Non-Hermitian Topology:
from Classical Optics to Quantum Matter

In the first part of this talk, I will discuss a quantum geometric approach to various Hermitian and non-Hermitian versions of the Su-Schrieffer-Heeger (SSH) model. We find that this method allows one to correctly identify different topological phases and topological phase transitions for all SSH models. Whereas the quantum geometry of Hermitian systems is Riemannian, introducing non-Hermiticity leads to pseudo-Riemannian and complex geometries, thus significantly generalizing from the quantum geometries studied thus far. We find a “dark” direction in some cases, such that within linear response, one can perturb the system by a particular change of parameters while maintaining a zero excitation rate [1]. In the second part of the talk, I will present a remarkably simple model and the experimental observation of topological monomodes generated dynamically. By focusing on non-Hermitian one-dimensional (1D) and 2D Su-Schrieffer-Heeger (SSH) models, we theoretically unveil the minimal configuration to realize a topological monomode upon engineering losses and breaking of lattice symmetries. Furthermore, we classify the systems in terms of the (non-Hermitian) symmetries that are present and calculate the corresponding topological invariants. To corroborate the theory, we present experiments in photonic lattices, in which a monomode is observed in the non-Hermitian 1D and 2D SSH models, thus breaking the paradigm that topological corner states should appear in pairs [2]. [1] Chen Chao Ye, W. L. Vleeshouwers, S. Heatley, V. Gritsev, and C. Morais Smith, arXiv: 2305.17675 [2] E. Slootman, W. Cherifi, L. Eek, R. Arouca, E. J. Bergholtz, M. Bourennane, and C. Morais Smith, arXiv:2304.05748

Although time is naturally a real variable, it is sometimes useful to consider it extended to the complex domain. After a general introduction, we will discuss two specific applications of such an extension within the framework of non-Hermitian quantum mechanics. The first application [1,2] concerns quantum tunneling. We will show that specific complex-time classical trajectories through multibarrier potentials yield correct semiclassical approximations of the smoothed transmission amplitudes. It turns out that the corresponding complex-extended continuum level density exhibits singularities analogous to excited-state quantum phase transitions in bound (discrete-energy) quantum systems. Potential wider consequences of complex-time tunneling and its link to the concept of weak measurements will be mentioned. The second application [3] is related to quantum quench dynamics. We will demonstrate that zeros of the after-quench survival amplitude of the initial state in complex-extended time represent computationally detectable precursors of dynamical quantum phase transitions in finite systems. The behavior of zeros with increasing system size makes it possible to distinguish true precursors of criticality from the false ones. This approach hints at similar descriptions of equilibrium phase transitions, particularly those in the thermodynamic domain. References: [1] P. Stránský, M. Sindelka, M. Kloc, P. Cejnar, Complex density of continuum states in resonant quantum tunneling, Phys. Rev. Lett. 125 (2020) 020401. [2] P. Stránský, M. Sindelka, P. Cejnar, Continuum analogs of excited-state quantum phase transitions, Phys. Rev. A 103 (2021) 062207. [3] Á.L. Corps, P. Stránský, P. Cejnar, Mechanism of dynamical phase transitions: The complex-time survival amplitude, Phys. Rev. B 107 (2023) 094307.

Many properties of a linear system are determined by the manner in which its eigenvalue spectrum can be tuned. Non-Hermitian systems offer unique features in this regard. Among the most striking is that tuning control parameters around a closed loop can cause the system's spectrum to return to itself in a topologically non-trivial manner. It is well-known that this behavior is related to the manner in which the control loop encircles points of degeneracy. The general relationship between control loops, eigenvalue spectra, and points of degeneracy (for any number of modes and for any degree of degeneracy!) maps exactly on to an elegant and easily visualizable piece of mathematics. The results of this are that: 1) degeneracies lie on knots in the space of control parameters; 2) a control loop causes the eigenvalue spectrum to trace out a braid; and 3) the specific braid is determined by the manner in which the control loop encircles the knot of degeneracies. I will give a pedagogical introduction to these results (with lots of images and animations). I will also present measurements that illustrate these structures, including the knot of degeneracies and the non-Abelian character of the resulting braids.

Majorana zero modes (MZMs) emerge as edge states in topological superconductors and are promising for topological quantum computation, but their detection has so far been elusive. Here we show that non-Hermiticity can be used to obtain dramatically more robust MZMs. The enhanced properties appear as a result of an extreme instability of exceptional points to superconductivity, such that even a vanishingly small superconducting order parameter already opens a large energy gap, produces well-localized MZMs, and leads to strong superconducting pair correlations. Our work thus illustrates the large potential of enhancing topological superconductivity using non-Hermitian exceptional points.

We study theoretically a junction consisting of a normal metal, PT-symmetric non-Hermitian superconductor, and an insulating thin layer between them. We calculate current-voltage characteristics for this junction using left-right and right-right bases and compare the results. We find that left-right basis gives the opposite current of Andreev-scattered particles compared to the right-right basis and conventional Andreev scattering. This leads to profound differences in current-voltage characteristics. Based on this and other signatures, we argue that left-right basis is not applicable in this case. Remarkably, we find that the quasiparticle current is conserved across the junction in both bases, and the growth and decay with time of the states with imaginary energies in right-right basis is equilibrated.

Open dissipative systems described by non-Hermitian Hamiltonians have recently attracted a lot of interests as they lead to a wide range of effects such as coherent-perfect absorption and lasing [1], directional emission [1], novel topological invariants [2] and edge states [3]. One of such systems is the exciton polaritons in an optical microcavity which arise from the strong coupling of excitons in a semiconductor and cavity photon modes. The inherent loss and gain in this system have already enabled measurements of non-Hermitian degeneracies [4] and topological invariants [2]. Inspired by recent studies [3, 5] showing the self-acceleration of wave packets in non-Hermitian systems, we theoretically investigate the time-evolution of wave packets in a non-Hermitian exciton-polariton system. In particular, we numerically study wavepacket dynamics in momentum space and observe self-acceleration. The wave packets tend to evolve into the eigenstate with the smaller decay rate (or the larger imaginary part of the eigenenergy), then propagate towards the momenta corresponding to the minima of the decay rate (or the maxima of the imaginary part of the eigenenergies). We also observe the generation of pseudospin defects (half skyrmions) along the imaginary Fermi arc in momentum space, where the decay rates of the two eigenstates coincide. All of these effects do not require an external potential and can be measured in an exciton-polariton system with optical anisotropy, e.g., perovskite [2], organics [6], or ZnO-based microcavities [7]. Our results highlight the excellent potential of exciton polaritons as a platform to study non-Hermitian dynamics. [1] Ş. K. Özdemir, et al., Natural Materials 18, 783-798 (2019). [2] R. Su, et al., Science Advances 7, eabi8905 (2021). [3] S. Longhi, Phys. Rev. B 105, 245143 (2022). [4] T. Gao, et al., Nature 526, 554–558 (2015). [5] D. D. Solnyshkov, et al., Phys. Rev. B 103, 125302 (2021). [6] Q. Liao, et al., Phys. Rev. Lett. 127, 107402 (2021). [7] S. Richter, et al., Phys. Rev. Lett. 123, 227401 (2019).

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. Reference: K. Kawabata, T. Numasawa, and S. Ryu, Phys. Rev. X 13, 021007 (2023).

In photonic systems, gain and loss can be used to induce intriguing effects that are linked to non-Hermitian and topological physics. Prominent examples are exceptional points and the non-Hermitian skin effect, which can be used for enhanced sensing and directed amplification, as well as symmetry-protected states, which can be addressed by topological mode selection. Many of these applications make explicit use of mode nonorthogonality, which becomes especially intriguing when the system is nonreciprocal. I describe how these effects can be probed in response theory, transport, and scattering, and highlight fundamental practical limits of the observability of some effects. [1] H. Schomerus, "Fundamental constraints on the observability of non-Hermitian effects in passive systems," Phys. Rev. A 106, 063509 (2022). [2] H. Schomerus, "Quantum Noise and Self-Sustained Radiation of PT-Symmetric Systems, " Phys. Rev. Lett. 104, 233601 (2010). [3] G. Yoo, H.-S. Sim, and H. Schomerus, "Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators, " Phys. Rev. A 84, 063833 (2011). [4] H. Schomerus, "Nonreciprocal response theory of non-Hermitian mechanical metamaterials: Response phase transition from the skin effect of zero modes", Phys. Rev. Research 2, 013058 (2020).

While topological phases of matter have predominantly been studied for isolated Hermitian systems, a recent shift has been made towards considering these phases in the context of non-Hermitian Hamiltonians. Non-Hermitian topological phenomena reveal an enrichment of the phenomenology of topological phases, and forms a rapidly growing new cross-disciplinary field. In particular, non- Hermiticity plays a central role in both classical and quantum systems. In the classical realm, this comes about due to, e.g., gain and loss processes in optics, while in the quantum realm, non-Hermiticity describes the dynamics of open quantum systems as well as scattering, decay, broadening and resonances due to, e.g., interactions and disorder. Non-Hermitian Hamiltonians may feature many exotic properties, which are radically different from their Hermitian counterparts, such as the generic appearance of exotic exceptional structures, a break down of the famed bulk-boundary correspondence, and the piling up of bulk states at the boundaries known as the non-Hermitian skin effect. In this talk, I will provide an overview of the field focussing on fundamental aspects, experimental realizations and I will briefly touch upon applications.

Beyond-hermitian physics has recently attracted a great deal of attention due to remarkable advances in experimental techniques and theoretical methods in AMO, condensed matter and nonequilibrium statistical physics. Complete knowledge about quantum jumps allows a description of quantum dynamics at the single-trajectory level. A subclass thereof without quantum jumps can be described by a non-hermitian Hamiltonian. Here, symmetry, topology and many-body effects are fundamentally altered. Importantly, transposition and complex conjugation, which are equivalent in hermitian physics, become inequivalent, leading to proliferation of new topological phases and symmetry classes. In random matrix theory, transposition symmetry leads to two new universality classes of level-spacing statistics other than the Ginibre ensemble. In many-body physics, non-hermiticity leads to the dynamical sign reversal of magnetism in dissipative Hubbard models, violation of the g-theorem in the Kondo problem, and quantum phase transitions without gap closing. In the lecture, I will provide an overview on the fundamentals and new frontiers about beyond-Hermitian quantum physics. Reference: Y. Ashida, Z. Gong, and M. Ueda, Adv. Phys. 69, 249 (2021)

The concept of dissipation engineering has enriched the methods available for state preparation, dissipative quantum computing and quantum information processing. Combining such engineered dissipative processes with coherent dynamics allows for new effects to emerge. For example, we found that any factorisable (coherent) Hamiltonian interaction can be rendered nonreciprocal if balanced with the corresponding dissipative interaction. This powerful concept can be exploited to engineer nonreciprocal devices for quantum information processing, computation and communication protocols, Abstract Body: e.g., to achieve control over the direction of propagation of photonic signals. However, although nonreciprocal concepts are realizable in quantum architectures, nonreciprocity itself holds up to the classical level and is not inherently quantum; the fundamental aspects of nonreciprocity in the quantum regime have yet to be fully investigated. In this talk we will address these aspects and discuss routes of how nonreciprocal concepts can find application beyond quantum information processing.

Non-Hermitian systems are featured by eigenmodes taking complex values. The exceptional points are described by braids of eigenmodes. The topology can be worked out systematically with the application of homotopy theory. We show that in the absence of symmetry, the non-Abelian topology admits unpaired exceptional points under periodic boundary condition. The unpaired exceptional points do not annhilate but fuse into a monopole. When PT symmetry is imposed, we show that homotopy groups reveal new topology from the interplay of eigenvalues and eigenvectors. This includes a novel type of fragile topology in the eigenvectors. The non-Abelian eigenvector topology transits into non-Abelian eigenvalue topology under simultaneously symmetry breaking. We further show that in PT symmetric systems, Chern numbers and Euler numbers can coexist. This opens the door to a variety of different topology in non-Hermitian systems.

Non-Hermitian topological features, such as the so-called Exceptional Points and concomitant Fermi arcs, can be observed in various different classes of physical systems, including strongly correlated materials at finite temperature, where non-hermiticity is provided by the finite scattering rate of quasiparticles. Usually, special requirements on the form of the Coulomb interaction are needed in order to provide a lifetime to the different quasiparticle flavors that does not just result in an uniform imaginary shift in eigenenergies. Here, we demonstrate how symmetry-broken phases, such as ferromagnetism, can cooperate with hermitian topology to generate robust and globally protected Exceptional Points. We consider a weak-topological finite-size geometry and assess the effects of interaction at finite temperature: a spontaneous symmetry-broken magnetic phase can be stabilized, in which the semimetallic surface states is ‘upgraded’ to a metallic dispersion featuring Exceptional Points, which are moreover robust against any perturbation.

At thermal equilibrium, we find that generalized susceptibilities encoding the static physical response properties of Hermitian many-electron systems possess inherent non-Hermitian (NH) matrix symmetries, emerging solely due to Fermi-Dirac statistics. This leads to the generic occurrence of exceptional points, i.e., NH spectral degeneracies, in the generalized susceptibilities of prototypical Fermi-Hubbard models, as a function of a single parameter such as chemical potential. By means of both analytic and numerical approaches, we demonstrate that these exceptional points promote correlation-induced thermodynamic instabilities, such as phase-separation occurring in the proximity of a Mott metal-to-insulator transition, to a topologically stable phenomenon.

The state of a quantum system may be steered towards a predesignated target state, employing a sequence of weak blind measurements (where the detector’s readouts are traced out). Here we analyze the steering of a two-level system using the interplay of a system Hamiltonian and weak measurements and show that any pure or mixed state can be targeted. We show that the optimization of such a steering protocol is underlain by the presence of Liouvillian exceptional points. More specifically, for high-purity target states, optimal steering implies purely relaxational dynamics marked by a second-order exceptional point, whereas for low-purity target states, it implies an oscillatory approach to the target state. The dynamical phase transition between these two regimes is characterized by a third-order exceptional point.

We show that interactions in quantum spin liquids can result in non-Hermitian phenomenology that differs qualitatively from mean-field expectations. We demonstrate this in two prominent cases through the effects of phonons and disorder on a Kitaev honeycomb model. Using analytic and numerical calculations, we show the generic appearance of exceptional points and rings depending on the symmetry of the system. Their existence is reflected in dynamical observables including the dynamic structure function measured in neutron scattering. The results point to different phenomenological features in realizable spin liquids that must be incorporated into the analysis of experimental data and also indicate that spin liquids could be generically stable to wider classes of perturbations.

Exceptional bound states represent an unique class of robust bound states protected by the de- fectiveness of non-Hermitian exceptional points. Conceptually distinct from the more well-known topological and non-Hermitian skin states, they were recently discovered as a novel source of negative entanglement entropy in the quan- tum entanglement context. Yet, arising as enig- matic negative probability eigenstates in the free- fermion propagator, they are previously thought to be elusive, existing only in non-Hermitian quantum liquids. In this talk, we generalize the existence of exceptional bound states to purely classical systems, and describe their first experi- mental observation in an electrical circuit. Surprisingly, they remain extremely robust in small local lattice networks, despite their mathematical origin in highly-local propagators. This work shows that exceptional bound states exist as a truly new class of robust bound states that are experimentally viable.

A point gap is a gap structure intrinsic to complex spectra in non-Hermitian systems. Corresponding to this unique structure, non-Hermitian systems may host distinct topological phases called point-gap topological phase that does not appear in Hermitian systems. In this talk, I will explain our current understanding of the physics of point-gap topological phases.

In my presentation, I will first speak about the topological aspects of encircling an exceptional point (EP) in a laser and the corresponding chiral state transfer [1]. Building on the insight that EPs can also be identified in the absorption spectra of anti-lasers [2], I will present experimental results on the chiral and degenerate perfect absorption at an EP [3]. Finally, I will show how not only two modes can be critically coupled, but more than a thousand – leading to a massively degenerate coherent perfect absorber for arbitrary incoming wavefronts [4]. [1] A. Schumer, Y. G. N. Liu, J. Leshin, L. Ding, Y. Alahmadi, A. U. Hassan, H. Nasari, S. Rotter, D. N. Christodoulides, P. LiKamWa, and M. Khajavikhan, Science 375, 884 (2022) [2] W. R. Sweeney, C. W. Hsu, S. Rotter, and A. D. Stone, Phys. Rev. Lett. 122, 093901 (2019) [3] S. Soleymani, Q. Zhong, M. Mokim, S. Rotter, R. El-Ganainy, and S. K. Özdemir, Nature Commun. 13, 599 (2022) [4] Y. Slobodkin, G. Weinberg, H. Hörner, K. Pichler, S. Rotter, and O. Katz, Science 377, 995 (2022)

The operator growth hypothesis (OGH) is a technical conjecture about the growth of operators (specifically their "Lanczos coefficients") under repeated action by a Liouvillian for a sufficiently generic many-body system. If OGH is obeyed, it gives bounds on the high frequency behavior of local correlation functions as well as the growth rates of various measures of chaos (like OTOCs). It is natural to ask how this hypothesis generalises to open quantum systems where the Liouvillian is replaced by a Lindbladian. In the talk I will describe the consequences of dissipation on operator growth and in particular on the Lanczos coefficients in an open quantum setting.

For many quantum mechanical applications dissipation is often regarded as an undesirable yet unavoidable consequence because it potentially degrades quantum coherences and renders the system classical. However, interactions with the environment can also be considered a fundamental resource for striking collective effects typically impossible in Hamiltonian systems. A hallmark of such collective behavior in nonequilibrium systems is the phenomenon of synchronization: in the complete absence of any time-dependent forcing from the outside, a group of oscillators adjusts their frequencies such that they spontaneously oscillate in unison. With the recent developments in quantum technology which allow one to exquisitely tailor both the system and environmental properties, synchronization has emerged in the quantum domain with various different examples ranging from nonlinear oscillators to spin-1 systems, superconducting qubits and optomechanics. However, to observe synchronization in large networks of classical or quantum systems demands both excellent control of the interactions between nodes and accurate preparation of the initial conditions due to the involved nonlinearities and dissipation. This limits its applicability for future devices. We present a potential route towards significantly enhancing the robustness of synchronized behavior in open nonlinear systems that utilizes the power of topological insulators, which exhibit an insulating bulk but conducting surface states, known as topological edge states. These edge states display a surprising immunity to a wide range of local deformations and even circumvent localization in the presence of disorder. By combining nontrivial topological lattices with nonlinear oscillators, we show that synchronized motion emerges at the lattice boundaries in the classical mean field as well as the quantum regime. Furthermore, the synchronized edge modes inherit the topological protection known from closed systems with remarkably robust dynamics against local disorder and even random initial conditions. Our work demonstrates a general advantage of topological lattices in the design of potential experiments and devices as fabrication errors and longterm degradation are circumvented in this way. This is especially important in networks where specific nodes need special protection.

Wave-packet and entanglement dynamics in a generalized Hatano-Nelson model (asymmetric hopping + on-site disorder + nearest-neighbor inter-particle interaction) is studied. In the regime of weak disorder, the eigen wave functions are extended, and corresponding eigen energies are complex under the periodic boundary. In the time evolution an initial wave packet with many eigenstates superposed tends to evolve into a single eigen state with a maximal imaginary part. This results in a non-monotonic time evolution of the entanglement entropy in the regime of intermediate disorder; in the regime, collapse of the superposition is not too fast but occurs in the time scale comparable to the growth of the entanglement entropy [1]. The effect of on-site disorder W is also non-monotonic. With increasing W, it first enhances the entanglement entropy then suppresses it. After a long enough time in the quench dynamics, the entanglement entropy in the clean and non-interacting limit obeys a logarithmic scaling [2], reminiscent of the one known (in the Hermitian case) in conformal field theory [3]. [1] T. Orito and K.-I. Imura, Phys. Rev. B 105, 024303 (2022). [2] T. Orito and K.-I. Imura, in preparation. [3] P. Calabrese, J. Cardy, J. Phys. A: Math. Theor. 42 504005 (2009).

The non-Hermitian skin effect, in which eigenstates exhibit localized behaviors drastically different from the extended Bloch waves of Hermitian systems, is among the most scrutinized dissipative phenomena. The localization of the eigenstates at a system's edge hints at non-reciprocal transport towards the latter. However, non-Hermitian Hamiltonians are obtained as an approximation that might wash out critical information on the timescales and conditions required for the non-reciprocal accumulation of bulk modes to take place. Using a symmetry-based approach, we derive a master equation that reduces, in the mean-field limit, to the non-Hermitian Hamiltonian under investigation and investigate the spin-wave dynamics. Our analysis uncovers the limitation of non-Hermitian approaches and clarifies the origin of the skin effect and the ensuing non-reciprocity in magnetic systems. Our results highlight the connection between the non-Hermitian skin effect and the classical magnetization dynamics, suggesting that our predictions can be tested in multilayered magnetic structures with interlayer Dzyaloshinskii-Moriya interactions.

In thermal equilibrium the dynamics of phase transitions is largely controlled by fluctuation-dissipation relations: On the one hand, friction suppresses fluctuations, while on the other hand the thermal noise is proportional to friction constants. Out of equilibrium, this balance dissolves and one can have situations where friction vanishes due to antidamping in the presence of a finite noise level. We study a wide class of O(N) field theories where this situation is realized at a phase transition, which we identify as a critical exceptional point. In the ordered phase, antidamping induces a continuous limit cycle rotation of the order parameter with an enhanced number of 2N−3 Goldstone modes. Close to the critical exceptional point, however, fluctuations diverge so strongly due to the suppression of friction that in dimensions d<4 they universally either destroy a preexisting static order, or give rise to a fluctuation-induced first order transition. This is demonstrated within a non-perturbative approach based on Dyson-Schwinger equations for N=2, and a generalization for arbitrary N, which can be solved exactly in the long wavelength limit. We show that in order to realize this physics it is not necessary to drive a system far out of equilibrium: Using the peculiar protection of Goldstone modes, the transition from an xy magnet to a ferrimagnet is governed by an exceptional critical point once weakly perturbed away from thermal equilibrium.

Exceptional points are ubiquitous in generic non-hermitian quantum many-body systems and imply an interesting topology of the entire spectrum of the Hamiltonian. Furthermore, interactions can have peculiar effects on exceptional points of the non-interacting system in the presence of a protecting symmetry. In the example of a system with two fermions we find that exceptional points protected by symmetry are generally stable in the presence of interactions. Furthermore, new many-body exceptional points can emerge from diagonalizable degeneracies, while it is also possible that exceptional points annihilate each other if they meet in parameter space, forming an endpoint of an "exceptional line". [1] D. J. Luitz and F. Piazza “Exceptional points and the topology of quantum many-body spectra” Phys. Rev. Research, 1, 033051 (2019) [2] R. Schäfer, J. C. Budich, D. J. Luitz “Symmetry protected exceptional points of interacting fermions” Phys. Rev. Research 4, 033181 (2022)

We explore anomalous skin effects at non-Hermitian impurities by studying their interplay with potential disorder and by exactly solving a minimal lattice model. A striking feature of the solvable single-impurity model is that the presence of anisotropic hopping terms can induce a scale-invariant accumulation of all eigenstates opposite to the bulk hopping direction, although the non-monotonic behavior is fine-tuned and further increasing such hopping weakens and eventually reverses the effect. The interplay with bulk potential disorder, however, qualitatively enriches this phenomenology leading to a robust non-monotonic localization behavior as directional hopping strengths are tuned. These phenomena persist even in the limit of an entirely Hermitian bulk with a single non-Hermitian impurity.

One of the most important practical hallmarks of topological matter is the presence of topologically protected, exponentially localised edge states at interfaces of regions characterised by unequal topological invariants. Here, we show that even when driven far from their equilibrium ground state, Chern insulators can inherit topological edge features from their parent Hamiltonian. In particular, we show that the asymptotic long-time approach of the non-equilibrium steady state, governed by a Lindblad Master equation, can exhibit edge-selective extremal damping. This phenomenon derives from edge states of non-Hermitian extensions of the parent Chern insulator Hamiltonian. The combination of (non-Hermitian) topology and dissipation hence allows to design topologically robust, spatially localised damping patterns.

Non-Hermitian (NH) lattice Hamiltonians display a unique kind of energy gap, known as point gap, and extreme sensitivity to changes of boundary conditions—-the so-called NH skin effect. Due to the NH skin effect, the separation between edge and bulk states is blurred and the bulk-boundary correspondence in its conventional form breaks down [1]. Despite considerable efforts to accommodate the NH skin effect into a modified bulk-boundary correspondence, a formulation for point-gapped spectra has remained elusive. In this contribution, I will show how to restore the bulk-boundary correspondence for the most paradigmatic class of NH lattice models, namely those with point-gapped spectra (single-band models without symmetries). This is achieved thanks to an alternative classification of NH topological phases, where the focus is shifted (i) from effective NH Hamiltonians to NH Hamiltonians obtained from the dynamics of driven-dissipative arrays of cavities, and (ii) from the eigen-decomposition to the singular value decomposition as the tool for studying their bandstructure. The target NH Hamiltonian is implemented in one-dimensional driven-dissipative cavity arrays, in which Hermiticity-breaking terms, in the form of non-reciprocal hopping amplitudes, gain and loss, are explicitly modelled via coupling to (engineered and non-engineered) reservoirs. I will show that this approach introduces extra constraints to the NH Hamiltonian and determine the following major change to the topology: topologically non-trivial Hamiltonians are only a strict subset of those with a point gap; this implies that the NH skin effect does not have a topological origin. I will then show how to reinstate the bulk-boundary correspondence in terms of the singular value decomposition, instead of the eigen-decomposition, and explain its physical significance. The NH bulk-boundary correspondence takes the following simple form: An integer value $\nu$ of the winding number defined on the complex spectrum of the system under periodic boundary conditions corresponds to $\vert \nu\vert$ exponentially small singular values associated with singular vectors that are exponentially localized at the system’s edge under open boundary conditions and vice versa; the sign of $\nu$ determines at which edge the vectors localize. Non-trivial topology manifests as directional amplification with gain exponential in system size, which is the hallmark of NH topology [2]. I will explain the physical relevance of this peculiar behaviour. More details in: M. Brunelli, C. C. Wanjura, and A. Nunnenkamp, arXiv:2207.12427 (2022). References: [1] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Rev. Mod. Phys. 93, 015005 (2021). 
[2] C. C. Wanjura, M. Brunelli, and A. Nunnenkamp, Nature Communications 11, 3149 (2020).

Exceptional points play a central role in linear non-Hermitian optics. In this talk, I will describe our recent efforts to extend the notion of exceptional points to optical systems that exhibit nonlinear interactions and will discuss some of the practical implications of the interplay between non-Hermiticity and nonlinearity.

Many aspects of non-Hermitian systems are not well described in the framework of Bloch band theory. The non-Bloch band theory, in which the concept of Brillouin zone is generalized, has been widely applied to study non-Hermitian systems in one spatial dimension. However, its generalization to higher dimensions has been challenging. Here, we introduce a formulation of non-Hermitian band theory in arbitrary spatial dimensions, which is based on a natural geometrical object known as the amoeba. This theory provides a general framework for studying non-Hermitian bands beyond one dimension. Key quantities of non-Hermitian bands, including the energy spectrum, eigenstates profiles, and the generalized Brillouin zone, can be efficiently obtained from this approach. (Reference: H.-Y. Wang, F. Song, Z. Wang, arXiv:2212.11743)

Physically, one tends to think of non-Hermitian systems in terms of gain and loss: the decay or amplification of a mode is given by the imaginary part of its energy. Here, we introduce an alternative avenue to the realm of non-Hermitian physics, which involves neither gain nor loss. Instead, complex eigenvalues emerge from the amplitudes and phase differences of waves backscattered from the boundary of insulators. We show that for any strong topological insulator in a Wigner-Dyson class, the reflected waves are characterized by a reflection matrix exhibiting the non-Hermitian skin effect. This leads to an unconventional Goos-Hänchen effect: due to non-Hermitian topology, waves undergo a lateral shift upon reflection, even at normal incidence. Going beyond systems with gain and loss vastly expands the set of experimental platforms that can access non-Hermitian physics and show signatures associated with non-Hermitian topology.

Linear response theory provides a general framework to describe the leading order effects on observables of physical systems due to small time-dependent perturbations. We perform an extension of the theory to the realm of quantum systems described by non-Hermitian Hamiltonians. It is shown that the time-dependence of the expectation of an observable in the latter case is described by a generalized Kubo formula which characterizes cases when either or both system and perturbation are non-Hermitian. We discuss two applications of the linear response theory. First we derive a finite optical conductivity for tachyons in one-dimensions. These are hypothetical particles moving faster than the speed of light. They can be simulated in optical lattices or cold attom systems as excitations moving faster than some intrinsic velocity in the experiment such as the Fermi velocity. Second we discuss the density-density response of a Hermitian Fermi gas subjected to a non-Hermitian perturbation, and show the absence of Friedel oscillations.

We consider non-Hermitian energy band theory in two-dimensional systems, and study eigenenergy braids on slices in the two-dimensional Brillouin zone. We show the consequences of reciprocity and geometric symmetry on such eigenenergy braids. The point-gap topology of the energy bands can be found from the projection of the eigenenergy braid onto the complex energy plane. We show that the conjugacy class transitions in the eigenenergy braid results in the changes in the number of bands in a complete point-gap loop. This transition occurs at exceptional points. We numerically demonstrate these concepts using two-dimensional reciprocal and nonreciprocal photonic crystals.

Low-energy excitations (magnons) of magnetically ordered systems have been demonstrated as suitable conduit for computing and information transfer at low-energy consumption [1]. This contribution presents a proposal for exploiting the features of spin dynamics at exceptional points in coupled magnonic waveguides. The system consists of two or more dipolarly or otherwise coupled magnetic stripes, both attached to a normal conductive substrate with a strong spin-orbit coupling. A charge current in the substrate acts on the adjacent magnetic parts with spin-orbit torques which result (depending on the magnetization direction in the waveguides) in damping or antidamping of the magnons, and hence in gain or loss in the magnon amplitudes in the individual waveguides. The gain and loss can be balanced (which yields a typical PT-symmetric configuration) and the system can be driven to the exceptional points by varying the current density in the normal metal. Using linearized analytical models for the spin dynamic supported with full-fledged numerical simulations in the non-linear region, it is shown [2] how the magnon propagation at the exceptional points can be steered, amplified, and exploited in functional magnonic devices, including reconfigurable magnonic diodes and logic devices. [1] Q. Wang, et al. Nat. Electron. 3, 765 (2020). [2] X.-g. Wang et al. Nat. Commun. 11, 5663, (2020); Phys Rev Appl. 18, 024080, (2022); Phys. Rev. Appl. 15, 034050 (2021); New J. Phys. 24, 023031, (2022); Yuan et al. Commun. Phys. 6, 95 (2023)

A remarkable phenomenon associated with Hermitian topology is the quantum Hall effect – the quantisation of the Hall resistance in terms of a topological invariant. However, so far, such a clear observable signature of non-Hermitian topology has been lacking. In this talk, I will show that non-trivial, non-Hermitian topology is in one-to-one correspondence with the phenomenon of directional amplification [1-2] in one-dimensional bosonic systems, such as cavity arrays. Directional amplification allows to selectively amplify signals depending on their propagation direction and has attracted much attention as key resource for applications, such as quantum information processing. Specifically, we have shown that there is a one-to-one correspondence between a non-zero invariant of non-Hermitian topology defined on the spectrum of the dynamic matrix and regimes of directional amplification, in which the end-to-end gain grows exponentially with the number of cavities. Our framework allows us to analytically compute the scattering matrix, the gain and reverse gain which explicitly depend on the value of the topological invariant. Parameter regimes achieving directional amplification can be elegantly obtained from a topological ‘phase diagram’. Furthermore, topological, directional amplification is robust against disorder [2] with the amount of tolerated disorder given by the separation between the complex spectrum and the origin (the size of the point gap). Together with our collaborators at AMOLF (Amsterdam), we experimentally implemented the bosonic version of the Majorana-Kitaev chain (BKC) introduced in [3] which displays phases of non-trivial non-Hermitian topology associated with directional amplification that can be understood with our framework [1]. We experimentally demonstrate the distinct behaviour of the BKC in the trivial and the non-trivial phase as well as at the phase transition. Furthermore, it was proposed that non-trivial topology can help us build better sensors [4,5] since they can achieve a sensitivity that scales exponentially in system size. Here, we confirm this in our experiment and measure the exponential sensitivity of a sensor based on the BKC. Our work opens up new routes for the design of both phase-preserving and phase-sensitive multimode robust directional amplifiers and sensors based on non-Hermitian topology that can be integrated in scalable platforms such as superconducting circuits, optomechanical systems, photonic crystals and nanocavity arrays. [1] C.C. Wanjura, M. Brunelli, A. Nunnenkamp, Topological framework for directional amplification in driven-dissipative cavity arrays. Nat Commun 11, 3149 (2020). [2] C.C. Wanjura, J.J. Slim, J. del Pino, M. Brunelli, E. Verhagen, A. Nunnenkamp. Quadrature nonreciprocity: unidirectional bosonic transmission without breaking time-reversal symmetry. arXiv:2207.08523 (2022). [3] A. McDonald, T. Pereg-Barnea, A. A. Clerk. Phase-Dependent Chiral Transport and Effective Non-Hermitian Dynamics in a Bosonic Kitaev-Majorana Chain. Phys Rev X 8, 041031 (2018). [4] A. McDonald, A.A. Clerk. Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics. Nat Commun 11, 5382 (2020). [5] J.C. Budich and E.J. Bergholtz. Non-Hermitian Topological Sensors. Phys Rev Lett 125, 180403 (2020).

Strong coupling of excitons (electron-hole pairs) in semiconductors and photons in a cavity leads to the formation of exciton-polaritons, which have both light and matter properties. Endowed with hybrid properties, these particles can form quantum fluids at room temperature, a useful platform for studying quantum effects on a macroscopic scale at ambient conditions. Here, we present how losses, which are inherent in this system, can lead to strong effects on the energy dispersion in momentum space without spatial modulations. Firstly, we demonstrate the existence of exceptional points when anisotropic losses are introduced using excitons in an inorganic perovskite crystal. We also measured the pseudospin and spectral windings close to the exceptional points and clearly distinguish between them. Secondly, we also provide strong evidence of dissipative light-matter coupling in this system using excitons in an atomically thin semiconductor. This dissipative coupling leads to an inverted dispersion, hence, negative-mass particles which we confirmed to propagate opposite to their momenta. When combined with spatial modulations, e.g. lattice potentials, we believe that the mentioned effects can lead to the formation of new states of quantum matter.

Reflection matrices describing waves reflected from the boundaries of topological insulators exhibit non-Hermitian topological signatures. Based on this, we propose ways to realize non-Hermitian topology in the conductance matrix of topological multi-terminal devices. Our work is based on the insight that, in the limit of maximal non-reciprocity, the Hamiltonian for the simplest topological non-Hermitian system – the Hatano-Nelson chain – effectively describes a one-dimensional, unidirectionally propagating mode. This is analogous to the unidirectional boundary mode of a fully Hermitian topological insulator: the quantum Hall system. We show that the multi-terminal conductance matrix of this system exhibits a topologically protected non-Hermitian skin effect. Our approach can be used to construct and study models that go beyond the Hatano-Nelson model, like the non-Hermitian SSH chain.