Renyi entropy of the interacting Fermi liquid

Entanglement properties, including the Renyi $\alpha$ entropies and scaling laws, are becoming increasingly important in condensed-matter physics because they can be used to determine physical properties and the “fingerprint” of quantum phases. In this work we use variational quantum Monte Carlo to compute the Renyi $\alpha$ entropies, their scaling laws, and the relationship between different $\alpha$ entropies for one of the most important phases in condensed matter, the interacting Fermi liquid. We also investigate the relationship between the scaling laws and the discontinuity in the momentum distribution at the Fermi surface. Contrary to recent theoretical predictions, we find that interactions increase the prefactor for the $\alpha$-entropy scaling laws for all particle interaction strengths and forms. We also show that a theory of these scaling laws for the interacting systems may be developed by extending the free theory to incorporate properties of the momentum distribution.

Figure 1

Scaling form for the spin-polarized electron gas. (Main plot) $S_2/L$ for the spin-polarized interacting electron gas at $r_s=1,5,10,20$ and the noninteracting Fermi liquid. $L$ is scaled so that without interactions all lines lie on top of ${\Psi }_{FG}$ results. The high-density $r_s=1$ data falls on top of the noninteracting liquid while the lower-density, more strongly correlated liquids have larger Renyi 2 entropies. The ${\Psi }_{FG}$ reference is the exact noninteracting $S_2$ computed using the same number of particles as the interacting system for each $L$. (Inset) $\Delta S_2/L=(S_2^{HEG}-S_2^{FG})/L$ plotted against $\log L$. The 2 entropy for the interacting case scales the same as the noninteracting case with a larger prefactor. The lines are from a fit to $\Delta S_2/L=l+m\log L$ shown in Table .

Figure 2

Relationship between $\alpha$ entropies for the spin-polarized electron gas. (Left) $\Delta S_{\alpha }/L=(S_{\alpha }^{HEG}-S_{\alpha }^{FG})/L$ for $\alpha =2,3,4$ for region A size $\log L\approx 1.5$, $\langle N_A\rangle =6.25$. $S_{\alpha }$ increases as correlations increase and decreases as $\alpha$ increases. (Right) $\frac{2\alpha }{1+\alpha }\frac{S_{\alpha }}{L}$ for $\alpha =2,3,4$. Besides some small $L$ oscillations, this scaling makes the noninteracting Renyi $\alpha$ entropies linear in $\alpha$. Interactions modify this relationship: the rescaled $\alpha$ entropies decrease as $\alpha$ gets larger.

Figure 3

Relationship between particle interaction strength and modification of leading scaling law for $S_2$. Slope difference, $m$: $\Delta S_2/L=(S_2^{HEG}-S_2^{FG})/L=l+m\log L$, as a function of $1-Z$ for the Coulomb and modified Pöschl-Teller potential at all correlation strengths considered. Lines are guides to the eye from linear fits. For all interaction forms and strengths the prefactor of the leading scaling law is increased. Beyond some minimal $Z\approx 0.8$ the relationship for $m$ and $1-Z$ is linear within error bars for the Coulomb potential.