Local time-reversal-invariant topological markers in two dimensions
Abstract
Local topological markers provide a real-space characterization of topological phases and are suited to noncrystalline models. We derive a local topological marker, the Chern-Simons descendant marker, for characterizing the topology of two-dimensional time-reversal-invariant near-Gaussian states from the one-particle density matrix ρ. We construct an auxiliary three-dimensional state characterized by the local Chern-Simons marker, and use dimensional reduction to obtain the Chern-Simons descendant marker for the original two-dimensional state. This derivation introduces a local Hermitian involution S that is odd under time reversal. When [S,ρ]=0, the local Chern-Simons descendant marker reduces to half the difference of the local Chern markers in the S=±1 sectors. Constructing an S that commutes exactly with ρ generally requires fine-tuning, but this is not needed---the local Chern-Simon marker remains well defined whenever Γ={ρ,S}−S has a spectral gap around zero, even when [S,ρ]≠0. This leaves the practical problem of identifying a suitable S for a given state, which, importantly, is straightforward even when spin is not a candidate. We show this by considering both strong Rashba and Dresselhaus spin-orbit coupling, for which knowledge of the physical origin of the state reduces the general problem of choosing S to a one-parameter family S(θ). We evaluate the optimal angle θ maximizing the spectral gap of Γ analytically assuming translation invariance, and find that the gap of Γ remains open for a broad range of angles around the optimum. The same value of θ remains valid in disordered systems, providing an easy way to obtain choice of S for evaluating the local Chern-Simons descendant marker of the state.
Related
- Toric codes all the way down: decoupling translation-invariant topological CSS codes in higher dimensions
- Hall conductance in a weakly time-reversal invariant open system
- The Ericson Transition in Time-Reversal Invariant Systems: Symplectically Invariant Hamiltonians
- The Ericson Transition in Time-Reversal Invariant Systems: Orthogonally Invariant Hamiltonians
- Time-Reversal-Invariant Altermagnetic Acoustic Crystals
BibTeX
@misc{kvorning2026local,
title = {{Local time-reversal-invariant topological markers in two dimensions}},
author = {Thomas Klein Kvorning and Miguel F. Martínez and Julia D. Hannukainen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28213},
eprint = {2609.28213},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28213},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local time-reversal-invariant topological markers in two dimensions
AU - Thomas Klein Kvorning
AU - Miguel F. Martínez
AU - Julia D. Hannukainen
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28213
UR - https://arxiv.org/abs/2609.28213
AB - Local topological markers provide a real-space characterization of topological phases and are suited to noncrystalline models. We derive a local topological marker, the Chern-Simons descendant marker, for characterizing the topology of two-dimensional time-reversal-invariant near-Gaussian states from the one-particle density matrix ρ. We construct an auxiliary three-dimensional state characterized by the local Chern-Simons marker, and use dimensional reduction to obtain the Chern-Simons descendant marker for the original two-dimensional state. This derivation introduces a local Hermitian involution S that is odd under time reversal. When [S,ρ]=0, the local Chern-Simons descendant marker reduces to half the difference of the local Chern markers in the S=±1 sectors. Constructing an S that commutes exactly with ρ generally requires fine-tuning, but this is not needed---the local Chern-Simon marker remains well defined whenever Γ={ρ,S}−S has a spectral gap around zero, even when [S,ρ]≠0. This leaves the practical problem of identifying a suitable S for a given state, which, importantly, is straightforward even when spin is not a candidate. We show this by considering both strong Rashba and Dresselhaus spin-orbit coupling, for which knowledge of the physical origin of the state reduces the general problem of choosing S to a one-parameter family S(θ). We evaluate the optimal angle θ maximizing the spectral gap of Γ analytically assuming translation invariance, and find that the gap of Γ remains open for a broad range of angles around the optimum. The same value of θ remains valid in disordered systems, providing an easy way to obtain choice of S for evaluating the local Chern-Simons descendant marker of the state.
ER -