双曲流形上活性连续体的精确标度律与非厄米拓扑相变
Exact Scaling Laws and Non-Hermitian Topological Phase Transitions of Active Continuum on Hyperbolic Manifolds
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中文总结 AI 辅助
本文建立代数框架研究双曲空间上活性连续体的拓扑相变，推导了极化临界阈值等精确结果，识别出宏观二阶例外点的标度律，为软物质力学的相关问题提供精确解。
中文摘要 AI 辅助
弯曲流形上活性连续体的宏观集体运动通常通过微扰动力学重整化或有限元模拟来研究，这些方法往往会掩盖其背后的几何机制。本文建立了一个精确的代数框架，对双曲空间$\boldsymbol{\rm H}^2$上的活性相变进行了重新表述。通过对协变类Navier-Stokes方程进行严格展开并应用Weitzenböck恒等式，我们推导得到宏观极化的精确临界阈值$\boldsymbol{\rm α_c = \frac{5}{4}Dκ^2}$，该阈值由Hodge-de Rham拉普拉斯算子的几何质量隙决定。我们严格定义了拓扑自由能达到Bogomolny-Prasad-Sommerfield（BPS）极限的参数子空间$\boldsymbol{\rm α= 2Dκ^2}$。这使得复杂速度场可以通过Möbius规范对称性简化为Blaschke乘积。平坦空间极限（$\boldsymbol{\rm κ\to 0}$）会精确退化为经典O(2)模型的拓扑相，表明常负曲率对非线性振幅饱和起到了非微扰红外正则化的作用。此外，将非变分活性对流模式映射到缺陷平移零模，得到了一个本质上非互易的相互作用矩阵。通过对动力学凝聚缺陷环的奇异积分算子进行解析延拓，我们识别出一个宏观二阶例外点（EP2），其特征为严格代数形式的动力学标度律$\boldsymbol{\rm τ\thicksim |Δν|^{-1/2}}$。这种闭式理论范式为软物质力学中的几何阻挫与非厄米拓扑问题提供了精确解。
英文摘要
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 $\mathbb{H}^2$. By rigorously expanding the covariant Navier-Stokes-like equations and applying the Weitzenböck identity, we derive the exact critical threshold $α_c = \frac{5}{4}Dκ^2$ 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 ($κ\to 0$) 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 $τ\sim |Δν|^{-1/2}$. This closed-form theoretical paradigm provides exact solutions for geometric frustration and non-Hermitian topology in soft mechanics.
发表机构
- Tianjin University（天津大学）
机构由 AI 辅助整理，请以论文原文为准。
