Liouvillian skin effects in two-dimensional electron systems at finite temperatures

Liouvillian skin effects, manifested as the localization of Liouvillian eigenstates around the boundary, are distinctive features of non-Hermitian systems and are particularly notable for their impact on system dynamics. Despite their significance, Liouvillian skin effects have not been sufficiently explored in electron systems. In this work, we demonstrate that a two-dimensional electron system on a substrate exhibits $\mathbb{Z}$ and $\mathbb{Z}_2$ Liouvillian skin effects due to the interplay among energy dissipations, spin-orbit coupling, and a transverse magnetic field. In addition, our analysis of the temperature dependence reveals that these Liouvillian skin effects become pronounced below the energy scale of band splitting induced by the spin-orbit coupling and the magnetic field. While our $\mathbb{Z}$ Liouvillian skin effect leads to charge accumulation under quench dynamics, its relaxation time is independent of the system size, in contrast to that of previously reported Liouvillian skin effects. This difference is attributed to the scale-free behavior of the localization length, which is analogous to non-Hermitian critical skin effects.

💡 Research Summary

In this work the authors investigate non‑Hermitian skin effects that arise in the Liouvillian super‑operator describing the open‑system dynamics of a two‑dimensional electron gas placed on a substrate. The model consists of a square‑lattice tight‑binding Hamiltonian with nearest‑neighbour hopping t_h, Rashba spin‑orbit coupling (SOC) of strength α, and an in‑plane magnetic field H = (H_x , H_y ,0). The kinetic term ξ_k = −2t_h (cos k_x + cos k_y) − μ together with the SOC vector g_k = (−sin k_y , sin k_x ,0) and the Zeeman term μ_B H produce a band splitting ε_{k1,2}= ξ_k ± |η_k|, where η_k = −α sin k_y − μ_B H_x − i(α sin k_x − μ_B H_y).

Energy exchange with the substrate is introduced via Lindblad jump operators L_{k,lm}= √Γ_{k,lm} α†{k,l} α{k,m}. The transition rates Γ_{k,lm}=γ_{lm} e^{−β ε_{k,l}} e^{−β ε_{k,m}} satisfy detailed balance, with γ_{lm}=γ (for l≠m) describing dissipative inter‑band processes and γ_{d} (for l=m) describing pure dephasing. This construction guarantees that the steady state of the Liouvillian is the Gibbs state of the Hamiltonian.

To analyse the Liouvillian topology the authors first vectorise the density matrix, mapping the super‑operator L to a matrix acting on the doubled Hilbert space. A mean‑field decoupling is then applied: quartic terms are replaced by their expectation values in the Gibbs state, which yields a quadratic Liouvillian that can be written in Nambu form. The resulting matrix has a block‑triangular structure L(k) → ( X†(k) 0 ; X(k) 0 ), where the spectrum is given by the eigenvalues of the damping matrix X(k) and its Hermitian conjugate. Consequently, the point‑gap topology of the Liouvillian is fully characterised by X(k).

Two topological invariants are considered. For a one‑dimensional subsystem at fixed k_y, the winding number w(Λ_0)= (1/2π i) ∮_{k_x=−π}^{π} d ln det