Nonlinear response theory in ultra-thin electron layers
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Perturbational response theory is one of the most successfully used tools in physics, its linear version being the standard procedure to fruitfully treat complex systems. Ongoing advances in laser power combined with increasing accuracy in material design drive the need to go beyond linearity. Here we present a general formalism for the arbitrary order response functions of any observable in Fermi systems and derive a closed form for the non-interacting case. Novel technological devices for radio- and electrical signals operating outside the linear regime, e.g. as detectors, mixers, or multipliers for terahertz waves, are often based on very thin nano-structures. Thus we chose the two dimensional uniform electron gas (2Deg) to verify our expressions by confirming the results of Lee  for the quadratic order Lindhard function. Our newly derived cubic density response can be cast into a notably lucid form.
Interactions are then accounted for using the random phase approximation (RPA), which is quite accurate in experimentally studied 2Degs. Aiming at an analysis of the collective modes, the `sheet plasmons', we extend the RPA up to third order in the perturbation. In the harmonic co-linear case we further accomplished to extend Mikhailov's  relations to more general input signals.
Compared to the linear case, the much richer particle-hole excitation spectrum causes stronger Landau damping, significantly reducing the plasmons' live-time, in good agreement with the electron-energy-loss-scattering data.