Absolute and relative invariants for low-dimensional topological insulators
Abstract
[An extended abstract is provided in-text] Motivated by topological insulators, this paper analyzes classification schemes for projection-valued maps on d-dimensional tori, d ≤ 2, subject to Altland-Zirnbauer-Cartan (AZC) symmetries (time-reversal, particle-hole, and chiral). The main objective is to compare three notions of equivalence for these maps: Murray-von Neumann, unitary, and homotopy equivalence. The analysis relies on constructing symmetric bases for the projection ranges under a pseudo-periodicity constraint, where select basis vectors pick up a phase when looped around the torus. If d=1, fully periodic symmetric bases exist across all AZC classes. If d=2, topological obstructions appear as non-trivial phases characterized by ℤ (Chern number) or ℤ2 (Fu-Kane-Mele) invariants. These serve as absolute invariants that classify both Murray-von Neumann and unitary equivalences. For homotopy equivalence, the behavior depends on the ambient Hilbert space: if this has non-minimal dimension, homotopy equivalence reduces directly to unitary equivalence, while in minimal ambient spaces additional relative homotopy invariants emerge. Their precise values change under unitary conjugation, leaving only relative topological phases between maps well-defined. Ultimately, this provides a unified geometric framework for gapped quantum systems in low dimensions, demonstrating how distinct equivalence relations generate specific absolute and relative topological invariants.
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BibTeX
@misc{manzoni2026absolute,
title = {{Absolute and relative invariants for low-dimensional topological insulators}},
author = {Federico Manzoni and Domenico Monaco and Gabriele Peluso},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32442},
eprint = {2609.32442},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32442},
}
RIS
TY - UNPB
TI - Absolute and relative invariants for low-dimensional topological insulators
AU - Federico Manzoni
AU - Domenico Monaco
AU - Gabriele Peluso
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32442
UR - https://arxiv.org/abs/2609.32442
AB - [An extended abstract is provided in-text] Motivated by topological insulators, this paper analyzes classification schemes for projection-valued maps on d-dimensional tori, d ≤ 2, subject to Altland-Zirnbauer-Cartan (AZC) symmetries (time-reversal, particle-hole, and chiral). The main objective is to compare three notions of equivalence for these maps: Murray-von Neumann, unitary, and homotopy equivalence. The analysis relies on constructing symmetric bases for the projection ranges under a pseudo-periodicity constraint, where select basis vectors pick up a phase when looped around the torus. If d=1, fully periodic symmetric bases exist across all AZC classes. If d=2, topological obstructions appear as non-trivial phases characterized by ℤ (Chern number) or ℤ_2 (Fu-Kane-Mele) invariants. These serve as absolute invariants that classify both Murray-von Neumann and unitary equivalences. For homotopy equivalence, the behavior depends on the ambient Hilbert space: if this has non-minimal dimension, homotopy equivalence reduces directly to unitary equivalence, while in minimal ambient spaces additional relative homotopy invariants emerge. Their precise values change under unitary conjugation, leaving only relative topological phases between maps well-defined. Ultimately, this provides a unified geometric framework for gapped quantum systems in low dimensions, demonstrating how distinct equivalence relations generate specific absolute and relative topological invariants.
ER -