Scaling behavior of eigenspectrum for entanglement from correlation matrices
Abstract
We study the scaling behavior of the eigenvalues of correlation matrices, which characterize the entanglement of a subsystem with its complement part of a total pure state. A most distinguishing feature of entanglement entropy is its logarithmic dependence on the subsystem size for the groundstate of one-dimensional critical systems. Despite its robust universal character and relevance to a wide range of topics, a thorough understanding of this result requires sophisticated mathematical physics techniques or conformal field theory. The aim of our work is to shed light on this from the underlying eigenvalue distribution perspective. The central object is the correlation matrix, which takes the form of Toeplitz or block-Toeplitz matrix. We develop a circulant matrix approximation in the large matrix dimension limit, thus allowing for the individual eigenvalues behavior to be analysed analytically. We find that for both free lattice fermions and transverse field Ising chain, eigenvalues in the bulk of the eigenspectrum scales as 1/LA with the subsystem size. Together with the extensivity of the entropy function, it explains the robust log2 LA scaling of entanglement at criticality. Perturbing from the entanglement-free limit of the Ising chain, we find a smooth crossover behavior to `non-critical' scaling that is characterized by a very slow logarithmic dependence rendering it seemingly a constant entanglement value expected of non-critical systems.
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BibTeX
@misc{lim2026scaling,
title = {{Scaling behavior of eigenspectrum for entanglement from correlation matrices}},
author = {Lih-King Lim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25608},
eprint = {2609.25608},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25608},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scaling behavior of eigenspectrum for entanglement from correlation matrices
AU - Lih-King Lim
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25608
UR - https://arxiv.org/abs/2609.25608
AB - We study the scaling behavior of the eigenvalues of correlation matrices, which characterize the entanglement of a subsystem with its complement part of a total pure state. A most distinguishing feature of entanglement entropy is its logarithmic dependence on the subsystem size for the groundstate of one-dimensional critical systems. Despite its robust universal character and relevance to a wide range of topics, a thorough understanding of this result requires sophisticated mathematical physics techniques or conformal field theory. The aim of our work is to shed light on this from the underlying eigenvalue distribution perspective. The central object is the correlation matrix, which takes the form of Toeplitz or block-Toeplitz matrix. We develop a circulant matrix approximation in the large matrix dimension limit, thus allowing for the individual eigenvalues behavior to be analysed analytically. We find that for both free lattice fermions and transverse field Ising chain, eigenvalues in the bulk of the eigenspectrum scales as 1/L_A with the subsystem size. Together with the extensivity of the entropy function, it explains the robust log_2 L_A scaling of entanglement at criticality. Perturbing from the entanglement-free limit of the Ising chain, we find a smooth crossover behavior to `non-critical' scaling that is characterized by a very slow logarithmic dependence rendering it seemingly a constant entanglement value expected of non-critical systems.
ER -