# Introduce a model with logarithmic dispersion relation

Having a model Hamiltonian on a lattice, one can compute the band structure of a system by employing the Bloch theorem. Here for simplicity, let's focus on noninteracting models. This procedure for a tight-binding model on a square lattice, for instance, results in the dispersion relation proportional to $$\cos(k_x) +\cos(k_y)$$, where $$(k_x, k_y)$$ denote the 2D momentum. If one relaxes the realness of the momentum values, then obtaining the dispersion relations in terms of hyperbolic functions, e.g., $$\cosh(k_x) + \cosh(k_y)$$, is also feasible.

Out of curiosity, I wondered whether one could find a logarithmic band structure, even when the Bloch theorem is relaxed or when an interaction is switched on. I understand that logarithmic functions are exotic; they are not defined on half of the plane and diverge. Hence, it is better not to call the spectrum of these models "band structure." Nevertheless, can one come up with a model which hosts logarithmic energy dispersion?

• To recover the square lattice from the cosine band structure, you just do the inverse Fourier transform back to real space. I suppose you could try something similar with a logarithmic dispersion and see what pops out. Looks kind of tricky though. Commented Sep 14, 2022 at 2:10
• @aRockStr Thanks for sharing your thoughts. Indeed I do agree that this is a tricky problem. Commented Sep 14, 2022 at 13:11