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We have exploited a variety of techniques to study the universality andstability of the scaling properties of Harper's equation, the equation for aparticle moving on a tight-binding square lattice in the presence of a gaugefield, when coupling to next nearest sites is added. We find, from numericaland analytical studies, that the scaling behavior of the total width of thespectrum and the multifractal nature of the spectrum are unchanged, providedthe 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 lineis found at the boundary between the region in which the nearest neighbor termsdominate and the region in which the next nearest neighbor terms dominate. Onthe bicritical line a different critical exponent for the width of the spectrumand different multifractal behavior are found. In the region in which the nextnearest neighbor terms dominate the behavior is still critical if theHamiltonian is invariant under reflection in the directions parallel to thesides of the square, but a new length scale enters, and the behavior is nolonger universal but shows strongly oscillatory behavior. For a flux per unitcell equal to $1/q$ the measure of the spectrum is proportional to $1/q$ inthis region, but if it is a ratio of Fibonacci numbers the measure decreaseswith a rather higher inverse power of the denominator.Comment: 39 pages; REVTeX 3.0; 14 figures available as hard-copies or  postscript file

Critical and bicritical properties of Harper’s equation with next-nearest-neighbor coupling