non-Hermitian Topological Phases Research Articles

Periodically nonreciprocal transmission and entanglement in a non-Hermitian topological phase

Periodically nonreciprocal transmission and entanglement in a non-Hermitian topological phase

Machine learning of knot topology in non-Hermitian band braids

The deep connection among braids, knots and topological physics has provided valuable insights into studying topological states in various physical systems. However, identifying distinct braid groups and knot topology embedded in non-Hermitian systems is challenging and requires significant efforts. Here, we demonstrate that an unsupervised learning with the representation basis of su(n) Lie algebra on n-fold extended non-Hermitian bands can fully classify braid group and knot topology therein, without requiring any prior mathematical knowledge or any pre-defined topological invariants. We demonstrate that the approach successfully identifies different topological elements, such as unlink, unknot, Hopf link, Solomon ring, trefoil, and so on, by employing generalized Gell-Mann matrices in non-Hermitian models with n=2 and n=3 energy bands. Moreover, since eigenstate information of non-Hermitian bands is incorporated in addition to eigenvalues, the approach distinguishes the different parity-time symmetry and breaking phases, recognizes the opposite chirality of braids and knots, and identifies out distinct topological phases that were overlooked before. Our study shows significant potential of machine learning in classification of knots, braid groups, and non-Hermitian topological phases.

Spectral properties and non-Hermitian skin effect in the Hatano–Nelson model

At present, non-Hermitian topological systems continue to be actively studed. In a rigorous approach, we study one of the key non-Hermitian systems — the Hatano–Nelson model $H$. We find the Green function for this Hamiltonian. Using the Green function, we analytically obtain the eigenvalues and eigenfunctions of $H$ for finite and semi-infinite chains, as well as for an infinite chain with a local potential. We discuss the non-Hermitian skin effect for the models mentioned above. We also describe the boundary between localized and resonant eigenfunctions (for the zero spectral parameter, this is the boundary between non-Hermitian topological phases).

Topological phase transitions in the non-Hermitian SSH model with long-range hopping terms induced by gains and losses

The latest progress in the one-dimensional chain SSH model provide the basic understanding of non-Hermitian topological phases. By periodically introducing the gains and losses of imaginary potential ig,-ig,-ig,ig with anti-PT symmetry, we have investigated the topological property of non-Hermitian SSH model with long-range hopping terms. We find that the gains and losses of imaginary potential causes the model to exhibit non-Hermiticity, and the non-Hermiticity induces system to undergo topological phase transitions from topologically trivial phase with non-Hermitian winding number 0 to topologically nontrivial phase with non-Hermitian winding number 1 and −1. The introduction of gains and losses greatly broadens the topological non-trivial region and play a non-trivial role in the model. We fully characterize the topological phase transitions by the band gap closing-reopening, the non-Hermitian winding numbers v and the hidden Chern number C.

True exponentially enhanced sensing in the non-Hermitian topological phase

Non-Hermitian systems have been employed to construct a high-sensitivity sensor. To evaluate the performance of the sensors, the quantum Fisher information per photon, or equivalently signal-to-noise ratio per photon, is provided as a “true” sensing criterion, which avoids the trivial contribution from the photon numbers. The specific properties of non-Hermitian systems, e.g., exceptional points and skin effect, have been connected to the true exponentially enhanced sensing performance. To date, the relation between the non-Hermitian topological phase and the true sensing performance has not been reported clearly. Here, we construct the high-sensitivity sensor based on the non-Hermitian Su–Schrieffer–Heeger lattice and establish the relationship between the exponentially enhanced sensing and the non-Hermitian topologically nontrivial phase. The saturation of sensing with the size emerges in the sense of one perturbation. Such a limitation can be surpassed through the change of incident positions of driving fields, and the exponentially enhanced sensing reappears.

Anomalous dynamical response of non-Hermitian topological phases

Anomalous dynamical response of non-Hermitian topological phases

Collective non-Hermitian skin effect: point-gap topology and the doublon-holon excitations in non-reciprocal many-body systems

Open quantum systems provide a plethora of exotic topological phases of matter that have no Hermitian counterpart. Non-Hermitian skin effect, macroscopic collapse of bulk states to the boundary, has been extensively studied in various experimental platforms. However, it remains an open question whether such topological phases persist in the presence of many-body interactions. Previous studies have shown that the Pauli exclusion principle suppresses the skin effect. In this study, we present a counterexample by demonstrating the presence of the skin effect in doublon-holon excitations. While the ground state of the spin-half Hatano-Nelson model shows no skin effect, the doublon-holon pairs, as its collective excitations, display the many-body skin effect even in strong coupling limit. We establish the robustness of this effect by revealing a bulk-boundary correspondence mediated by the point gap topology within the many-body energy spectrum. Our findings underscore the existence of non-Hermitian topological phases in collective excitations of many-body interacting systems.

Third-order exceptional line in a nitrogen-vacancy spin system.

Exceptional points (EPs) are singularities in non-Hermitian systems, where k (k ≥ 2) eigenvalues and eigenstates coalesce. High-order EPs exhibit richer topological characteristics and better sensing performance than second-order EPs. Theory predicts even richer non-Hermitian topological phases for high-order EP geometries, such as lines or rings formed entirely by high-order EPs. However, experimental exploration of high-order EP geometries has hitherto proved difficult due to the demand for more degrees of freedom in the Hamiltonian's parameter space or a higher level of symmetries. Here we observe a third-order exceptional line in an atomic-scale system. To this end, we use a nitrogen-vacancy spin in diamond and introduce multiple symmetries in the non-Hermitian Hamiltonian realized with the system. Furthermore, we show that the symmetries play an essential role in the occurrence of high-order EP geometries. Our approach can in future be further applied to explore high-order EP-related topological physics at the atomic scale and, potentially, for applications of high-order EPs in quantum technologies.

Topological phase transitions of generalized Brillouin zone

It has been known that the bulk-boundary correspondence (BBC) of the non-Hermitian skin effect is characterized by the topology of the complex eigenvalue spectra, while the topology of the wave function gives rise to Hermitian BBC with conventional boundary modes. In this work, we go beyond the known description of the non-Hermitian topological phase and find a different type of BBC that appears in generalized boundary conditions. The generalized Brillouin zone (GBZ) possesses non-trivial topological structures in the intermediate boundary condition between open and periodic boundary conditions. Unlike the conventional BBC, the topological phase transition is characterized by the generalized momentum touching of GBZ, which manifests as exceptional points. As a realization of our proposal, we suggest the non-reciprocal Kuramoto oscillator lattice, where phase slips accompany exceptional points as a signature of such topological phase transition. Our work establishes an understanding of non-Hermitian topological matter by complementing the non-Hermitian BBC as a general foundation of the non-Hermitian topological systems.

Inhibition of non-Hermitian topological phase transitions in sliding photonic quasicrystals.

Non-Hermitian (NH) quasicrystals have been a topic of increasing interest in current research, particularly in the context of NH topological physics and materials science. Recently, it has been suggested and experimentally demonstrated using synthetic photonic lattices that a class of NH quasicrystals can feature topological spectral phase transitions. Here we consider a NH quasicrystal with a uniformly-drifting (sliding) incommensurate potential and show that, owing to violation of Galilean invariance, the topological phase transition is washed out and the quasicrystal is always in the delocalized phase with an entirely real-energy spectrum. The results are illustrated by considering quantum walks in synthetic photonic lattices.

Non-Hermitian topological phases and skin effects in kagome lattices

Non-Hermitian topological phases and skin effects in kagome lattices

Non-Hermitian topological phase transitions controlled by nonlinearity

Non-Hermitian topological phase transitions controlled by nonlinearity

Realization of non-Hermitian Hopf bundle matter

Non-trivial linking invariant encodes robust information of topological matter. It has been recently shown that the linking and winding of complex eigenenergy strings can classify one-dimensional non-Hermitian topological matter. However, in higher dimensions, bundles of linked strings can emerge such that every string is mutually linked with all the other strings. To the best of our knowledge, a non-Hermitian Hopf bundle has not been experimentally clarified. Here, we attempt to explore the non-Hermitian Hopf bundle by visualizing the global linking structure of spinor strings in the momentum space of a two-dimensional electric circuit. By exploiting the flexibility of reconfigurable couplings between circuit nodes, we study the non-Hermitian topological phase transition by exploring the intricate structure of the Hopf bundle. Furthermore, we find that the higher-order skin effect in real space is accompanied by the linking of spinor strings in momentum space, revealing bulk-boundary correspondence between the two domains.

Magnetic flux response of non-Hermitian topological phases

We derive the response of non-Hermitian topological phases with intrinsic point gap topology to localized magnetic flux insertions. In two spatial dimensions, we identify the necessary and sufficient conditions for a flux skin effect that localizes an extensive number of in-gap modes at a flux core. In three dimensions, we furthermore establish the existence of: a flux spectral jump, where flux tube insertion fills up the entire point gap only at a single parallel crystal momentum; a higher-order flux skin effect, which occurs at the ends of flux tubes in presence of pseudo-inversion symmetry; and a flux Majorana mode that represents a spectrally isolated mid-gap state in the complex energy plane. We uniquely associate each non-Hermitian symmetry class with intrinsic point gap topology with one of these cases or a trivial flux response, and discuss possible experimental realizations.

Entanglement Phase Transition Induced by the Non-Hermitian Skin Effect

Recent years have seen remarkable development in open quantum systems effectively described by non-Hermitian Hamiltonians. A unique feature of non-Hermitian topological systems is the skin effect, anomalous localization of an extensive number of eigenstates driven by nonreciprocal dissipation. Despite its significance for non-Hermitian topological phases, the relevance of the skin effect to quantum entanglement and critical phenomena has remained unclear. Here, we find that the skin effect induces a nonequilibrium quantum phase transition in the entanglement dynamics. We show that the skin effect gives rise to a macroscopic flow of particles and suppresses the entanglement propagation and thermalization, leading to the area law of the entanglement entropy in the nonequilibrium steady state. Moreover, we reveal an entanglement phase transition induced by the competition between the unitary dynamics and the skin effect even without disorder or interactions. This entanglement phase transition accompanies nonequilibrium quantum criticality characterized by a nonunitary conformal field theory whose effective central charge is extremely sensitive to the boundary conditions. We also demonstrate that it originates from an exceptional point of the non-Hermitian Hamiltonian and the concomitant scale invariance of the skin modes localized according to the power law. Furthermore, we show that the skin effect leads to the purification and the reduction of von Neumann entropy even in Markovian open quantum systems described by the Lindblad master equation. Our work opens a way to control the entanglement growth and establishes a fundamental understanding of phase transitions and critical phenomena in open quantum systems far from thermal equilibrium.

Non-Hermitian topological quantum states in a reservoir-engineered transmon chain

Dissipation in open systems enriches the possible symmetries of the Hamiltonians beyond the Hermitian framework, allowing the possibility of novel non-Hermitian topological phases which exhibit long-living end states that are protected against disorder. So far, non-Hermitian topology has been explored in settings where probing genuine quantum effects has been challenging. We theoretically show that a non-Hermitian topological quantum phase can be realized in a reservoir-engineered transmon chain. The spatial modulation of dissipation is obtained by coupling each transmon to a quantum circuit refrigerator, allowing in situ tuning of dissipation strength in a wide range. By solving the many-body Lindblad master equation using a combination of the density matrix renormalization group and Prosen-Seligman third quantization approaches, we show that the topological end modes and the associated phase transition are visible in simple reflection measurements with experimentally realistic parameters. Finally, we demonstrate that genuine quantum effects are observable in this system via robust and slowly decaying long-range quantum entanglement of the topological end modes, which can be generated passively starting from a locally excited transmon.

Advances and applications on non-Hermitian topological photonics

Abstract Non-Hermitian photonics and topological photonics, as new research fields in optics, have attracted much attention in recent years, accompanying by a great deal of new physical concepts and novel effects emerging. The two fields are gradually crossed during the development process and the non-Hermitian topological photonics was born. Non-Hermitian topological photonics not only constantly produces various novel physical effects, but also shows great potential in optical device applications. It becomes an important part of the modern physics and optics, penetrating into different research fields. On one hand, photonics system can introduce artificially-constructed gain and loss to study non-Hermitian physics. Photonics platform is an important methods and ways to verify novel physical phenomena and promote the development of non-Hermitian physics. On the other hand, the non-Hermitian topological photonics provides a new dimension for manipulating topological states. Active and dissipate materials are common in photonic systems; therefore, by using light pump and dissipation of photonic systems, it is expected to promote further development of topological photonics in device applications. In this review article, we focus on the recent advances and applications on non-Hermitian topological photonics, including the non-Hermitian topological phase transition and skin effect, as well as the applications emerging prosperously in reconfigurable, nonlinear and quantum optical systems. The possible future research directions of non-Hermitian topological photonics are also discussed at the end. Non-Hermitian topological photonics can have great potential in technological revolution and have the capacity of leading the development of both physics and technology industry.

Critical and noncritical non-Hermitian topological phase transitions in one-dimensional chains

In this work we investigate non-Hermitian topological phase transitions using real-space edge states as a paradigmatic tool. We focus on the simplest non-Hermitian variant of the Su-Schrieffer-Heeger model, including a parameter that denotes the degree of non-Hermiticity of the system. We study the behavior of the zero-energy edge states in the nontrivial topological phases with integer and semi-integer topological winding numbers, according to the distance to the critical point. We find that, depending on the parameters of the model, the edge states may penetrate into the bulk, as expected in Hermitian topological phase transitions. We also show that, using the topological characterization of the exceptional points, we can describe the intricate chiral behavior of the edge states across the whole phase diagram. Moreover, we characterize the criticality of the model by determining the correlation length critical exponent directly from numerical calculations of the penetration length of the zero-mode edge states.

Interconversion of exceptional points between different orders in non-Hermitian systems

Singularities of non-Hermitian systems typified by exceptional points (EPs) are critical for understanding non-Hermitian topological phases and trigger many intriguing phenomena. However, it remains unexplored what happens when EPs meet one another. Here, in a typical four-level model with both touching and crossing intersections of EP hypersurfaces, we report the interconversion mechanisms between EPs of different orders. By examining both the eigenvalues and eigenvectors, we show analytically that all EPs of higher orders are formed at the touching intersections of two different types of EP hypersurfaces of lower orders. Contrarily, the crossing intersection of EP structures lowers the order of EPs. The mechanisms of the increase and decrease in defectiveness discovered here are expected to hold for EPs of any order in various non-Hermitian systems, providing a comprehensive understanding of EPs and inspiration toward advanced applications such as biosensing and information processing.

Anomalous Floquet non-Hermitian skin effect in a ring resonator lattice

We present a one-dimensional coupled ring resonator lattice exhibiting a variant of the non- Hermitian skin effect (NHSE) that we call the anomalous Floquet NHSE. Unlike existing approaches to achieving the NHSE by engineering gain and loss on different ring segments, our design uses fixed on-site gain or loss in each ring. The anomalous Floquet NHSE is marked by the existence of skin modes at every value of the Floquet quasienergy, allowing for broadband asymmetric transmission. Varying the gain/loss induces a non-Hermitian topological phase transition, reversing the localization direction of the skin modes. An experimental implementation in an acoustic lattice yields good agreement with theoretical predictions, with a very broad relative bandwidth of around 40%.