Entanglement Entropy of Free Fermions on Random Fractal Lattices

Technical Summary: Entanglement Entropy of Free Fermions on Random Fractal Lattices

Problem Statement
Quantum entanglement is a fundamental resource for many-body physics and quantum technologies. While the scaling of entanglement entropy (EE) in Euclidean systems is well-understood—following an area law ($S \sim L^{d-1}$) for gapped systems and a logarithmically enhanced area law ($S \sim L^{d-1} \log L$) for gapless free fermions due to the presence of a Fermi surface—the behavior of EE in geometrically disordered, non-integer dimensional systems remains largely unexplored. Specifically, it is unclear how the absence of translational invariance and the presence of fractal geometry affect ground-state entanglement structures and the dynamical spreading of quantum information following a quench. This work investigates free fermions on random fractal lattices (RFLs) to determine if geometric randomness alone can generate nontrivial entanglement scaling and slow dynamics distinct from disorder-induced localization.

Methodology
The authors generate RFLs using a stochastic growth algorithm based on the dielectric breakdown model. Starting from a seed, new sites are added to the cluster from a set of boundary candidates with a probability $P(i \to j) \propto (\phi_j)^\eta$, where $\phi$ is a local potential satisfying the discrete Laplace equation on the underlying Euclidean grid. This process yields a single connected, loop-free graph.

To independently tune the geometric and transport properties of these lattices, the authors introduce a parameter $p$, representing the probability of adding "missing" nearest-neighbor links. This allows for the independent variation of:

• Hausdorff dimension ($d_H$): Characterizing the geometric mass and volume scaling ($V(r) \sim r^{d_H}$).
• Spectral dimension ($d_s$): Characterizing diffusion and connectivity, determined via unbiased classical random walks ($\Omega(t) \sim t^{d_s/2}$).

On these lattices, the authors study noninteracting spinless fermions described by a tight-binding Hamiltonian with hopping amplitude $J$ between connected sites. They analyze two regimes:

1. Ground State: The bipartite EE of subregions defined by graph radius $r$ is computed using the von Neumann entropy of the reduced density matrix, derived from the restricted one-body correlation matrix.
2. Dynamical Evolution: A global quantum quench is performed starting from an uncorrelated checkerboard product state. The time-dependent EE $S(r, t)$ is monitored to study the spreading of entanglement.

Numerical simulations are performed on lattices with $N=5000$ sites, averaging over $\sim 100$ independent realizations to suppress sample-to-sample fluctuations.

Key Results

• Ground-State Entanglement:

  • The disorder-averaged EE exhibits a robust power-law scaling $S(r) \propto r^\alpha$.
  • Away from half-filling, the exponent is consistently $\alpha \approx d_H - 1$. This indicates a generalized area law governed by the Hausdorff dimension.
  • Crucially, the logarithmic enhancement ($S \sim L^{d-1} \log L$) characteristic of Euclidean free fermions is absent. The authors attribute this to the lack of a well-defined, extended Fermi surface on fractal lattices lacking translational invariance.
  • Near half-filling, the scaling becomes anomalous, showing a stronger dependence on connectivity and spectral properties, likely due to high spectral degeneracy at the band center.

• Entanglement Dynamics (Post-Quench):

  • The entanglement growth follows a factorized scaling form: $S(r, t) \propto a(r; d_H) s(t; d_s)$.
  • Spatial Scaling: The subsystem-size dependence is controlled by the Hausdorff dimension, following the fractal area law $a(r) \sim r^{d_H-1}$.
  • Temporal Scaling: The time evolution is governed by the spectral dimension. In the intermediate-time regime (after microscopic short-time effects but before finite-size saturation), the growth is logarithmically slow, $s(t) \sim [\log(t)]^\gamma$.
  • The dynamical exponent $\gamma$ depends nonlinearly on the spectral dimension $d_s$ but is essentially independent of $d_H$.
  • Unlike clean Euclidean systems which show ballistic linear growth ($S \sim t$) or Anderson-localized systems which saturate quickly, these fractal systems exhibit slow dynamics without onsite disorder or interactions.

Significance and Claims
The paper establishes that geometric randomness alone, in the absence of onsite disorder or interactions, is sufficient to generate:

1. Nontrivial Ground-State Structure: A generalized area law determined by the Hausdorff dimension, devoid of the logarithmic enhancements seen in Euclidean gapless systems.
2. Slow Quantum Information Spreading: Logarithmically slow entanglement growth following a quench, driven purely by the hierarchical connectivity and anomalous diffusion intrinsic to the fractal geometry.

The results demonstrate a clear separation of roles between the two fractal dimensions: the Hausdorff dimension dictates the spatial extent of entanglement (the "boundary" available for correlations), while the spectral dimension governs the temporal efficiency of excitations exploring the graph. This provides a distinct mechanism for slow dynamics, separate from many-body localization or Fermi-surface physics, and positions random fractal lattices as a promising arena for exploring non-Euclidean quantum phenomena.