Scaling laws of entanglement entropy in fractal dimensions

Abstract

Entanglement entropy scaling links geometry to quantum correlations and diagnoses quantum criticality. Fractal lattices extend this link to noninteger dimensions, where boundary and bulk dimensions are independent. It remains open which dimension governs entropy scaling and how strongly gap closing can enhance it. Nonlocal occupied-state projectors and the lack of translation symmetry obstruct standard methods. We overcome these obstacles and classify free-fermion entropy scaling with bounds set by the hopping across the subsystem boundary: an upper bound from its rank and the spectral gap, and a lower bound from how much it lowers the ground-state energy. With only negative-energy states filled and this energy lowering proportional to the boundary, the entanglement entropy obeys the boundary law $S_X=Θ\left(L_X^{d_{\mathrm b}}\right)$ under a uniform gap and satisfies $L_X^{d_{\mathrm b}}\lesssim S_X\lesssim L_X^{d_{\mathrm b}}\ln L_X$ when the gap closes no faster than algebraically. Here $L_X$ is the subsystem linear size and $d_{\mathrm b}$ the boundary dimension. Half occupation of the chiral zero modes gives a mixed-state entropy $S_X^{\mathrm{mix}}$ with $\max\left\{S_X,z_X\ln2\right\}\leqslant S_X^{\mathrm{mix}}\leqslant S_X+z_X\ln2$, where $z_X$ is the local zero-mode weight; an extensive $z_X$ thus makes $S_X^{\mathrm{mix}}$ follow the bulk dimension.

The paper asks which geometric dimension controls free-fermion entanglement on fractal lattices, where boundary and bulk dimensions are independent, and how gap closure modifies scaling. It derives bounds without translation symmetry and applies them to Sierpiński-carpet and Vicsek graphs. For pure states filling only negative-energy levels, the upper bound is $S_X\leq8(\ln2)r_X[1+\pi^{-2}\ln(4+8M/\delta_g)]$, where $r_X$ is the cut-hopping rank, $M=\|h\|$, and $\delta_g$ is the occupied–unoccupied separation. Its derivation uses a Sylvester equation and singular-value estimates. Lower bounds use the cut-hopping energy expectation or density anticorrelations across boundary bonds. Assuming bounded $M$, rank proportional to boundary size, and a positive boundary-averaged lower bound, a uniform gap gives $S_X=\Theta(L_X^{d_{\mathrm b}})$; at most algebraic gap closure gives $cL_X^{d_{\mathrm b}}\leq S_X\leq CL_X^{d_{\mathrm b}}\ln L_X$. For the nearest-neighbor Sierpiński carpet, recursive dimer elimination and boundary-pattern estimates establish $\delta_g=\Theta(L_X^{-1})$ and a boundary law without a leading logarithm. Numerical finite-generation calculations test the predictions. Separately, half-occupying chiral zero modes produces mixed-state subsystem entropy satisfying $\max\{S_X,z_X\ln2\}\leq S_X^{\mathrm{mix}}\leq S_X+z_X\ln2$. On the Vicsek lattice, $z_X/|X|\to7/25$, yielding bulk scaling with exponent $\log_3 5$; this contribution is thermodynamic occupation entropy, not pure-state entanglement. The general gapless bounds do not determine whether a leading logarithm occurs. Separating quantum correlations from mixed-state entropy, and extending the analysis to nonchiral, random, or multifractal systems, remain open.