Violation of an f -sum rule with generalized kinetic energy

Abstract

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 α. 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 α−1 α where T is the temperature.At low temperatures and high densities, the sum rule is proportional to n with d being the number of spatial dimensions. The result in the low temperature and high density limit, when α < 1, can be used to qualitatively explain the behavior of the effective number of charge carriers in the cuprates at various doping concentrations.

Failure of Kohn's theorem and the apparent failure of the f -sum rule in intrinsic Dirac-Weyl materials in the presence of a filled Fermi sea

“…Thus, the f -sum rule must be restored by virtue of the number conservation in a full band description of the system, but it is violated in the low-energy Dirac description, as we show explicitly in the current paper. We mention here that the fact that the naïve f -sum rule is violated in various low-energy effective model Hamiltonians has been demonstrated in the literature in various contexts [12][13][14] , and our current work focuses explicitly on the low-energy models of graphenelike systems of great current interest, showing explicitly that the low-energy effective Hamiltonians here indeed violate the naïve f -sum rule defined by Eq. (1).…”

Generalized $f$-Sum Rules and Kohn formulas on Non-linear Conductivities

“…(Throughout the text we set e = = 1.) For more general models of the form Ĥ = K + Î, the fsum rule still holds although with a modified right-hand side [13][14][15][16][17][18][19] . The Kohn formula 20 is an analytic expression of the Drude weight, also called the charge stiffness, that characterizes the ballistic transport of the system.…”