We show that lasing in flat-band lattices can be stabilized by means of the geometrical properties of the Bloch states, in settings where the single-particle dispersion is flat in both its real and imaginary parts. We illustrate a general projection method and compute the collective excitations, which display a diffusive behavior ruled by quantum geometry through a peculiar coefficient involving gain, losses and interactions, and entailing resilience against modulational instabilities. Then, we derive an equation of motion for the phase dynamics and identify a Kardar-Parisi-Zhang term of geometric origin. This term is shown to exactly cancel whenever the real and imaginary parts of the laser nonlinearity are proportional to each other, or when the uniform-pairing condition is satisfied. We confirm our results through numerical studies of the $\pi$-flux diamond chain. This Letter highlights the key role of Bloch geometric effects in nonlinear dissipative systems and KPZ physics, with direct implications for the design of laser arrays with enhanced coherence.

中文翻译：

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非厄米平带中的激光：量子几何、相干性和卡达尔-帕里西-张物理学的命运

我们证明，在单粒子色散的实部和虚部都是平坦的情况下，平带晶格中的激光可以通过布洛赫态的几何特性来稳定。我们说明了一种通用的投影方法并计算了集体激励，该集体激励通过涉及增益、损耗和相互作用的特殊系数显示出由量子几何控制的扩散行为，并且需要抵抗调制不稳定性的弹性。然后，我们推导了相动力学的运动方程，并确定了几何起源的 Kardar-Parisi-Zhang 项。只要激光非线性的实部和虚部彼此成比例，或者当满足均匀配对条件时，该项就会完全抵消。我们通过数值研究证实了我们的结果$\pi$-助焊剂金刚石链。这封信强调了布洛赫几何效应在非线性耗散系统和 KPZ 物理中的关键作用，对增强相干性的激光阵列的设计具有直接影响。