Nouveaux régimes de fluctuations comme signatures de la transition de phase vers un état FFLO.

1. Nouveaux régimes de fluctuations comme signatures de la transition de phase vers un état FFLO. Anomalous fluctuation regimes at the FFLO transition (submitted to PRL 13.12.2006) Alexandre BUZDIN, Jérôme CAYSSOL and François KONSCHELLE Condensed Matter Theory Group, University of Bordeaux - CPMOH, Talence, France GDR NEEM Supraconductivité 18-19 décembre 2006 Laboratoire Léon Brillouin, Orsay

2. Inhomogeneous FFLO phases B Superconducting phase induced by splitting effects : • Zeeman pair breaking effect • Orbital pair breaking effect neglected • Singlet pairing • External field (heavy fermions) • Internal exchange field • 2D superconductors Non-zero momentum of Cooper Pairs : Spatial modulation. Refs. How to detect modulated phase ? B

3. Modified Ginzburg-Landau approach for FFLO phases Spatial modulation of Cooper pair density and with : Homogeneous phase : Ginzburg-Landau approach : Non-Gaussian term never changes sign Microscopic theory with Zeeman effect : Higher gradient terms in Ginzburg-Landau approach Gaussian approximation of modified Ginzburg-Landau functional Refs. How does this functional affect the fluctuation behaviours ?

4. Properties of modulated phases Tricritical point = vanishing stiffness Negative stiffness of the condensate Strong anisotropy between and for quasi-2D systems behaviours Changes in order of phase transition Many mean field studies Few fluctuational studies More complex phase diagram : Motivation : Mean field vs Fluctuations Large region / experimental evidence facilities of Gaussian fluctuations • - Is it possible to provide a powerful tool to characterize FFLO phase ? • Role(s) of vanishing stiffness in a general discussion ? Refs.

5. Gaussian fluctuations near homogeneous phase Free energy : Critical region Specific heat : C Cv Paraconductivity : Tc for 2D case Very small critical region and great Gaussian fluctuational region Physical viewpoint : Fluctuation around an isolated point : low energy modes : Levanyuk-Ginzburg criterion Quadratic spectrum : Magnetic susceptibility : Refs.

6. Gaussian fluctuations near isotropic FFLO state Fluctuations on a ring near FFLO wave vector Unidimensional fluctuations Enhancement of fluctuations for higher dimension : 2D and 3D adding terms in fluctuation spectrum Mean field treatment : - enhanced critical temperature : - wave vector of modulation : Canonical form : Specific heat : Paraconductivity : Physical viewpoint : Presence of quartic terms in the fluctuation spectrum : Fluctuations behaviours : Stiffness dependence on dimension No temperature dependence on dimension Do these specific behaviours resist to an anisotropy ?

7. Anisotropy considerations The most general form for anisotropy in a modified Ginzburg-Landau approach : Two origins for the anisotropy : d-waves superconductivity : Anisotropy of Fermi surface : Oriented modulation along higher symetrical lines] Nesting region for FFLO stability conditions Q : Nesting vector q : FFLO vector Microscopic view point : Equivalent forms of the anisotropy Anisotropy appears only in quartic term Elliptical anisotropy can be easily reduced in isotropic form by a rescalling process Refs.

8. Gaussian fluctuations near anisotropic phase : weight of anisotropy with Mean field treatment : Favouring wave vector depends on the anisotropy weight near Diagonal development near to each equilibrium point : Paraconductivity : Specific heat : Same behaviours as for the homogeneous state ! Expected anomalous behaviours when : Exemple of cubic anisotropy : Fluctuations for an anisotropic FFLO state : What become these fluctuations close to the tricritical point ?

9. Expected characteristic behaviours Near both homogeneous and inhomogeneous + anisotropic cases : Vanishing stiffness causes some anomalous behaviours Higher divergences for specific heat and strong dependence on dimension for paraconductivity Complete schematic phase diagram : Very specific behaviours of fluctuations close to the tricritical point / line are expected to reveal FFO states…

10. Fluctuations close to the tricritical point Specific heat : Pure quartic dependence of fluctuation spectrum : Paraconductivity : : anisotropic numerical terms General anisotropic spectrum : Always higher than region 1 behaviour • In comparison with region 1 : • Higher in 3D • Same in 2D • Smaller in 1D Very specific behaviours of fluctuations may reveal FFLO phase !

11. Discussions of fluctuation magnetic susceptibility Vanishing diamagnetic contribution for Close to the FFLO phase : Paramagnetic and non divergent contribution ! Manifestation of FFLO state… Close to the tricritical point : Diamagnetic contribution for 3D systems Vanishing contribution for 2D systems ! From uniform phase : Experimental signature of FFLO phase !

12. Conclusion Fluctuation near each phase : Isotropic FFLO one-dimensional behaviour Anisotropic FFLO = Homogeneous phase Proximity of tricritical point : very specific of vanishing stiffness in a general context Fluctuations as a signature of FFLO state : T = Cte Experimental crossovers beween : h = Cte FFLO state II : Tricritical regime III : Submitted to PRL, http://arxiv.org/abs/cond-mat/0612387

13. Interplay between Abrikosov, Josephson and FFLO effects

14. Complete schematic phase diagram

15. Levanyuk-Ginzburg criterion Critical region C Cv Tc for 2D case Very small critical region and great Gaussian fluctuationnal region Levanyuk-Ginzburg criterion for homogeneous case Due to a specific heat jump : For FFLO state (2D for II° order phase transition) [18] : greater jump ! at the same point : the tricritical point ! Compensation by modulation ! is very small close to the tricritical point ! Ginzburg-Levanyuk criterion is not strongly affected by FFLO state !

16. Anisotropic factors Specific heat : Paraconductivity :