the scatteringcorrections. For the density of the dissociated states we have

(26)

Now, let us go in more detail concerning the products of the spin discrete delta-functions.

So, the products of the discrete delta-functions appearing in the above expression for the pair-correlation function can again be reduced to the sum of the products of the spin wavefunctions.

So, the pair-correlation function can be represented as

Now, let us take into account that one can make the following replacements:

Thus, in the Hartree-Fock approximation, we get

(28)

Comparing Eq. (28) to Eq. (13), one can see that in the Hartree-Fock approximation, the 2-matrix does not include the sector of bound pair states either. However, the internal pair-wave functions are now symmetrized and antisymmetrized plane waves,

3. Hartree-Fock-Bogoliubov approximation:

so, the quantum-statistical correlations are included. Yet, there are no scattering corrections to the plane waves.

(29)

Using the results of the previous paragraph for the Hartree-Fock approximation, we can write

According to the well-known Wick theorem, one can express any averaged product of the field operators in terms of the pair contractions. Based on the results of the previous paragraph for the Hartree-Fock approximation and on the discussion about the spontaneous U(1) symmetry breaking, one can generalize the Hartree-Fock approximation to the superconducting (superfluid) case as follows:

Here we can use Eq. (17) (for the spin-singlet pairing), which can now be rewritten as

because there is only one sort of the bound pair states. This allows us to find

(30)

Comparing Eq. (30) to Eq. (13), we can find that in the Hartree-Fock-Bogoliubov approximation, the 2-matrix includes the sector of condensed bound pairs. These pairs are called the Cooper pairs. We remark that there are no uncondensed bound pairs with nonzero center-of-mass momenta in Eq. (30) . As to the sector of the dissociated (scattering) states, it is of the same form as in the Hartree-Fock approximation (however, the fermion momentum distribution appearing in Eq. (30) is not the same as in Eq. (26) due to the influence of the Cooper pairing).

How to construct the effective Hamiltonian based on the Hartree-Fock-Bogoliubovapproximation? It is of importance to note that the Hartree-Fock part has a minor effecton the results because there exists nearly the same contribution in the normal state. The main difference is due to the presence of the anomalous averages. This is why the Hartree-Fock mean field can be neglected in most applications. So, one can start with the approximation

Bogoliubov mean-field theory and its effective Hamiltonian.

The question arises whether or not it is possible to interpret anomalous averages in Eq. (30) as the manifestation of the momentum condensation in the scattering sector? This is possible but only for bosons. In the case of the Bose-Einstein condensation we also have pair anomalous averages related to the scattering states with zero center-of-mass and zero relative momentum.

Now, the question arises how one can construct the mean-field approach based on the above approximation. This should be done according to the general recipe: if we have the following approximation for the averaged product of a couple of operators, say,

then we can approximate the product of the same operators (but not the average of this product) as

So, the mean-field approximation is given by

(31)

Based on Eq. (31), we can introduce the following approximation, the basic point of the Bogoliubov mean-field theory:

The next usual step (for the s-wave pairing) is to use the delta-function electron-electron interaction

which results in

where

(32)

As seen, we conserve the quantum-dynamics of in the presence of the averaged valuefor , and vice versa. Correlations in quantum dynamics are ignored.

AB

It is worth noting that the delta-function interaction is not here the point-like interaction. This is a kind of the pseudopotential, and it is used not to go in much detail about a complex structure of the electron-electron pair interaction. The payment is the well-known ultraviolet divergence and the cut-off (at the Debye frequency) needed to remove this divergence.

Thus, the effective Hamiltonian of the Bogoliubov mean-field theory (s-wave) reads

with the superconducting order parameter

and kinetic term

This is for a bulk superconductor. What about the nanosized superconductors? There is only one change in the mean-field Hamiltonian, namely,

Another interesting question is to what extent the mean-field theory is applicable on nanoscale? But this is another story.

(33)