Quantum Corrections to Bosonization of 2+1D Fermi Liquid

Corrections to density response functions provide crucial insights into the low-energy behavior of strongly interacting systems. We compute the non-analytic correction to the density response of a 2+1D Fermi liquid by bosonizing the Fermi surface using the co-adjoint orbit formalism. The non-analytic correction arises naturally as the one-loop correction within the bosonization framework, highlighting the simplicity and efficiency of this approach compared to the conventional fermionic methods, particularly for Fermi liquids with generic Landau parameters. We employ a patch prescription to evaluate the one-loop correction and match it with the result obtained from the fermionic approach for free Fermi gas. Our method can be straightforwardly extended to compute the response function in non-Fermi liquid systems, where strong quantum fluctuation dominates the low-energy regime, making the non-analytic the leading contribution.