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 SX=Θ(LXdb) under a uniform gap and satisfies LXdb≲ SX≲ LXdbln LX when the gap closes no faster than algebraically. Here LX is the subsystem linear size and db the boundary dimension. Half occupation of the chiral zero modes gives a mixed-state entropy SXmix with max{SX,zXln2}leqslant SXmixleqslant SX+zXln2, where zX is the local zero-mode weight; an extensive zX thus makes SXmix follow the bulk dimension.
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BibTeX
@misc{cheng2026scaling,
title = {{Scaling laws of entanglement entropy in fractal dimensions}},
author = {Zheyu Cheng and Kailin Song and Jian-Wei Liu and Hanxi Sun and Hongyu Zou and Yuanjin Zheng and Yihao Yang and Baile Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41761},
eprint = {2610.41761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41761},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scaling laws of entanglement entropy in fractal dimensions
AU - Zheyu Cheng
AU - Kailin Song
AU - Jian-Wei Liu
AU - Hanxi Sun
AU - Hongyu Zou
AU - Yuanjin Zheng
AU - Yihao Yang
AU - Baile Zhang
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41761
UR - https://arxiv.org/abs/2610.41761
AB - 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(L_X^d_b) under a uniform gap and satisfies L_X^d_b≲ S_X≲ L_X^d_bln L_X when the gap closes no faster than algebraically. Here L_X is the subsystem linear size and d_b the boundary dimension. Half occupation of the chiral zero modes gives a mixed-state entropy S_X^mix with max{S_X,z_Xln2}leqslant S_X^mixleqslant S_X+z_Xln2, where z_X is the local zero-mode weight; an extensive z_X thus makes S_X^mix follow the bulk dimension.
ER -