Jorge Duelsy
Brief History Cooper pair and BCS Theory (1956-57) Richardson exact solution (1963). Gaudin magnet (1976). Proof of Integrability. CRS (1997). Recovery of the exact solution in applications to ultrasmall grains (000). SU() Richardson-Gaudin models (001). Rational and Hyperbolic families. Applications of rational RG model to superconducting grains, atomic nuclei, cold atoms, quantum dots, etc Generalized RG Models for r>1 (006-009). SO(6) Color pairing. SO(5) T=1 and SO(8) T=0,1 p-n pairing model and spin 3/ cold atoms. Realization of the hyperbolic family in terms of a p-wave integrable pairing Hamiltonian (010). Applications to nuclear structure (011).
The Cooper Problem Problem : A pair of electrons with an attractive interaction on top of an inert Fermi sea. F 1 1 1 c c FS, E G E F
Bound pair for arbitrary small attractive interaction. The FS is unstable against the formation of these pairs
Bardeen-Cooper-Schrieffer e v 0, u c c
Richardson s Exact Solution
Exact Solution of the BCS Model P Eigenvalue equation: H P H n g c c c c, ' E ' ' Ansatz for the eigenstates (generalized Cooper ansatz) M 0, 1 1 E c c
Richardson equations M M 1 1 1 g g 0, E E E E E Properties: 0 1 1 This is a set of M nonlinear coupled equations with M unnowns (E ). The pair energies are either real or complex conjugated pairs. There are as many independent solutions as the dimension of the Hilbert space. The solutions can be classified in the wea coupling limit (g0). Exact solvability reduces an exponential complexity of the many-body problem to an algebraic problem.
i 1,0 0,8 0,6 0,4 0, 0,0 1,0 0,8 0,6 0,4 0, 0,0 1,0 0,8 0,6 0,4 0, 0,0 E=1.7+0.0i 4 6 8 10 1 14 16 18 0 4 6 8 30 3 E=1.0+4.0i 4 6 8 10 1 14 16 18 0 4 6 8 30 3 E=0.0+.0i 4 6 8 10 1 14 16 18 0 4 6 8 30 3 i Evolution of the real and imaginary part of the pair energies with g. L=16, M=8
Pair energies E for a system of 00 equidistant levels at half filling. Hilbert space dimension: 9.06x10 58. Exact solution 100 non-linear coupled equation. Solid lines represent the continuous limit of the Richardson equations. μ+iδ μ-iδ
Integrals of motion of the Richardson-Gaudin Models L. Amico, A. Di Lorenzo, and A. Osterloh, Phys. Rev. Lett. 86, 5759 (001). J. D., C. Esebbag and P. Schuc, Phys. Rev. Lett. 87, 066403 (001). Pair realization of the SU() algebra z 1 j 1 S a a -, S a a 4 j jm jm j jm jm m m The most general combination of linear and quadratic generators, with the restriction of being hermitian and number conserving, is X ij R S g S S S S Z S S ji z z z i i i j i j ij i j Ri, Rj 0 The integrability condition leads to Z X X Z X X 0 ij j j i i ij These are the same conditions encountered by Gaudin (J. de Phys. 37 (1976) 1087) in a spin model nown as the Gaudin magnet.
Gaudin (1976) found three solutions XXX (Rational) X ij Z ij 1 i j XXZ (Hyperbolic) 1 i j i j X ij, Zij Cothxi x j Sinh x x i j i j i j Exact solution R i r i Eigenstates of the Rational Model : Richardson Ansatz 1 i XXX Si 0, XXZ Si 0 i i E i i E
Any function of the R operators defines an exactly solvable Hamiltonian.. Within the pair representation two body Hamiltonians can be obtained by a linear combination of R operators: L H R, g C l1 l The parameters g, s and s are arbitrary. There are L+1 free parameters to define an integrable Hamiltonian in each of the families. (L number of single particle levels) l The BCS Hamiltonian solved by Richardson can be obtained from the XXX family by choosing =. An important difference between RG models and any other ES model is the large number of free parameters. They can be used to define physical interactions. They can even be chosen randomly. z H S g S S BCS ï i i j i ij
Some models derived from rational (XXX) RG BCS Hamiltonian (Fermion and Boson). Generalized Pairing Hamiltonians (Fermion and Bosons). The Universal Hamiltonian of quantum dots. Central Spin Model. Generalized Gaudin magnets. Lipin Model. Two-level boson models (IBM, molecular, etc..) Atom-molecule Hamiltonians (Feshbach resonances), or Generalized Jaynes-Cummings models, Breached superconductivity (Sarma state). Pairs with finite center of mass momentum, FFLO superconductivity. Review: J.Duelsy, S. Pittel and G. Sierra, Rev. Mod. Phys. 76, 643 (004).
What is a Cooper pair in the superfluid is medium? G. Ortiz and J. Duelsy, Phys. Rev. A 7, 043611 (005) Cooper pair wavefunction From MF BCS: A r r r 1 1 N/ N/ 1 ir r e V BCS C BCS v u E From pair correlations: From Exact wavefuction: r C E e r E/ r P BCS c c BCS C Puv E E CE E E real and <0, bound eigenstate of a zero range interaction parametrized by a. E complex and R (E) < 0, tightly bound molecule. E complex and R (E) > 0, wealy bound Cooper pair. E Real and >0 free two particle state.
BCS-BEC Crossover diagram f=1 Re(E)<0 f pairs with Re(E) >0 1-f unpaired, E real >0 = -1, f = 0.35 (BCS) = 0, f = 0.87 (BCS) = 0.37, f = 1 (BCS-P) = 0.55, f = 1 (P-BEC) = 1,, f=1 (BEC) f=1 some Re(E)>0 others Re(E) <0
Cooper pair wave function Wea coupling BCS Strong coupling BCS BEC r (r) 6 5 4 3 1 x 10-0 0 4 6 8 10 10 8 E 6 P 4 0 0 1 3 4 5 5 0 15 10 5 0 0.0 0.5 1.0 1.5.0 r BCS
Sizes and fraction of the condensate 1/ Im E E r dr dr N 3 16 Im i r, r 1 P 1
The Hyperbolic Richardson-Gaudin Model A particular RG realization of the hyperbolic family is the separable pairing Hamiltonian: With eigenstates: Richardson equations: H R S G S S z i i i i i j i j i i i, j M M i M Si 0, E M E 1 i i E 1 si Q 1 1 0, Q L M E E E E G i i ' ' The physics of the model is encoded in the exact solution. It does not depend on any particular representation of the Lie algebra
(p x +ip y ) exactly solvable model In D one can find a representation of the SU() algebra in terms of spinless fermions. z 1 i S c c c c S c c S x y 1, Choosing = we arrive to the p x +ip y Hamiltonian H c c c c G i i c c c c x x y x y ' ' 0, ', 0 x ' x M. I. Ibañez, J. Lins, G. Sierra and S. Y. Zhao, Phys. Rev. B 79, 180501 (009). C. Dunning, M. I. Ibañez, J. Lins, G. Sierra and S. Y. Zhao,, J. Stat. Mech. P08005 (010). S. Rombouts, J. Duelsy and G. Ortiz, Phys. Rev. B. 8, 4510 (010).
Why p-wave pairing? p x +ip y paired phase has been proposed to describe the A1 superfluid phase of 3 He. N. Read and D. Green (Phys. Rev. B 61, 1067 (000)), studied the p x +ip y model. They showed that p-wave pairing has a QPT (º order?) separating two gapped phases: a) a non-trivial topological phase. Wea pairing; b) a phase characterized by tightly bound pairs. Strong pairing. Moreover, there is a particular state in the phase diagram (the Moore- Read Pfaffian) isomorphic to the =5/ fractional quantum Hall state. In polarized (single hyperfine state) cold atoms p-wave pairing is the most important scattering channel (s-wave is suppressed by Pauli). p-wave Feshbach resonances have been identified and studied. However, a p-wave atomic superfluid is unstable due to atom-molecule and molecule-molecule relaxation processes. Current efforts to overcome these difficulties. The great advantage is that the complete BCS-BEC transition could be explored.
1) The Cooper pair wavefunction From the exact solution i cc x y E E real positive uncorrelated pair E complex Correlated Cooper pair E real negative Bound state ) All pair energies converge to zero (Moore-Read line) 1 1 G, g E 0 ; L M 1 1 1 cc 0 x M Exact M, 0x i y 3) All pair energies real and negative (Phase transition) 1 1 G, g L M 1 1 M 1 for g 1/ 1 E 0 PBCS Density Coupling
Quantum phase diagram of the hyperbolic model M / L The phase diagram can be parametrized in terms of the density and the rescaled coupling g GL In the thermodynamic limit the Richardson equations BCS equations E 0 11/ g 0 1/ 1/ g
Exact solution in a D lattice with dis geometry of R=18 with total number of levels L=504 and M=16. (quarter filling) D 10 1 g=0.5 wea pairing g=1.33 Moore-Read g=1.5 wea pairing g=.0 QPT g=.5 strong pairing
Higher order derivatives of the GS energy in the thermodynamic limit 3º order QPT
Characterization of the QPT In the thermodynamic limit the pair wavefunction in -space is: c c u v Moore-Read QPT, 0 Size of the pair wavefunction r rms d d lim 0 r rms Ln Accessible experimentally by quantum noise interferometry and time of flight analysis???
A similar analysis can be applied to the pairs in the exact solution, x S E c c i E y The root mean square or real and negative. r rms, exact of the pair wavefunction is finite for E complex However, r for E real and 0 rms, exact In strong pairing all pairs are bound and have finite radius. At the QPT one pair energy becomes real an positive corresponding to a single deconfined Cooper pair on top of an ensemble of bound molecules.
Exactly Solvable Pairing Hamiltonians 1) SU(), Ran 1 algebra ) SO(5), Ran algebra H S g S S z R i i i j i ij H n g P P i i i j i ij J. Duelsy, V. G. Gueorguiev, P. Van Isacer, S. Dimitrova, B. Errea y S. Lerma H. PRL 96 (006) 07503. 3) SO(6), Ran 3 algebra 4) SO(8), Ran 4 algebra H n g P P i i i j i ij B. Errea, J. Duelsy and G. Ortiz, PRA 79 05160 (009) H n g P P g D D ST i i i j i j i ij ij H3/ ini gpi 00Pj 00 Pi mpj m i ij m S. Lerma H., B. Errea, J. Duelsy and W. Satula. PRL 99, 03501 (007).
Summary For finite systems, the exact solution incorporates mesoscopic fluctuations absent in BCS and PBCS. From the analysis of the exact Richardson wavefunction we proposed a new view to the nature of the Cooper pairs in the BCS-BEC transition for s-wave and p-wave pairing. The hyperbolic RG offers a unique tool to study a rare 3º order QPT in the p x +ip y paired superfluid. We found that the root mean square size of the pair wave function diverges at the critical point. It could be a clear experimental signature of the QPT. Extensions to larger ran algebras open new horizons for the RG models.