Unsupervised Classification of Non-Hermitian Topological Phases under Symmetries

Abstract

The integration of artificial intelligence (AI) into fundamental science has opened new possibilities to address long-standing scientific challenges rooted in mathematical limitations. For example, topological invariants are used to characterize topology, but there is no universally applicable one. This limitation explains why, in the past decades-long classification of topological phases of matter -- mainly focused on Hermitian systems -- many phases initially classified ``trivial" were later identified as topological. Recently, the discovery of non-Hermitian band topology has spurred substantial efforts in non-Hermitian topological classification, including the development of new topological invariants. However, such classifications similarly risk overlooking key topological features. Here, without relying on any topological invariant, we develop an AI-based unsupervised classification of symmetry-protected non-Hermitian topological phases. This algorithm distinguishes topological differences among non-Hermitian Hamiltonians with symmetries, and constructs, in an unsupervised manner, a topological periodic table for non-Hermitian systems. Additionally, it can account for the boundary effects, enabling the exploration of open-boundary effects on the topological phase diagram. These results introduce an unsupervised approach for classifying symmetry-protected non-Hermitian topological phases without omission and provide valuable guidance for the development of theories and experiments.

Figures (5)

• Figure 1: Gap types in non-Hermitian systems and symmetry-preserving continuous deformation. (a) Typical three gap types : point gap, real line gap, and imaginary line gap. The gray regions denote the regions covered by the eigen-energies of the Hamiltonian on the complex-energy plane. (b) Symmetry-preserving continuous deformation between two Hamiltonians $H_1$ and $H_2$. When $H_1$ and $H_2$ are topologically equivalent, one can find a path to realize the continuous deformation between them without closing the gap. While $H_1$ and $H_2$ are topologically distinct, any continuous deformation between them will close the gap. The purple arrows denote the continuous deformation.
• Figure 2: Unsupervised learning of non-Hermitian point-gap topological phases. The left plots represent the system settings. The central plots represent the number of samples $M$ for the topologically distinct phases. $c$ denotes the custom label of phase. The right plots show the topological phase diagrams obtained by the similarity between the Hamiltonian with the given parameters and Hamiltonians in $\mathcal{G}$. The different colors and labels denote the topologically distinct phases, but not the topological invariants. (a) 1D Hatano-Nelson system. We set $J_L=1$, $J_R \in [0,2]$, $E_f=0$. (b) 1D non-Hermitian system with twisted winding in the complex-energy plane. We set $\kappa=1$, $\kappa_2 \in [0,3]$, $E_f=i$. (c) 1D non-Hermitian topological point-gap phase induced by onsite losses and gains. We set $t_1=\gamma=2$, $t_2=\mu=1$ and $\tau \in [0,3]$, $E_f=0$. Here, we generate 100 samples for each case.
• Figure 3: Unsupervised learning of non-Hermitian real line-gap topological phases. The left plots represent the system settings. The central plots represent the number of samples $M$ for the topologically distinct phases. $c$ denotes the labels of phases. The right plots show the topological phase diagrams obtained by the similarity between the Hamiltonian with the given parameters and Hamiltonians in $\mathcal{G}$. The different colors and labels denote the topologically distinct phases but not the topological invariants. Note that $c=0$ denotes the gapless system. (a) 1D non-Hermitian SSH system. We set $t_2=1$, $t_1\in [0,3]$, $\gamma\in [0,3]$ (b) 1D topological insulator phase solely induced by on-site gains and losses. We set $\kappa=1$, $g_1,g_2\in [-3,3]$. (c) 1D topological insulator phase solely induced by non-reciprocal couplings. We set $t_0=1$, $\varepsilon \in [0,2]$. (d) 2D non-Hermitian Chern insulator. We set $t_x=t_y=v_x=v_y=1$, $m\in [1,3]$, $\gamma \in [0,0.5]$. (e) 2D non-Hermitian topological Möbius insulator. We set $\kappa=0.25$, $t_2=1$, $t_1 \in [0,2]$, $\gamma \in [0,2]$. Here, for obtaining the central plots, we generate 500 samples for each case and filter out the gapless systems. $E_f=0$ for all cases.
• Figure 4: Unsupervised learning of topological classifications of 1D non-Hermitian Hamiltonians in different symmetry classes. (a) Classification results for non-Hermitian symmetry classes. Here, we demonstrate 6 symmetry classes for each type of gaps. (b) Classification results for non-Hermitian symmetry classes with considering the parity transformation. Here, we randomly generate 500 Hamiltonian samples for each symmetry class according to the 0D $n\times n$ Hamiltonians Note1. The number of phases $N_c$ can reflect the topological classification, e.g., $N_c=2$ corresponds to $\mathbb{Z}_2$ and $N_c =n+1$ corresponds to $\mathbb{Z}$. Here, we set $n=8$ and $E_f=0$.
• Figure 5: Unsupervised learning of non-Hermitian real line-gap topological phases under GBZ. The left plots represent the number of samples $M$ for the topologically distinct phases. $c$ denotes the label of phases. The right plots show the topological phase diagrams obtained by the similarity between the Hamiltonian with the given parameters and Hamiltonians in $\mathcal{G}$. The different colors and labels denote the topologically distinct phases, but not the topological invariants. Note that $c=0$ denotes the gapless system. (a) 1D non-Hermitian SSH system. We set $t_2=1$, $t_1\in [0,3]$, $\gamma \in [0,3]$. (b) 2D non-Hermitian Chern insulator. We set $t_x=t_y=v_x=v_y=1$, $m\in [1,3]$, $\gamma \in [0, 0.5]$. (c) 2D non-Hermitian topological Möbius insulator. We set $\kappa=0.25$, $t_2=1$, $t_1\in [0,2]$, $\gamma \in [0,2]$. Here, for obtaining the left plots, we randomly generate 500 samples for each case and filter out the gapless systems. $E_f=0$ for all cases.