Mixed-State Topology in Non-Hermitian Systems

TL;DR

This work develops a finite-temperature framework for mixed-state topology in non-Hermitian systems using the Uhlmann phase and thermal Uhlmann-Chern numbers. It demonstrates and characterizes topological structures—an exceptional ring in 2D, an exceptional surface in 3D, and an exceptional hypersphere in 4D non-Abelian settings—through temperature-dependent invariants such as $\Phi_U$, $C_U$, $\mathcal{DD}_B$, and their non-topological counterparts. The results reveal temperature- and geometry-dependent transitions absent in pure-state NH topology and extend to higher dimensions with a second thermal Chern number, highlighting richer mixed-state geometry. An experimental feasibility pathway is outlined, connecting eigenstate reconstruction and tomography to the measurement of thermal topological invariants, thus offering practical routes to explore NH topology at finite temperature.

Abstract

Non-Hermitian (NH) systems, due to the existence of exceptional point (or ring, surface), exhibit exotic topological features which are inaccessible with the Hermition ones. Current studies on NH topology mainly focus on pure states at zero temperature, while those on mixed states remain largely unexplored. In this work, we investigate the topological properties of mixed states in two-dimentional NH systems, by use of the Uhlmann phase and the thermal Uhlmann-Chern number which are structured via the Uhlmann connection at specific temperatures, revealing distinct topological features compared to their pure state counterparts. We further extend our study to the mixed states in the three-dimensional Abelian and four-dimentional non-Abelian NH systems and verify the high-order mixed-state topology. Our study provides a conceptual and practical pathway for exploring topological properties in the mixed-state regime of NH physics.

Figures (7)

• Figure 1: The Uhlmann phase $\Phi_U$ as a function of temperature $T$, which scales with $\gamma$.
• Figure 2: The Uhlmann phase $\Phi_U$ as a function of the parameter loop displacement $d$, which scales with $\gamma$. $\Phi_U$ accumulates 2$\pi$, $\pi$, and 0 after one cycle with two loops of mixed-state evolution when the parameter loop encloses two EPs, encircles the ER while enclosing one EP and disengages from the ER, respectively.
• Figure 3: The thermal Uhlmann-Chern number $C_U$ (blue solid line) and NT thermal Uhlmann-Chern number $C_U^{nt}$ (yellow dashed line) as functions of temperature $T$, which scales with $\gamma$.
• Figure 4: The thermal Uhlmann-Chern number $C_U$ (blue solid line) and NT thermal Uhlmann-Chern number $C_U^{nt}$ (yellow dashed line) as functions of the parameter sphere radius R, which scales with $\gamma$.
• Figure 5: The NT thermal DD invariant $\mathcal{DD}_B^{nt}$ with respect to temperature $T$ (scales with $\gamma$) in NH system. The blue solid line represents the NT thermal DD invariant when the 4D parameter space encloses the ES, and the yellow dash line denotes the case while the parameter space does not enclose the ES. The inset figure is the Hermitian case, where the blue solid and yellow dash lines are the $\mathcal{DD}_B$ and $\mathcal{DD}_B^{nt}$ as functions of temperature $T$, respectively.