Emergent Macroscopic Nonreciprocity from Identical Active Particles via Spontaneous Symmetry Breaking

Abstract

Nonreciprocity is known to generate a wide range of exotic phenomena in multi-species many-body systems, where different species influence one another through couplings that violate Newton's third law. In contrast, in the absence of explicitly imposed macroscopic nonreciprocal processes, single-species nonreciprocity -- another distinct form of nonreciprocity -- typically plays only a limited role in shaping macroscopic physics. Here, using a single-species Vicsek model with a vision cone and extrinsic noise, we show that spontaneous symmetry breaking (SSB) can dramatically enhance the macroscopic consequences of microscopic single-species nonreciprocity. In the ordered phase, this enhancement gives rise to an emergent macroscopic nonreciprocity that induces the system of identical active particles to admit an effective description with a "two-species" non-Hermitian structure. The resulting SSB-enhanced nonreciprocity substantially promotes traveling-band formation and, more strikingly, drives a novel real-space condensation of identical active particles, characterized by a "traveling line" with vanishing longitudinal width. Our findings uncover a fundamental mechanism by which microscopic single-species nonreciprocity can exert strong macroscopic influences in complex systems.

Figures (4)

• Figure 1: (a) Typical steady-state configurations at high noise, $\eta=0.75>\eta_{c}$, where the system is disordered. In this regime, microscopic front--back nonreciprocity does not accumulate macroscopically, and no qualitative differences are observed for different view angles. (b) Typical steady-state configurations at low noise, $\eta=0.1<\eta_{c}$, where SSB induces collective motion. In this ordered phase, nonreciprocity is enhanced and produces pronounced macroscopic effects. For sufficiently small view angle (leftmost panel), the system exhibits real-space condensation, in which all particles collapse to the same longitudinal position along the collective velocity. (c) Distribution of the band width $w_{b}$ in the $(\eta,\phi)$ parameter space. The lower-left blue region corresponds to real-space condensation with vanishing $w_{b}$. For $\eta<\eta_{c}$, decreasing $\phi$ stabilizes traveling bands and eventually drives condensation. In contrast, for $\eta>\eta_{c}$, varying $\phi$ produces no observable effect (upper red region), demonstrating that macroscopic nonreciprocity emerges only in the SSB phase. See text for details.
• Figure 2: (a) Representative single-trajectory time evolution of the band width $w_{b}$ at noise level $\eta=0.6$, slightly below the flocking transition $\eta_{c}$ ($N=10^{4}$, $L=50$). For $\phi=5\pi/3$, temporal fluctuations of $w_{b}$ are strongly suppressed, demonstrating the stabilization of the traveling band by SSB-enhanced nonreciprocity. (b) Typical steady-state configurations for $\phi=5\pi/3$, showing robust traveling bands over a broad range of noise strengths ($N=2500$, $L=25$). See text for details.
• Figure 3: (a) Normalized density profile $\rho(x)$ along the collective velocity $\mathbf{v}_{c}$ for different view angles at low noise $\eta=0.1$. For clarity, the peak of each distribution is shifted to $x=0$. When $\phi$ becomes slightly smaller than $\pi$, the profile collapses into a sharp peak, signaling real-space condensation. (b) $\phi$-dependence of the peak value $\max[\rho(x)]$ at different noise levels. For low noise ($\eta=0.1,\,0.2$), $\max[\rho(x)]$ reaches unity once $\phi\apprle\pi$, indicating complete longitudinal condensation. (c) $\phi$-dependence of the band width $w_{b}$ at different noise levels. For $\eta=0.1,\,0.2$, and $0.6<\eta_{c}$, $w_{b}$ decreases as $\phi$ is reduced, reflecting enhanced longitudinal compression. In contrast, for $\eta=0.7>\eta_{c}$, $w_{b}$ is essentially independent of $\phi$, demonstrating that macroscopic nonreciprocity is operative only in the SSB phase. See text for details.
• Figure 4: The view angle $\phi$ dependence of the traveling band width $w_{b}$ in different cases. These results show that the particle density and the system size do not assume strong influences on the physical effects of the SSB-enhanced single-species nonreciprocity. See text for more details.