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

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.

💡 Research Summary

In this paper the authors investigate the linear thermal noise generated by a finite Berry curvature dipole (BCD) in a four‑terminal mesoscopic device using the nonequilibrium Green’s function (NEGF) formalism. The study begins with a two‑dimensional tilted massive Dirac Hamiltonian, H(k)=A k²+(B k²+δ)σ_z+v_y k_y σ_y+D σ_x, where the term D σ_x produces a BCD vector pointing along the x‑direction. The model respects time‑reversal symmetry but breaks the mirror symmetry M_y, thereby allowing a non‑zero BCD while keeping the system overall T‑invariant. The Hamiltonian is discretized on a square lattice (a₀=1) and a central scattering region of size 30 × 30 sites is coupled to four metallic leads, forming the four‑terminal geometry shown in Fig. 1.

The authors define the current‑fluctuation operator ΔĨ_α at each terminal α and the symmetrized noise correlator S_{αβ}=½⟨ΔĨ_αΔĨ_β+ΔĨ_βΔĨ_α⟩. Using the Fisher–Lee relation they express the scattering matrix s_{αβ}=δ_{αβ}+i Γ_α^{1/2} G^r Γ_β^{1/2} in terms of the retarded Green’s function G^r, the linewidth matrices Γ_α, and the self‑energies of the leads. The zero‑frequency noise S_{αβ}(0) is written in the standard NEGF form (Eq. 10). By expanding the bias voltages V_α to linear order, they isolate the linear thermal noise term S^{(1)}{αβ}=∂S{αβ}/∂V|_{V→0}. Crucially, they incorporate the internal Coulomb response through a characteristic potential u_α, which is obtained from a quasi‑neutrality approximation to the Poisson equation. This ensures gauge invariance and current conservation (∑β S^{(1)}{αβ}=∑α S^{(1)}{αβ}=0).

Two bias configurations are considered. Setup I applies V=(0,0,V/2,−V/2), generating an electric field along the y‑direction (E_y). Setup II uses V=(V/2,−V/2,0,0), producing a field along x (E_x). For each setup the authors compute the auto‑correlation functions S^{(1)}{11} (Setup I) and S^{(1)}{33} (Setup II) as well as the cross‑correlations S^{(1)}{13}, S^{(1)}{14}, S^{(1)}{31}, and S^{(1)}{32}. The numerical results reveal a set of clear selection rules that mirror the semiclassical bulk theory:

1. When the driving field is perpendicular to the BCD (E_y ⟂ D_x), the auto‑correlation S^{(1)}{11} scales as 2 k_B T, exactly matching the bulk result S{xx}^{(1)}=2 k_B T D_x E_y. The noise peaks near the band edge p₁, where the BCD is maximal, and decays deeper into the band.

2. When the field is parallel to the BCD (E_x ∥ D_x), the corresponding auto‑correlation S^{(1)}{33} vanishes, reproducing the bulk prediction S{yy}^{(1)}=0.

3. All cross‑correlation functions scale linearly with k_B T. They also display pronounced peaks at the band edges (p₁ and p₂) and change sign across the gap, reflecting the underlying Berry‑dipole physics. The asymmetry between S^{(1)}{13} and S^{(1)}{14} (or S^{(1)}{31} and S^{(1)}{32}) originates from the explicit breaking of y‑direction symmetry in the four‑terminal layout.

4. Temperature dependence shows that at low T (≈10 K) the linear thermal noise grows roughly linearly with T, as expected from the fluctuation–dissipation theorem. At higher temperatures, dephasing—modeled by the internal potential U—suppresses the noise, indicating that phase‑breaking processes diminish the geometric contribution.

The authors verify current conservation analytically (∑β S^{(1)}{αβ}=0) and numerically (the sum of auto‑ and cross‑terms vanishes in each setup). They also demonstrate that the internal‑potential term is essential for maintaining gauge invariance; neglecting it would lead to spurious voltage‑dependent noise.

Overall, the paper establishes a one‑to‑one correspondence between direction‑resolved linear thermal noise in bulk systems and terminal‑resolved noise in multi‑terminal devices. By embedding Berry curvature dipole effects into a fully quantum NEGF framework, the work bridges semiclassical geometric transport theory and quantum multi‑terminal transport, providing a concrete set of symmetry‑selection rules for linear thermal noise. The findings suggest that measuring terminal‑specific current noise in appropriately designed four‑terminal devices offers a viable experimental route to detect and quantify Berry curvature dipoles in time‑reversal‑invariant materials.