2025/03/14 by Steven Khang Truong, Fan Yang, Truong, Steven Khang +3 · 1 citation
Materials Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics #Topological Materials and Phenomena
paper · pdf · doi:10.48550/arxiv.2503.11382
openalex publication_date 2025/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a general class of random block Schrödinger operators (RBSOs) in dimensions 1 and 2, which naturally extend the Anderson model by replacing the random potential with a random block potential. Specifically, we focus on two RBSOs -- the block Anderson and Wegner orbital models -- defined on the d-dimensional torus (\mathbb Z/L\mathbb Z)d. They take the form H=V + λΨ, where V is a block potential with i.i.d. Wd× Wd Gaussian diagonal blocks, Ψ describes interactions between neighboring blocks, and λ>0 is a coupling parameter. We normalize the blocks of Ψ so that each block has a Hilbert-Schmidt norm of the same order as the blocks of V. Assuming W≥ Lδ for a small constant δ>0 and λ≫ W-d/2, we establish the following results. In dimension d=2, we prove delocalization and quantum unique ergodicity for bulk eigenvectors. Combined with the localization result from arXiv:1608.02922, which holds under the condition λ≪ W-d/2, this provides a rigorous proof of the Anderson localization-delocalization transition as λ crosses the critical threshold W-d/2. In dimension d=1, we show that the localization length of bulk eigenvectors is at least of order (Wλ)2, which is believed to be the correct scaling.