Multifractal superconductivity Vladimir Kravtsov, ICTP (Trieste) Collaboration: Michael Feigelman (Landau Institute) Emilio Cuevas (University of Murcia) Lev Ioffe (Rutgers) Seattle, August 27, 2009
Download ReportTranscript Multifractal superconductivity Vladimir Kravtsov, ICTP (Trieste) Collaboration: Michael Feigelman (Landau Institute) Emilio Cuevas (University of Murcia) Lev Ioffe (Rutgers) Seattle, August 27, 2009
Multifractal superconductivity Vladimir Kravtsov, ICTP (Trieste) Collaboration: Michael Feigelman (Landau Institute) Emilio Cuevas (University of Murcia) Lev Ioffe (Rutgers) Seattle, August 27, 2009 Superconductivity near localization transition in 3d 3d lattice tight-binding model with diagonal disorder Local tunable attraction f ( i ) H r r ',r V r ,r aˆ Hint H int 0 r (r ) (r ) (r ) (r ) W / 2 NO Coulomb interaction Relevant for cold atoms in disordered optical lattices W / 2 i Cold atoms trapped in an optical lattice Fermionic atoms trapped in an optical lattice speckles Disorder is produced by: Other trapped atoms (impurities) New possibilities for superconductivity in systems of cold atoms 1d localization is experimentally observed [J.Billy et al., Nature 453, 891, (2008); Roati Strong controllable disorder, et al., Nature 453, 895 (2008)] of the isAnderson realization 2d and 3d localization on the way model Tunable short-range interaction between atoms: superconductive properties vs. dimensionless attraction constant No long-range Coulomb interaction Q: What does the strong disorder do to superconductivity? A1: disorder gradually kills superconductivity A2: eventually disorder kills superconductivity but before killing it enhances it Why superconductivity is possible when singleparticle states are localized Single-particle conductivity: only states in the energy strip ~T near Fermi energy contribute Superconductivity: states in the energy strip ~D near the Fermi-energy contribute Interaction sets in a new scale D which stays constant as T->0 R(T) 1 R (T ) 0T x x R(Tc ) 1/ 3 x R(Tc ) R(D) R(D) ~ R(Tc ) R(T 0) Weak and strong disorder Anderson transition disorder W /V L Extended states Critical states ~ F Localized states Multifractality of critical and off-critical states | (r ) | i r W>Wc W<Ec 2n 1 Ld n ( n1) Matrix elements Interaction comes to play via matrix elements M nm V d r (r ) (r ) d 2 n 2 m Ideal metal and insulator M nm Vd r n (r ) m (r ) 2 d Metal: Insulator: 1 V V V V x d 1 xd 1 1 V xd d x V 1 1 2 Small amplitude 100% overlap Large amplitude but rare overlap Critical enhancement of correlations | E E'|1d2 / d Amplitude higher than in a metal but almost full overlap States rather remote (\E-E’|<E0) in energy are strongly correlated Simulations on 3D Anderson model ( E0 030 ) 1 ~ D Wc Wc x F Wc W x ( 0x C(E E' ) V d d r n (r ) m (r ) ( En E ) ( Em E ' ) n,m 2 2 (E n E ) ( Em E ' ) n,m ) d 1 Wc=16.5 W=10 Multifractal metal: x l 0 Critical power law persists 1 d 2 / d 0 ~ FWc1/ 3 E0 E E' W=5 W=2 Ideal metal: x l 0 x E0 Superconductivity in the vicinity of the Anderson transition Input: statistics of multifractal states: scaling (diagrammatics and sigma-model do not work) How does the superconducting transition temperature depend on interaction constant and disorder? Mean-field approximation and the Anderson theorem D(r ) dr' D(r ' ) K (r , r ' ; T ) K (r , r ' ; T ) U ij * * ( T ) ( r ) ( r ) ( r ' ) ij 1/V i j j i (r ' ) ij ij (T ) tanh(Ei / 2T ) tanh(E j / 2T ) Ei E j Wavefunctions drop out of the equation Anderson theorem: Tc does not depend on properties of wavefunctions What to do at strong disorder? D(r) cannot be averaged independently of K(r,r’) Fock space instead of the real space Single-particle states, strong disorder included Heff 2 i Si U M ij (Si S j Si S j ) z i EF i j S a a EF 1 S z (a a a a 1) 2 Superconducting phase Normal phase Six, y 0 Six, y 0 Why the Fock-space mean field is better than the real-space one? H eff 2 i Si M ij (Si S S S ) z x x j i j i M ij dr i2 (r )j2 (r ) i y i y j j Infinite or large coordination number for extended and weakly localized states Weak fluctuations of Mij due to space integration Di D j j tanh(E j / 2T ) Ej M ij MF critical temperature close to At a small critical disorder parametrically large enhancement of Tc tanh(E ' / 2T ) D( E ) dE' D( E ' ) M (E E' ) E' 1 d 2 / d E0 M ( E E ' ) | E E '| Tc ~ E0 1/(1d 2 / d ) ~ E0 1.78 >> D exp 1 / M.V.Feigelman, L.B.Ioffe, V.E.K. and E.Yuzbshyan, Phys.Rev.Lett. v.98, 027001 (2007); Thermodynamic phase fluctuations Only possible if off-diagonal terms like M ijkl i (r )j (r )k (r )l (r ) dr are taken into account Ginzburg parameter Gi~1 Expecting DTc / Tc ~ 1 Cannot kill the parametrically large enhancement of Tc The phase diagram Tc Mobility edge Extended states BCS 0.2 Localized states (Disorder) BUT… Sweet life is only possible without Coulomb interaction 2 e r Virial expansion method of calculating the superconducting transition temperature Heff 2 i Si U M ij (Si S jx Siy S jy ) z i j i (Tc ) n (Tc ) n Replacing by: x limn n 1 (Tc ) 1 n (Tc ) 2 (Tc ) 3 (Tc ) Operational definition of Tc for numerical simulations Tc at the Anderson transition: MF vs virial expansion M.V.Feigelman, L.B.Ioffe, V.E.K. and E.Yuzbshyan, Phys.Rev.Lett. v.98, 027001 (2007); | E E' | 1/ Tcrit c E0 , 1 d2 / d MF result Virial expansion 2d Analogue: the MayekawaFukuyama-Finkelstein effect The diffuson diagrams The cooperon diagrams Superconducting transition temperature Virial expansion on the 3d Anderson model metal insulator Anderson localization transition (disorder) Conclusion: Enhancement of Tc by disorder metal insulator Maximum of Tc in the insulator Direct superconductor to insulator transition BUT Fragile superconductivity: Small fraction of superconducting phase Critical current decreasing with disorder Two-eigenfunction correlation in 3D Anderson model (insulator) NC ( ) ln d 1 x Mott’s resonance physics Ideal insulator limit only in one-dimensions | E E'|1d2 / d x critical, multifractal physics Superconductor-Insulator transition: percolation without granulation Coordination number K>>1 SC Tc Tc (crit ) x x Only states in the strip ~Tc near the Fermi level take part in superconductivity Coordination number K=0 Tc (crit ) x INS Tc x First order transition? K (Tc )M (Tc )U ~ Tc x T Tc KMU ~ (T / x ) f (T / x ) ( E0 / T ) x ~ ~ T0 (T / T0 ) d2 / d f (T / x ) T T f(x)->0 at x<<1 Conclusion Fraclal texture of eigenfunctions persists in metal and insulator (multifractal metal and insulator). Critical power-law enhancement of eigenfunction correlations persists in a multifractal metal and insulator. Enhancement of superconducting transition temperature due to critical wavefunction correlations. T M IN BCS limit K (Tc )M (Tc )U ~ Tc SC SC or IN SC disorder Anderson localization transition KMU x T T KMU ~ (T / x )(E0 / T ) x ~ T0 (T / T0 )d2 / d Corrections due to off-diagonal terms Dij U D kl kl (T ) M ijkl kl D i U D l l (T ) M il U km M iikmM llkm l k , k m Average value of the correction term increases Tc Average correction is small when 3d 4 2 d 2 d d 2 1 Melting of phase by disorder D(r ) D (r ) (r ) ij ij ij In the diagonal D(r ) approximation i j 2 D t anh( E / 2 T ) i i i (r ) i The sign correlation <D(r)D(r’)> is perfect : solutions Di>0 do not lead to a global phase destruction Beyond the D U D (T ) l l l i diagonal approximation: M il U km M iikmM llkm k , k m stochastic term destroys phase correlation How large is the stochastic term? Qil U km M iikmM llkm k ,k m Stochastic term: d2<d/2 2 il Q ~N 1 2 d 2 / d Qi2l Qil Qil 2 ~ N 2 2 d 2 d2>>d/2 weak oscillations d d2 < d/2 strong oscillations / but still too small to support 2 the glassy solution More research is needed Conclusions Mean-field theory beyond the Anderson theorem: going into the Fock space Diagonal and off-diagonal matrix elements Diagonal approximation: enhancement of Tc by disorder. Enhancement is due to sparse single-particle wavefunctions and their strong correlation for different energies Off-diagonal matrix elements and stochastic term in the MF equation The problem of “cold melting” of phase for d2 <d/2