Total energy for fermions on a two-dimensional lattice in a magnetic field.

Abstract

We consider the dependence on the magnetic field of the total energy of noninteracting spinless fermions on a two-dimensional square lattice. We show how a Green-function method can be used to develop a perturbation theory when the Fermi energy is in a gap of the unperturbed system. This method is used to calculate the cusp in the energy when the filling is one-third and the flux per unit cell is varied uniformly in the neighborhood of one-third, and it is explained how a cusplike minimum can occur whenever the Fermi energy lies in a sufficiently large gap of the Hofstadter spectrum. Close to a filling of one-half the gaps go to zero, and the perturbation theory must be partly replaced by a WKB argument for the discrete Dirac equation. This leads to an energy variation proportional to the 3/2 power of the deviation from flux one-half. We show how the theory of the magnetic translation group can be used to treat small deviations of the field from uniformity, and carry out an explicit calculation for the case of one-half filling and one-half flux per square to show that in this case uniform flux gives a local minimum of the energy.