Study of the LOFF phase diagram in a Ginzburg-Landau approach G. Tonini, University of Florence, Florence, Italy R. Casalbuoni,INFN & University of Florence, Florence, Italy We explained the two tricritical points behavior in the two plane waves case. We provided a general Ginzburg-Landau approach which can be applied to more general cases Different crystalline structures have been analyzed by other authors (Bowers-Rajagopal, Combescot-Mora). The scenario coming from all these studies, is a cascade of different structures with the temperature. In particular the two plane waves case seems to be the most probable near the tricritical point while the face centered cube near T=0. But more work is needed. CONCLUSIONS Initially studied as a pairing mechanism for an electron superconductor with Zeeman splitting between spin-up and spin-down Fermi surfaces. Applied also to dense QCD with three flavors. Cooper pairs with non zero total momentum p -p+2q Fixed |q|~  pF arbitrary directions Rotational and traslational symmetry breaking ”Crystalline structure Two flavors problem: critical value Paring not allowed everywhere: → blocking and pairing regions The LOFF pairing geometry for a Cooper pair with momentum 2q. Green/red sphere=up/down quark Fermi surface. An up quark with momentum p near its Fermi surface coupled with a down quark with momentum -p+2q. Strongest coupling for up and down quark near the pink rings. Applications Compact stars  Explication of pulsar glitches: jumps of the rotational frequency due to the angular momentum stored and then suddenly released by the superfluid neutrons → from the interaction between the rigid crust and the vortices in the neutron superfluid → pinning of the vortices in the crust  Vortices in the nodes of the LOFF crystal  Existence of strange stars Ultracold Fermi gases  BEC: cold bosons → cold fermions (lithium-6 or potassium-40)  Feshbach resonance provides an attractive interaction between two different hyperfine states  Control the two different atomic densities  Expansion of the gas when the trap is switched off →spatial distribution of momenta  Observation of LOFF phase by the periodic modulation of the atom densities in the crystalline superfluid Single plane wave case Gap equation: Integrating for T=0:  blocking region for Minimizing the granpotential respect to q LOFF phase diagram. The transition between LOFF and normal phase is always second order. The transition between BCS and LOFF is first order. There is one tricritical point at T  0.32  0 Free energies for normal, BCS and LOFF Phase. The LOFF interval is [  1,  2 ]. THE PHASE DIAGRAM IN THE TWO PLANE WAVE CASE First order transition near T=T tric one more tricritical point! Second order near T=0  one more tricritical point! Ginzburg-Landau expansion Gap equation with propagator expansion = + + Introduction of the Matsubara frequencies Expansion around T=0 All the possible vectors configurations From symmetry considerations Study of the minima of the granpotential  three dimensional space! From the type of solution we find the nature (symmetric or broken) of the phase in every octant Suppose  and discuss the results in function of . Sum over all the vectors configurations From the study of the second derivative and the equation  =0 we find the first order and second order surfaces SECOND ORDER SURFACE SECOND ORDER SURFACE : a=0 between octants 1-2 and, in part, 4-3 FIRST ORDER SURFACE FIRST ORDER SURFACE : D=0, where D is the discriminant of the cubic equation  =0 Two tricritical lines: Second tricritical point: when the second order line on the plane  =0 meets the tricritical line given by D=0  the position of the tricritical point in the phase space is  -dependent Temperature, momentum and chemical potential of the tricritical point respect to  What is LOFF phase? hep-ph/ to be published on Phys. Rev. B