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

Abstract

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 1 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.73.40. Dx,03.65.Sq

Generalized Aubry-André-Harper model withp-wave superconducting pairing

“…The Aubry-André-Harper (AAH) model has been extensively used as a quasiperiodic model to theoretically study the phase transition between extended, critical and localized phases [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. With the realization of the AAH model in photonic crystals [15][16][17] and ultracold atoms [18,19], this model has gained attention in recent years.…”

Topological pumping over a photonic Fibonacci quasicrystal

“…The Harper and Fibonacci models have different physical properties [12,23], and until recently only partial success has been achieved in attempts to combine them under the same general framework [24][25][26]. A recent paper presented a smooth deformation between the two models, which preserves the topological properties of their energy spectra and enables the definition of a generalized family of topologically equivalent quasiperiodic models [19]:…”

“…While the bulk properties of the Harper and Fibonacci models have been studied extensively in the past [12,23], their boundary states have received less attention. Our first experiment is designed to observe boundary states in the generalized model presented in Eq.…”

“…This model has a long-range order which originates from the sampling of the cosine function, where the modulation frequency b determines the bulk properties of the chain [12]. A quasiperiodic chain is produced whenever the hopping modulation is incommensurate with the underlying lattice (i.e., b is irrational).…”

“…The 2D QHE is deeply related to the 1D Harper model and its off-diagonal variant [10][11][12]. Whenever the cosine modulation of the model is incommensurate with the underlying lattice, the Harper model describes a quasiperiodic model, i.e., a lattice which is ordered but nonperiodic [13].…”