Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory
| Autor: | Maarten Hoogerland, Thomas A. Haase, Dylan J. Brown, Sophie S. Shamailov |
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| Rok vydání: | 2021 |
| Předmět: |
Physics
QC1-999 General problem FOS: Physical sciences Context (language use) Disordered Systems and Neural Networks (cond-mat.dis-nn) Condensed Matter - Disordered Systems and Neural Networks Connection (mathematics) symbols.namesake Landscape theory Quantum Gases (cond-mat.quant-gas) symbols Statistical physics Exponential decay Condensed Matter - Quantum Gases Hamiltonian (quantum mechanics) Quantum tunnelling Eigenvalues and eigenvectors |
| Zdroj: | SciPost Physics Core, Vol 4, Iss 2, p 017 (2021) |
| ISSN: | 2666-9366 |
| DOI: | 10.21468/scipostphyscore.4.2.017 |
| Popis: | 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where the tunnelling-in-the-effective-potential picture breaks down, and explicitly demonstrate that our method is no longer applicable in this regime.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. Comment: 38 pages, 10 figures. Minor changes compared to previous version. arXiv admin note: substantial text overlap with arXiv:2003.00149 |
| Databáze: | OpenAIRE |
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