1. Chaos and interactions in nano-size metallic grains: the competition between superconductivity and ferromagnetism Yoram Alhassid (Yale) Introduction Universal Hamiltonian for a chaotic grain: the competition between superconductivity (pairing correlations) and ferromagnetism (exchange correlations). Quantum phase diagram (ground-state spin). Transport: mesoscopic fluctuations of Coulomb blockade conductance Conclusion Sebastian Schmidt (Yale, ETH Zurich)

2. Introduction: metallic grains (nanoparticles) Discrete energy levels extracted from non- linear conductance measurements. Superconducting at low temperatures [von Delft and Ralph, Phys. Rep. 345, 61 (2001)] A pairing gap was observed in spectra of size ~ 10 nm grains. Explained by BCS theory: valid in the bulk limit = single-particle level spacing = pairing gap However, in grains smaller than ~ 3 nm, , the fluctuations dominate and “superconductivity would no longer be possible” ( Anderson ).

3. Universal Hamiltonian for a chaotic grain An isolated chaotic grain with a large number of electrons is described by the universal Hamiltonian [ Kurland, Aleiner, Altshuler, PRB 62, 14886 (2000) ] A ferromagnetic exchange interaction with exchange constant ( is the total spin of the grain). Discrete single-particle levels (spin degenerate) and wave functions that follow random matrix theory (RMT). BCS-like pairing interaction with coupling ( creates pairs of spin up/down electrons). Charging energy term describing a grain with capacitance and electrons : constant interaction (CI) model. Competition : Pairing correlations and one-body term favor minimal ground- state spin, while spin exchange interaction favors maximal spin polarization.

4. A derivation from symmetry principles [ Y. Alhassid, H.A. Weidemuller, A. Wobst, PRB 72, 045318 (2005)] Hamiltonian of interacting electrons in a dot:  The randomness of the single-particle wave functions induces randomness in the two-body interaction matrix elements. Cumulants of the interaction matrix elements are determined by requiring invariance under a change of the single-particle basis (single-particle dynamics are chaotic). Averages: There are three (two) invariants in the orthogonal (unitary) symmetry:

5. The eigenstates factorizes into two parts: (i) are zero-spin eigenstates of the reduced BCS Hamiltonian in a subset of doubly-occupied and empty levels. Eigenstates of the universal Hamiltonian: (ii) are eigenstates of, obtained by coupling spin-1/2 singly-occupied levels in to total spin and spin projection. Example of 8 electrons in 7 levels: 3 pairs plus 2 singles = blue levels = red levels

6. Quantum phase diagram (ground-state spin) Exact solution: there is a coexistence regime of superconducting and ferromagnetic correlations ( ). Mean field (S-dependent BCS): lowest solutions with do not have pairing correlations (gap is zero). S. Schmidt, Y.A., K. van Houcke, Europhys. Lett. 80, 47004 (2007) [ Ying et al, PRB 74, 012503 (2006)] Ground-state spin in the plane (for an equally-spaced single-particle spectrum)

7. A Zeeman field broadens the coexistence regime and makes it accessible to typical values of For a fixed the spin increases by discrete steps as a function of Controling the coexistence regime: a Zeeman field Stoner staircase Spin jumps: the first step can have (Ground-state spin versus ) Experiments: it is difficult to measure the ground-state spin.

8. Quantum dots: CI model Conductance peak height (for  <<T<<  ) is the partial width of the single-electron resonance to decay into the left (right) lead:    r c  2 where r c is the point contact. follows RMT wave function statistics. Exp: Chang et al. PRL 76 1695 (1996) Exp: Folk et al., PRL 76 1699 (1996) [ R. Jalabert, A.D. Stone, Y. Alhassid, PRL 68, 3468 (1992)]. Transport: Coulomb blockade conductance Peak height distributions

9. Quantum dots: charging + exchange correlations [ [ Y. Alhassid and T. Rupp, PRL 91, 056801 (2003)] Excellent agreement of theory and experiment for the peak spacing width  (  2 ) Conductance Peak spacings  2 Conductance Peak spacings  2 Conductance peak heights g max Better quantitative agreement for the ratio at Excellent agreement for peak height distribution at Experiments: C.M. Marcus et al. (1998) (Exchange constant = )

10. Nano-size metallic grains: charging, exchange + pairing correlations For a grain weakly-coupled to leads we can use the rate equation formalism plus linear response in the presence of interactions [ Alhassid, Rupp, Kaminski, Glazman, PRB 69, 115331 (2004)]. The linear conductance is calculated from the many-body energies of the dot and the lead-grain tunneling rates between many-body eigenstates of the N-electron grain and of the (N+1)-electron grain. Only a single level contributes: (i)The electron tunnels into an empty level and blocks it: (ii)The electron tunnels into a singly-occupied level : S. Schmidt and Y. Alhassid, arXive: 0802.0901, PRL, in press (2008)

11. Mesoscopic fluctuations of the conductance peaks (i) Peak-spacing statistics ( ) Peak-spacing distributions Exchange suppresses bimodality while pairing enhances it. Average peak spacing Single-particle energies and wave functions described by random matrix statistics (GOE).

12. (ii) Peak-height statistics ( ) Peak-height distributions Exchange interaction suppresses the peak-height fluctuations. Mesoscopic signatures of coexistence of pairing and exchange correlations for and : bimodality of peak spacing distribution and suppression of peak height fluctuations. Peak height fluctuation width

13. Conclusion A nano-size chaotic metallic grain is described by the universal Hamiltonian a competition between superconductivity and ferromagnetism in a finite-size system. Quantum phase diagram (ground-state spin): coexistence regime of superconductivity and ferromagnetism. Transport : signatures of coexistence between pairing and exchange correlations in the mesoscopic conductance fluctuations. Effects of spin-orbit scattering in the presence of pairing and exchange correlations: g-factor statistics,… (time-reversal remains a good symmetry). [Spin-orbit + exchange: D. Gorokhov and P. Brouwer, PRB 69, 155417 (2004).] Open problems Experimental candidates : platinum ( ), vanadium ( ).