Cooper pairs and BCS ( ) Richardson exact solution (1963). Ultrasmall superconducting grains (1999). SU(2) Richardson-Gaudin models (2000). Cooper pairs and pairing correlations from the exact solution. BCS-BEC crossover in cold atoms (2005). Generalized Richardson-Gaudin Models for r>1 ( ). 3-color pairing. T=0,1 p-n pairing model and spin 3/2 atoms. p-wave integrable pairing Hamiltonian (2009). Jorge Dukelsky. IEM. CSIC.

The Cooper Problem Problem : A pair of electrons with an attractive interaction on top of an inert Fermi sea.

“Bound” pair for arbitrary small attractive interaction. The FS is unstable against the formation of these pairs

Richardson’s Exact Solution

Exact Solution of the BCS Model Eigenvalue equation: Ansatz for the eigenstates (generalized Cooper ansatz)

Richardson equations Properties: This is a set of M nonlinear coupled equations with M unknowns (E  ). The first and second terms correspond to the equations for the one pair system. The third term contains the many body correlations and the exchange symmetry. The pair energies are either real or complex conjugated pairs. There are as many independent solutions as states in the Hilbert space. The solutions can be classified in the weak coupling limit (g  0).

Pair energies E for a system of 200 equidistant levels at half filling In the thermodynamic limit Richardson → BCS + an equation for the arc in the complex plane.

Condensation energy for even and odd grains PBCS versus Exact J. Dukelsky and G. Sierra, PRL 83, 172 (1999)

r s : Interparticle distance,  : Size of the “Cooper” pair The Structure of the Cooper pairs in BCS-BEC Quasibound molecules Pair resonaces + Free fermions+ Quasibound molecules Pair resonances

What is a Cooper pair in the superfluid is medium? G. Ortiz and J. Dukelsky, Phys. Rev. A 72, (2005) “Cooper” pair wavefunction From MF BCS: From pair correlations: From Exact wavefuction: E real and <0, bound eigenstate of a zero range interaction parametrized by a. E complex and R (E) < 0, quasibound molecule. E complex and R (E) > 0, molecular resonance. E Real and >0 free two particle state.

BCS-BEC Crossover diagram f pairs with Re(E) >0 1-f unpaired, E real >0 f=1 Re(E)<0 f=1 some Re(E)>0 others Re(E) <0  = -1, f = 0.35 (BCS)  = 0, f = 0.87 (BCS)  = 0.37, f = 1 (BCS-P)  = 0.55, f = 1 (P-BEC)  = 1,2, f=1 (BEC) =1/k f a s

“Cooper” pair wave function Weak coupling BCS Strong coupling BCS BEC

Sizes and Fraction of the condensate

Nature 454, (2008) A spectroscopic pair size can be defined from the threshold energy of the pair dissociation spectrum as Cooper wavefunction in the BEC region

The Hyperbolic Richardson-Gaudin Model A particular RG realization of the hyperbolic family is the separable pairing Hamiltonian : With eigenstates: Richardson equations: 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 ) SU(2) spinless fermion representation Choosing  k = k 2 we arrive to the p x +ip y Hamiltonian M. I. Ibañez, J. Links, G. Sierra and S. Y. Zhao, Phys. Rev. B 79, (2009). C. Dunning, M. I. Ibañez, J. Links, G. Sierra and S. Y. Zhao,, J. Stat. Mech. P (2010). S. Rombouts, J. Dukelsky and G. Ortiz, ArXiv: It is know that p-wave pairing has a QPT separating two gapped phases: A non-trivial topological phase. Weak pairing. A phase characterized by tightly bound quasi-molecules. Strong pairing. N. Read and D. Green, Phys. Rev. B 61, (2000). Moreover, there is a particular state ( the Moore-Read Pfafian) isomorphic to the a fractional quantum Hall GS.

From the Richardson equations the necessary condition to have N pairons converging to zero, E α -> 0, is: 1) No pairons converge to zero 2) All pairons converge to zero (Moore-Read line) QPT The presence of a zero energy level with variable degeneracy determines the physics of the model

Quantum phase diagram of the hyperbolic model p x +ip y

Pairons distribution in a Disk of R=18 with total degeneracy L=504 and M=126. (quarter filling) g=0.5 weak coupling g=1.5 weak pairing g=2.5 strong pairing

Momentum density profiles for L=504 and M= 126. Exact versus BCS BCS

Higher order derivatives of the GS energy in the thermodynamic limit 3º order QPT Possible third-order phase transition in the large-N lattice gauge theory D. J. Gross and E. Witten, Phys. Rev. D 21, 446–453 (1980)

Characterization of the QPT Accessible experimentally by quantum noise interferometry and time of flight analysis? In the thermodynamic limit the condensate wavefunction in k-space is: The length scale can be calculated as:

A similar analysis can be applied to the pairs in the exact solution The root mean square of the pair wavefunction is finite for E complex or real and negative. However, for In strong pairing all pairs have finite radius. At the QPT one pairon becomes real an positive corresponding to a single deconfined Cooper pair on top of an ensemble of quasi-bound molecules.

Exactly Solvable Pairing Hamiltonians 1) SU(2), Rank 1 algebra 2) SO(5), Rank 2 algebra 4) SO(8), Rank 4 algebra J. Dukelsky, V. G. Gueorguiev, P. Van Isacker, S. Dimitrova, B. Errea y S. Lerma H. PRL 96 (2006) S. Lerma H., B. Errea, J. Dukelsky and W. Satula. PRL 99, (2007). 3) SO(6), Rank 3 algebra B. Errea, J. Dukelsky and G. Ortiz, PRA (2009)

Phase diagram L=500, N=150, P=(N G -N B )/(N G +N B ) Breached, Unbreached configurations 3-color Pairing

Occupation probabilities Density profiles 3-color Pairing

Summary For finite system, the exact solution incorporates mesoscopic correlations absent in BCS and PBCS. From the analys is 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 for p-wave pairing. 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 higher rank algebras include: The SO(6) model for three-component systems describing color superconductivity and exotic phases with two condensates. The SO(8) model for four-components systems ( 3/2 fermions or T=0,1 proton- neutron pairing). Competence between pair and quartet correlations.

Construction of the Integrals of Motion The most general combination of linear and quadratic generators, with the restriction of being hermitian and number conserving, is The integrability condition leads to These are the same conditions encountered by Gaudin (J. de Phys. 37 (1976) 1087) in a spin model known as the Gaudin magnet.

Gaudin (1976) found three solutions Hamiltonianos XXX (Rational) XXZ (Hyperbolic)

Exactly Solvable RG models for simple Lie algebras Cartan classification of Lie algebras rankA n su(n+1)B n so(2n+1)C n sp(2n)D n so(2n) 1 su(2), su(1,1) pairing so(3)~su(2)sp(2) ~su(2)so(2) ~u(1) 2 su(3) Three level Lipkins so(5), so(3,2) pn-pairing sp(4) ~so(5)so(4) ~su(2)xsu(2) 3su(4) Wigner so(7) FDSM sp(6) FDSM so(6)~su(4) color superconductivity 4su(5)so(9)sp(8) so(8) pairing T=0,1. Ginnocchio. 3/2 fermions