Quantum Ising transition on the Cayley tree: boundary criticality, universality, and an effective Bethe line
Abstract
We investigate quantum criticality in the transverse-field Ising model on finite Cayley trees and demonstrate that their extensive boundary fundamentally changes the nature of the quantum phase transition compared with the infinite, boundary-free Bethe lattice. Using tree tensor-network simulations for systems with up to N=12286 sites, we identify a boundary-driven continuous transition at hxc≈1.76, distinct from the mean-field Bethe-lattice transition at hxc≈2.23. We characterize this transition through finite-size scaling of the magnetization and magnetic susceptibility and extract a consistent set of critical exponents. By introducing next-nearest-neighbor interactions, we demonstrate the robustness of the extracted critical exponents, providing numerical argument for the Ising boundary universality class on the Cayley tree. Furthermore, we show that the physics of the boundary-free Bethe lattice can be efficiently recovered on a finite Cayley tree by fine-tuning a local boundary field. The resulting \textit{effective Bethe line}, whose existence is tied to the mean-field nature of the bulk transition, effectively screens the extensive boundary and restores both the critical point and the mean-field critical scaling of the infinite Bethe lattice.
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BibTeX
@misc{richelli2026quantum,
title = {{Quantum Ising transition on the Cayley tree: boundary criticality, universality, and an effective Bethe line}},
author = {Pietro Richelli and Natalia Chepiga},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16774},
eprint = {2610.16774},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16774},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Ising transition on the Cayley tree: boundary criticality, universality, and an effective Bethe line
AU - Pietro Richelli
AU - Natalia Chepiga
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16774
UR - https://arxiv.org/abs/2610.16774
AB - We investigate quantum criticality in the transverse-field Ising model on finite Cayley trees and demonstrate that their extensive boundary fundamentally changes the nature of the quantum phase transition compared with the infinite, boundary-free Bethe lattice. Using tree tensor-network simulations for systems with up to N=12286 sites, we identify a boundary-driven continuous transition at h_x^c≈1.76, distinct from the mean-field Bethe-lattice transition at h_x^c≈2.23. We characterize this transition through finite-size scaling of the magnetization and magnetic susceptibility and extract a consistent set of critical exponents. By introducing next-nearest-neighbor interactions, we demonstrate the robustness of the extracted critical exponents, providing numerical argument for the Ising boundary universality class on the Cayley tree. Furthermore, we show that the physics of the boundary-free Bethe lattice can be efficiently recovered on a finite Cayley tree by fine-tuning a local boundary field. The resulting \textit{effective Bethe line}, whose existence is tied to the mean-field nature of the bulk transition, effectively screens the extensive boundary and restores both the critical point and the mean-field critical scaling of the infinite Bethe lattice.
ER -