Lieb-Robinson causality and non-Fermi liquids

Quantum mechanical lattice models with local, bounded interactions obey Lieb-Robinson causality. We show that this implies a domain of analyticity of the retarded Green's function $G^R(\omega,{\bf k})$ of local lattice operators as a function of complex frequency $\omega$ and momentum ${\bf k}$, similar to the lightcone analyticity property of relativistic field theories. Low-energy effective descriptions of the dynamics must be consistent with this microscopic analyticity constraint. We consider two canonical low-energy fermionic Green's functions describing non-Fermi liquids, the marginal Fermi liquid and the `Hertz liquid'. The pole in these Green's functions must be outside of the Lieb-Robinson domain of analyticity for all complex momenta captured by the low-energy theory. We show that this constraint upper bounds the magnitude of the dimensionless non-Fermi liquid coupling in certain Hertz liquids. We furthermore obtain, from causality, an upper bound on the magnitude $|G^R(\omega,{\bf k})|$ within the analytic domain. We use this bound to constrain the quasiparticle residue of the non-Fermi liquids.