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Quantum Condensation: Disorder and Instability

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Quantum Condensation: Disorder and Instability Quantum Condensation: Disorder and Instability Boris Altshuler Physics Department, Columbia University The 6th Windsor Summer School Low-Dimensional Materials, Strong Correlations and Quantum Technologies Great Park, Windsor, UK, August 14 - 26, 2012 1.Bose Condensation in the presence of disorder Bose-Einstein Condensation Macroscopic occupation of a single quantum state 0K 100nK 200 nK Size of this preview: 800 × 526 pixels. Other resolutions: 320 × 210 pixels | 640 × 421 pixels | 1,024 × 673 pixels | 1,280 × 841 pixels. Bose–Einstein condensation: 2D velocity distribution of Rb atoms at different temperatures, JILA Science, 1995 Bose-Einstein Condensation Macroscopic occupation of a single quantum state 1. External Potential Q:2. Interaction between the particles ? Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. Hänsch & Immanuel Bloch “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms” Nature 415, 39-44 (3 January 2002) Superfluid Mott Insulator In the After release of lattice the potential Bose-Einstein Condensation Macroscopic occupation of a single quantum state ? 1. External Potential 2. Interaction between Q: the particles ? In general the problem of bosons subject to an external potential is pathological: at zero temperature all of them will find themselves in the one-particle ground state even if it is a localized one. Even weak interaction is relevant! Bose-Einstein Condensation Disorder – Localized one-particle states Weak Interaction Need tunneling Macroscopic occupation of a single quantum state Global Phase Coherence – single wave function Bose-Einstein Condensation rt, Wave function of the condensate Gross-Pitaevskii equation 2 rt, 2 2 i V r g r,, t r t tm2 VrExternal potential g Coupling constant - smooth Short Range interaction Localized one-particle states? Condensation centers Discrete version of the GP equation Gross-Pitaevskii equation Discrete version: 2 rt, 2 2 ir, t t“Wave V functions” r g at r,, tdifferent r t tmi 2 condensation centers t 2 ii r g t t J t i i i ij i t ji “Josephson coupling” between the Jij condensation centers i and j One-particle energy at the condensation center i i Gross-Pitaevskii equation Discrete version: “Wave functions” at i t different condensation centers (CC) “Josephson coupling” between CC i and j One-particle energies t 2 i 2 iiSmall r g quantum i t i t J ij Only i t phase Large t i t fluctuations of ji matters J ij ii iit t e XY spin model i Superfluid – Insulator transition Small quantum Only phase Large fluctuations of matters XY spin model Ordered phase – Superfluid Disordered phase – Insulator 2 Reason fori tthe transition quantum fluctuations of the phase due to the onsite interaction Energy scales in theii problem: • iiCouplingt t e • Dispersion in the one-particle energies • Interaction energy Two well problem I 2 1 Hamiltonian I E 0 ˆ 1 ˆ 1 H diagonalize H I 2 0 E2 2 2 E2 E1 2 1 I diagonalize 2 2 2 1 2 1 I E2 E1 2 1 I I 2 1 I I E 0 ˆ von1 Neumann & Wigner “noncrossingˆ rule”1 H H Level repulsion I 2 0 E2 v. Neumann J. & Wigner E. 1929 Phys. Zeit. v.30, p.467 What about the eigenfunctions ? What about the eigenfunctions ? I ,;,,;, 2 2 2 EE1 2 1 E2 E1 1 1 2 21 2 I 1 1 2 2 I 2 1 I 2 1 I 2 1 I 1 I I Hˆ O 1,2 1,2I 2,1 22 1 1,2 1,2 2,1 Off-resonance Resonance Eigenfunctions are In both eigenstates the close to the original on- probability is equally site wave functions shared between the sites Anderson • Lattice - tight binding model • Onsite energies - random Model i • Hopping matrix elements I j i ij Iij I i and j are nearest neighbors -W < <W I = i ij { uniformly distributed 0 otherwise Anderson Transition Ic f dW I < I c I > Ic Insulator Metal All eigenstates are localized There appear states extended Localization length x all over the whole system Anderson insulator Anderson metal Few isolated resonances There are many resonances and they overlap Transition: Typically each site is in the resonance with some other one Energy scales (Thouless, 1972) 1 d Mean level spacing 1. 1 L L L system size d # of dimensions 2 2. Thouless energy ET 2 DL D diffusion constant ET has a meaning of the inverse diffusion time of the traveling through the system or the escape rate. (for open systems) dimensionless 2 Thouless g2 e G R G gE T 1 conductance Q Superfluid – Insulator transition Tunneling amplitude J Energy needed for the tunneling Ec “charging energy” JE c Ordered phase – Superfluid JE c Disordered phase – Insulator Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. Hänsch & Immanuel Bloch “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms” Nature 415, 39-44 (3 January 2002) Superfluid JE c Mott JE c Insulator In the After release of lattice the potential 2. Superconductor-Insulator Transition in two dimensions Superconductor-Insulator Transition R Bi R h RRQ cQge4 2 ? c ? Rc D. B. Haviland, Y. Liu and A. M. Goldman, Phys. Rev. Lett., 62, 2180-2183, (1989). T ˆ H BCS aa BCS aaa a , , , P.W. Anderson: Anderson spin chain Phys. Rev. 112,1800, 1958 1 Kˆ z a a 1 Kˆ a a Kˆ a a SU2 algebra 2 , spin 1/2 ˆ ˆ z ˆ ˆ H BCS K BCS K K , , , z x normal 0 BCS (paired) 0 Ovchinnikov 1973, Disorder-caused corrections to Tc, D Suppression of Superconductivity Maekawa & Fukuyama 1982 Finkelshtein 1987 by Coulomb interaction 1.Quantum corrections in normal metals in 2D Weak localization, e-e interactions g 1 ln 2 Dimensionless gg g 2 Thouless 4e conductance mean free time frequency, infrared cutoff 2. Correction to T c due to the Coulomb repulsion 3 Tc Dexp 1 BCS TTc BCS1 c 2 ln Tgc0 BCS 1 BCS ln BCS g } D 1 Anderson Theorem Neither superconductor order parameter D nor Provided that transition temperature D is homogenous in space Tc depend on disorder Corrections to the Anderson theorem – due to inhomogeneity in D 3 Tc 1 log Tc Tc g Tc 1 Interpretation : Tc D exp BCS Dimensionless SC temperature BCS coupling Debye temperature constant 1 Tc BCS ln D # 1 Pure BCS interaction 1 ln (no Coulomb repulsion): eff BCS g Tc Interpretation continued: “weak localization” logarithm Q: Why logarithm? A: Return and interference O # 1 1 ln eff BCS g Tc •In the universal ( g ) limit the effective coupling constant equals to the bare one - Anderson theorem •If there is only BCS attraction, then disorder increases T and D by optimizing1 spatial c T ! dependence of D. c BCS ln D Tc and D reach maxima at the point of Anderson localization # 1 1 ln Problem in conventionaleff BCSsuperconductors: Coulomb Interaction g Tc Interpretation continued: Coulomb Interaction Anderson theorem - the gap is homogenous in space. Without Coulomb interaction adjustment of the gap to the random potential strengthens superconductivity. Homogenous gap in the presence of disorder violates electroneutrality; Coulomb interaction tries to restore it and thus suppresses superconductivity # eff BCS ln g Tc Perturbation theory: 2 TcD11 2 ln ln Tc BCS g T c T c Ovchinnikov 1973, Disorder-caused corrections to Tc, D Suppression of Superconductivity Maekawa & Fukuyama 1982 Finkelshtein 1987 by Coulomb interaction 2. Correction to T c due to the Coulomb repulsion 3 TT 1 c BCSln c Tg2 c0 BCS } D 1 3. One can sum up triple log corrections neglecting single log terms, i.e. neglecting effects of Anderson localization. ! Tc Dexp 1 BCS BCS 1 ln BCS g Ovchinnikov 1973, Disorder-caused corrections to Tc, D Suppression of Superconductivity Maekawa & Fukuyama 1982 Finkelshtein 1987 by Coulomb interaction 3 TT 1 One can sum up triple log c BCS c corrections neglecting 2 ln Tglocalization effects. c0 BCS 4. g 2 Finkelshtein (1987) g ln renormalization group T c0 Tc Aleiner (unpublished) g ln BCS-like mean field Tc0 5. 2 1 gTln 0 T 2 c c0 BCS Ovchinnikov 1973, Disorder-caused corrections to Tc, D Suppression of Superconductivity Maekawa & Fukuyama 1982 Finkelshtein 1987 by Coulomb interaction 4. 5. # gT2 c 0 BCS g 2 g ln 1 T RQ g c10 R R Tc c2 c Q BCS g c g ln Tc0 Conclusion: critical resistance is much smaller than the RR quantum resistance cQ! QUANTUM PHASE TRANSITION Theory of Dirty Bosons Fisher, Grinstein and Girvin 1990 Wen and Zee 1990 Fisher 1990 Only phase fluctuations of the order parameter are important near the superconductor - insulator transition CONCLUSIONS 1. Exactly at the transition point and at T 0 conductance 2 tends to a universal value gqc 4e gKqc 6 ?? h 2. Close to the transition point magnetic field and temperature dependencies demonstrate universal scaling Granular films. 0 Single grain: charging energy E c one-particle mean level spacing 1 SC transition temperature Tc0 SC gap D 2D array: tunneling conductance gt 1 dwell time esc g t 1 N RQ normal state sheet resistance R gt Below T : Josephson coupling EgDAmbegoakar & c0 JtBaratoff (1963) Fisher, Grinstein & Girvin 1990 Quantum Phase Transition Wen and Zee 1990 Dirty Boson Theory D ∞ Fisher 1990 Only phase fluctuations 1. Exactly at the transition point and at T 0 conductance tends to a universal value gc 1 h Rc g c R Q R Q 2 6 K ?? 4e 2. Close to the transition point magnetic field and temperature dependencies demonstrate universal scaling Superconductor-insulator transition in granular films.
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