Euler Topological Metals in 1D
Abstract
We give a rigorous one-dimensional formulation of Kane's transport proposal for probing the Euler characteristic of a Fermi sea. For a clean, translation-invariant continuum Hamiltonian with real-analytic dispersion, a Fermi sea confined to a finite momentum range, and nonzero Fermi velocity at every Fermi point, we analyze a particular Abel-regularized transport response (obtained by tracing over the Fermi sea) and prove that its large-time limit equals the Euler characteristic. We derive an explicit finite-time formula, show that quantization requires the large-time limit, and obtain a convergence rate under additional nonstationarity assumptions. We also prove the lattice analog, where the sharp commutator is trace-class and a completely filled band contributes zero. Finite-volume calculations illustrate the prescribed order of the thermodynamic and large-time limits.
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BibTeX
@misc{hu2026euler,
title = {{Euler Topological Metals in 1D}},
author = {Yichen Hu and Jacob Shapiro},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27554},
eprint = {2608.27554},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27554},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euler Topological Metals in 1D
AU - Yichen Hu
AU - Jacob Shapiro
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27554
UR - https://arxiv.org/abs/2608.27554
AB - We give a rigorous one-dimensional formulation of Kane's transport proposal for probing the Euler characteristic of a Fermi sea. For a clean, translation-invariant continuum Hamiltonian with real-analytic dispersion, a Fermi sea confined to a finite momentum range, and nonzero Fermi velocity at every Fermi point, we analyze a particular Abel-regularized transport response (obtained by tracing over the Fermi sea) and prove that its large-time limit equals the Euler characteristic. We derive an explicit finite-time formula, show that quantization requires the large-time limit, and obtain a convergence rate under additional nonstationarity assumptions. We also prove the lattice analog, where the sharp commutator is trace-class and a completely filled band contributes zero. Finite-volume calculations illustrate the prescribed order of the thermodynamic and large-time limits.
ER -