CRITICAL AND BICRITICAL PROPERTIES OF HARPERS EQUATION WITH NEXT-NEAREST-NEIGHBOR COUPLING

We have exploited a variety of techniques to study the universality an d stability of the scaling properties of Harper's equation, the equati on for a particle moving on a tight-binding square lattice in the pres ence of a gauge held, when coupling to next-nearest sites is added. We find, from numerical and analytical studies, that the scaling behavio r of the total width of the spectrum and the multifractal nature of th e 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 diago nals should remain as reflection lines. A bicritical line is found at the boundary between the region in which the nearest-neighbor terms do minate and the region in which the next-nearest-neighbor terms dominat e. 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 strong ly oscillatory behavior. For a flux per unit cell equal to 1/q the mea sure 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 hi gher inverse power of the denominator.