Lasing in Non-Hermitian Flat Bands: Quantum Geometry, Coherence, and the Fate of Kardar-Parisi-Zhang Physics

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.