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Mott-Berezinsky formula, instantons, and integrability Ilya A. Gruzberg In collaboration with Adam Nahum (Oxford University) Euler Symposium, Saint Petersburg, Russia, July 8 th, 2011 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

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Anderson localization Single electron in a random potential (no interactions) Ensemble of disorder realizations: statistical treatment Possibility of a metal-insulator transition (MIT) driven by disorder Nature and correlations of wave functions Transport properties in the localized phase: - DC conductivity versus AC conductivity - Zero versus finite temperatures Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

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Weak localization Qualitative semi-classical picture Superposition: add probability amplitudes, then square Interference term vanishes for most pairs of paths D. Khmelnitskii ‘82 G. Bergmann ‘84 R. P. Feynman ‘48

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Weak localization Paths with self-intersections - Probability amplitudes - Return probability - Enhanced backscattering Reduction of conductivity

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Strong localization As quantum corrections may reduce conductivity to zero! Depends on nature of states at Fermi energy: - Extended, like plane waves - Localized, with - localization length P. W. Anderson ‘58 Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

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Localization in one dimension Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 All states are localized in 1D by arbitrarily weak disorder Localization length = mean free path All states are localized in a quasi-1D wire with channels with localization length Large diffusive regime for allows to map the problem to a 1D supersymmetric sigma model (not specific to 1D) Deep in the localized phase one can use the optimal fluctuation method or instantons (not specific to 1D) N. F. Mott, W. D. Twose ‘61 D. J. Thouless ‘73 D. J. Thouless ‘77 K. B. Efetov ‘83

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Optimal fluctuation method for DOS Tail states exist due to rare fluctuations of disorder Optimize to get DOS in the tails Prefactor is given by fluctuation integrals near the optimal fluctuation I. Lifshitz, B. Halperin and M. Lax, J. Zittartz and J. S. Langer

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Mott argument for AC conductivity Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 Apply an AC electric field to an Anderson insulator Rate of energy absorption due to transitions between states (in 1D) Need to estimate the matrix element N. F. Mott ‘68

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Mott argument Consider two potential wells that support states at The states are localized, and their overlap provides mixing between the states Diagonalize Minimal distance Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

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Mott-Berezinsky formula Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 Finally In dimensions the wells can be separated in any direction which gives another factor of the area: First rigorous derivation has been obtained only in 1D For large positive energies (so that ) Berezinsky invented a diagrammatic technique (special for 1D) and derived Mott formula in the limit of “weak disorder” V. L. Berezinsky ‘73

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Supersymmetry and instantons Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 Write average DOS and AC conductivity in terms of Green’s functions, represent them as functional integrals in a field theory with a quartic action For large negative energies (deep in the localized regime) the action is large, can use instanton techniques: saddle point plus fluctuations near it Many degenerate saddle points: zero modes Saddle point equation is integrable, related to a stationary Manakov system (vector nonlinear Schroedinger equation) Integrability is crucial to find exact two-instanton saddle points, to control integrals over zero modes, and Gaussian fluctuations near the saddle point manifold Reproduced Mott formula in the “weak disorder” limit R. Hayn, W. John ‘90

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Other results in 1D and quasi 1D Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 Other correlators involving different wave functions Correlation function of local DOS in 1D Correlation function of local DOS in quasi1D from sigma model Something else? L. P. Gor’kov, O. N. Dorokhov, F. V. Prigara ‘83 D. A. Ivanov, P. M. Ostrovsky, M. A. Skvortsov ‘09

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Our model Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 Hamiltonian (in units ) Disorder Same model as used for derivation of DMPK equation Assumptions: - saddle point technique requires - small frequency - “weak disorder”

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Some features and results Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 Saddle point equations remain integrable, related to stationary matrix NLS system Two-soliton solutions are known exactly (Two-instanton solutions that we need can also be found by an ansatz) The two instantons may be in different directions in the channel space, hence there is no minimal distance between them! Nevertheless, for we reproduce Mott-Berezinsky result Specifically, we show F. Demontis, C. van der Mee ‘08

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Calculation of DOS: setup Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 Average DOS Green’s functions as functional integrals over superfields is a vector (in channel space) of supervectors

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Calculation of DOS: disorder average Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 After a rescaling (In the diffusive case (positive energies) one proceeds by decoupling the quartic term by Hubbard-Stratonovich transformation, integrating out the superfields, and deriving a sigma model)

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Calculation of DOS: saddle point Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 Combine bosons into Rotate integration contour The saddle point equation Saddle point solutions (instantons) The centers and the directions of the instantons are collective coordinates (corresponding to zero modes) The classical action does not depend on them

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Calculation of DOS: fluctuations Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 Expand around a classical configuration: has a zero mode corresponding to rotations of has a zero mode corresponding to translations of, and a negative mode with eigenvalue

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Calculation of DOS: fluctuation integrals Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 Integrals over collective variables Integrals over modes with positive eigenvalues give scattering determinants Grassmann integrals give the square of the zero mode of Integral over the negative mode of gives Collecting everything together gives given above

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Calculation of the AC conductivity Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 is much more involved due to appearance of nearly zero modes Need to use the integrability to determine exact two-instanton solutions and zero modes Surprising cancelation between fluctuation integrals over nearly zero modes and the integral over the saddle point manifold In the end get the Mott-Berezinsky formula plus ( -dependent) corrections with lower powers of

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Conclusions Euler Symposium, Saint Petersburg, Russia, July 8th, 2011 We present a rigorous and conreolled derivation of Mott-Berezinsky formula for the AC conductivity of a disordered quasi-1D wire in the localized tails Generalizations to higher dimensions Generalizations to other types of disorder (non-Gaussian) Relation to sigma model

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