Unsupervised classification of non-Hermitian topological phases under symmetries

The integration of machine learning into fundamental science has opened new avenues for addressing long-standing challenges rooted in mathematical limitations. For instance, while topological invariants are essential for characterizing topological phases of matter, no single invariant is universally applicable. This limitation explains why, over decades of classifying topological phases-primarily in Hermitian systems-many phases initially deemed 'trivial' were later recognized as topological. Recently, the discovery of non-Hermitian band topology has driven substantial efforts in non-Hermitian topological classification, leading to the development of new topological invariants. However, these invariants still fail to capture all non-Hermitian topological features. Here, without relying on any topological invariant, we develop a machine-learning algorithm for the unsupervised classification of symmetry-protected non-Hermitian topological phases. By utilizing random Hamiltonians, we unsupervisedly construct a topological periodic table without requiring advanced mathematical knowledge. Furthermore, based on the learning results, we derive a formula that reveals the impact of parity transformation on periodicity. Our algorithm can also account for boundary effects, enabling the exploration of open-boundary influences on the topological phase diagram. These findings establish an unsupervised approach for classifying symmetry-protected non-Hermitian topological phases, uncover previously unnoticed topological features in non-Hermitian systems, and provide valuable guidance for both theoretical advancements and experimental realizations.