Area law and vacuum reordering in harmonic networks

We review a number of ideas related to area-law scaling of the geometric entropy from the point of view of condensed matter, quantum field theory, and quantum information. An explicit computation in arbitrary dimensions of the geometric entropy of the ground state of a discretized scalar free field theory shows the expected area law result. In this case, area-law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations. Furthermore, the explicit control on all the eigenvalues of the reduced density matrix allows for a verification of entropy loss along the renormalization group trajectory driven by the mass term. A further result of our computation shows that single-copy entanglement also obeys area law scaling, majorization relations, and decreases along renormalization group flows.

Figure 1

The entropy $S$ and the single-copy entanglement $E_1$ resulting from tracing the ground state of a massless scalar field in three dimensions, over the degrees of freedom inside a sphere of radius $R$.

Figure 2

Coefficient for the area law entropy the area law in $D=3$ as a function of size of the system. Good stability is reached for $N=600$. In the inset, the corresponding coefficient for the single-copy area law is plotted.

Figure 3

Dependence of the geometric entropy and single-copy entanglement slopes, $k_S$ and $k_E$, on the dimension $D$ for a massless scalar field. Note the divergence at $D=5$ due to the insufficient radial regularization of the original field theory.

Figure 4

Evolution of entropy to single-copy entanglement ratio $S∕E_1$ as a function of the dimension $D$. The line starts at a value of 2, as demonstrated analytically in [9] and grows monotonically. The higher the dimension is, the less entanglement is carried by a single copy of the system as compared to many copies.

Figure 5

Geometric entropy $S$ for a sphere of radius $R$ in $D=3$ as a function of the mass $\mu$. Note that larger masses produce a smaller coefficient in the scaling area law.

Figure 6

Entanglement loss along the RG trajectories seen in the space spanned by the eigenvalues of the reduced density matrix.