Supersymmetry and kinetic properties of one-dimensional disordered systems

Antsygina, T. N.; Slyusarev, V. A.

doi:10.1007/bf01017680

Cited by 15 publications

(59 citation statements)

References 5 publications

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“…In other terms the Lyapunov exponent gives a good estimate of the localization length of ϕ n (x) if the statistical properties of the envelope of the solution of the Schrödinger equation is not affected when imposing the second boundary condition. In the high energy limit where processes θ(x) and ξ(x) rapidly decorrelate [15] this is not a problem, however it is not obvious that this holds in any situation (in particular for the supersymmetric Hamiltonian H susy , the 6 This picture suggests that the wave function behaves roughly as ψ(x) ∼ e ±γx × (oscillations), however one should keep in mind that such a simple picture is dangerous since it forgets the important fact that the argument of the exponential, ξ(x), presents large fluctuations increasing like √ x (fluctuations of ξ(x)/x vanish for x → ∞, but not those of ξ(x)). The envelope of the wave function is an exponential of a drifted Brownian motion, what can have important consequences [58] ; neglecting this important feature can lead to wrong conclusion, like in Ref.…”

Section: Localizationmentioning

confidence: 99%

“…, where V (x) is a random function, have been studied in great detail [6] and their properties are rather generic under the asumption that V (x) is correlated on a small length scale and dx V (x)V (0) remains finite 1 : exponential tail in the density of states 2 at low energies (Lifshits singularity) [2,13,14,3,11,15,6] and decreasing Lyapunov exponent (inverse localization length) at high energy [15,6] γ ∝ 1/E for E → +∞. The situation can be quite different if the Hamiltonian possesses some symmetry preserved by the introduction of the random potential.…”

Section: Introductionmentioning

confidence: 99%

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Breaking supersymmetry in a one-dimensional random Hamiltonian

Hagendorf¹,

Texier²

2008

J. Phys. A: Math. Theor.

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The one-dimensional supersymmetric random Hamiltonian H susy = − d 2 dx 2 +φ 2 +φ ′ , where φ(x) is a Gaussian white noise of zero mean and variance g, presents particular spectral and localization properties at low energy : a Dyson singularity in the integrated density of states (IDoS) N (E) ∼ 1/ ln 2 E and a delocalization transition related to the behaviour of the Lyapunov exponent (inverse localization length) vanishing like γ(E) ∼ 1/| ln E| as E → 0. We study how this picture is affected by breaking supersymmetry with a scalar random potential : H = H susy + V (x) where V (x) is a Gaussian white noise of variance σ. In the limit σ ≪ g 3 , a fraction of states N (0) ∼ g/ ln 2 (g 3 /σ) migrate to the negative spectrum and the Lyapunov exponent reaches a finite value γ(0) ∼ g/ ln(g 3 /σ) at E = 0. Exponential (Lifshits) tail of the IDoS for E → −∞ is studied in detail and is shown to involve a competition between the two noises φ(x) and V (x) whatever the larger is. This analysis relies on analytic results for N (E) and γ(E) obtained by two different methods : a stochastic method and the replica method. The problem of extreme value statistics of eigenvalues is also considered (distribution of the n−th excited state energy). The results are analyzed in the context of classical diffusion in a random force field in the presence of random annihilation/creation local rates.

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Section: Localizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Breaking supersymmetry in a one-dimensional random Hamiltonian

Hagendorf¹,

Texier²

2008

J. Phys. A: Math. Theor.

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“…A powerful approach to handle such problems is the phase formalism 6 (presented in refs. [15,157] for instance). This formalism leads to a broad variety of stochastic processes.…”

Section: Overview Of the Papermentioning

confidence: 99%

“…We have used the equality in law (15) which implies the following property : given a Brownian bridge (x(τ ), 0 τ t | x(0) = x(t) = 0), for any even function f (x) we…”

Section: Historical Perspectivementioning

confidence: 99%

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Functionals of Brownian motion, localization and metric graphs

Comtet¹,

Desbois²,

Texier³

2005

J. Phys. A: Math. Gen.

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We review several results related to the problem of a quantum particle in a random environment.In an introductory part, we recall how several functionals of the Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization).Two aspects of the physics of the one-dimensional strong localization are reviewed : some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues).Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schrödinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated.Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of the planar Brownian motion.

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Conformal field theory approach to gapless 1D fermion systems and application to the edge excitations of ν=1/(2p+1) quantum Hall sequences

Degiovanni

Mélin²,

Chaubet

1998

Theor Math Phys

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We present a comprehensive study of the effective Conformal Field Theory (CFT) describing the low energy excitations of a gas of spinless interacting fermions on a circle in the gapless regime (Luttinger liquid). Functional techniques and modular transformation properties are used to compute all correlation functions in a finite size and at finite temperature. Forward scattering disorder is treated exactly. Laughlin experiments on charge transport in a Quantum Hall Fluid on a cylinder are reviewed within this CFT framework. Edge excitations above a given bulk excitation are described by a twisted version of the Luttinger effective theory. Luttinger CFTs corresponding to the ν = 1/(2p + 1) filling fractions appear to be rational CFTs (RCFT). Generators of the extended symmetry algebra are identified as edge fermions creators and annihilators, thus giving a physical meaning to the RCFT point of view on edge excitations of these sequences.

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Supersymmetry and kinetic properties of one-dimensional disordered systems

Cited by 15 publications

References 5 publications

Breaking supersymmetry in a one-dimensional random Hamiltonian

Breaking supersymmetry in a one-dimensional random Hamiltonian

Functionals of Brownian motion, localization and metric graphs

Conformal field theory approach to gapless 1D fermion systems and application to the edge excitations of ν=1/(2p+1) quantum Hall sequences

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