Entanglement entropy from charge statistics: Exact relations for noninteracting many-body systems

We present an exact expression for the entanglement entropy generated at a quantum point contact between noninteracting electronic leads in terms of the full counting statistics of charge fluctuations, which we illustrate with examples from both equilibrium and nonequilibrium transport. The formula is also applicable to ground-state entanglement entropy in systems described by noninteracting fermions in any dimension, which in one dimension include the critical spin-1/2 $\mathit{XX}$ and Ising models where conformal field theory predictions for the entanglement entropy are reproduced from the full counting statistics. These results may play an important role in experimental measurements of entanglement entropy in mesoscopic structures and cold atoms in optical lattices.

Figure 1

(a) The entanglement entropy characterizes nonlocal quantum correlations between subsystem $A$ and the remainder $B$, such as (b) two electrodes upon opening of a quantum point contact (QPC) and (c) strongly repulsive bosons hopping in a one-dimensional opticallattice.

Figure 2

Exact entanglement entropy (dashed) and approximation by cumulants (solid) for a QPC with transmission $D$ for (a) zero bias voltage and (b) bias voltage $V$, scaled to the maximum value at $D=1$ and 0.5, respectively. The cutoff $K$ for the number of cumulants included, from bottom to top, is indicated in each plot.

Figure 3

The exact entanglement entropy and approximation by cumulants for (a) the critical spin-1/2 $\mathit{XX}$ chain (upper set of curves) and critical Ising chain in transverse field (lower set of curves) with periodic boundary conditions, $L=100$, and (b) free fermions on $L\times L$ square lattices with periodic boundary conditions, with the partition taken to be half the lattice. Solid curves are the exact entanglement entropy, while the cutoff number $K$ for both figures is 2 (circles), 4 (squares), 8 (crosses), and 40 (pluses).