Categorifying Quantum Spin Systems
Abstract
We develop a homotopical framework for understanding the space of quantum spin systems based on topological n-nets, which assign associative unital algebras to local regions in an n-dimensional space in a manner that captures the locality structure of the underlying lattice theories. The net-theoretic perspective admits a natural categorification, in which the category of topological n-nets carries the structure of a higher Morita category. Moreover, the nets of greatest physical interest form a Karoubi-complete subcategory that shares many of the characteristic structural features of the category of finite TQFTs. Our main result identifies the group of locality-preserving symmetries of (3+1)-dimensional quantum spin systems (i.e. quantum cellular automata modulo stabilization, finite-depth quantum circuits, and translations) with the full Witt group of nondegenerate braided fusion categories, thereby answering a question posed by Haah.
Related
- Superselection theory for 2D braided quantum spin systems via Connes fusion
- Response theory at low temperature for weakly interacting quantum spin systems
- Dynamical Conservation Laws in Quantum Spin Systems
- Extremal States of Multipartite Quantum Spin Systems
- Rapid mixing for Gibbs samplers for quantum spin systems with all-to-all interactions
BibTeX
@misc{yu2026categorifying,
title = {{Categorifying Quantum Spin Systems}},
author = {Matthew Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14165},
eprint = {2610.14165},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14165},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Categorifying Quantum Spin Systems
AU - Matthew Yu
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14165
UR - https://arxiv.org/abs/2610.14165
AB - We develop a homotopical framework for understanding the space of quantum spin systems based on topological n-nets, which assign associative unital algebras to local regions in an n-dimensional space in a manner that captures the locality structure of the underlying lattice theories. The net-theoretic perspective admits a natural categorification, in which the category of topological n-nets carries the structure of a higher Morita category. Moreover, the nets of greatest physical interest form a Karoubi-complete subcategory that shares many of the characteristic structural features of the category of finite TQFTs. Our main result identifies the group of locality-preserving symmetries of (3+1)-dimensional quantum spin systems (i.e. quantum cellular automata modulo stabilization, finite-depth quantum circuits, and translations) with the full Witt group of nondegenerate braided fusion categories, thereby answering a question posed by Haah.
ER -