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Level statistics and self-induced decoherence in disordered spin-1/2 systems

Level statistics and self-induced decoherence in disordered spin-1/2 systems. Mikhail Feigel’man L.D.Landau Institute, Moscow. In collaboration with: Lev Ioffe Rutgers University Marc Mezard Orsay University

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Level statistics and self-induced decoherence in disordered spin-1/2 systems

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  1. Level statistics and self-induced decoherence in disordered spin-1/2 systems Mikhail Feigel’man L.D.Landau Institute, Moscow In collaboration with: Lev Ioffe Rutgers University Marc Mezard Orsay University Emilio Cuevas University of Murcia Previous publications on the related subjects: Phys Rev Lett.98, 027001(2007) (M.F.,L. Ioffe,V. Kravtsov, E.Yuzbashyan) Annals of Physics325, 1368 (2010) (M.F.,L.Ioffe, V.Kravtsov, E.Cuevas) Phys.Rev. B 82, 184534 (2010)(M.F.,L. Ioffe, M. Mezard) Nature Physics, 7, 239 (2011) (B.Sacepe,T.Doubochet,C.Chapelier,M.Sanquer, M.Ovadia,D.Shahar, M. F., L..Ioffe)

  2. Plan of the talk • Superconductivity with pseudogap and effective spin-1/2 model 2.Bethe lattice model of quantum phase transition. Critical lines from the analitical solution 3. Level statistics on small random graph: exact numerical diagonalization. 4. Summary of results

  3. On insulating side (far enough): Kowal-Ovadyahu 1994 D.Shahar & Z.Ovadyahu amorphous InO 1992 T0 = 15 K R0 = 20 kW

  4. Amorphous InOx films Nature Physics, 7, 239 (2011) SC side: local tunneling conductance Superconductive state with a pseudogap: Fermi-level in the localized band Superconductive state near SIT is very unusual: The spectral gap appears much before (with T decrease) than superconductive coherence does Coherence peaks in the DoS appear together with resistancevanishing Distribution of coherence peaks heights is very broad near SIT

  5. Single-electron states suppressed bypseudogap ΔP >> Tc “Pseudospin” approximation 2eV1 = 2Δ Andreev point-contact spectroscopy eV2 = Δ+ ΔP

  6. arXiv:1011.3275 Nature Physics 2011

  7. S-I-T: Third Scenario • Bosonic mechanism: preformed Cooper pairs + competition Josephson v/s Coulomb – S I T in arrays • Fermionic mechanism: suppressed Cooper attraction, no pairing – S M T • Pseudospin mechanism: individually localized pairs - S I T in amorphous media SIT occurs at small Z and lead to paired insulator How to describe this quantum phase transition ? Bethe lattice model is solved Phys.Rev. B 82, 184534 (2010) M. Feigelman, L.Ioffe, M. Mezard

  8. Distribution function for the order parameter General recursion: Linear recursion (T=Tc) Diverging 1st moment Solution in the RSB phase: T=0

  9. Vicinity of the Quantum Critical Point << 1

  10. Order parameter: scaling near transition Typical value near the critical point: (at T=0) Near KRSB:

  11. K = 4

  12. Insulating phase: continuous v/s discrete spectrum ? Consider perturbation expansion over Mij in H below: Within convergence region the many-body spectrum is qualitatively similar to the spectrum of independent spins No thermal distribution, no energy transport, distant regions “do not talk to each other” What will happen when Mij are increasing ?

  13. Recursion relations for level widths Spectral function of external noise We look for the distribution function of the form

  14. Threshold energy at T=0 Low-energy limit ω << 1 Full band localization ω = 1 Now set T>0. What happens to level width at low excitation energies ?

  15. Threshold for activated transport Nonzero line-width appears above threshold frequency only: This is T = 0 result ! Nonzero activation energy for transport of pairs is due to the absence of thermal bath at low ω Nonzero but low temperatures: Activation law

  16. What else could one expect? Phase diagram Major feature: green and red line meet at zero energy Temperature Energy Hopping insulator Superconductor Full localization: Insulator with Discrete levels MFA line RSB state g gc

  17. Phase diagram-version 2 Here green and red line do not meet at zero energy Temperature Do gapless delocalized excitations exist WITHOUT Long-range order ? Energy Hopping insulator with Mott (or ES) law Full localization: Insulator with Discrete levels Superconductor g gc2 gc1

  18. Phase diagram-version 3 Here green and red line cross at non-zero energy: first-order transition?? Temperature Energy Full localization: Insulator with Discrete levels Superconductor g gc

  19. Major results from Bethe lattice study - Full localization of eigenstates with E ~ W at weakest coupling between spins, g < g*(or K < K*(g)) -No intermediate phase without bothorder parameter and localization of low-energy modes Questions: • what about highly excited states with E >> W • how universal is the absence of intermediate phase ? • How to avoid the use of Bethe lattice ?

  20. Different definitions for the fully many-body localized state • 1.    No level repulsion (Poisson statistics of the full system spectrum) • 2.    Local excitations do not decaycompletely • 3.    Global time inversion symmetry is not broken (no dephasing, no irreversibility) • 4.   No energy transport (zero thermal conductivity) • 5. Invariance of the action w.r.t. local time transformations t → t + φ(t,r): d φ(t,r)/dt = ξ (t,r) – Luttinger’s gravitational potential

  21. Level statistics: Poisson v/s WD • Discrete many-body spectrum with zero level width: Poisson statistics • Continuous spectrum (extended states) : Wigner-Dyson ensemble with level repulsion V.Oganesyan & D.Huse Phys. Rev. B 75, 155111 (2007) Model of interacting fermions (no-conclusive concerning sharp phase transition)

  22. Numerical results for random Z=3 graphE.Cuevas (Univ. of Murcia, Spain) Low-lying excitations, Jc=0.10 Middle of the band, Jc = 0.07

  23. Density of States T = (dS/dE)-1 = # (E/Ns)1/2 E ~ Ns T2 (at T << 1)

  24. Phase diagram Red line: analytical theory Orange ovals: numerical data Green line: correction of analytical result for finite-size effect

  25. Role of Jz Siz Sjz interaction J = 0.1 0.07 → 0.02 (midband) 0.10 → 0.075 (low E) What is the reason for such a strong effect?

  26. Original model: XY exchange + transverse field Full model with Sz-Sz coupling Summation over large number of configurations with different makes it easier to meet resonant conditions Conclusion: critical coupling g depends on temperature, i.e. on E/Ns

  27. SzSz interaction results in temperature-controlled transition to the state with zero level widths and zero conductivity Basko et al 2006, Mirlin et al 2006 T T ~ (E/Ns)1/2 Г > 0 Г = 0 g g1 g2

  28. Conclusions New type of S-I phase transition is described On insulating side activation of pair transport is due to ManyBodyLocalization threshold Resultsfrom level statistics studies support general shape of the phase diagram, but the possibility of intermediate phase cannot be excluded this way Interaction in the “density channel” is crucially important for the shape of the phase diagram at extensive energies E ~ Ns

  29. Open problems - Analitical study of energy localization in Euclidean space or RGM: order parameter ? anything to do with compactification of space and black holes ? - Is it possible to modify the model in a way to find an intermediate phase or 1st order? - How to calculate electric and thermal conductivities directly within recursion relations approach?

  30. The End

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