Linear thermal noise in a four-terminal system with a Berry curvature dipole

In this work, we numerically investigate linear thermal noise in a four-terminal system with a finite Berry curvature dipole (BCD) using the nonequilibrium Green's function formalism. By comparing with the semiclassical results for bulk systems, we establish a one-to-one correspondence between terminal-resolved linear noise in multi-terminal systems and direction-resolved noise in bulk transport. Specifically, the auto-correlation function scales as $2 k_B T$ when the driving field is perpendicular to the BCD and vanishes when they are parallel, whereas the cross-correlation scales as $k_B T$. Both the auto- and cross-correlation functions exhibit pronounced peaks near the band edges, consistent with BCD-induced features. In addition, the linear thermal noise increases approximately linearly with $T$ at low temperatures and is suppressed by dephasing effect at high temperatures. Our work bridges semiclassical bulk theory and quantum multi-terminal theory for linear thermal noise, highlighting the symmetry(geometry)-selection rule in quantum transport.