Area law and vacuum reordering in harmonic networks

TL;DR

The paper addresses how geometric entropy in local quantum systems scales with subsystem size and how the vacuum reorganizes under renormalization-group flows, focusing on harmonic networks across $D$ dimensions. It provides an explicit radial discretization of a scalar free field, yielding a full entanglement spectrum and proving area-law scaling for both the geometric entropy $S$ and the single-copy entanglement $E_1$, with detailed control over the eigenvalues of the reduced density matrix. A central finding is vacuum reordering, captured by majorization relations, which also underlie entanglement loss along RG trajectories driven by the mass term; this behavior holds across dimensions and is complemented by perturbative tail analyses that establish convergence for $D<5$. The work argues for the relevance of finite-χ PEPS as a natural framework for encoding area-law states in regulated QFTs and connects these entanglement properties to the locality structure of the vacuum, with implications for classical simulability and tensor-network approaches to quantum field theories.

Abstract

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.

Figures (6)

• 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 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 OLEC05 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 are law.