Topological Dimensions from Disorder and Quantum Mechanics?

Area of Science:

• Condensed Matter Physics
• Quantum Mechanics
• Disordered Systems

Background:

• The critical Anderson electron in 3D space effectively occupies a region with an infrared (IR) scaling dimension (dIR) of approximately 8/3.
• Understanding the dimensional substructure and the distribution of relevant dimensions is crucial for characterizing electron behavior in disordered systems.

Purpose of the Study:

• To investigate the dimensional substructure of the critical Anderson electron in 3D.
• To determine the probability density function, p(d), for the scaling dimension 'd' accessed by the electron.
• To explore the potential emergence of integer dimensions from quantum mechanics and disorder.

Main Methods:

• Spatial partitioning of 3D space into regions of equal quantum occurrence probabilities.
• Calculation of the IR scaling dimension 'd' for each partitioned region.
• Inference of the probability density p(d) based on calculated dimensions.

Main Results:

• The probability density p(d) exhibits a strong peak near dimension d=2.
• p(d) is non-zero over the interval [dmin, dmax] ≈ [4/3, 8/3].
• A discrete component (delta-function) at d=2 may emerge in the infinite-volume limit.

Conclusions:

• The study suggests that quantum mechanics and disorder can lead to the emergence of integer (topological) dimensions.
• The upper bound of the accessed dimensions (dmax ≈ 8/3) is consistent with the effective IR scaling dimension (dIR).
• A potential connection exists between these findings and recent observations of dIR ≈ 2 in Dirac near-zero modes of thermal quantum chromodynamics.