Topological Dimensions from Disorder and Quantum Mechanics?

Abstract

We have recently shown that critical Anderson electron in $D=3$ dimensions effectively occupies a spatial region of infrared (IR) scaling dimension $d_{\text{IR}}\approx 8/3$. Here we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probability, such that points comprising a region are of similar relevance, and calculate the IR scaling dimension $d$ of each. This allows us to infer the probability density $p(d)$ for dimension $d$ to be accessed by electron. We find that $p(d)$ has a strong peak at $d$ very close to 2. In fact, our data suggests that $p(d)$ is non-zero on the interval $[d_{\text{min}},d_{\text{max}}]\approx [4/3,8/3]$ and may develop a discrete part ($\delta$-function) at $d=2$ in infinite-volume limit. The latter invokes the possibility that combination of quantum mechanics and pure disorder can lead to emergence of topological dimensions. Although $d_{\text{IR}}$ is based on effective counting of which $p(d)$ has no a priori knowledge, $d_{\text{IR}}\ge d_{\text{max}}$ is an exact feature of the ensuing formalism. Possible connection of our results to recent findings of $d_{\text{IR}}\approx 2$ in Dirac near-zero modes of thermal quantum chromodynamics is emphasized.

arXiv comment5 pages, 6 figures

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