1 Introduction
The past decade has witnessed a paradigm shift in our understanding of topological matter, driven by the recognition that non-Hermiticity arising from gain, loss, or nonreciprocal coupling can fundamentally reshape topological phases. Unlike Hermitian systems, these open systems exhibit a rich repertoire of phenomena with no analogs in Hermitian physics: exceptional points where eigenvalues and eigenvectors coalesce [
1], the non-Hermitian skin effect where bulk modes collapse onto boundaries [
2,
3], and complex spectral topology that links complex eigenvalues to the skin effect [
4,
5].
These fundamental insights have stimulated significant theoretical advances. Key developments include the discovery of parity-dependent skin effects [
6], symmetry-guided nonreciprocal transport [
7], and the experimental measurement of non-Hermitian left eigenvectors in acoustic systems [
8], which collectively reshape our understanding of bulk–boundary correspondence in open systems.
On the experimental and technological front, these theoretical breakthroughs have enabled a diverse range of platforms and applications. Electrical circuits have emerged as a powerful testbed, hosting realizations of the non-Hermitian Haldane model [
9], Möbius insulators and graphene-like topological semimetals [
10], as well as impurity-induced scale-free localization [
11] and robust wireless power transfer [
12-
15]. Acoustic platforms have revealed topological monomodes that defy the conventional “pairing” rule of topological states [
16] and demonstrated loss-induced higher-order topology [
17] and non-Hermitian topological Anderson insulators [
18]. Photonic systems have enabled the observation of edge solitons in non-Hermitian lattices [
19] and the development of topological biosensors for multiplexed disease detection [
20] and exceptional-point-enhanced optomechanical sensing [
21]. This perspective aims to provide a snapshot of the rapidly evolving field of non-Hermitian topology, and identify outstanding challenges and emerging opportunities that we believe will shape the next frontier of this interdisciplinary field.
2 Fundamental advances in non-Hermitian topological physics
Non-Hermitian physical dynamics exhibit remarkable distinctions from their Hermitian counterparts. A central classification in non-Hermitian systems is the dramatic dependence on boundary conditions: under periodic boundary conditions, where conventional Bloch theorem applies, non-Hermitian band theory reveals phenomena such as exceptional points and biorthogonal eigenstructures; under open boundary conditions, where Bloch theorem no longer holds, the non-Hermitian skin effect emerges as a hallmark phenomenon that fundamentally reshapes the topological classification.
2.1 Periodic boundary conditions: Exceptional points and left eigenvectors
Under periodic boundary conditions, where the conventional Bloch theorem applies, non-Hermitian systems exhibit exceptional points and biorthogonal eigenvector structures, features with no Hermitian analogs.
Exceptional points, where two or more eigenvalues and their corresponding eigenstates simultaneously coalesce, represent the most distinctive feature of the non-Hermitian eigen-space compared to Hermitian counterparts. Their detection and numerical diagonalization algorithms, however, has long been challenging due to the defective nature of the Hamiltonian at the exceptional points. Among various theoretical attempts, a promising solution has been proposed by employing the Hilbert-Schmidt speed, a measure of quantum statistical speed that does not require diagonalization of the density matrix [
1]. This method established a universal criterion: across an exceptional point, the temporal behavior of the Hilbert-Schmidt speed exhibits a qualitative transition from oscillatory to non-oscillatory dynamics. This criterion was validated across a wide spectrum of models from two-level systems to interacting many-body systems, demonstrating its remarkable versatility as a universal tool for characterizing exceptional points.
Left eigenvectors represent another remarkable distinction that sets non-Hermitian systems apart from their Hermitian counterparts. However, these eigenvectors have long been conjectured as purely mathematical constructs without physical correspondence. This conjecture has recently been overturned by a foundational advance in acoustic systems [
8]: the experimental measurement of left eigenvectors. By exploiting the fact that Green’s function columns encode right eigenvectors while rows encode left eigenvectors, the experiment directly measured left eigenvectors through a simple yet powerful method of fixing the detector while varying the source position. This work conclusively demonstrated that left eigenvectors are not mere mathematical abstractions but physically observable entities, paving the way for exploring biorthogonal quantum mechanics in realistic settings.
2.2 Open boundary conditions: Robustness induced by non-Hermitian skin effect
Under open boundary conditions, where the conventional Bloch theorem no longer applies, the non-Hermitian skin effect emerges as a hallmark topological phenomenon [
2,
3,
22]: an extensive number of eigenstates localize at boundaries, driven by point-gap topology and the non-Bloch Brillouin zone. This fundamentally alters the bulk–boundary correspondence familiar from Hermitian systems [
23,
24].
Recent work has revealed its unexpectedly rich interplay with flat-band physics in the context of non-Hermitian Aharonov–Bohm cages [
25]. Random antisymmetric disorder induces delocalization accompanied by a reentrant non-Hermitian skin effect, while Bernoulli antisymmetric disorder, the type known to induce inverse Anderson localization in Hermitian systems — can generate the non-Hermitian skin effect even with infinitesimal nonreciprocal hopping.
Another intriguing extension is scale-free localization, where the localization length of the non-Hermitian skin effect scales with system size, as experimentally observed in disordered systems with a single impurity [
11]. The impurity controls the localization direction opposite to the bulk hopping direction, with localization length proportional to system size, which is in stark contrast to the conventional non-Hermitian skin effect.
A systematic investigation of multilayer nonreciprocal SSH structures further revealed a striking parity effect [
6]: the skin effect and topological phase transitions depend on both the parity of the layer number and the parity of the band index. Odd-layer structures host robust zero-energy edge modes resilient to disorder, while even-layer structures exhibit only nonzero-energy edge modes that are fragile to perturbations, providing a design principle for engineering robust topological states in multilayer architectures.
Counterintuitively, the non-Hermitian skin effect is not synonymous with nonreciprocal transport, and the two concepts can be clearly distinguished via typical physical examples. Notably, parity−time (
PT)-symmetric non-Hermitian systems can support prominent non-Hermitian skin effect while maintaining fully reciprocal wave transmission, directly proving that skin localization does not inherently lead to nonreciprocity. A comprehensive symmetry analysis [
7] revealed that parity−time-symmetric non-Hermitian systems, despite being non-Hermitian, still exhibit reciprocal transmission. In stark contrast, breaking reciprocity requires the interplay of magnetic flux (breaking time-reversal symmetry), non-Hermitian phases, and spatial asymmetry — no single symmetry-breaking mechanism suffices to induce directional nonreciprocal transport. Together, these findings establish a unified picture: non-Hermiticity provides robustness via the non-Hermitian skin effect, while reciprocity is governed by the interplay between symmetry and non-Hermitian phases.
The robust physical manifestations arising from the non-Hermitian skin effect are far from accidental. As established in Refs. [
4,
5], both the non-Hermitian skin effect and its inherent robustness originate from the topological nature encoded in the spectral winding number of the periodic-boundary eigenspectrum. Nonzero spectral winding with respect to some reference complex energy necessarily implies the emergence of open-boundary skin modes, and vice versa. This result unveils the topological root of the non-Hermitian skin effect within the point-gap topological framework, offering a direct bulk topological criterion to judge whether skin localization occurs in one-dimensional non-Hermitian lattices.
The dual role of non-Hermiticity in topological systems — simultaneously breaking and enhancing robustness — appears paradoxical at first glance. On one hand, non-Hermitian disorder can drive topological phase transitions [
17,
18], seemingly undermining the topological protection inherited from Hermitian counterparts. On the other hand, the non-Hermitian skin effect provides unprecedented robustness against disorder and environmental fluctuations [
12,
14]. The key to resolving this paradox lies in the distinction between local and global topology: phase transitions are decided by whether a specific local gap in the complex spectrum is open or closed — which can be easily altered by adjusting gain, loss, or disorder strength because these perturbations act locally on the complex energy landscape — while the skin effect is governed by a global winding number, which is immune to such local changes. This distinction suggests a practical design principle: functional non-Hermitian devices should be engineered to exploit global spectral topology for robustness, while treating local gaps as tunable knobs for reconfigurable phase switching.
3 Experimental platforms and applications
The richness of non-Hermitian topological phenomena has been explored across a diverse range of physical platforms, each offering distinct advantages. Electrical circuits provide outstanding flexibility in engineering coupling strengths and on-site potentials, acoustic platforms allow precise control over gain and loss profiles, and photonic systems enable direct observation of wave dynamics in real space.
3.1 Electrical circuits and wireless power transfer
Electrical circuits have emerged as a particularly powerful platform for realizing non-Hermitian topological phenomena, owing to their exquisite control over coupling strengths and on-site potentials through lumped components [
26]. The non-Hermitian Haldane model was experimentally realized in a circuit lattice with asymmetric next-nearest-neighbor hopping [
9]. Two previously uncovered phases are observed: a non-Hermitian Chern insulator and a non-Hermitian semimetal phase, both exhibiting boundary-dependent amplifying or dissipative chiral edge states, marking the first experimental realization of the non-Hermitian Haldane model. Beyond fundamental verification, circuit-based non-Hermitian topologies bear promising prospects for communication engineering. Apart from wireless power transfer, the robust signal localization and intrinsic anti-interference capability enable designs of radio-frequency and terahertz communication channels with enhanced immunity, as well as multiport multiplexers, which align closely with the on-chip signal routing paradigm.
Non-Hermitian Möbius insulators and graphene-like topological semimetals were realized in electric circuits through projective symmetry [
10]. By introducing nonreciprocal hopping via negative impedance converters, the non-Hermitian skin effect enhances the energy localization of topological edge states. The circuit platform’s flexibility, achieving different topological structures on the same printed circuit board through jumper caps, highlights its potential for rapid prototyping of non-Hermitian topological devices. A particularly striking demonstration was the observation of impurity-induced scale-free localization in disordered non-Hermitian electrical circuits [
11], where a single impurity controls the localization direction with localization length scaling proportionally to system size. It is notable that this phenomenon does not have a Hermitian analog.
Beyond fundamental topological phase realizations, the exquisite control afforded by circuit platforms has enabled a particularly compelling application: robust wireless power transfer that operates without frequency tracking. The concept of bound-state-in-the-continuum (BIC) enhanced wireless power transfer was demonstrated in both second- and third-order non-Hermitian systems [
12,
14]. Unlike conventional
PT-symmetric wireless power transfer requiring strict gain-loss balance, BIC-assisted wireless power transfer maintains stable real eigenfrequencies regardless of gain, loss, or coupling strength, enabling efficient power transfer even in weak coupling regimes while reducing field leakage and standby power consumption. This approach effectively transforms a practical engineering challenge, maintaining efficiency under varying coupling conditions, into a topological protection problem, where the BIC provides built-in robustness against environmental fluctuations.
Higher-order anti-parity−time symmetry was exploited for efficient wireless power transfer in multiple-transmitter systems [
13]. By leveraging interference between shared sources to construct virtual coupling, the authors achieved frequency-stable, broadband power transfer without complex frequency tracking, maintaining high efficiency even when transmitter and receiver resonant frequencies were mismatched. Compared with conventional single-resonator wireless power transfer schemes, this multi-transmitter anti-
PT design achieves a substantially broader effective operating bandwidth, significantly alleviating efficiency degradation induced by frequency mismatch. This represents a significant advance over conventional resonant wireless power transfer schemes, which suffer from severe efficiency degradation upon frequency detuning. Large-area topological wireless power transfer was achieved through topological large-area defect states [
15], where energy uniformly distributes over multiple target sites, enabling simultaneous multi-load charging with strong robustness against positional perturbations.
3.2 Acoustic and mechanical devices
Acoustic platforms have enabled a series of foundational observations that challenge and expand the conventional understanding of non-Hermitian topology. One striking example is the realization of topological monomodes in coupled acoustic cavities with non-Hermitian losses [
16]. This counterintuitive phenomenon originates from non-Hermitian dissipation that selectively annihilates one state of the conventional topological mode pair, rather than fundamentally breaking the intrinsic pairing constraint of topological states. By introducing loss into Su–Schrieffer–Heeger chains, it can be demonstrated that only a single topological mode survives in the open system, a phenomenon that fundamentally revises the conventional understanding of paired topological states inherited from Hermitian systems. These monomodes were characterized through fractional mode charges derived from acoustic local density of states.
Acoustic platforms have also served as testbeds for dissipation in topological phase transitions. In a landmark experiment [
17], a higher-order topological insulator was realized in an acoustic crystal by deliberately introducing losses into an otherwise trivial configuration. The loss-induced quadrupole topology was experimentally verified through the observation of gapped edge states and in-gap corner states, demonstrating that non-Hermiticity can actively generate topological phases.
Moreover, the interplay between non-Hermitian disorder and topology has been explored in acoustic lattices [
18], where purely non-Hermitian disorder was shown to drive topological phase transitions, realizing the non-Hermitian topological Anderson insulator. The topological edge states induced by non-Hermitian disorders exhibited robustness against both weak Hermitian and non-Hermitian perturbations.
While finite-frequency acoustic waves have yielded rich physics when combined with non-Hermiticity, the zero-frequency static limit of mechanical systems also hosts striking emergent phenomena enabled by non-Hermitian ingredients [
27]. This regime is governed by odd elasticity, a paradigm first established in Ref. [
23]. The core feature of odd elasticity lies in the antisymmetric off-diagonal entries of the elasticity tensor within the constitutive relation of continuum mechanics, which introduces nonreciprocal mechanical couplings without requiring dynamical gain or loss. Building upon this fundamental framework, Ref. [
24] further bridged odd elasticity with topological metamaterials, revealing unconventional non-Hermitian topological behaviors in static mechanical lattices stabilized by odd elastic responses.
The interplay between (non-Hermitian) odd elasticity and higher-dimensional mechanical structures is of especially significant challenge [
28], as the gauge choice of the direction of odd elasticity can be arbitrary [
29]. Such ambiguous gauge dependence hinders the standardized construction and topological classification of high-dimensional odd-elastic metamaterials. Therefore, establishing a rigorous gauge-invariant framework is essential for extending odd elasticity toward practical multi-dimensional mechanical topological devices.
3.3 Photonic and optoelectronic devices
Photonic systems have provided rich ground for observing non-Hermitian topological phenomena and translating them into functional devices. In photonic platforms, it has been widely studied that edge solitons emerge on Hermitian systems that preserve energy. Interestingly, a recent study [
19] reveals that when systems are fully non-Hermitian, edge solitons can still emerge from photonic lattices that exhibit type-II Dirac cones. By constructing a domain wall and introducing gain and loss at different sublattice sites, stable edge solitons with real propagation constants are observed. These solitons propagate stably over distances far exceeding experimental sample lengths (~900 diffraction lengths).
The versatility of photonic platforms extends beyond fundamental observations to practical sensing and detection. A photonic biosensor based on cascade-coupled SSH boundary modes was demonstrated [
20]. By exploiting three topological boundary modes in an asymmetric photonic crystal architecture, the authors achieved multiplexed detection of three disease markers in a single transmission measurement. Each binding configuration produces distinct signatures in frequency shifts and transmittance changes. This work exemplifies how non-Hermitian topology can address the critical need for point-of-care diagnostic devices that combine high sensitivity with operational reliability.
Furthermore, an optomagnomechanical mass sensor enhanced by a third-order exceptional point was theoretically proposed [
21], achieving a hundred-fold enhancement in frequency sensitivity through cube-root scaling of eigenfrequency shifts with mass perturbations — breaking the linear scaling limit of conventional sensors.
4 Outlook and discussion
While significant progress has been made in understanding non-Hermitian topological phenomena in non-interacting systems, several fundamental questions and technical challenges remain open. Here we discuss what we perceive as the most pressing directions — ranging from quantum many-body physics to experimental scalability and interdisciplinary opportunities — that will likely define the next frontier of this field.
The many-body regime remains largely uncharted territory. The variational quantum algorithm for scanning the complex spectrum of non-Hermitian systems [
30] provides a promising route for studying non-Hermitian many-body physics on near-term quantum computers, yet the exponential growth of Hilbert space poses fundamental limitations. At the mean-field level, many-body interactions naturally manifest as nonlinearities, which bridges quantum many-body physics with classical nonlinear dynamics [
31-
34]. The study of nonlinear topological pumping with time-dependent interactions [35] reveals new possibilities for controlling soliton dynamics through temporal modulation of nonlinearity, restoring quantized transport even where time-independent nonlinearity would lead to breakdown. Whether such nonlinearity-induced topological protection extends to fully quantum regimes remains an open question.
Complementing these many-body challenges, the classification of non-Hermitian topological phases under different symmetry classes remains incomplete, particularly in higher dimensional platforms. Recent discoveries of parity-dependent skin effects [
6], symmetry-guided nonreciprocal transport [
7], and topological monomodes [
16] point toward a rich landscape of unconventional phases unique to non-Hermitian systems. Developing a comprehensive framework for understanding and classifying these phases and their experimental signatures represents one of the major challenges for the field.
Finally, it is intriguing to study the connection between non-Hermitian topology across vastly different scales. The same mathematical structures, namely exceptional points, non-Hermitian skin effect, and point-gap topology, appear in electrical circuits, acoustic metamaterials, and quantum optical systems. The demonstration of non-Hermitian left eigenvectors in acoustic systems [
8] and the observation of scale-free localization in electrical circuits [
11] exemplify this cross-fertilization. We envision that the future of non-Hermitian topology will be increasingly interdisciplinary, drawing on concepts from quantum information, condensed matter physics, photonics, and mechanical engineering to address both fundamental questions and societal challenges in energy, health, and communication technologies.
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