Anderson Beta Flows on Hierarchical Weighted Tori
Year: 2026
Type: preprint
Abstract: We construct a deterministic family of finite hierarchical weighted tori whose \emph{finite-scale effective} spectral dimension can be tuned continuously over approximately $d_s^{eff}=1.65$--$1.90$, and study a symplectic Anderson model on this family. The hierarchy is carried by deterministic bond weights on a torus with periodic boundary conditions, while quenched disorder is introduced independently as a scalar onsite potential. The use of a finite-scale effective spectral dimension is deliberate: asymptotic spectral dimension is a coarse-geometric invariant under broad classes of finite-scale graph deformations, whereas the present construction is designed to control path statistics over the same finite size window probed electronically. A small boundary twist defines a level-curvature Thouless number, from which we estimate the finite-size beta flow $\beta_x(L\!\to\!2L)=\frac{<\ln g_x(2L)>-<\ln g_x(L)>}{\ln 2}.$ At $d_s{eff}\simeq1.75$, the $L=64\to128$ data exhibit multiple zero/fixed-point-like candidates and a finite disorder interval with positive or near-zero beta. Two-zero and critical-phase flows are known in symplectic fractals; the present result is therefore interpreted as their possible realization in a qualitatively different, periodically closed weighted-hierarchy architecture rather than as the first observation of such a flow. The data do not establish an asymptotic fourth phase. A matched-$d_s^{eff}$ comparison between layered and de-layered hierarchies shows that the finite-size Anderson response is not fixed by $d_s^{eff}$ alone. This is consistent with earlier topology dependence found on bifractals, while the present controls expose specific accompanying quantities---gap scaling, transverse conductivity, and weakest-cut bypassability. A third, intermediate hierarchy can be recalibrated to the same $d_s^{eff}$ while changing the weakest-cut ratio by orders of magnitude, establishing path geometry as an independently designable coordinate. A fixed-Lanczos, same-seed 16-realization curvature audit on these three geometries does not yet resolve a monotonic dependence of $\beta$ or conductance variance on bypassability, so transport-design claims are kept exploratory. Replacing deterministic spin-orbit hopping directions by bond-random Haar-$\mathrm{SU}(2)$ matrices strongly alters the flow on the same weighted graph. Finally, a dimension sweep identifies a low-disorder near-critical candidate at $d_s^{eff}\simeq1.85$ and $W/B_{\rm clean}\simeq0.109$. The individual ingredients of this program have important precedents; the targeted literature audit performed for this draft does not, however, identify a single prior study combining the same weighted-torus construction, disorder separation, matched-$d_s^{eff}$ geometry controls, bond-randomness control, and dimension-resolved closed-system beta-flow analysis. The resulting platform therefore complements self-similar fractal and long-range hierarchical Anderson models at the level of the integrated numerical design rather than by claiming novelty for each ingredient separately.
Author: Ueoka, Yoshiki
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208751416
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