Theories of electrons in one-dimensional disordered systems

Erdös, Paul; Herndon, R.C.

doi:10.1080/00018738200101358

Cited by 317 publications

(127 citation statements)

References 189 publications

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“…In this case KE represents the (symmetrized) lattice equivalent of the second derivative Kd 2 /dx 2 and P E is the interaction potential between the head and lattice systems. In this form H is seen to be similar to that used in the tight binding model (with off-diagonal potentials) to describe one dimensional particle motion in solids [22]. This similarity will be exploited in future work.…”

Section: Quantum Ballistic Evolutionmentioning

confidence: 90%

Quantum ballistic evolution in quantum mechanics: Application to quantum computers

Benioff

1996

Phys. Rev. A

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Quantum computers are important examples of processes whose evolution can be described in terms of iterations of single step operators or their adjoints. Based on this, Hamiltonian evolution of processes with associated step operators T is investigated here. The main limitation of this paper is to processes which evolve quantum ballistically, i.e. motion restricted to a collection of nonintersecting or distinct paths on an arbitrary basis. The main goal of this paper is proof of a theorem which gives necessary and sufficient conditions that T must satisfy so that there exists a Hamiltonian description of quantum ballistic evolution for the process, namely, that T is a partial isometry and is orthogonality preserving and stable on some basis. Simple examples of quantum ballistic evolution for quantum Turing machines with one and with more than one type of elementary step are discussed. It is seen that for nondeterministic machines the basis set can be quite complex with much entanglement present. It is also proved that, given a step operator T for an arbitrary deterministic quantum Turing machine, it is decidable if T is stable and orthogonality preserving, and if quantum ballistic evolution is possible. The proof fails if T is a step operator for a nondeterministic machine.It is an open question if such a decision procedure exists for nondeterministic machines. This problem does not occur in classical mechanics. Also the definition of quantum Turing machines used here is compared with that used by other authors.

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Section: Quantum Ballistic Evolutionmentioning

confidence: 90%

Quantum ballistic evolution in quantum mechanics: Application to quantum computers

Benioff

1996

Phys. Rev. A

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“…The resistance is related to a product of transfer matrices, and is thus relatively easy to disorder average, while the transmission is its inverse, which is harder to average. Group-theoretical approaches [37,38] were suggested to perform the disorder average of the transmission, with some success in the weak-disorder limit. We present here an approach that enables obtaining disorder averaging in a simpler fashion, and can possibly be generalized to spatially correlated disorder distribution.…”

Section: Transmission Coefficient: Tmentioning

confidence: 99%

Nonperturbative expression for the transmission through a leaky chiral edge mode

2014

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Chiral edge modes of topological insulators and Hall states exhibit nontrivial behavior of conductance in the presence of impurities or additional channels. We present a simple formula for the conductance through a chiral edge mode coupled to a disordered bulk. For a given coupling matrix between the chiral mode and bulk modes, and a Green's function matrix of bulk modes in real space, the renormalized Green's function of the chiral mode is expressed in closed form as a ratio of determinants. We demonstrate the usage of the formula in two systems: (i) a 1d wire with random on-site impurity potentials for which we found that the disorder averaging is made simpler with the formula, and (ii) a quantum Hall fluid with impurities in the bulk for which the phase picked up by the chiral mode due to the scattering with the impurities can be conveniently estimated.

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“…II, solve Eq. (5) by the transfer matrix method, 16 and obtain from Eq. (4) the resistance of a single wire.…”

Section: Microscopic Modelingmentioning

confidence: 99%

Coherent resistance of a disordered one-dimensional wire: Expressions for all moments and evidence for non-Gaussian distribution

et al. 2003

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We study coherent electron transport in a one-dimensional wire with disorder modeled as a chain of randomly positioned scatterers. We derive analytical expressions for all statistical moments of the wire resistance ρ. By means of these expressions we show analytically that the distribution P (f ) of the variable f = ln(1 + ρ) is not exactly Gaussian even in the limit of weak disorder. In a strict mathematical sense, this conclusion is found to hold not only for the distribution tails but also for the bulk of the distribution P (f ).

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Theories of electrons in one-dimensional disordered systems

Cited by 317 publications

References 189 publications

Quantum ballistic evolution in quantum mechanics: Application to quantum computers

Quantum ballistic evolution in quantum mechanics: Application to quantum computers

Nonperturbative expression for the transmission through a leaky chiral edge mode

Coherent resistance of a disordered one-dimensional wire: Expressions for all moments and evidence for non-Gaussian distribution

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