Renyi entropy, mutual information, and fluctuation properties of Fermi liquids

TL;DR

This work develops a universal, geometry-driven description of quantum information in Fermi liquids by mapping low-energy excitations onto a framework of patchwise 1+1D conformal field theories. It derives the Rényi entropy $S_\alpha$ for regions of linear size $L$, revealing a universal boundary-law-violating term tied to the Fermi surface geometry and a finite-temperature crossover function that connects $T=0$ and thermal behavior, with explicit results in arbitrary dimensions. By extending to disjoint regions, the paper provides a method to compute entanglement entropy and mutual information via a multi-interval 1D CFT approach integrated over the Fermi surface, while also predicting the scaling of number fluctuations $\Delta N_L^2$ and their finite-temperature crossovers, all governed by Landau parameters such as $F_0$ and the renormalized Fermi velocity $v_F$. Collectively, these results offer a comprehensive, universal characterization of the low-energy quantum information content of Fermi liquids and reveal the central role of Fermi-surface geometry in governing entanglement and fluctuations, with potential experimental realizations in clean, tunable systems. $S_\alpha$, $I(A,B)$, and $\Delta N_L^2$ are expressed through geometry-dependent integrals over real-space boundaries and the Fermi surface, reflecting an elegant interplay between higher-dimensional CFT structure and condensed-mmatter physics.

Abstract

I compute the leading contribution to the ground state Renyi entropy $S_α$ for a region of linear size $L$ in a Fermi liquid. The result contains a universal boundary law violating term simply related the more familiar entanglement entropy. I also obtain a universal crossover function that smoothly interpolates between the zero temperature result and the ordinary thermal Renyi entropy of a Fermi liquid. Formulas for the entanglement entropy of more complicated regions, including non-convex and disconnected regions, are obtained from the conformal field theory formulation of Fermi surface dynamics. These results permit an evaluation of the quantum mutual information between different regions in a Fermi liquid. I also study the number fluctuations in a Fermi liquid. Taken together, these results give a reasonably complete characterization of the low energy quantum information content of Fermi liquids.

Figures (3)

• Figure 1: Geometry of the entanglement entropy for a non-convex real space region. The shaded grey region surrounded by the thick black line represents a particular non-convex real space region. Dotted lines labeled $V_i$ ($i=1,2,3$) represent particular one dimensional cuts experienced by a patch of the Fermi surface with vertical Fermi velocity. Similarly, dashed lines labeled $H_i$ ($i=1,2$) represent particular one dimensional cuts experienced by a patch of the Fermi surface with horizontal Fermi velocity. Notice how the vertically moving fermion switches from an effectively single interval geometry in $V_1$ to a two interval geometry in $V_2$ and back to a single interval in $V_3$. On the other hand, $H_1$ and $H_2$ are both effectively single interval geometries.
• Figure 2: Multiple interval geometry and flux factors. A sequence of intervals similar to the situation shown in Fig. 1 $V_2$ with the local Fermi velocity vertical. The dashed grey line encloses the real space boundary segments of interest; the segments themselves are the black lines which continue into dotted lines outside the dashed grey enclosing line. The dark grey shaded regions are interior regions of the real space region, while arrows at the surfaces of the these regions indicate local normals to the real space boundary.
• Figure 3: Geometry of the mutual information for disconnected real space regions. The three one dimensional cuts labeled A, B, and C represent three different patches on the Fermi surface for different angles of the Fermi velocity. Cut A comes from a mode with vertical Fermi velocity; it always cuts both spheres and hence always contributes to the mutual information in this geometry. Cut B also connects both spheres, at least for some choices of real space boundary point. However, cut C never intersects both spheres simultaneously and hence does not contribute to the mutual information. For the sphere geometry shown here, there is always a maximum angle from vertical such that cuts beyond that angle never intersect both spheres. As described in the text, this maximum angle approaches zero as the spheres are taken far apart relative to their size.