Condensed Matter

2603 Submissions

[2] ai.viXra.org:2603.0075 [pdf] submitted on 2026-03-16 21:04:09

Resolution of the Nematic-to-Smectic A Universality Problem

Authors: Thomas A. Husmann
Comments: 11 pages, 6 tables, 14 references. CC BY-NC-SA 4.0. Code: github.com/thusmann5327/Unified_Theory_Physics

The nematic-to-smectic A (N-SmA) phase transition has remained one of the principalunsolved problems in statistical physics of condensed matter for over four decades. Experimental measurements of the heat capacity exponent α vary continuously from approximately 0 (3D-XY universality) to approximately 0.5 (tricritical mean-field), depending on the McMillan ratio r = TNA/TNI. No theoretical framework has explained this crossover from first principles. We present a resolution in two parts. Part I (model-independent) shows that the N-SmA transition maps exactly onto the Aubry—André—Harper (AAH) metal—insulator transition at the self-dual critical point V = 2J. This mapping uses only the de Gennes free energy (1972), lattice discretization, and the generic incommensurability of smectic layer spacing and molecular length. At criticality, the energy spectrum is a Cantor set with Hausdorff dimension Ds = 1/2 (Süt˝o 1989), giving ν = 2/3 and predicting that α varies continuously from 0 to 2/3. This qualitative result holds for any irrational frequency α and requires no new physics. Part II (quantitative) identifies the AAH frequency as α = 1/ϕ (golden ratio), producing the five-band Cantor partition. This yields a zero-free-parameter formula:α(r) = 23u2000r−rc1−rcu20004, r > rc; α = 0 otherwisewhere rc = 1−1/ϕ4 = 0.8541 and ϕ = (1+√5)/2. The formula fits 11 experimental compoundsspanning 40 years of published calorimetry with RMS = 0.033, reduced χ2 = 0.47, and all points within 2σ. Zero free parameters.
Category: Condensed Matter

[1] ai.viXra.org:2603.0071 [pdf] submitted on 2026-03-15 21:13:36

Hofstadter’s Golden Butterfly: The Metallic Mean Hierarchy in Moiré Superlattices

Authors: Thomas A. Husmann
Comments: 16 pages, 7 tables, 36 references. CC BY-NC-SA 4.0. Code: github.com/thusmann5327/Unified_Theory_Physics

The Hofstadter butterfly—the fractal energy spectrum of a two-dimensional electron ina magnetic field on a lattice—is shown to possess a natural hierarchy parameterized by themetallic means, the roots of x2 = nx + 1. The Harper equation that generates each horizontalslice of the butterfly is mathematically identical to the Aubry—André—Harper (AAH) Hamiltonian at the self-dual critical point V= 2J. Each irrational flux ratio α produces a Cantor-set spectrum with Hausdorff dimension Ds = 1/2.We show that two experimentally significant systems in graphene moiré physics correspondto specific metallic means: (1) the graphene/hBN lattice mismatch (δ= 1.68%) corresponds to metallic mean n = 60, with golden-ratio quasiperiodicity nested inside the n = 60 shell via continued fraction structure [0; 59, 1, 1, 1, . . .]; (2) the magic angle of twisted bilayer graphene (θ= 1.08◦) corresponds to metallic mean n = 53, matching to 0.06%.At golden flux (α = 1/ϕ), the five-band Cantor partition carries Chern numbers +2,−1, +1,−2.The outer pair (+2,−2) annihilates via topological pair annihilation, collapsing five bands to three—the 5→3 mechanism supported by Liu, Fulga & Asbóth (2020). The first three metallic mean discriminants ∆n = n2 + 4 are consecutive Fibonacci numbers (5, 8, 13), forming a Pythagorean triple (√5)2 + (√8)2 = (√13)2 that closes at exactly three spatial dimensions.36 supporting references span Hofstadter spectroscopy, moiré physics, Floquet topology, and metallic mean quasicrystals.
Category: Condensed Matter