Topological Band Theory for High-Dimensional Parameter Spaces
Abstract
We study the topological band theory of quantum Hall systems in which the adiabatic flux parameters of an external gauge potential form a high-dimensional manifold. This situation arises both for strongly correlated matter on spatial Riemann surfaces of genus g > 1, as well as in the formal study of matter on compact oriented manifolds of dimension 4ℓ+2 = 2p coupled to p-form gauge potentials. In these cases, the underlying topological invariants defined over the parameter space are significantly richer than the low-dimensional (i.e., two-dimensional) case and involve both the curvature (i.e., first Chern class) as well as higher order curvature invariants. These higher curvature invariants correspond to topologically protected contributions to Kubo-like formulae, constructed from correlation functions of the physical current operators. We illustrate these general considerations with an explicit example from hyperbolic band theory based on the genus-two Bolza Riemann surface, a case which has recently been simulated on a synthetic-dimension platform.
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BibTeX
@misc{chakrabhavi2026topological,
title = {{Topological Band Theory for High-Dimensional Parameter Spaces}},
author = {Vivek Chakrabhavi and Walt Chavarria and Jonathan J. Heckman and Steven Rayan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35977},
eprint = {2609.35977},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35977},
}
RIS
TY - UNPB
TI - Topological Band Theory for High-Dimensional Parameter Spaces
AU - Vivek Chakrabhavi
AU - Walt Chavarria
AU - Jonathan J. Heckman
AU - Steven Rayan
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35977
UR - https://arxiv.org/abs/2609.35977
AB - We study the topological band theory of quantum Hall systems in which the adiabatic flux parameters of an external gauge potential form a high-dimensional manifold. This situation arises both for strongly correlated matter on spatial Riemann surfaces of genus g > 1, as well as in the formal study of matter on compact oriented manifolds of dimension 4ℓ+2 = 2p coupled to p-form gauge potentials. In these cases, the underlying topological invariants defined over the parameter space are significantly richer than the low-dimensional (i.e., two-dimensional) case and involve both the curvature (i.e., first Chern class) as well as higher order curvature invariants. These higher curvature invariants correspond to topologically protected contributions to Kubo-like formulae, constructed from correlation functions of the physical current operators. We illustrate these general considerations with an explicit example from hyperbolic band theory based on the genus-two Bolza Riemann surface, a case which has recently been simulated on a synthetic-dimension platform.
ER -