Role of noise on defect formation and correlations in a long-range Ising model under adiabatic driving

We study an exactly solvable long-range (LR) transverse-field Ising model with a power-law decaying interaction characterized by the decay exponentα. For1<α<2, the model exhibits two quantum critical points with distinct critical exponents. In the thermodynamic limit, the system is adiabatically driven across these critical points in the presence of noise, from the paramagnetic phase with all spins down to one with all spins up. In the noiseless case, within the LR regime, the steady-state properties are primarily influenced by the modes near thekc=πregion, while the low-energy modes aroundkc=0remain nearly adiabatic and contribute negligibly. The defect density follows Kibble-Zurek (KZ) scaling law and scales asn∝τQ-1/2, implying KZ scaling exponent independent of decay exponent. However, the magnitude of the defect density increases with increasing the range (i.e. decreasing the value ofα). On the other hand in the presence of noise, the low-energy modes aroundkc=0become highly susceptible to excitation, making them the dominant source of excitations, with their contribution gradually diminishing as the interaction range increases. This differs from the short-range model, where previous studies have shown that under noisy condition modes aroundk=π/2play a significant role. The contrasting influence of LR interactions under noisy and noiseless driving, is also observed in spin-spin correlation function and defect distribution, highlighting how LR interactions fundamentally reshape the interplay between adiabatic driving and noise.