Lecture 6. Fermion Pairing. WS2010/11: Introduction to Nuclear and Particle Physics

Transcription

1 Lecture 6 Fermion Pairing WS2010/11: Introduction to Nuclear and Particle Physics

2 Experimental indications for Cooper-Pairing Solid state physics: Pairing of electrons near the Fermi surface with antiparallel momentum and spin (k up; -k down) Nuclear physics: Pairing of nucleons to total I Z =0 : (m i, -m i ) Experimental observations: Odd-even effect in nuclear physics The nuclear mass for given odd A: -> Binding energy for odd nuclei: Energy gap in nuclear physics for low excitations: E( even even) > E( odd even) Superconductivity in solid state physics

3 Pairing: BCS model Bardeen-Cooper-Schrieffer (BCS) model: 1957 Microscopic theory of superconductivity (received the Nobel Prize in Physics in 1972) : there is some attraction between electrons, which can overcome the Coulomb repulsion electrons near the Fermi surface become unstable against the formation of Cooper pairs due to an attractive potential = pairing force Consider two time-reversed single-particle states k and k, where k is the angular momentum projection, coupled by the pairing force: The Hamiltonian: Single-particle part residual interaction acting only on pairs Assume a constant matrix element: -G => (1) (2)

4 Pairing: BCS model Approximate solution Bardeen-Cooper-Schrieffer (BCS) state: (3) In this state each pair of single-particle levels (k,-k) is : occupied with a probability υ k 2 and remains empty with probability u k 2. The parameters υ k and u k will be determined through the variational principle. We will assume that they are real numbers. In case of Hartree-Fock HF> states : υ k 1, k Fermi level = 0, k > Fermi level occupied probability - hole - particle 0, k Fermi level u k = 1, k > Fermi level unoccupied probability In case of BCS> states: states may be occupied above Fermi level!

5 Pairing: BCS model Examine a few properties of the BCS state: Normalization: the norm is given by (4) The terms in parentheses all commute for different indices, so only the product of two such terms with the same index (k =k) needs to be considered: (5) the norm is (5b) For normalization we thus must require : (6)

6 Pairing: BCS model Particle number: this is not a good quantum number for the BCS state! Its expectation value is This fits the interpretation of υ k2 as the probability for having the pair (k,-k) occupied. (8) (7) Particle-number uncertainty: the mean square deviation of the particle number is given by (9)

7 Variational method Consider the variational condition with the Hamiltonian (2): (10) and considering the free parameters υ k (11) The u k depend on the υ k via the normalization (12) The evaluation of the matrix elements in (11) gives: (13)

8 The pairing matrix element now reads : Variational method (14) The expectation value of the Hamiltonian becomes: (15) Now differentiate (15) according to (12): υ k (16)

9 Pairing: BCS model All the equations for the different values of k are coupled through the term (17) Introduce the abbreviation (λ corresponds to the Fermi energy ): (18) Rewrite eq. (16) and using (17) and (18), we obtain the BCS equation: (19) Let s assume that is known and express u k and υ k via. Squaring equation (19) allows to replace u k by υ k ; then - solve for υ k : (20)

10 Pairing: BCS model Choose the correct sign such that for very large single-particle energies the occupation probabilities υ k must go to zero; this is achieved by taking the negative sign. The final result is thus (21) for ε k =0, i.e. when u k and υ k =1/2 for large negative ε k : u k2 0 and υ k 2 1 for large positive ε k : u k2 1 and υ k 2 0

11 Gap equation The unknown parameter can now be determined by inserting the explicit forms for u k and υ k, i.e. (21), into its definition : (22) Gap equation: (23) How to solve the gap eq. (23)? Unknown parameters in (23): G, ε k0, λ + extra condition (cf. (8)) for the total particle number Assume that we know G, ε k0, then fix λ from (8) + in (23) neglect the term Gυ k 2 with the argument that it corresponds only to a renormalization of the single-particle energy solve (23) iteratively!!

12 The Bogolyubov transformation The BCS model may be formulated in a more elegant way by a transformation to new quasiparticle operators, the so-called Bogolyubov transformation developed by Bogolyubov and Valatin (1958) a simple method of constructing the excited states of the nucleus as quasiparticle excitations. The basic idea is to look for operators vacuum state, i.e., for which the BCS ground state is the (24) Analogy: quasiparticle operators for the particle-hole Hartree-Fock states HF> 1) k>f : unoccupied Hartree-Fock states - particles (above the Fermi level) 2) k<f: occupied Hartree-Fock states - holes (below the Fermi level) the creation of a hole k implies the destruction of a particle with angular momentum projection k, so that its index should be denoted as -k.

13 BCS states BCS> : The Bogolyubov transformation The particle number is no longer sharp and it appears reasonable to try the more general transformation quasiparticle operator: (25) Apply (25) to (acts only on index k) (26) (27) Solution of (27): where s is an arbitrary real factor (28)

14 The Bogolyubov transformation Substitute (28) into (25) (with parameter t for k states): (29) The unknown factors s and t can be determined by requiring the usual fermion commutation rules, for example, (30) (30) can be fulfilled by setting s = t = 1 and demanding that The Bogolyubov transformation > quasiparticle operators: (31)

15 The Bogolyubov transformation The inverse Bogolyubov transformation is given by (32) and the Hermitian conjugate for the creation operators. Now transform the Hamiltonian: (33) kinetic energy + two-body interaction where is the antisymmetrized matrix element Replace the operators a, a + in (33) by the quasiparticle operators (32) many terms!

16 The Bogolyubov transformation Consider only kinetic-energy term: = k k > ( + + ak + a ) ( ) 1 k a 1 k + a 2 k 2 (34) Rearrange the terms in the Hamiltonian using commutation relations, e.g.: +

17 The Bogolyubov transformation Rearrange the terms in the Hamiltonian using commutation relations: the operator products should be brought into normal order, i.e. all creation operators to the left of all annihilation operators, since in this case they will not contribute in the BCS ground state. Doing the commutation also generates terms with fewer operators like, as in the example above, one with no operators at all. Treating all terms in this manner finally leads to a natural decomposition of the Hamiltonian according to the number of operators in the terms. Subtracting the term used to constrain the particle number, we write it as (35) where the two indices H ij denote the number of creation and annihilation operators in the terms: U is the energy of the BCS ground state with zero quasiparticles, H 11 indicates the dependence of the energy of quasiparticle-quasihole excitations, H 20 violates quasiparticle number conservation and even implies that the BCS state will not be the true ground state. The other terms contain higher-order couplings and may be ignored for the moment.

18 The Bogolyubov transformation A reasonable interpretation of a BCS ground state with quasiparticle excitations requires H 20 = 0, and we can use this as the condition for determining the υ k (and u k ), which have so far been arbitrary. H 20 turns out to be a sum of terms in and requiring the coefficients to vanish leads to (36) pairing term This set of equations is a generalization of the Hartree-Fock equations to which they reduce if the occupation numbers are restricted to 1 or 0. The second term is denoted as the pairing term.

19 The Bogolyubov transformation Introduce abbreviations for the Hartree-Fock-Bogolyubov potential: (37) The sum is now over both positive and negative values of k", allowing the combination of the two terms in parentheses), and for the pairing potential (38) in terms of which the Hartree-Fock-Bogolyubov equations read (39)

20 For the other terms: The Bogolyubov transformation (40) Reduce the above equations for the assumption of a diagonal pairing potential In order to simplify the problem let s choose the single-particle states as eigenstates of a suitably selected single-particle Hamiltonian h. In this case the natural choice is (41) (42)

21 The Bogolyubov transformation inserting (41) leads to the simplified form of the Hartree-Fock-Bogolyubov equations (39) (for the diagonal pairing potential ): (43) Compare (43) to the BCS eq. (19): Eqs. (43) and (19) are identical if (44) Matrix element is constant: (45) Solving (43) obtain: (46)

22 The Bogolyubov transformation Gap equation for the diagonal pairing potential: (47) Difference with simple BCS theory: the coupling of the occupation numbers and the self-consistency problem. => The single-particle Hamiltonian (42) depends on the occupation numbers υ k, which have to be determined by solving the gap equation (38) simultaneously with the iterations of the self-consistent field. Finally we can insert the results for the pure pairing force into the other parts of the Hamiltonian. The ground-state energy becomes (48)

23 The Bogolyubov transformation for the quasiparticle-quasihole part : (49) which may be simplified further using (50) (51) with the quasiparticle energy (51) has the form of a Hamiltonian of noninteracting quasiparticles. (52) Thus, the problem of pairing correlations has been simplified considerably: the ground state (48) now contains correlations between the nucleons via fractional occupation numbers and the excited states can be approximated as consisting of noninteracting quasiparticles - (51) with their energies related to the underlying single-particle Hartree-Fock eigenenergies via (52).