1994/06/08 by John Hwan Han, J. H. Han, D. J. Thouless +4 · 5 citations
Mathematics · Physics and Astronomy · #Combinatorics #Condensed matter physics #Criticality #Exponent #Fibonacci number #Fractal #Geometry #Hamiltonian (control theory) #Ising model #Mathematical analysis #Mathematics #Multifractal system #Nonlinear Photonic Systems #Photonic Crystals and Applications #Physics #Quantum mechanics #Scale invariance #Scaling #Self-organized criticality #Square lattice #Statistical physics #Theoretical and Computational Physics #cond-mat #k-nearest neighbors algorithm
paper · pdf · doi:10.1103/physrevb.50.11365
39 pages; REVTeX 3.0; 14 figures available as hard-copies or postscript files
arxiv created 1994/06/08 · openalex publication_date 1994/10/15 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We have exploited a variety of techniques to study the universality and stability of the scaling properties of Harper's equation, the equation for a particle moving on a tight-binding square lattice in the presence of a gauge field, when coupling to next-nearest sites is added. We find, from numerical and analytical studies, that the scaling behavior of the total width of the spectrum and the multifractal nature of the spectrum are unchanged, provided the next-nearest-neighbor coupling terms are below a certain threshold value. The full square symmetry of the Hamiltonian is not required for criticality, but the square diagonals should remain as reflection lines. A bicritical line is found at the boundary between the region in which the nearest-neighbor terms dominate and the region in which the next-nearest-neighbor terms dominate. On the bicritical line a different critical exponent for the width of the spectrum and different multifractal behavior are found. In the region in which the next-nearest-neighbor terms dominate, the behavior is still critical if the Hamiltonian is invariant under reflection in the directions parallel to the sides of the square, but a new length scale enters, and the behavior is no longer universal but shows strongly oscillatory behavior. For a flux per unit cell equal to 1/q the measure of the spectrum is proportional to 1/q in this region, but if it is a ratio of Fibonacci numbers the measure decreases with a rather higher inverse power of the denominator.