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arXiv:2609.03360

Nonlinear Dissipation and Hopf Criticality in Driven Dissipative Collective Spins

Shu Yang, Jun Wang, Weidong Li, Cangtao Zhou, Jian-Song Pan, Jianwen Jie

quant-phSubmitted Sep 3, 2026• Announced Sep 4, 2026

v1 · Sep 3, 2026 v2 · Sep 6, 2026

Abstract

Self-sustained oscillations in open many-body systems can be organized by two distinct dynamical ingredients: the selection of a finite-amplitude background attractor and the subsequent control of its neutral phase. We develop this bifurcation-based picture for driven-dissipative collective spins and show how the microscopic dissipative structure determines the available background and its response to explicit U(1) breaking. In the thermodynamic-limit mean-field dynamics, a single linear U(1)-covariant jump produces only polar fixed-point backgrounds, whereas nonlinear covariant dissipation provides amplitude-dependent saturation and stabilizes a finite-latitude self-sustained-oscillator manifold through a supercritical Hopf bifurcation. Under coherent U(1) breaking, exact resonance leads to a reversible double-zero degeneracy with vanishing critical frequency rather than a standard Hopf onset. Finite detuning unfolds this singularity into a genuine finite-frequency Hopf boundary, which exists only on the self- sustained-oscillator side and can be either supercritical or subcritical. By contrast, a single linear dissipative U(1)-breaking jump cannot generate a standard Hopf instability: when its phase-pinning invariant vanishes the azimuthal direction remains neutral, whereas otherwise the phase-locked fixed points have a purely real Jacobian spectrum. These results establish a general design principle: nonlinear covariant dissipation selects the self-sustained background, while the structure of the symmetry-breaking channel determines whether the resulting local response is double-zero, genuinely Hopf, or non-Hopf.

Comments: 20 pages, 7 figures

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