Effective equations for disordered one-dimensional systems

Hayn, R.; John, W.

doi:10.1007/bf01303977

Cited by 22 publications

(26 citation statements)

References 9 publications

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“…Its DOS and localization length have a long history of analysis [8,28,29,30]. The spectrum of U , together with the asymptotic behavior of the corresponding eigenstates Ψ z , which as we shall see is determined by b z , have the same information content as the input signal u(t).…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

Kazakopoulos

Moustakas

2008

Phys. Rev. E

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We calculate the Lyapunov exponent for the non-Hermitian Zakharov-Shabat eigenvalue problem corresponding to the attractive non-linear Schrödinger equation with a Gaussian random pulse as initial value function. Using an extension of the Thouless formula to non-Hermitian random operators, we calculate the corresponding average density of states. We also calculate the distribution of a set of scattering data of the Zakharov-Shabat operator that determine the asymptotics of the eigenfunctions. We analyze two cases, one with circularly symmetric complex Gaussian pulses and the other with real Gaussian pulses. We discuss the implications in the context of information transmission through non-linear optical fibers.

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Section: Introductionmentioning

confidence: 99%

“…The proof of the Thouless formula for the ZS eigenproblem proceeds similarly to the proofs in [25,29]. We consider the system of equations (2) on the interval [0, T ].…”

Section: Appendix A: Thouless Formulamentioning

confidence: 99%

Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

Kazakopoulos

Moustakas

2008

Phys. Rev. E

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“…According to the Thouless formula the mean localization length ℓ(ω) can be obtained from the real part of the disorder-averaged single-particle Green's function. Originally the Thouless formula was derived for a one-band model with quadratic energy dispersion [23], but it can be shown to hold also for the FGM, where it can be written as [24,25] ∂ ∂ω…”

Section: Lyapunov Exponent and Localization Lengthmentioning

confidence: 99%

Exactly solvable toy model for the pseudogap state

Bartosch

Kopietz

2000

Eur. Phys. J. B

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Abstract. We present an exactly solvable toy model which describes the emergence of a pseudogap in an electronic system due to a fluctuating off-diagonal order parameter. In one dimension our model reduces to the fluctuating gap model (FGM) with a gap ∆(x) that is constrained to be of the form ∆(x) = Ae iQx , where A and Q are random variables. We explicitly calculate the average density of states, the localization length, the average single-particle Green's function, and the real part of the average conductivity. In our model phase fluctuations generate delocalized states at the Fermi energy, which give rise to a finite Drude peak in the conductivity. We also find that the interplay between phase and amplitude fluctuations leads to a weak logarithmic singulatity in the single-particle spectral function at the bare quasi-particle energies. In higher dimensions our model might be relevant to describe the pseudogap state in the underdoped cuprate superconductors.

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“…In [6] Balents and Fisher consider a random mass Dirac equation where Φ = V = 0 so only M is non-zero. This is a commensurate Dirac equation [31,4]. The commensurate Dirac equation is equivalent to the XY spin chain in a random transverse magnetic [29,30].…”

Section: The Modelmentioning

confidence: 99%

“…Supersymmetry has been used to evaluate exactly the disorder averaged density of states for a one-dimensional Schrödinger equation with a random white-noise potential [3], various random one-dimensional Dirac equations [4][5][6][7], and a number of models associated with the lowest Landau level of a two-dimensional electron gas in a high magnetic field [8]. The replica trick has been used to evaluate exactly the disorder averaged density of states of the same one-dimensional models involving the Schrödinger equation [9], and Dirac equation [10,7].…”

Section: Introductionmentioning

confidence: 99%

Derivation of the probability distribution function for the local density of states of a disordered quantum wire via the replica trick and supersymmetry

Bunder

McKenzie

2001

Nuclear Physics B

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We consider the statistical properties of the local density of states of a one-dimensional Dirac equation in the presence of various types of disorder with Gaussian white-noise distribution. It is shown how either the replica trick or supersymmetry can be used to calculate exactly all the moments of the local density of states. Careful attention is paid to how the results change if the local density of states is averaged over atomic length scales. For both the replica trick and supersymmetry the problem is reduced to finding the ground state of a zero-dimensional Hamiltonian which is written solely in terms of a pair of coupled "spins" which are elements of u(1, 1). This ground state is explicitly found for the particular case of the Dirac equation corresponding to an infinite metallic quantum wire with a single conduction channel. The calculated moments of the local density of states agree with those found previously by Al'tshuler and Prigodin [Sov. Phys. JETP 68 (1989) 198] using a technique based on recursion relations for Feynman diagrams.

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Effective equations for disordered one-dimensional systems

Cited by 22 publications

References 9 publications

Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

Exactly solvable toy model for the pseudogap state

Derivation of the probability distribution function for the local density of states of a disordered quantum wire via the replica trick and supersymmetry

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