We study the phase diagram of the tight-binding model for an electron on an anisotropic square lattice with a four-dimensional parameter space defined by two nearest-neighbour and two next-nearest-neighbour couplings. Using a renormalization scheme, we show that the inequality of the two next-nearest-neighbour couplings destroys the fat critical regime found in the isotropic case above the bicritical line and replaces it with another re-entrant extended phase. The scaling properties of the model are those of the corresponding tight-binding models on the nearest-neighbour square and triangular lattices. The triangular universality class also describes the quantum Ising chain in a transverse field with the only exception being the conformally invariant state of the Ising model which has no analogue in the triangular-lattice case.