Chiral Color Ice: Exact Local Handedness Constraints and Möbius Zero Modes in Frustrated Magnets
Abstract
Local constraints govern the low-energy physics of frustrated matter, but familiar ice-type rules constrain flux-like quantities and are insensitive to handedness. Here we show that handedness itself can be imposed as an exact local quantum constraint without selecting an axis in spin space. We construct positive-semidefinite, SU(2)-invariant parent Hamiltonians whose complete zero-energy space on a tetrahedron has a prescribed chirality sign, rather than selecting a particular chiral wave function. For spin-1/2 the local term is a rank-one projector onto a chiral tetrahedral singlet, while for arbitrary spin it factorizes as B† B through a singlet-annihilation operator, with a completely characterized kernel given by the span of the globally rotated chiral color-ice states. For coherent states, the same zero-energy condition becomes an S-independent nonlinear constraint in which three spin directions determine the fourth through a Möbius transformation; compositions of these maps define constraint holonomies on extended lattices. Connecting the same local constraint in different ways produces qualitatively different collective regimes: corner-sharing lattices retain exponentially large quantum ground-state kernels, with rigorous lower bounds exceeding conventional ice benchmarks; edge-sharing lattices support subdimensional plane or line zero modes; while triangular constructions suppress nonuniform coherent deformations and contain the complete Anderson tower of tetrahedral magnetic order at exactly zero energy. Two inequivalent triangular coverings further show that harmonic zero-mode counting does not determine the size of the quantum kernel. These results establish a tractable setting in which local handedness, nonlinear constraint geometry, and quantum degeneracy can be disentangled and related directly to the connectivity of the constraint network.
Related
- Existence and large-field asymptotics of Hopfions in a continuum model of frustrated magnets
- Quaternionic Möbius invariant Laplacian and quaternionic Möbius harmonic functions on the unit ball
- Mobius invariant weighted reduced quaternionic Qp-modules
- A Möbius function computing h-polynomials of nestohedra
- Möbius Maps, Reflections and Lipschitz Constants
BibTeX
@misc{kranitz2026chiral,
title = {{Chiral Color Ice: Exact Local Handedness Constraints and Möbius Zero Modes in Frustrated Magnets}},
author = {Péter Kránitz and Yasir Iqbal and Karlo Penc},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30802},
eprint = {2608.30802},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30802},
}
RIS
TY - UNPB
TI - Chiral Color Ice: Exact Local Handedness Constraints and Möbius Zero Modes in Frustrated Magnets
AU - Péter Kránitz
AU - Yasir Iqbal
AU - Karlo Penc
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30802
UR - https://arxiv.org/abs/2608.30802
AB - Local constraints govern the low-energy physics of frustrated matter, but familiar ice-type rules constrain flux-like quantities and are insensitive to handedness. Here we show that handedness itself can be imposed as an exact local quantum constraint without selecting an axis in spin space. We construct positive-semidefinite, SU(2)-invariant parent Hamiltonians whose complete zero-energy space on a tetrahedron has a prescribed chirality sign, rather than selecting a particular chiral wave function. For spin-1/2 the local term is a rank-one projector onto a chiral tetrahedral singlet, while for arbitrary spin it factorizes as B^† B through a singlet-annihilation operator, with a completely characterized kernel given by the span of the globally rotated chiral color-ice states. For coherent states, the same zero-energy condition becomes an S-independent nonlinear constraint in which three spin directions determine the fourth through a Möbius transformation; compositions of these maps define constraint holonomies on extended lattices. Connecting the same local constraint in different ways produces qualitatively different collective regimes: corner-sharing lattices retain exponentially large quantum ground-state kernels, with rigorous lower bounds exceeding conventional ice benchmarks; edge-sharing lattices support subdimensional plane or line zero modes; while triangular constructions suppress nonuniform coherent deformations and contain the complete Anderson tower of tetrahedral magnetic order at exactly zero energy. Two inequivalent triangular coverings further show that harmonic zero-mode counting does not determine the size of the quantum kernel. These results establish a tractable setting in which local handedness, nonlinear constraint geometry, and quantum degeneracy can be disentangled and related directly to the connectivity of the constraint network.
ER -