Density of states and localization length in weakly disordered peierls chains

Mertsching, J.

doi:10.1016/0379-6779(93)90789-y

Cited by 3 publications

(7 citation statements)

References 10 publications

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“…[76] developed a renormalised perturbation expansion for the self energy. Recursion formulae encoding the exact solution [78,79] can also sometimes allow one to calculate the localisation length (and the density of states [79]).…”

Section: Literature Reviewmentioning

confidence: 99%

“…Out of the studies above, 1D [52,63,75,58,72,64,59,65,76,73,78,60,79,61] and 2D [52,39,63,64,55,65,66,67,68,50,69,57,70,77,62,38] models have been numerically explored far more thoroughly than three-dimensional (3D) [52,75,69], simply because of the increased computational requirements of higher-dimensional spaces. Possibly the most heavily studied model of localisation is the Anderson model, also known as the tight-binding Hamiltonian [4,75,58,72,64,55,66,65,76,67,68,78,69,57,77,…”

Section: Literature Reviewmentioning

confidence: 99%

“…While reflection at the entrance to the channel depends somewhat on the presence and nature of the ramp potential, the similarity between the blue and green (red and black) lines proves that it is the kinetic energy part only that governs localisation properties. 79 decays to zero within the length of the channel with a flat background potential. Then we add on an acceleration, and indeed a large fraction of the atoms now transmits to the second reservoir.…”

Section: Accelerationmentioning

confidence: 99%

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Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations

Shamailov¹,

Brown²,

Haase³

et al. 2020

Preprint

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Motivated by rapid experimental progress in ultra-cold atomic systems, we aim to provide a simple, intuitive description of Anderson localisation that allows for a direct quantitative comparison to experimental data, as well as yielding novel insights. To this end, we advance, employ and validate a recently-discovered theory -Localisation Landscape Theory (LLT) -which has unparalleled strengths and advantages, both computational and conceptual, over alternative methods. We focus on two-dimensional systems with point-like random scatterers, although an analogous study in other dimensions and with other types of disorder would proceed similarly. We begin by showing that exact eigenstates cannot be efficiently used to extract the localisation length. We then provide a comprehensive review of known LLT, and show that the effective potential of LLT can, to some degree, replace the real potential in the Hamiltonian. Next, we use LLT to compute the localisation length and test our method against exact diagonalisation. Furthermore, we propose a transmission experiment that optimally detects Anderson localisation and link the simulated observations of such an experiment to the predictions of LLT. In addition, we study the dimensional crossover from one to two dimensions, providing a new explanation to the established trends. The prediction of a mobility edge coming from LLT is tested by direct Schrödinger time evolution and is found to be unphysical. Moreover, we investigate expanding wavepackets, and find interesting differences between wavepackets that are initiated within and outside the disorder. We explain these differences using LLT combined with multidimensional tunnelling. Then, we utilise LLT to uncover a connection between the Anderson model for discrete disordered lattices and continuous two-dimensional disordered systems, which provides powerful new insights. From here, we demonstrate that localisation can be distinguished from other effects by a comparison to dynamics in an ordered potential with all other properties unchanged. Finally, we thoroughly investigate the effect of acceleration and repulsive interparticle interactions, as relevant for current experiments.

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Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Section: Accelerationmentioning

confidence: 99%

See 1 more Smart Citation

Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations

Shamailov¹,

Brown²,

Haase³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…[35] developed a renormalised perturbation expansion for the self energy. Recursion formulae encoding the exact solution [37,38] can also sometimes allow one to calculate the localisation length (and the density of states [38]).…”

Section: Introductionmentioning

confidence: 99%

“…Out of the studies above, one-dimensional (1D) [6, 13, 17-19, 21-23, 30, 31, 34, 35, 37, 38] and two-dimensional (2D) [6-8, 10, 12, 14, 15, 20-28, 36] models have been numerically explored far more thoroughly than three-dimensional (3D) [6,27,34], simply because of the increased computational requirements of higher-dimensional spaces. Possibly the most heavily studied model of localisation is the Anderson model, also known as a tight-binding Hamiltonian [4, 10, 12, 13, 22-27, 29, 30, 32, 34-37, 39-44], but other examples include the kicked rotor [19] (formally equivalent to the Anderson model), the Lloyd model [13,21], the Peierls chain [38], a quantum walker [31], and the continuous Schrödinger equation [13,14,17], with either a speckle potential [7,16], delta-function point scatterers [6,10], or more realistic Gaussian scatterers [8,15].…”

Section: Introductionmentioning

confidence: 99%

Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory

et al. 2021

View full text Add to dashboard Cite

While Anderson localisation is largely well-understood, its description has traditionally been rather cumbersome. A recently-developed theory -Localisation Landscape Theory (LLT) -has unparalleled strengths and advantages, both computational and conceptual, over alternative methods. To begin with, we demonstrate that the localisation length cannot be conveniently computed starting directly from the exact eigenstates, thus motivating the need for the LLT approach. Then, we confirm that the Hamiltonian with the effective potential of LLT has very similar low energy eigenstates to that with the physical potential, justifying the crucial role the effective potential plays in our new method. We proceed to use LLT to calculate the localisation length for very low-energy, maximally localised eigenstates, as defined by the length-scale of exponential decay of the eigenstates, (manually) testing our findings against exact diagonalisation. We then describe several mechanisms by which the eigenstates spread out at higher energies where the tunnelling-in-the-effective-potential picture breaks down, and explicitly demonstrate that our method is no longer applicable in this regime. We place our computational scheme in context by explaining the connection to the more general problem of multidimensional tunnelling and discussing the approximations involved. Our method of calculating the localisation length can be applied to (nearly) arbitrary disordered, continuous potentials at very low energies.

show abstract

Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory

Shamailov¹,

Brown²,

Haase³

et al. 2020

Preprint

View full text Add to dashboard Cite

While Anderson localisation is largely well-understood, its description has traditionally been rather cumbersome. A recently-developed theory -Localisation Landscape Theory (LLT) -has unparalleled strengths and advantages, both computational and conceptual, over alternative methods. To begin with, we demonstrate that the localisation length cannot be conveniently computed starting directly from the exact eigenstates, thus motivating the need for the LLT approach. Then, we reveal the physical significance of the effective potential of LLT, justifying the crucial role it plays in our new method. We proceed to use LLT to calculate the localisation length, as defined by the length-scale of exponential decay of the eigenstates, (manually) testing our findings against exact diagonalisation. We place our computational scheme in context by explaining the connection to the more general problem of multidimensional tunnelling and discussing the approximations involved. The conceptual approach behind our method is not restricted to a specific dimension or noise type and can be readily extended to other systems.

show abstract

Density of states and localization length in weakly disordered peierls chains

Cited by 3 publications

References 10 publications

Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations

Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations

Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory

Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory

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