in-medium pair wave functions the Cooper pair wave function the superconducting order parameter anomalous averages of the field operators

Transcription

1 (by A. A. Shanenko) in-medium wave functions in-medium pair-wave functions and spatial pair particle correlations momentum 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 pair from the Hartree and the Hartree-Fock to the Hartree-Fock-Bogoliubov approximations Bogoliubov mean-field theory and effective Hamiltonian

2 The initial point of most textbooks on the microscopic superconductivity is the BCS ansatz for 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 of the 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 Cooperpair wave function. This is why we are going to discuss the following important links: in-medium pair wave functions the Cooper pair wave function anomalous averages of the field operators the superconducting order parameter spontaneous breakdown of U(1) symmetry

3 The concept of in-medium wave functions. Generally, two particles in medium have no wave function. So, the concept of an in-medium wave functions looks controversial at first sight (the same concerns the Cooper pair wave function). However, such a concept is not a fiction but a very useful tool introduced and developed by several theorists. The most important contribution is due to Bogoliubov (1958). To go in more detail, let us start with a gas of superconducting (superfluid) fermions being in the ground-state: All the physical information about the system properties is contained in the N-particle density matrix (1) However, the N-particle density matrix contains too much information, which significantly complicates any investigation (this is why constructing an ansatz for the N-particle wave function is not always the most efficient way). A more elegant procedure involves the so-called reduced density matrices. For instance, the reduced density matrix of the m-order (the m-matrix) is defined as

4 So, we keep only information about m-particle cluster, say, embedded into a system N-m other particles. It turns out that working with the reduced density matrices allows one to elegantly introduce the concept of the in-medium wave functions. The m-matrix is hermitian, and we can write Then, the m-matrix can be represented as (2) ( m) (m) where ξ ν ( x 1,.., xm ) stands for the eigenfunction of the m-matrix and w ν is the eigenvalue. The orthogonality condition for the eigenfunctions reads and using this condition together with results in (3) These eigenfunctions are a natural generalization of a wave function for a complex of m particles out of the medium, and the eigenvalues of the m-matrix control the probability to find an in-medium complex of m particles in the state ν. The most interesting situations for the many-body problem are m=1,2. Working with the Bose-Einstein condensation,

5 the BEC condensate is considered in the context of the 1-matrix. When investigating the superconducting (superfluid) fermions, we should study the 2-matrix and the in-medium pair wave functions: For illustrative purposes, let us assume that a 2-particle in-medium complex has no any correlations (even quantum) with the other N-2 particles in the system. In such an extreme 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 function rather 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-medium situation. In particular, the 2-matrix and its eigenfunctions are a promising way to treat the in-medium bound states, e.g., the Cooper pairs. Recall that our initial point was the ground-state wave function for the system of N superconducting (superfluid) fermions. The finite temperature generalization is straightforward. For finite temperatures one should construct the N-particle density matrix with the help of the Gibbs statistical operator (the statistical ensemble of N-particle wave functions). Then, one can invoke the same procedure as above. Now, let us turn to the problem of the Cooper pairing and in-medium pair waves.

6 In-medium pair waves. 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 first consider such eigenfuctions in bulk (the Bogoliubov-de Gennes equations being helpful in the presence of quantum confinement are discussed in the next lecture). In the homogeneous case the total system momentum, the total system spin and its z-projection are the conserved quantities. The pair momentum hqˆ, the pair spin Ŝ and its z-projection Ŝ z commute with the total system momentum, the total system spin and its z-projection, respectively. Thus, we can expect that ν = { λ, Q r, S, ms }, with λ the set of other, additional quantum numbers. Therefore, the in-medium pair wave function can be written in the form [with the center-of-mass (Wigner) coordinates] (4) In most cases we can ignore any correlations between the spin and spatial coordinates, and, so, we get the product (5) The spin part is given by

7 For the spin-singlet states we get (6) and and for the spin-triplet states we obtain (7) and Another important thing about the in-medium pair wave functions concerns the additional set of quantum numbers λ. We have two options: the scattering pair states and bound pair 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 the 2-matrix are in-medium bound states!

8 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 first is from one bound pair and the second is from another, are in a scattering pair state). Thus, starting from the general expansion of the 2-matrix in terms of the pair wave functions one can find [with Eqs. (4)-(7)] the following expression (8) Spatial particle correlations. Now the question arises how the in-medium pair-wave functions can be related to the basic thermodynamic quantities. A good avenue for such a relation is through the pair-correlation function (9)

9 that is connected with the 2-matrix as (10) Here the creation and annihilation field operators obey the fermionic permutation relations with { Aˆ, Bˆ} = AB ˆ ˆ + BA ˆ ˆ. The diagonal pair-correlation function determines the mean interaction energy To prove Eq. (10), one should first derive a similar relation between the 1-matrix and one-particle correlation function where (11) 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 the field operators

10 and So, for the one-particle correlation function we have This is the product of the bra-vector N ψ + (y) and ket-vector ψ ( y ') N. Let us consider the ket-vector and rearrange it as follows:

11 Hence, we get N steps

12 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 two field operators should move as a single file towards the vacuum vector. Note that the proof is readily generalized to finite temperatures when we have a statistical ensemble of N-particle wave functions rather than the ground-state one.

13 Then, the pair correlation function can be written as (12) Here double number of the in-medium bound pairs in the state n, Q r, S,. double number of pairs r in the scattering (dissociated) state q, Q r, S, m S. In the system of N fermions the total number of pairs is N(N -1)/2. In the thermodynamic limit N,V, we can rewrite Eq. (12) as m S (13)

14 3 where Vρ n, S, m S ( Q r ) d Q stands for the number of the in-medium bound pairs with the quantum numbers n, S, m S situated in d around the point Q r 3 2 r r 3 3 Q ; and V ρ S, m S ( q, Q) d qd Q is the 3 3 number r of dissociated pairs with the spin coordinates S, m S situated in d Q d q around the point Q q r,. Correlation-weakening principle and ODLRO (Off-Diagonal Long Range Order). To proceed further, we need to employ a basic principle of the quantum statistical mechanics, 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 (15) The system Hamiltonian commutes with the particle-number operator [ Hˆ, Nˆ ] = 0. It means that the eigenstates of the Hamiltonian are at the same time the eigenstates of the particle number operator Nˆ N = N N, Hˆ N = EN N. Thus, one can expect that the anomalous + + averages appearing in Eq. (15) should be equal to zero: N ψ ψ N = 0!!!

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 r by virtue r of the Riemann theorem (when we have reasonable behavior of ρ n, S, m S ( Q) and ρ n, S, m S ( Q) ). 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: (16)

16 with ( j) = δ. Inserting Eq. (16) into Eq. (13), we can get i i, j 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, Nˆ stands for the particle-number operator. The system Hamiltonian under investigation is of the form where (19) (18)

17 We can show that such a Hamiltonian is invariant under the unitary transformation (18). First, let us show (for an arbitrary ket-vector χ ) that ~ Here N N, however Nˆ ~ ~ N = N N, Nˆ N = N N. On the other side, So, we get (20) Now, let us consider the transformed Hamiltonian Using Eq. (20), one can rearrange the above expression as follows:

18 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

19 and, in turn, As seen, subtracting the second result from the first one, we obtain Thus, we have Fluctuations. Now the question arises why this breaking is called spontaneous? And what is 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 different nature which can contribute to the Hamiltonian. At first sight, these extremely small fluctuating terms (they should be small if we are based on a reasonable approximation for the Hamiltonian!) can not produce any effect,

20 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.

21 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 particlenumber 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 theory of 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 particlenumber 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.

22 Order parameter. In the presence of the superconducting (superfluid) phase transition associated with the pairmomentum 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) Thus, the off-diagonal superconducting order parameter reads (g the coupling constant) (23) r r where, recall, ϕ n ( is the wave function for the in-medium condensed pair of fermions 0,0,0 1 2) with the quantum number n0 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): For the d-wave pairing the diagonal order parameter is exactly zero and, so, only the offdiagonal 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!

23 Hartree-Fock-Bogoliubov approximation. 1. Hartree approximation: ( ) It is interesting to express F H x, x ; x', ' ) in terms of the pair-wave functions. 2 ( x 2 momentum and spin conservation Now, let us introduce the Wigner coordinates (the center-of-mass and relative wave vectors)

24 Taking into account this rearrangement, we can write In the thermodynamic limit, this expression reduces to which, with the help of the completeness relation (24) can be represented in the following form (25) Compare it to Eq. (13), the general form of the pair correlation function,

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: (26) no quantum-statistical effects (symmetrization or antisymmetrization); r r no the scattering corrections. For the density of the dissociated states ρ S, ( q, Q) we have (27) 2 r r 3 3 where, recall, V ρ S, m S ( q, Q) d qd Q stands for the number of r the scattering states with the 3 3 spin specifications S, m S and situated in d Q d q around Q q r,. As there are no bound pairs, one can expect m S 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

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 wave functions.

27 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)

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, (29) so, the quantum-statistical correlations are included. Yet, there are no scattering corrections to the plane waves. 3. Hartree-Fock-Bogoliubov approximation: 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: Using the results of the previous paragraph for the Hartree-Fock approximation, we can write

29 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).

30 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. Bogoliubov mean-field theory and its effective Hamiltonian. How to construct the effective Hamiltonian based on the Hartree-Fock-Bogoliubov approximation? It is of importance to note that the Hartree-Fock part has a minor effect on 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 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)

31 As seen, we conserve the quantum-dynamics of  in the presence of the averaged value for Bˆ, and vice versa. Correlations in quantum dynamics are ignored. Based on Eq. (31), we can introduce the following approximation, the basic point of the Bogoliubov mean-field theory: (32) The next usual step (for the s-wave pairing) is to use the delta-function electron-electron interaction which results in where 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.

32 Thus, the effective Hamiltonian of the Bogoliubov mean-field theory (s-wave) reads (33) 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.