One-dimensional non-Hermitian quasicrystals with parity and time-reversal (PT) symmetry can simultaneously exhibit localization-delocalization transition, topological phase transition, and PT-symmetry-breaking transition. This motivates this work to investigate how the absence of PT symmetry impacts topological metal-insulator transitions in non-Hermitian quasicrystals. We propose a non-Hermitian quasiperiodic model that generally does not preserve PT symmetry and demonstrate that, in most parameter regions, such a system supports triple phase transitions that encompass localization, topology, and degeneracy-breaking. The system may also exhibit a particular type of localization-delocalization transition analogous to the Hermitian case, namely, without activating topological phase transitions or degeneracy-breaking transitions. Our work extends the topological metal-insulator transitions previously studied in PT-symmetric systems to a more general class of non-Hermitian setting, and further reveals that non-Hermitian systems can host distinct types of localization behavior.One-dimensional non-Hermitian quasicrystals with parity and time-reversal (PT) symmetry can simultaneously exhibit localization-delocalization transition, topological phase transition, and PT-symmetry-breaking transition. This motivates this work to investigate how the absence of PT symmetry impacts topological metal-insulator transitions in non-Hermitian quasicrystals. We propose a non-Hermitian quasiperiodic model that generally does not preserve PT symmetry and demonstrate that, in most parameter regions, such a system supports triple phase transitions that encompass localization, topology, and degeneracy-breaking. The system may also exhibit a particular type of localization-delocalization transition analogous to the Hermitian case, namely, without activating topological phase transitions or degeneracy-breaking transitions. Our work extends the topological metal-insulator transitions previously studied in PT-symmetric systems to a more general class of non-Hermitian setting, and further reveals that non-Hermitian systems can host distinct types of localization behavior.

cond-mat.dis-nnCross-listedarXiv:2605.25072May 26, 2026

Point-gap topology of non-Hermitian many-body systems

和樹山本, Yamamoto, Kazuki

Dec 1, 20230 citations

Non-Hermitian topological Euler insulators

Longwen Zhou

Topological Euler insulators emerge in multiband systems with real Bloch Hamiltonians and wavefunctions. Their fragile topologies are characterized by the Euler class of degenerate bands and protected by the $PT$ or $C_2T$ symmetry in two dimensions, which go beyond the tenfold $K$ -theory classification of topological matter. In this work, we extend the conception of topological Euler insulators to non-Hermitian systems and propose a theoretical framework to unlock their nontrivial Euler topology. Focusing on two-dimensional, three-band non-Hermitian lattice models with symmetric Hamiltonians, we formulate a comprehensive description of their topological Euler bands, entanglement spectrum and bulk-boundary correspondence. Three typical models of non-Hermitian Euler insulators are constructed and investigated explicitly to illustrate our theory. Unique topological phase transitions and anomalous edge-band overlaps with non-Hermitian origins are further identified. Our study establishes the presence of topological Euler bands in non-Hermitian systems and unveils their intriguing physical characteristics, thereby broadening the existing territory of topological matter in non-Hermitian open systems.Topological Euler insulators emerge in multiband systems with real Bloch Hamiltonians and wavefunctions. Their fragile topologies are characterized by the Euler class of degenerate bands and protected by the $PT$ or $C_2T$ symmetry in two dimensions, which go beyond the tenfold $K$-theory classification of topological matter. In this work, we extend the conception of topological Euler insulators to non-Hermitian systems and propose a theoretical framework to unlock their nontrivial Euler topology. Focusing on two-dimensional, three-band non-Hermitian lattice models with symmetric Hamiltonians, we formulate a comprehensive description of their topological Euler bands, entanglement spectrum and bulk-boundary correspondence. Three typical models of non-Hermitian Euler insulators are constructed and investigated explicitly to illustrate our theory. Unique topological phase transitions and anomalous edge-band overlaps with non-Hermitian origins are further identified. Our study establishes the presence of topological Euler bands in non-Hermitian systems and unveils their intriguing physical characteristics, thereby broadening the existing territory of topological matter in non-Hermitian open systems.

cond-mat.mes-hallCross-listedarXiv:2608.25411Aug 27, 2026

Unsupervised classification of non-Hermitian topological phases under symmetries

Yang Long, Haoran Xue, Baile Zhang

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.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.

arXiv:2412.20882National Science ReviewNov 26, 20250 citations

Universal Eigenvector Statistics of Non-Hermitian Random Matrices

Ze Chen, Zhenyu Xiao, Shinsei Ryu

Eigenvector overlaps quantify nonorthogonality and govern the response and dynamics of non-Hermitian systems. We extend the universality of non-Hermitian random matrices to the statistics of these overlaps. We obtain analytical expressions for eigenvector overlaps in the spectral bulk and near the origin, covering ten symmetry classes in the limit of large matrix size. Using fermionic replica nonlinear $σ$ models, we relate these overlaps to Hermitian level statistics, symmetry class by symmetry class and topological sector by topological sector. Numerical calculations in various physical models support the universality of the normalized overlaps in the regimes studied. Our work establishes a duality between Hermitian level statistics and non-Hermitian eigenvector overlaps.Eigenvector overlaps quantify nonorthogonality and govern the response and dynamics of non-Hermitian systems. We extend the universality of non-Hermitian random matrices to the statistics of these overlaps. We obtain analytical expressions for eigenvector overlaps in the spectral bulk and near the origin, covering ten symmetry classes in the limit of large matrix size. Using fermionic replica nonlinear $σ$ models, we relate these overlaps to Hermitian level statistics, symmetry class by symmetry class and topological sector by topological sector. Numerical calculations in various physical models support the universality of the normalized overlaps in the regimes studied. Our work establishes a duality between Hermitian level statistics and non-Hermitian eigenvector overlaps.

cond-mat.stat-mechCross-listedarXiv:2609.22079Sep 21, 2026

PT

C_2T

symmetry in two dimensions, which go beyond the tenfold

K

-theory classification of topological matter. In this work, we extend the conception of topological Euler insulators to non-Hermitian systems and propose a theoretical framework to unlock their nontrivial Euler topology. Focusing on two-dimensional, three-band non-Hermitian lattice models with symmetric Hamiltonians, we formulate a comprehensive description of their topological Euler bands, entanglement spectrum and bulk-boundary correspondence. Three typical models of non-Hermitian Euler insulators are constructed and investigated explicitly to illustrate our theory. Unique topological phase transitions and anomalous edge-band overlaps with non-Hermitian origins are further identified. Our study establishes the presence of topological Euler bands in non-Hermitian systems and unveils their intriguing physical characteristics, thereby broadening the existing territory of topological matter in non-Hermitian open systems.

σ

models, we relate these overlaps to Hermitian level statistics, symmetry class by symmetry class and topological sector by topological sector. Numerical calculations in various physical models support the universality of the normalized overlaps in the regimes studied. Our work establishes a duality between Hermitian level statistics and non-Hermitian eigenvector overlaps.