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We present detailed theoretical study on zero-frequency Drude weight andoptical conductivity of a two-dimensional heavy-hole gas(2DHG) with $k$-cubicRashba and Dresselhaus spin-orbit interactions. The presence of $k$-cubicspin-orbit couplings strongly modifies the Drude weight in comparison to theelectron gas with $k$-linear spin-orbit couplings. For large hole density andstrong $k$-cubic spin-orbit couplings, the density dependence of Drude weightdeviates from the linear behavior. We establish a relation between opticalconductivity and the Berry connection. Unlike two-dimensional electron gas with$k$-linear spin-orbit couplings, we explicitly show that the opticalconductivity does not vanish even for equal strength of the two spin-orbitcouplings. We attribute this fact to the non-zero Berry phase for equalstrength of $k$-cubic spin-orbit couplings. The least photon energy needed toset in the optical transition in hole gas is one order of magnitude smallerthan that of electron gas. Types of two van Hove singularities appear in theoptical spectrum are also discussed.

Drude weight and optical conductivity of a two-dimensional heavy-hole gas with $k$-cubic spin-orbit interactions