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Fermion Pairing LLeeccttuurree 66 FFeerrmmiioonn PPaaiirriinngg WS2010/11: ‚Introduction to Nuclear and Particle Physics‘ EExxppeerriimmeennttaall iinnddiiccaattiioonnss ffoorr CCooooppeerr--PPaaiirriinngg • 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 IZ=0 : (mi , -mi) 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 PPaaiirriinngg:: BBCCSS mmooddeell 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: (1) Single-particle part residual interaction acting only on pairs Assume a constant matrix element: -G => (2) PPaaiirriinngg:: BBCCSS mmooddeell Approximate solution Bardeen-Cooper-Schrieffer (BCS) state: (3) In this state each pair of single-particle levels (k,-k) is : 2 2 occupied with a probability |υk| and remains empty with probability |uk| . The parameters υk and uk will be determined through the variational principle. We will assume that they are real numbers. °In case of Hartree-Fock |HF> states : À À1, k ≤ Fermi level - hole 0, k ≤ Fermi level uk = Ã υk = > 0, k > Fermi level - particle Õ1, k Fermi level occupied probability unoccupied probability °In case of |BCS> states: states may be occupied above Fermi level ! PPaaiirriinngg:: BBCCSS mmooddeell 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) Ï the norm is (5b) For normalization we thus must require : (6) PPaaiirriinngg:: BBCCSS mmooddeell °Particle number: this is not a good quantum number for the BCS state! Its expectation value is (7) 2 This fits the interpretation of υk as the probability for having the pair (k,-k) occupied. (8) °Particle-number uncertainty: the mean square deviation of the particle number is given by (9) VVaarriiaattiioonnaall mmeetthhoodd Consider the variational condition with the Hamiltonian (2): (10) and considering the free parameters υk (11) The uk depend on the υk via the normalization (12) The evaluation of the matrix elements in (11) gives: (13) VVaarriiaattiioonnaall mmeetthhoodd The pairing matrix element now reads : (14) The expectation value of the Hamiltonian becomes: (15) ∂ Now differentiate (15) according to (12): ∂υk (16) PPaaiirriinngg:: BBCCSS mmooddeell 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 uk and υk via ∆. Squaring equation (19) allows to replace uk by υk ; then - solve for υk : (20) PPaaiirriinngg:: BBCCSS mmooddeell Choose the correct sign such that for very large single-particle energies the occupation probabilities υk must go to zero; this is achieved by taking the negative sign. The final result is thus (21) ° for εk =0, i.e. when uk and υk =1/2 ° for large negative εk : 2 2 uk ‰0 and υk ‰ 1 ° for large positive εk : 2 2 uk ‰1 and υk ‰ 0 GGaapp eeqquuaattiioonn The unknown parameter ∆ can now be determined by inserting the explicit forms for uk and υk , i.e. (21), into its definition : (22) Gap equation: (23) How to solve the gap eq. (23) ? 0 Unknown parameters in (23): G, εk , λ + extra condition (cf. (8)) for the total particle number 0 Assume that we know G, εk , then fix λ from (8) 2 + in (23) neglect the term –Gυk with the argument that it corresponds only to a renormalization of the single-particle energy solve (23) iteratively !! TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn 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 for which the BCS ground state is the vacuum state, i.e., (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. TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn BCS states |BCS> : 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): (28) where s is an arbitrary real factor TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn 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) TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn 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! TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn Consider only kinetic-energy term: = (a%+ + a%+ ) (a% + a% ) k1 −k1 k2 −k2 k1k2 >0 (34) Rearrange the terms in the Hamiltonian using commutation relations, e.g.: + TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn 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 Hij denote the number of creation and annihilation operators in the terms: °U is the energy of the BCS ground state with zero quasiparticles, °H11 indicates the dependence of the energy of quasiparticle-quasihole excitations, ° H20 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. TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn A reasonable interpretation of a BCS ground state with quasiparticle excitations requires H20 = 0, and we can use this as the condition for determining the υk (and uk), which have so far been arbitrary. H20 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. TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn 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) TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn For the other terms: (40) Reduce the above equations for the assumption of a diagonal pairing potential (41) 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 (42) TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn 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) TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn 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 υk , 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) TThhee BBooggoollyyuubboovv ttrraannssffoorrmmaattiioonn for the quasiparticle-quasihole part : (49) which may be simplified further using (50) (51) with the quasiparticle energy (52) (51) has the form of a Hamiltonian of noninteracting quasiparticles. 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). .
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