On the localization length of finite-volume random block Schr\"odinger operators
On the localization length of finite-volume random block Schr\"odinger operators
物理学
Steven Khang Truong,Fan Yang,Jun Yin.On the localization length of finite-volume random block Schr\"odinger operators[EB/OL].(2025-03-14)[2025-10-25].https://arxiv.org/abs/2503.11382.点此复制
We study a general class of random block Schr\"odinger 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 $H=V +
\lambda \Psi$, where $V$ is a block potential with i.i.d. $W^d\times W^d$
Gaussian diagonal blocks, $\Psi$ describes interactions between neighboring
blocks, and $\lambda>0$ is a coupling parameter. We normalize the blocks of
$\Psi$ so that each block has a Hilbert-Schmidt norm of the same order as the
blocks of $V$. Assuming $W\ge L^\delta$ for a small constant $\delta>0$ and
$\lambda\gg W^{-d/2}$, 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 $\lambda\ll W^{-d/2}$, this provides a rigorous proof of the
Anderson localization-delocalization transition as $\lambda$ 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 $(W\lambda)^2$,
which is believed to be the correct scaling.
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