Higher-order localization landscape theory of Anderson localization
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912170549182464 |
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| author | Skipetrov, Sergey E. |
| author_facet | Skipetrov, Sergey E. |
| contents | For a Hamiltonian ${\hat H}$ containing a position-dependent (disordered) potential, we introduce a sequence of landscape functions $u_n(\vec{r})$ obeying ${\hat H} u_n(\vec{r}) = u_{n-1}(\vec{r})$ with $u_0(\vec{r}) = 1$. For $n \to \infty$, $1/v_n(\vec{r}) = u_{n-1}(\vec{r})/u_{n}(\vec{r})$ converges to the lowest eigenenergy $E_1$ of ${\hat H}$ whereas $u_{\infty}(\vec{r})$ yields the corresponding wave function $ψ_1(\vec{r})$. For large but finite $n$, $v_n(\vec{r})$ can be approximated by a piecewise constant function $v_n(\vec{r}) \simeq v_n^{(m)}$ for $\vec{r} \in Ω_m$ and yields progressively improving estimations of eigenenergies $E_m = 1/v_n^{(m)}$ of locally fundamental eigenstates $ψ_m(\vec{r}) \propto u_{n}(\vec{r})$ in spatial domains $Ω_m$. These general results are illustrated by a number of examples in one dimension: box potential, sequence of randomly placed infinite potential barriers, smooth and spatially uncorrelated random potentials, quasiperiodic potential, as well as for the uncorrelated random potential in two dimensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_05078 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher-order localization landscape theory of Anderson localization Skipetrov, Sergey E. Disordered Systems and Neural Networks Mathematical Physics For a Hamiltonian ${\hat H}$ containing a position-dependent (disordered) potential, we introduce a sequence of landscape functions $u_n(\vec{r})$ obeying ${\hat H} u_n(\vec{r}) = u_{n-1}(\vec{r})$ with $u_0(\vec{r}) = 1$. For $n \to \infty$, $1/v_n(\vec{r}) = u_{n-1}(\vec{r})/u_{n}(\vec{r})$ converges to the lowest eigenenergy $E_1$ of ${\hat H}$ whereas $u_{\infty}(\vec{r})$ yields the corresponding wave function $ψ_1(\vec{r})$. For large but finite $n$, $v_n(\vec{r})$ can be approximated by a piecewise constant function $v_n(\vec{r}) \simeq v_n^{(m)}$ for $\vec{r} \in Ω_m$ and yields progressively improving estimations of eigenenergies $E_m = 1/v_n^{(m)}$ of locally fundamental eigenstates $ψ_m(\vec{r}) \propto u_{n}(\vec{r})$ in spatial domains $Ω_m$. These general results are illustrated by a number of examples in one dimension: box potential, sequence of randomly placed infinite potential barriers, smooth and spatially uncorrelated random potentials, quasiperiodic potential, as well as for the uncorrelated random potential in two dimensions. |
| title | Higher-order localization landscape theory of Anderson localization |
| topic | Disordered Systems and Neural Networks Mathematical Physics |
| url | https://arxiv.org/abs/2408.05078 |