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Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model

Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model,10.1007/s00220-010-1117-5,Communications in Mathematical Physics,Margherita Disertori,To

Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model   (Citations: 9)
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We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a ‘diffusive’ phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green’s functions and a family of Ward identities coming from internal supersymmetry.
Journal: Communications in Mathematical Physics - COMMUN MATH PHYS , vol. 300, no. 2, pp. 435-486, 2010
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    • ...They proved a quasi-diffusive estimate for the two-point correlation functions in a three dimensional supersymmetric hyperbolic nonlinear sigma model at low temperatures [13]...

    László Erdoset al. Quantum Diffusion and Eigenfunction Delocalization in a Random Band Ma...

    • ...Finally, we mention a recent and exciting development: in [17], the existence of a delocalized phase in three dimensions is proven for a supersymmetric model which is interpreted as a toy version of the Anderson model...

    W. De Roecket al. Diffusion of a Massive Quantum Particle Coupled to a Quasi-Free Therma...

    • ...We refer to [2] for a historical introduction and motivations...
    • ...More recently the existence of a ‘diffusive’ phase at low temperatures (β large) has been proved for the H (2|2) model in three or more dimensions, see [2]...
    • ...Moreover, for a one dimensional chain we recover localization for all values of β. Localization is also expected in 2D (see [2] Sect...
    • ...The techniques employed in this work to prove localization are quite different from the ones used in [2] to prove extended states...
    • ...4 and Appendix C in [2] for an introduction to supersymmetric Ward identities...
    • ...Remark. Note that e t D ε e t corresponds to the symbol Dβ,� introduced in [2] Eq. (1.1)...
    • ...where A is the generator of a random walk in a random environment in [2 ]E q. (1.1)...
    • ...For technical reasons the above representation is more convenient when we want to prove diffusion as in [2] while the other is more practical when we study localization (except in the proof of Theorem 2 where we will go back to the “diffusion” representation for a while)...
    • ...where ξ , η are odd elements and z, xy are even elements of a real Grassmann algebra (see [2] for more details)...
    • ...Note that the change of coordinates (1.12) is different from the one introduced in Eq. (2.7) of [2]...
    • ...There the covariance of the gaussian fields in the action is (1.7) (“diffusion” representation) and the measure (see (2.12) in [2] ) has a � e −t factor...
    • ...We can go back from (1.12) to (2.7) of [2] by performing the change of coordinates: s → se +t ,ψ → ψe +t , ¯ ψ → ¯ ψe +t ...
    • ...We refer to [2] for a more detailed exposition...
    • ...Normalization and choice of ε j . By internal supersymmetry (see [2] Sect...

    M. Disertoriet al. Anderson Localization for a Supersymmetric Sigma Model

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