Lieb-Robinson causality and non-Fermi liquids

Abstract

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 $G^R(ω,{\bf k})$ of local lattice operators as a function of complex frequency $ω$ 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 $|G^R(ω,{\bf k})|$ within the analytic domain. We use this bound to constrain the quasiparticle residue of the non-Fermi liquids.

Figures (2)

• Figure 1: The retarded Green's function must be analytic in the shaded region, given by (\ref{['eq:cons']}). Recall that $\omega_2$ and $q$ are the imaginary parts of the frequency and momentum as defined in (\ref{['eq:im']}). The dashed line shows the Lieb-Robinson lightcone with the velocity defined in (\ref{['eq:v']}).
• Figure 2: Motion of the pole (\ref{['eq:sol']}) for various models over the range $0 \leq \omega_2 \leq \Lambda$. All models are shown with the illustrative values $\Lambda = \frac{1}{2} h$ and $v_F^\star = \frac{1}{3}v$. The dashed line is a MFL with $\lambda = 1$. The solid black lines are Hertz liquids. From top to bottom these have $\{\alpha = \frac{3}{2}, \lambda = -1\}$, $\{\alpha = \frac{3}{2}, \lambda = -\frac{1}{2}\}$ and $\{\alpha = \frac{1}{2}, \lambda = 1\}$. The top curve enters the shaded region, which is excluded by the causality bound (\ref{['eq:cons']}), and therefore does not define a consistent theory.