Topological phases of many-body non-Hermitian systems

We show that many-body fermionic non-Hermitian systems require two distinct sets of topological invariants to describe the topology of energy bands and quantum states, respectively, with the latter yet to be explored. We identify 10 symmetry classes—determined by particle-hole, linearized time-reversal, and linearized chiral symmetries. Each class has a topological invariant associated with each dimension, dictating the topology of quantum states. These findings pave the way for a deeper understanding of the topological phases of many-body non-Hermitian systems.

Figure 1

Schematic diagram of the non-Hermitian topological insulator.

Figure 2

Phase diagram for the non-Hermitian topological insulator at $T=0.2, t=0.5$, and $J=1$. Region I: Both quantum states and band structure are nontopological. Region II: Topological quantum states with nontopological band structure. Region III: Nontopological quantum states with topological band structure. Region IV: Both quantum states and band structure are topological.

Figure 3

(a) Numerical spectra for the non-Hermitian topological insulator with 50 lattice cells ($L=50$) under open boundary conditions, featuring the zero-mode line depicted in red. This line denotes a twofold-degenerate state, with any inconspicuous splitting being neglected. (b) Numerical spectra of the corresponding effective model for the non-Hermitian topological insulator, also with 50 lattice cells ($L=50$) and subjected to open boundary conditions. Here, the zero-mode line is illustrated in blue, indicating a twofold-degenerate state while disregarding any negligible splitting. For these calculations, we set $T=1, t=1, J=1$, and $\gamma =0.5$.

Figure 4

(a) Particle distributions of the system. The upper plot illustrates the average quantity of particles per cell (the number of cells is $L$, the number of particles is $L+1$). Parameters are set to $U=1.2, T=0.1, \gamma =J-\delta$. (b) Particle distributions given by $U=0, T=0.15$, and $\gamma =-(J-\delta )$. All figures are examined under parameters $L=500, t=1, J^2=1.6\times {10}^4$, and ${\delta }^2=2.5\times {10}^{-10}$.

Figure 5

(a) Numerical result of the topological invariant $W$. (b) Numerical result of the topological invariant $w$. We set $T=1, t=1, J=1$, and $\gamma =0.5$.