Non-Bloch band theory for finite banded non-Hermitian Toeplitz matrices
Abstract
Non-Bloch band theory recovers the open-boundary spectrum of a non-Hermitian chain from its Bloch Hamiltonian evaluated along the generalised Brillouin zone rather than the unit circle. We develop its finite-size counterpart for banded Toeplitz matrices with real open-boundary spectrum. Reality of the spectrum forces the generalised Brillouin zone to be parametrised by an analytic polar curve, so the known sufficient condition for a real spectrum is also necessary under some mild non-degeneracy conditions. At finite size n, the eigenvalues of the original and the symmetrised Bloch Hamiltonians are given by samples at n quantised momenta from the same limiting band, although separated by a phase mismatch of order 1/n which we obtain in closed form. Iterating this correction K times yields a quantisation rule on the generalised Brillouin zone reproducing the finite spectrum with an error 𝒪(n−K−1) for any fixed K. At finite size, the generalised Brillouin zone itself is unchanged and only its parametrisation depends on n.
Related
- Constructive recurrences for determinants and permanents of banded Toeplitz matrices
- Eigenvalues of Hermitian Toeplitz matrices with Fisher--Hartwig symbols
- Finite-rank multiplicative perturbations of rotationally invariant non-Hermitian random matrices
- Symmetry reductions and recurrence degrees for banded Toeplitz determinants and permanents
- The Fourier Uncertainty Principle Through the Lens of Infinite Banded Matrices
BibTeX
@misc{bruijn2026nonbloch,
title = {{Non-Bloch band theory for finite banded non-Hermitian Toeplitz matrices}},
author = {Yannick de Bruijn and Erik Orvehed Hiltunen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13715},
eprint = {2610.13715},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13715},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-Bloch band theory for finite banded non-Hermitian Toeplitz matrices
AU - Yannick de Bruijn
AU - Erik Orvehed Hiltunen
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13715
UR - https://arxiv.org/abs/2610.13715
AB - Non-Bloch band theory recovers the open-boundary spectrum of a non-Hermitian chain from its Bloch Hamiltonian evaluated along the generalised Brillouin zone rather than the unit circle. We develop its finite-size counterpart for banded Toeplitz matrices with real open-boundary spectrum. Reality of the spectrum forces the generalised Brillouin zone to be parametrised by an analytic polar curve, so the known sufficient condition for a real spectrum is also necessary under some mild non-degeneracy conditions. At finite size n, the eigenvalues of the original and the symmetrised Bloch Hamiltonians are given by samples at n quantised momenta from the same limiting band, although separated by a phase mismatch of order 1/n which we obtain in closed form. Iterating this correction K times yields a quantisation rule on the generalised Brillouin zone reproducing the finite spectrum with an error 𝒪(n^−K−1) for any fixed K. At finite size, the generalised Brillouin zone itself is unchanged and only its parametrisation depends on n.
ER -