Nejvíce citovaný článek - PubMed ID 36112458
Super-Universality in Anderson Localization
We have recently shown that the critical Anderson electron in D=3 dimensions effectively occupies a spatial region of the infrared (IR) scaling dimension dIR≈8/3. Here, we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probabilities, such that the 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 the electron. We find that p(d) has a strong peak at d very close to two. In fact, our data suggest that p(d) is non-zero on the interval [dmin,dmax]≈[4/3,8/3] and may develop a discrete part (δ-function) at d=2 in the infinite-volume limit. The latter invokes the possibility that a combination of quantum mechanics and pure disorder can lead to the emergence of integer (topological) dimensions. Although dIR is based on effective counting, of which p(d) has no a priori knowledge, dIR≥dmax is an exact feature of the ensuing formalism. A possible connection of our results to the recent findings of dIR≈2 in Dirac near-zero modes of thermal quantum chromodynamics is emphasized.
- Klíčová slova
- Anderson transition, dimension content, effective counting dimension, effective number theory, effective support, emergent space, localization,
- Publikační typ
- časopisecké články MeSH
Fractal-like structures of varying complexity are common in nature, and measure-based dimensions (Minkowski, Hausdorff) supply their basic geometric characterization. However, at the level of fundamental dynamics, which is quantum, structure does not enter via geometric features of fixed sets but is encoded in probability distributions on associated spaces. The question then arises whether a robust notion of the fractal measure-based dimension exists for structures represented in this way. Starting from effective number theory, we construct all counting-based schemes to select effective supports on collections of objects with probabilities and associate the effective counting dimension (ECD) with each. We then show that the ECD is scheme-independent and, thus, a well-defined measure-based dimension whose meaning is analogous to the Minkowski dimension of fixed sets. In physics language, ECD characterizes probabilistic descriptions arising in a theory or model via discrete "regularization". For example, our analysis makes recent surprising results on effective spatial dimensions in quantum chromodynamics and Anderson models well founded. We discuss how to assess the reliability of regularization removals in practice and perform such analysis in the context of 3d Anderson criticality.
