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arXiv:2606.29304

Volume Law and Universality of Entanglement Entropy in Random Graph Fermi Systems

Saikat Sur

quant-phmath-phSubmitted Jun 28, 2026• Announced Jun 30, 2026• 0 citations• Revised Jul 6, 2026 (v2)
v1 · Jun 28, 2026v2 · Jul 6, 2026
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Abstract

We study the ground-state entanglement entropy of free fermions on random graphs. We first establish a general criterion for the volume law on random graphs, based on the spectral characteristics of the Hamiltonian. We then apply this criterion to the Erdős--Rényi random graph, where each of the possible edges is present independently with some probability. Using random matrix theory and asymptotic freeness, we show that the ground-state entanglement entropy obeys an exact volume law in Erdős--Rényi graphs in the thermodynamic limit, with a universal coefficient that is independent of the edge probability of the graph. This coefficient is confirmed numerically to take the value approximately $0.3863$ nats, strictly below the Page value. The volume law therefore reflects the absence of geometric locality in the random graph.

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We study the ground-state entanglement entropy of free fermions on random graphs. We first establish a general criterion for the volume law on random graphs, based on the spectral characteristics of the Hamiltonian. We then apply this criterion to the Erdős--Rényi random graph, where each of the possible edges is present independently with some probability. Using random matrix theory and asymptotic freeness, we show that the ground-state entanglement entropy obeys an exact volume law in Erdős--Rényi graphs in the thermodynamic limit, with a universal coefficient that is independent of the edge probability of the graph. This coefficient is confirmed numerically to take the value approximately 0.38630.38630.3863 nats, strictly below the Page value. The volume law therefore reflects the absence of geometric locality in the random graph.
We consider a quantum system of large size N and its subsystem of size L, assuming that N is much larger than L, which can also be sufficiently large, i.e., 1≪L≲N. A widely accepted mathematical version of this inequality is the asymptotic regime of successive limits: first the macroscopic limit N→∞, then an asymptotic analysis of the entanglement entropy as L→∞. In this paper, we consider another version of the above inequality: the regime of asymptotically proportional L and N, i.e., the simultaneous limits L→∞,N→∞,L/N→λ>0. Specifically, we consider a system of free fermions that is in its ground state, and such that its one-body Hamiltonian is a large random matrix, which is often used to model long-range hopping. By using random matrix theory, we show that in this case, the entanglement entropy obeys the volume law known for systems with short-range hopping but described either by a mixed state or a pure strongly excited state of the Hamiltonian. We also give streamlined proof of Page’s formula for the entanglement entropy of black hole radiation for a wide class of typical ground states, thereby proving the universality and the typicality of the formula.
We show how the area law for entanglement entropy may be violated by free fermions on a lattice, and we look for conditions leading to the emergence of a volume law. We give an explicit construction of the states with maximal entanglement entropy based on the fact that, once a bipartition of the lattice in two complementary sets AAA and A‾\overline{A}A is given, the states with maximal entanglement entropy (volume law) may be factored into Bell pairs (BPs) formed by two states with support on AAA and A‾\overline{A}A. We then exhibit, for translational invariant fermionic systems on a lattice, a Hamiltonian whose ground state is such that it yields an exact volume law. As expected, the corresponding Fermi surface has a fractal topology. We also provide some examples of fermionic models for which the ground state may have an entanglement entropy SA{S}_{A}SA​ between the area and the volume law, building an explicit example of a one-dimensional free fermion model where ${S}_{A}(L)\ensuremath{\propto}{L}^{\ensuremath{\beta}}$, with $\ensuremath{\beta}$ being intermediate between $\ensuremath{\beta}=0$ (area law) and $\ensuremath{\beta}=1$ (BP state inducing volume law). For this model, the dispersion relation has a ``zigzag'' structure leading to a fractal Fermi surface whose counting box dimension equals, for large lattices, $\ensuremath{\beta}$. Our analysis clearly relates the violation of the area law for the entanglement entropy of the ground state to the emergence of a nontrivial topology of the Fermi surface.
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Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability ppp, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.
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We investigate multipartite information and entanglement measures in the ground state of a free-fermion model defined on a Hamming graph. Using the known diagonalization of the adjacency matrix, we solve the model and construct the ground-state correlation matrix. Moreover, we find all the eigenvalues of the chopped correlation matrix when the subsystem consists of n disjoint Hamming subgraphs embedded in a larger one. These results allow us to find an exact formula for the entanglement entropy of disjoint graphs, as well as for the mutual and tripartite information. We use the exact formulas for these measures to extract their asymptotic behavior in two distinct thermodynamic limits, and find excellent match with the numerical calculations. In particular, we find that the entanglement entropy admits a logarithmic violation of the area law which decreases the amount of entanglement compared to the area law scaling.