Non-Hermitian Topology:
from Classical Optics to Quantum Matter

Quantum matter and its novel manifestations are often characterized by an intricate interplay of quantum fluctuations, ideas of band topology and entanglement [1]. In case of open systems the system might exchange energy/particles (or other degrees of freedom) among the subsystems. In that case, each subsystem encounters an overall growth or decay in energy or probability norm. Such energy non-conserving or dissipative systems can be modeled through the adoption of complex energy eigenvalues and a non-Hermitian Hamiltonian [2]. Interestingly, non-Hermiticity enriches as well as offers unique topological phases considering the interplay between ramified symmetry and topology [3]. Particularly, the spectral topology becomes more interesting with the introduction of non-Hermiticity, which leads to a generalized band diagram in the complex plane and features two different types of complex-energy gaps. The direct relevance between the single-particle states and the many-body state is absent for non-Hermitian systems, given that the system may not reach an equilibrium state but rather a non-equilibrium steady state at long times. The appearance of complex eigen-spectra requires us to restructure the framework for characterising the many body phases in a non-Hermitian system. In this work we study many-body ‘steady states’ that arise in the non-Hermitian generalisation of the non-interacting Su-Schrieffer-Heeger model at a finite density of fermions [4]. We find that the hitherto known phase diagrams for this system, derived from the single-particle gap closings, in fact correspond to distinct non-equilibrium phases, which either carry finite currents or are dynamical insulators where particles are entrapped. Each of these have distinct quasi-particle excitations and steady state correlations and entanglement properties. Looking at finite-sized systems, we further modulate the boundary to uncover the topological features in such steady states – in particular the emergence of leaky boundary modes. Using a variety of analytical and numerical methods we develop a theoretical understanding of the various phases and their transitions, and uncover the rich interplay of non-equilibrium many-body physics, quantum entanglement and topology in a simple looking, yet a rich model system. References: [1] Xiao-Gang Wen, Rev. Mod. Phys. 89, 041004 (2017). [2] N. Moiseyev, Non-Hermitian quantum mechanics (Cambridge University Press, 2011). [3] Kohei Kawabata, Ken Shiozaki, Masahito Ueda, and Masatoshi Sato, Phys. Rev. X 9, 041015 (2019). [4] Ayan Banerjee, Suraj Hegde, Adhip Agarwala, and Awadhesh Narayan, Phys. Rev. B 105, 205403 (2022).

Resonances in open quantum systems have been actively studied since the very birth of quantum mechanics [1,2]. Their linewidth broadenings caused by the finite lifetimes can be analytically estimated only in a few particular cases [3]. In this sense, the analytically estimated linewidth broadenings of the electron-impurity resonant states by Monozon and Schmelcher [4] is a remarkable theoretical result which can be used as a reliable reference for qualitative estimations [5]. In the current report, using the complex-scaling calculations as a quantitative insight, we show how the qualitative theoretical estimations made by Monozon and Schmelcher for the electron-impurity in very narrow quantum wells (QWs) can be improved and generalized to more practical case of the QW widths of order of the electron-impurity's Bohr radius [6]. In particular, we show that discovered by Fano [7] and confirmed by Monozon and Schmelcher the fourth-power scaling of the linewidth broadenings with respect to QW width holds only for very narrow QWs which are hardly be practically used in the spectroscopy of heterostructures [5]. In contrast to [4], we analytically and numerically demonstrate that for the real QWs the scaling of the linewidths with respect to the QW width appears to be linear. As a result, our studies shed light to the linewidth broadenings in the regimes inaccessible by the Fano theory of resonances. Moreover, many calculated resonant states of electron-impurity and electron-hole pairs in semiconductor QWs as well as their dependencies on the QW width as a parameter allow us to study formation of the exceptional points as a degeneracy of resonances in such systems [8,9]. [1] G. A. Gamow, Z. Physik 51, 204 (1928). [2] N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University Press, Cambridge, 2011). [3] K. Rapedius, Eur. J. Phys. 32, 1199 (2011). [4] B. S. Monozon and P. Schmelcher, Phys. Rev. B 71, 085302 (2005). [5] P. A. Belov, Physica E, 112, 96 (2019). [6] P. A. Belov, Phys. Rev. B 105, 155417 (2022). [7] U. Fano, Phys. Rev. 124, 1866 (1961). [8] M. Feldmaier et al J. Phys. B: At. Mol. Opt. Phys. 49, 144002 (2016). [9] P. A. Belov, Semiconductors 53, 2049 (2019).

Non-Hermitian systems attract a lot of attention in recent years as effective description of open quantum systems. A prominent example in this context is the Hatano-Nelson model. While historically the model has short-range non-reciprocal hoppings, long-range hopping has not been systematically studied. In this talk, I will present our results on the extended Hatano-Nelson model. Using analytical techniques, we demonstrate how the underlying physics of the original Hatano-Nelson model is enriched when longer-range hoppings are also included. I will discuss how the crucial elements of the Hatano-Nelson model, namely, the non-Hermitian skin effect and the exceptional points, are modified for the generalized model.

Certain aspects of open quantum and wave systems can be described via non-Hermitian Hamiltonians. The non-Hermiticity can lead to an interesting type of degeneracy the so called exceptional point (EP$_n$) of order $n$ where $n$ eigenvalues and the corresponding eigenstates coalesce. Generic perturbations result in a $n$-root scaling of the eigenvalue changes. For non-generic perturbations the topological behavior is highly dependent on the structure of the perturbation matrix. In order to get a deeper and intuitive understanding of the EP$_n$ topology a graph-theoretical approach is beneficial. The idea is to comprehend the Hamiltonian of the perturbed system as the adjacency matrix for the graph representation. Through consideration of sub graphs consisting of closed loops the qualitative behavior of the eigenvalues of the complete system is represented. Furthermore the graph-theoretical picture offers an easy way to get an understanding for the exotic topological behavior of non-generic perturbations. A comparison between the suggested approach and the eigen solutions for a photonic system is performed to point out the benefits of a graph-theoretical perspective on higher-order EPs.

Quadratic Majorana Hamiltonians appear in various situations in physics. For example, the BdG Hamiltonain of superconductivity is expressed by the quadratic form of Majorana operators. Another example is the Kitaev honeycomb model [1] that can be attributed to a free fermion system by rewriting the spin operator with the Majorana operators. Once the quadratic Majorana Hamiltonian has been obtained, it is possible to derive information on many-body eigensystems from information on one-body eigensystems. On the other hand, non-Hermitian Hamiltonians appear in various situations such as effective models of dissipation. The problem that arises when treating non-Hermitian quadratic Majorana systems is that there can appear non-diagonalizable points called exceptional points. There is a previous study by Prosen [2] that has addressed this issue, but the important lemma to develop a general theory was just conjectured. In this study, we proved this conjecture, and we develop an argument valid for the non-Hermitian case using the Jordan canonical form. At an exceptional point, it is necessary to consider an extended concept of eigenstates called generalized eigenstates. We will see the dimensions of the generalized eigenspaces can be expressed in an explicit form known as q-binomial coefficients. Besides, we will give an procedure for obtaining a concrete expression of the generalized eigenstates. [1] A. Kitaev, Ann. Phys. 321, 2 (2006). [2] T. Prosen, J. Stat. Mech. 2010, P07020 (2010).

Non-Hermitian physics has been developing especially actively in recent years. Starting with a significant number of works indicating the inapplicability to non-Hermitian systems of the concepts familiar to us from Hermitian physics, this area has acquired new concepts and theory to date. One of the characteristic effects peculiar to non-Hermitian systems is the non-Hermitian skin effect (NHSE), the localization of bulk states at the boundary of a finite lattice. There are a large number of theoretical papers either considering abstract non-Hermitian lattices, or proposing complex experimental schemes for implementing the non-Hermitian skin effect, but experimental implementations of NHSE in real space are still few. In our work [1], we propose a new generic way of implementing non-reciprocity based on a combination of the Rashba-Dresshlauss spin-orbit coupling, existing for electrons, cold atoms, and photons, and a lifetime imbalance between two spin components. We show that one can realize the Hatano-Nelson model, the non-Hermitian Su-Schrieffer-Heeger model, and even observe the NHSE in a 1D potential well without the need for a lattice. We further demonstrate the practical feasibility of this proposal by considering the specific example of a photonic liquid crystal microcavity. This platform allows one to switch on and off the NHSE by applying an external voltage to the microcavity. [1] Kokhanchik P, et al. arXiv.2303.08483

We demonstrate the existence of topologically stable unpaired exceptional points, and construct simple non-Hermitian tight-binding models exemplifying such remarkable nodal phases. While Fermion doubling, i.e. the necessity of compensating the topological charge of a stable nodal point by an anti-dote, rules out a direct counterpart of our findings in the realm of Hermitian semimetals, here we derive how non-commuting braids of complex energy levels may stabilize unpaired EPs. Drawing on this insight, we reveal the occurrence of a single, unpaired EP, manifested as a non-Abelian monopole in the Brillouin zone of a minimal three-band model. This third-order degeneracy cannot be fully gapped by any local perturbation. Instead, it may split into simpler (second-order) degeneracies that can only gap out by pairwise annihilation after having moved around inequivalent large circles of the Brillouin zone. Our results imply the incompleteness of a topological classification based on winding numbers, due to non-Abelian representations of the braid group intertwining three or more complex energy levels.

A striking manifestation of the non-Hermitian physics in open systems are non-Hermitian degenceracies called exceptional points (EPs) at which not only the eigenvalues but also the corresponding eigenvectors coalesce. Such EPs lead to fascinating properties of the system like chiral eigenstates or an enhanced sensitivity against external perturbations. Although the research on EPs has grown enormously in the last years, especially, the construction of high-order EPs where more than two eigenvalues and eigenvectors coalesce is still an ongoing challenge. We tackle this challenge by an intuitive and robust implementation of high-order EPs in microring cavities. To do so we combine the known mirror-induced asymmetric backscattering with the unidirectional coupling between microcavities via adjacent waveguides. We investigate the topology of eigenvalues around the EPs and show that cavity-selective perturbations realize interesting non-generic perturbation schemes. Our results are motivated by an effective Hamiltonian description and verified with full numerical simulations of the photonic structure.

Dynamical quantum phase transition (DQPT) is the temporal analog of quantum phase transition where non-analyticities in rate function i.e., the temporal analog of free energy, with respect to time give rise to the phase transition. Its topological nature is characterized by the dynamical winding number which acts as an order parameter. We explore DQPT both in Hermitian and Non-Hermitian systems at zero temperature. In addition, the finite temperature DQPT (FTDQPT) in the context of mixed state is also investigated. We generalize the FTDQPT formalism for Non-Hermitian systems in the context of the non-unitary evolution of density Matrix. All the studies refer to the occurrence of DQPT for a gapped system. We find DQPT for gapless systems for both Hermitian and non-Hermitian cases at zero and finite temperatures. Our study paves the way to analyze the non-Hermitian DQPT and FTDQPT for gapless systems.

The non-Hermitian skin effects are representative phenomena intrinsic to non-Hermitian systems: the energy spectra and eigenstates under the open boundary condition (OBC) drastically differ from those under the periodic boundary condition (PBC). Whereas a non-trivial topology under the PBC characterizes the non-Hermitian skin effects, their proper measure under the OBC has not been clarified yet. This paper reveals that topological enhancement of non-normality under the OBC accurately quantifies the non-Hermitian skin effects. Correspondingly to spectrum and state changes of the skin effects, we introduce two scalar measures of non-normality and argue that the non-Hermitian skin effects enhance both macroscopically under the OBC. We also show that the enhanced non-normality correctly describes phase transitions causing the non-Hermitian skin effects and reveals the absence of non-Hermitian skin effects protected by average symmetry. The topological enhancement of non-normality governs the perturbation sensitivity of the OBC spectra and the anomalous time-evolution dynamics through the Bauer-Fike theorem.

The bulk-boundary correspondence (BBC) is a fundamental principle in Hermitian topological phases. It, however, has been known that the BBC may break down in non-Hermitian systems. The key is the difference between the bulk spectra under open boundary conditions (OBCs) and that under periodic boundary conditions (PBCs) due to the non-Hermitian skin effect. Since non-Hermitian Hamiltonians take the complex energy, we can define two types of topological phases, line-gap and point-gap topological phases. Many previous works have shown that the BBC becomes valid again for the line-gap topological phases if the bulk spectra under OBCs are treated properly. On the other hand, the BBC for the point-gap topological phases has remained unclear. Here, we clarify that the classification of bulk point-gap topological phases in OBCs can be different from those in PBCs. Moreover, we show that the BBC also exists for the point-gap topological phases by introducing real-space point-gap topological numbers defined under OBCs. We also discuss an experimental realization of point-gap topological phases based on Hermitian topological insulators and superconductors. [1] D. Nakamura, T. Bessho, and M. Sato, arXiv:2205.15635. [2] D. Nakamura, K. Inaka, N. Okuma, and M. Sato, arXiv:2304.08110.

A latent symmetry is a novel type of symmetry which, in general, is not apparent from a geometric inspection of the system. Instead, it becomes visible after a suitable dimensional reduction: The so-called isospectral reduction, which is akin to an effective Hamiltonian. Latent symmetries can appear in many discrete systems, including networks of acoustic waveguides, transmission line networks, electric circuits, or coupled oscillators. Besides providing a new viewpoint on seemingly asymmetric systems, latent symmetries can also be used to design certain properties of the system. For instance, it is possible to design lattices with seemingly accidental degeneracies, which are in fact caused by non-abelian latent symmetries. In this talk, I will give an introduction to the topic and showcase some of our recent results.

We discuss classifications of multiple arbitrary-order exceptional points by invoking the permutation group and its conjugacy classes. We classify topological structures of Riemann surfaces generated by multiple states around multiple arbitrary-order exceptional points, using the permutation properties of stroboscopic encircling exceptional points. The results are realized in non-Hermitian effective Hamiltonian based on Jordan normal forms and fully desymmetrized optical microcavities. Additionally, we reveal the relation between the spectral topology originating from complex eigenvalues in non-Hermitian systems and wavefunction topology related to the additional geometric phases. Finally, we discuss the topology of one-dimensional multi-bands systems based on exceptional points.

Non-Abelian phenomena and non-Hermitian systems have both been widely explored in recent years. As a bridge between the two, we introduce and develop non-Abelian gauge engineering for realizing multi-fold spectral topology. As an example of our proposal, we engineer non-Hermiticity in the paradigmatic Su-Schrieffer-Heeger (SSH) model by introducing a generalized non-Abelian gauge, leading to an emergent two-fold spectral topology that governs the decoupled behaviour of the corresponding non-Hermitian skin effect. As a consequence of the non-Abelian gauge choice, our model exhibits a rich phase diagram consisting of distinct topological phases, which we characterize by introducing the notion of paired winding numbers, which, in turn, predict the direction of skin localization under open boundaries. We demonstrate that the choice of gauge parameters enables control over the directionality of the skin effect, allowing for it to be unilateral or bilateral. Furthermore, we discover non-dispersive flat bands emerging within the inherent SSH model framework, arising from the non-Abelian gauge. We also introduce a simplified toy model to capture the underlying physics of the emergent flat bands and direction-selective skin effect. Our findings pave way for the exploration of unconventional spectral topology through non-Abelian gauges.

When a Hamiltonian is effectively non-Hermitian, its eigenstates can be localized at the boundary of the system, a phenomenon known as the non-Hermitian skin effect(NHSE) [1,2]. Yao and Wang characterized the NHSE as a phenomenon such that ``all the eigenstates of an open chain are found to be localized near the boundary.'' Through the study of the NHSE in higher dimensions, it has been recognized that the NHSE can be realized in two-dimensional non-Hermitian systems of size LxL where the number of eigenstates localized at the boundary is of order L [3,4,5]. How many localized eigenstates are needed to characterize NHSE is a question of interest. It is also well-known that the NHSE in free systems originates from a non-Hermitian topology protected by the point gap of the complex energy spectrum [3,6]. However, many-body systems exhibiting the NHSE do not always possess the point gap [7], which requests a characterization of many-body NHSE using eigenstates. In this talk, we propose a formulation of the NHSE in free systems only using some eigenstates, applicable to one- or higher-dimensional systems and even the generic geometry systems [8]. Furthermore, we extend it to many-body systems and introduce the NHSE in the many-body Fock space (dobbed ``Fock space skin effect''), discussing the relation with the Liouvillian skin effect, slowing down of relaxation in dissipative systems [9,10,11]. For this purpose, we define ``localization'' and ``localization length'' of state vectors in our manner. We rigorously show that if the number of given state vectors localized at the same location exceeds a certain threshold, a Hamiltonian such that all of their state vectors are right eigenstates of it must be non-Hermitian. If the NHSE is characterized by the fact that more states than the threshold are localized at the same location, then we can conclude that the NHSE in this sense cannot occur in Hermitian systems. [1] S. Yao and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018). [2] F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Phys. Rev. Lett. 121, 026808 (2018). [3] N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Phys. Rev. Lett. 124, 086801 (2020). [4] K. Kawabata, M. Sato, and K. Shiozaki, Phys. Rev. B 102, 205118 (2020). [5] R. Okugawa, R. Takahashi, and K. Yokomizo, Physical Review B 102, 241202 (2020). [6] K. Zhang, Z. Yang, and C. Fang, Physical Review Letters 125, 126402 (2020). [7] K. Kawabata, K. Shiozaki, and S. Ryu, Physical Review B 105, 165137 (2022). [8] K Zhang, Z Yang, and C Fang, Nature communications 13, 2496 (2022). [9] F. Song, S. Yao, and Z. Wang, Physical review letters 123, 170401 (2019). [10] T. Mori, and T. Shirai, Physical Review Letters 125, 230604 (2020). [11] T. Haga, M. Nakagawa, R. Hamazaki, and M. Ueda, Physical Review Letters 127, 070402 (2021).

After a quench, isolated thermalizing quantum many-body systems relax locally to an equilibrium state that is universally determined by conservation laws and the principle of maximum entropy. In contrast, open quantum systems, subjected to Markovian drive and dissipation, typically evolve toward nonequilibrium steady states that are highly model-dependent. However, focusing on a driven-dissipative Kitaev chain, we show that relaxation after a quantum quench can be described by a maximum entropy ensemble, if the Liouvillian governing the dynamics has parity-time (PT) symmetry. We dub this ensemble, which is determined by the biorthogonal eigenmodes of the adjoint Liouvillian, the PT-symmetric generalized Gibbs ensemble (PTGGE). Resembling isolated systems, thermalization becomes manifest in the growth and saturation of entanglement, and the relaxation of local observables. In contrast, the directional pumping of fermion parity represents a phenomenon that is unique to relaxation dynamics in driven-dissipative systems. We expect that our results apply rather generally to integrable, driven-dissipative bosonic and fermionic quantum many-body systems with PT symmetry.

One of the main drivers of the research on non-Hermitian systems is the occurence of special degeneracies in the spectrum of non-Hermitian Hamiltonians called Exceptional Points (EPs). At these points, two or more eigenvalues and corresponding eigenvectors overlap rendering the Hamiltonian defective. EPs can be utilized for quantum sensing and for adiabatic state conversion, to name a few applications. Experimental realization of devices based on EPs is complicated by the need to tune the system to the vicinity of an EP. At the same time, the number of independent parameters required for tuning is diminished in the presence of symmetries. I consider the systems with Pseudo-Hermitian symmetry, which is closely related to the usually employed PT-symmetry. I characterize separate non-degenerate levels with a Z2 topological index, corresponding to the signs of the pseudometric operator eigenvalues in the absence of Hermiticity-breaking terms. After that I show that the formation of second-order EPs is governed by this topological index: EPs are provided only by pairs of levels with opposite indices. To demonstrate the approach, I consider transverse-field Ising chain with longitudinal staggered gain and loss, which is pseudo-Hermitian with respect to parity. Using the integrability of the model in the absence of Hermiticity-breaking terms, I compute all the topological indices analytically and then use them to analyze the formation of second- and third-order Exceptional points. As a side note, I also consider the ground state quantum Phase transitions in the thermodynamic limit of the model.

We explore the geometrical structure of Quantum State Manifolds for parametric driven Hamiltonian systems, which have significant applications in areas such as quantum phase transitions theory or decohering quantum state driving. We conduct a geometrical analysis of a single-qubit and fully-connected multi-qubit model, with a special interest in diabolic points and the ground state manifold geodesics.

We devise a generic and experimentally accessible recipe to prepare boundary states of topological or non-topological quantum systems through an interplay between coherent Hamiltonian dynamics and local dissipation. Intuitively, our recipe harnesses the spatial structure of boundary states which vanish on sublattices where losses are suitably engineered. This yields unique non-trivial steady states that populate the targeted boundary states with infinite life times while all other states are exponentially damped in time. Remarkably, applying loss only at one boundary can yield a unique steady state localized at the very same boundary. We detail our construction and rigorously derive full Liouvillian spectra and dissipative gaps in the presence of a spectral mirror symmetry for a one-dimensional Su-Schrieffer-Heeger model and a two-dimensional Chern insulator. We outline how our recipe extends to generic non-interacting systems (arXiv:2305.00031).

In a class of non-Hermitian quantum walk in lossy lattices with open boundary conditions, an unexpected peak in the distribution of the decay probabilities appears at the edge, dubbed edge burst. It is proposed that the edge burst is originated jointly from the non-Hermitian skin effect (NHSE) and the imaginary gaplessness of the spectrum [Wen-Tan Xue et al., Phys. Rev. Lett. 128, 120401 (2022)]. Using a particular one-dimensional lossy lattice with a nonuniform loss rate, we show that the edge burst can occur even in the absence of NHSE. Furthermore, we discuss that the edge burst may not appear if the spectrum satisfies the imaginary gaplesness condition. Aside from its fundamental importance, by removing the restrictions on observing the edge burst effect, our results open the door to broader design space for future applications of the edge burst effect. C Yuce, H Ramezani, arXiv:2212.02879

Non-Hermitian matrices arise in the formalisms of open quantum systems. We look at several quantum models coupled to a bath and search for non-hermitian phenomena in the spectra of corresponding Liouvillians and other matrices characteristic of open quantum systems.