Exact Scaling Laws and Non-Hermitian Topological Phase Transitions of Active Continuum on Hyperbolic Manifolds
Abstract
The macroscopic collective motion of active continuum on curved manifolds is conventionally addressed through perturbative dynamic renormalization or finite-element simulations, often obscuring the underlying geometric mechanisms. Here, an exact algebraic framework is established to reformulate the active phase transition on hyperbolic spaces ℍ2. By rigorously expanding the covariant Navier-Stokes-like equations and applying the Weitzenböck identity, we derive the exact critical threshold αc = 5/4Dκ2 for macroscopic polarization, which is dictated by the geometric mass gap of the Hodge-de Rham Laplacian. We strictly define the parameter subspace α= 2Dκ2 where the topological free energy reaches the Bogomolny-Prasad-Sommerfield (BPS) limit. This enables the reduction of the complex velocity field to Blaschke products via Möbius gauge symmetry. The flat-space limit (κ→ 0) exactly degenerates to the topological phase of the classical O(2) model, demonstrating that the constant negative curvature acts as an un-perturbative infrared regularization for non-linear amplitude saturation. Furthermore, mapping the non-variational active convective modes onto the defect translational zero-modes yields an intrinsically non-reciprocal interaction matrix. By analytically extending the singular integral operator of the dynamically condensed defect ring, we identify a macroscopic second-order exceptional point (EP2) characterized by a strictly algebraic dynamic scaling law τ∼ |Δν|−1/2. This closed-form theoretical paradigm provides exact solutions for geometric frustration and non-Hermitian topology in soft mechanics.
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BibTeX
@misc{xie2026exact,
title = {{Exact Scaling Laws and Non-Hermitian Topological Phase Transitions of Active Continuum on Hyperbolic Manifolds}},
author = {Yu-Xin Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03500},
eprint = {2609.03500},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03500},
note = {CC0},
}
RIS
TY - UNPB
TI - Exact Scaling Laws and Non-Hermitian Topological Phase Transitions of Active Continuum on Hyperbolic Manifolds
AU - Yu-Xin Xie
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03500
UR - https://arxiv.org/abs/2609.03500
AB - The macroscopic collective motion of active continuum on curved manifolds is conventionally addressed through perturbative dynamic renormalization or finite-element simulations, often obscuring the underlying geometric mechanisms. Here, an exact algebraic framework is established to reformulate the active phase transition on hyperbolic spaces ℍ^2. By rigorously expanding the covariant Navier-Stokes-like equations and applying the Weitzenböck identity, we derive the exact critical threshold α_c = 5/4Dκ^2 for macroscopic polarization, which is dictated by the geometric mass gap of the Hodge-de Rham Laplacian. We strictly define the parameter subspace α= 2Dκ^2 where the topological free energy reaches the Bogomolny-Prasad-Sommerfield (BPS) limit. This enables the reduction of the complex velocity field to Blaschke products via Möbius gauge symmetry. The flat-space limit (κ→ 0) exactly degenerates to the topological phase of the classical O(2) model, demonstrating that the constant negative curvature acts as an un-perturbative infrared regularization for non-linear amplitude saturation. Furthermore, mapping the non-variational active convective modes onto the defect translational zero-modes yields an intrinsically non-reciprocal interaction matrix. By analytically extending the singular integral operator of the dynamically condensed defect ring, we identify a macroscopic second-order exceptional point (EP2) characterized by a strictly algebraic dynamic scaling law τ∼ |Δν|^−1/2. This closed-form theoretical paradigm provides exact solutions for geometric frustration and non-Hermitian topology in soft mechanics.
ER -