Violation of f-sum Rule with Generalized Kinetic Energy

Motivated by the normal state of the cuprates in which the f-sum rule increases faster than a linear function of the particle density, we derive a conductivity sum rule for a system in which the kinetic energy operator in the Hamiltonian is a general function of the momentum squared. Such a kinetic energy arises in scale invariant theories and can be derived within the context of holography. Our derivation of the f-sum rule is based on the gauge couplings of a non-local Lagrangian in which the kinetic operator is a fractional Laplacian of order $\alpha$. We find that the f-sum rule in this case deviates from the standard linear dependence on the particle density. We find two regimes. At high temperatures and low densities, the sum rule is proportional to $nT^{\frac{\alpha-1}{\alpha}}$ where $T$ is the temperature. At low temperatures and high densities, the sum rule is proportional to $n^{1+\frac{2(\alpha-1)}{d}}$ with $d$ being the number of spatial dimensions. The result in the low temperature and high density limit, when $\alpha < 1$, can be used to qualitatively explain the behavior of the effective number of charge carriers in the cuprates at various doping concentrations.