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

We compute the leading contribution to the ground-state Rényi entropy $S_{\alpha }$ for a region of linear size $L$ in a Fermi liquid in $d$ dimensions. The result contains a universal boundary law violating term simply related to the more familiar entanglement entropy. We also obtain a universal crossover function that smoothly interpolates between the zero-temperature result and the ordinary thermal Rényi entropy of a Fermi liquid. Formulas for the entanglement entropy of more complicated regions, including nonconvex 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. We 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.

Figure 1

Geometry of the entanglement entropy for a nonconvex real-space region. The shaded gray region surrounded by the thick black line represents a particular nonconvex 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 gray line encloses the real-space boundary segments of interest; the segments themselves are the black lines which continue into dotted lines outside the dashed gray enclosing line. The dark gray shaded regions are interior regions of the real-space region, while arrows at the surfaces of 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.