Search

We investigate the low-energy physics of non-Hermitian quantum spin models with $PT$-symmetry. To this end we consider the one-dimensional Ising chain and the two-dimensional toric code in a non-Hermitian staggered field. For both systems dual descriptions in terms of non-Hermitian staggered Ising interactions in a conventional transverse field exist. We perform high-order series expansions about the high- and low-field limit for both systems to determine the ground-state energy per site and the one-particle gap. The one-dimensional non-Hermitian Ising chain is known to be exactly solvable. Its ground-state phase diagram consists of second-order quantum phase transitions, which can be characterized by logarithmic singularities of the second derivative of the ground-state energy and, in the symmetry-broken phase, the gap closing of the low-field gap. In contrast, the gap closing from the high-field phase is not accessible perturbatively due to the complex energy and the occurrence of exceptional lines in the high-field gap expression. For the two-dimensional toric code in a non-Hermitian staggered field we study the quantum robustness of the topologically ordered phase by the gap closing of the low-field gap. We find that the well-known second-order quantum phase transition of the toric code in a uniform field extends into a large portion of the non-Hermitian parameter space. However, the series expansions become unreliable for a dominant anti-Hermitian field. Interestingly, the analysis of the high-field gap reveals the potential presence of an intermediate region., Comment: 15 pages, 9 figures

We introduce a continuous one-dimensional non-Hermitian matrix gauge potential and study its effect on dynamics of a two-component field. The model is emulated by a system of evanescently coupled nonlinear waveguides with distributed gain and losses. The considered gauge fields lead to a variety of unusual physical phenomena in both linear and nonlinear regimes. In the linear regime, the field may undergo superexponential convective amplification. A total power of an input Gaussian beam may exhibit a finite-distance blowup, which manifests itself in absolute delocalization of the beam at a finite propagation distance, where the amplitude of the field remains finite. The defocusing Kerr nonlinearity initially enhances superexponential amplification, while at larger distances it suppresses the growth of the total power. The focusing nonlinearity at small distances slows down the power growth and eventually leads to the development of the modulational instability. Complex periodic gauge fields lead to the formation of families of stable fundamental and dipole solitons., Comment: accepted for Phys. Rev. A (Letters)

We investigate the effects of non-Hermitian driving potential on quantum coherence in a bipartite system. The results show that the dynamical localization will revive after being destroyed by the Hermitian interaction, which provides evidence of the restoration of quantum coherence by a non-Hermitian driving potential. Besides, the entanglement between the two subsystems also decays with the boosting of non-Hermitian driving strength, which gives more evidence that non-Hermitian driving potential will protect quantum coherence. The physics behind this phenomenon is the domination of the quasieigenstate with maximum imaginary value of the quasieigenvalue over the dynamics of the non-Hermitian system. Our discovery establishes a restoration mechanism of quantum coherence in interacting and dissipative quantum systems which is highly expectable in updated experiments from many-body physics to quantum information.

Although it originated in quantum physics, the concept of $n\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\ensuremath{-}H\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}y$ (particularly involving a Hamiltonian with $b\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}d$ gain and loss, and thus $P\phantom{\rule{0}{0ex}}T$ symmetry and an exceptional point in frequency space) can also play a key role in classical systems, including those in acoustics. By incorporating suitably engineered gain or loss media, both cavity and scattering acoustic systems can produce a series of intriguing wave phenomena, with prospects for application in the design of innovative functional devices. This review aims to introduce the pedagogical models and recent achievements in this field, in the hope of being useful to a diverse audience.

Eigenmode coalescence imparts remarkable properties to non-Hermitian time evolution, culminating in a purely non-Hermitian spectral degeneracy known as an exceptional point (EP). Here, we revisit time evolution around EPs, looking at both static and periodically modulated non-Hermitian Hamiltonians. We connect a Möbius group classification of two-level non-Hermitian Hamiltonians with the theory of Hill's equation, which unlocks a large class of analytical solutions. Together with the classification, this allows us to investigate the impact of the shape of the temporal modulation on the long-term dynamics of the system. In particular, we find that EP encircling does not predict the temporal class, and that the elaborate interplay between non-Hermitian and modulation instabilities is better understood through the lens of parametric resonance. Finally, we identify specific signatures of complex parametric resonance by exhibiting stability diagrams with features that cannot occur in traditional parametric resonance.

Exceptional points (EPs), at which both eigenvalues and eigenvectors coalesce, are ubiquitous and unique features of non-Hermitian systems. Second-order EPs are by far the most studied due to their abundance, requiring only the tuning of two real parameters, which is less than the three parameters needed to generically find ordinary Hermitian eigenvalue degeneracies. Higher-order EPs generically require more fine-tuning, and are thus assumed to play a much less prominent role. Here, however, we illuminate how physically relevant symmetries make higher-order EPs dramatically more abundant and conceptually richer. More saliently, third-order EPs generically require only two real tuning parameters in the presence of either a parity-time (PT) symmetry or a generalized chiral symmetry. Remarkably, we find that these different symmetries yield topologically distinct types of EPs. We illustrate our findings in simple models, and show how third-order EPs with a generic $\sim k^{1/3}$ dispersion are protected by PT symmetry, while third-order EPs with a $\sim k^{1/2}$ dispersion are protected by the chiral symmetry emerging in non-Hermitian Lieb lattice models. More generally, we identify stable, weak, and fragile aspects of symmetry-protected higher-order EPs, and tease out their concomitant phenomenology., minor typos fixed

We demonstrate that crystal defects can act as a probe of intrinsic non-Hermitian topology. In particular, in point-gapped systems with periodic boundary conditions, a pair of dislocations may induce a non-Hermitian skin effect, where an extensive number of Hamiltonian eigenstates localize at only one of the two dislocations. An example of such a phase are two-dimensional systems exhibiting weak non-Hermitian topology, which are adiabatically related to a decoupled stack of Hatano-Nelson chains. Moreover, we show that strong two-dimensional point-gap topology may also result in a dislocation response, even when there is no skin effect present with open boundary conditions. For both cases, we directly relate their bulk topology to a stable dislocation non-Hermitian skin effect. Finally, and in stark contrast to the Hermitian case, we find that gapless non-Hermitian systems hosting bulk exceptional points also give rise to a well-localized dislocation response., 6 pages, 4 figures, supplement included, accepted manuscript

Non-Hermitian systems show a non-Hermitian skin effect, where the bulk states are localized at a boundary of the systems with open boundary conditions. In this paper, we study the dependence of the localization length of the eigenstates on a system size in a specific non-Hermitian model with a critical non-Hermitian skin effect, where the energy spectrum undergoes discontinuous transition in the thermodynamic limit. We analytically show that the eigenstates exhibit remarkable localization, known as scale-free localization, where the localization length is proportional to a system size. Our result gives theoretical support for the scale-free localization, which has been proposed only numerically in previous works.

Recently, the interplay between the non-Hermitian skin effect (NHSE) and topological phases has attracted a lot of attention. When the NHSE occurs in a topological system, the conventional bulk-boundary correspondence (BBC) based on the Bloch band theory is destroyed. In this work, we investigate a non-Hermitian nodal-ring semimetal with the $s$-wave superconducting (SC) term. We find that the SC term introduces the intrinsic coupling between the different general Brillouin zones (GBZs) of the particle and hole subbands in our model, which drives the total GBZ to the BZ. Furthermore, when the SC term is finite, we show a possibility that the GBZ is the same as the BZ. We also find that the Bloch winding number of the quasi-one-dimensional model can predict the number of Majorana zero states faithfully in this case, which implies that the conventional BBC is reconstructed in our model. The localization strengths of the Majorana zero states are also calculated to show the reconstruction of conventional BBC. The finite-size effect, related experimental regimes in the electric circuit, and the fragility of this reconstructed conventional BBC are also discussed in this work.

The simple update (SU) and full update (FU) are the two paradigmatic time evolution algorithms for a tensor network known as the infinite projected entangled pair state (iPEPS). They differ by an error measure that is either, respectively, local or takes into account full infinite tensor environment. In this paper we test an intermediate neighborhood tensor update (NTU) accounting for the nearest neighbor environment. This small environment can be contracted exactly in a parallelizable way. It provides an error measure that is Hermitian and non-negative down to machine precision. In the 2D quantum Ising model NTU is shown to yield stable unitary time evolution following a sudden quench. It also yields accurate thermal states despite correlation lengths that reach up to 20 lattice sites. The latter simulations were performed with a manifestly Hermitian purification of a thermal state. Both were performed with reduced tensors that do not include physical (and ancilla) indices. This modification naturally leads to two other schemes: a local SVD update (SVDU) and a full tensor update (FTU) being a variant of FU., Comment: 11 pages, 12 figures

Non-Hermitian skin effect exhibits the collapse of the extended bulk modes into the extensive number of localized boundary states in open boundary conditions. Here we demonstrate the disorder-driven phase transition of the trivial non-Hermitian system to the higher-order non-Hermitian skin effect phase. In contrast to the clean systems, the disorder-induced boundary modes form an arc in the complex energy plane, which is the manifestation of the disorder-driven dynamical phase transition. At the phase transition, the localized corner modes and bulk modes characterized by trivial Hamiltonian coexist within the single-band but are separated in the complex energy plane. This behavior is analogous to the mobility edge phenomena in the disordered Hermitian systems. Using effective medium theory and numerical diagonalizations, we provide a systematic characterization of the disorder-driven phase transitions., 4+7 pages

The effects of gain and loss on the band structures of a bulk topological dielectric photonic crystal (PC) with $C_{6v}$ symmetry and the PC-air-PC interface are studied based on first-principle calculation. To illustrate the importance of parity-time (PT) symmetry, three systems are considered, namely the PT-symmetric, PT-asymmetric, and lossy systems. We find that the system with gain and loss distributed in a PT symmetric manner exhibits a phase transition from a PT exact phase to a PT broken phase as the strength of the gain and loss increases, while for the PT-asymmetric and lossy systems, no such phase transition occurs. Furthermore, based on the Wilson loop calculation, the topology of the PT-symmetric system in the PT exact phase is demonstrated to keep unchanged as the Hermitian system. At last, different kinds of edge states in Hermitian systems under the influences of gain and loss are studied and we find that while the eigenfrequencies of nontrivial edge states become complex conjugate pairs, they keep real for the trivial defect states., 6 pages, 6 figures

The quantum speed limit is important in determining the minimum evolution time of a quantum system, and thus is essential for quantum community. In this Letter, we derive a novel unified quantum speed limit bound for Hermitian and non-Hermitian quantum systems. The bound is quantified by the changing rate of phase of the quantum system, which represents the transmission mode of the quantum states over their evolution. The bound leads to further insights beyond the previous bounds on concrete evolution modes of the quantum system, such as horizontal or parallel transition or horizontal joining of the two quantum states in Hilbert space. The bound is linked to the feasibility of the evolutions of the state vectors, and provides a tighter upper bound. In addition, the generalized Margolus-Levitin bound is discussed.

A striking feature of non-Hermitian tight-binding Hamiltonians is the high sensitivity of both spectrum and eigenstates to boundary conditions. Indeed, if the spectrum under periodic boundary conditions is point gapped, by opening the lattice the non-Hermitian skin effect will necessarily occur. Finding the exact skin eigenstates may be demanding in general, and many methods in the literature are based on ansatzes and on recurrence equations for the eigenstates' components. Here we devise a general procedure based on the Green's function method to calculate the eigenstates of non-Hermitian tight-binding Hamiltonians under open boundary conditions. We apply it to the Hatano-Nelson and non-Hermitian SSH models and finally we contrast the edge states localization with that of bulk states., Comment: 8 pages, 4 figures

Time-periodic driving fields could endow a system with peculiar topological and transport features. In this work, we find dynamically controlled localization transitions and mobility edges in non-Hermitian quasicrystals via shaking the lattice periodically. The driving force dresses the hopping amplitudes between lattice sites, yielding alternate transitions between localized, mobility edge and extended non-Hermitian quasicrystalline phases. We apply our Floquet engineering approach to five representative models of non-Hermitian quasicrystals, obtain the conditions of photon-assisted localization transitions and mobility edges, and find the expressions of Lyapunov exponents for some models. We further introduce topological winding numbers of Floquet quasienergies to distinguish non-Hermitian quasicrystalline phases with different localization nature. Our discovery thus extend the study of quasicrystals to non-Hermitian Floquet systems, and provide an efficient way of modulating the topological and transport properties of these unique phases., 17 pages, 7 figures, 6 tables, version accepted by Phys. Rev. Res

Non-Hermitian topological phases exhibit a number of exotic features that have no Hermitian counterparts, including the skin effect and breakdown of the conventional bulk-boundary correspondence. Here, we implement the non-Hermitian Su-Schrieffer-Heeger (SSH) Hamiltonian, which is a prototypical model for studying non-Hermitian topological phases, with a solid-state quantum simulator consisting of an electron spin and a $^{13}$C nuclear spin in a nitrogen-vacancy (NV) center in a diamond. By employing a dilation method, we realize the desired non-unitary dynamics for the electron spin and map out its spin texture in the momentum space, from which the corresponding topological invariant can be obtained directly. Our result paves the way for further exploiting and understanding the intriguing properties of non-Hermitian topological phases with solid-state spins or other quantum simulation platforms., 10 pages, 6 figures

We investigate the properties of two anti-parity-time (anti-PT )-symmetric four-site scattering centers. The anti-PT -symmetric scattering center may have imaginary couplings, real couplings, and real on-site potentials. The only difference between the two scattering centers is the coupling between two central sites of the scattering center, which plays a crucial role in determining the parity of anti-PT symmetry and significantly affects the scattering properties. For the imaginary coupling, the even-parity anti-PT -symmetric scattering center possesses nonunitary scattering and the difference between the reflection and transmission is unity; for the real coupling, the odd-parity anti-PT -symmetric scattering center possesses unitary scattering and the sum of the reflection and transmission is unity. The coupling-induced different scattering behaviors are verified in the numerical simulations. Our findings reveal that a significant difference in the dynamics can be caused by a slight difference between two similar anti-PT -symmetric non-Hermitian scattering centers., Comment: 8 pages, 6 figures

We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied. The problem is mapped to a Markovian process and proved that there are large spectral equivalence classes -- different configurations have the same spectrum. Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified. Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity. In particular, non-Hermiticity can cause the rate to be either faster, or, even more interestingly, slower than predicted by the matrix eigenvalue gap. This is caused by an exponential in system size explosion of expansion coefficient sizes resulting in a 'phantom' eigenvalue, and is due to non-orthogonality of non-Hermitian eigenvectors. We numerically demonstrate that the phenomenon occurs also in random circuits with non-optimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance also in other non-Hermitian settings, including correlations., 22 pages; v2: 27 pages, new data on generic gates, Haar random gates, and on fluctuations and non-random systems

In the context of dynamical control, non-Hermitian systems offer unique opportunities and challenges. In the simplest case of two interacting levels with gain and/or loss, the existence of an exceptional point (EP) of degeneracy in the chiral complex eigenvalue landscape fundamentally changes how the system responds to adiabatic changes in the coupling $\ensuremath{\gamma}$ and energy separation $\ensuremath{\delta}$ of the bare levels. In particular, previous studies have shown that selective population transfer can be achieved by adiabatically steering the system around closed paths that encircle the EP with the proper helicity in the $\ensuremath{\gamma}$ vs $\ensuremath{\delta}$ parameter space. Here we show that efficient state-selective population transfer can, in some cases, be achieved even with control loops that do not enclose the EP.

Recently, topological phases in non-Hermitian systems have attracted much attention because non-Hermiticity sometimes gives rise to unique phases with no Hermitian counterparts. Non-Hermitian Bloch Hamiltonians can always be mapped to doubled Hermitianized Hamiltonians with chiral symmetry, which enables us to utilize the existing framework for Hermitian systems into the classification of non-Hermitian topological phases. While this strategy succeeded in the topological classification of non-Hermitian Bloch Hamiltonians in the presence of internal symmetries, the generalization of symmetry indicators -- a way to efficiently diagnose topological phases -- to non-Hermitian systems is still elusive. In this work, we study a theory of symmetry indicators for non-Hermitian systems. We define space group symmetries of non-Hermitian Bloch Hamiltonians as ones of the doubled Hermitianized Hamiltonians. Consequently, symmetry indicator groups for chiral symmetric Hermitian systems are equivalent to those for non-Hermitian systems. Based on this equivalence, we list symmetry indicator groups for non-Hermitian systems in the presence of space group symmetries. We also discuss the physical implications of symmetry indicators for some symmetry classes. Furthermore, explicit formulas of symmetry indicators for spinful electronic systems are included in appendices., 12+25 pages

Dispersionless bands -- flatbands -- have been actively studied thanks to their interesting properties and sensitivity to perturbations, which makes them natural candidates for exotic states. In parallel non-Hermitian systems have attracted much attention in the recent years as a simplified description of open system with gain or loss motivated by potential applications. In particular, non-Hermitian system with dispersionless energy bands in their spectrum have been studied theoretically and experimentally. Flatbands require in general fine-tuning of Hamiltonian or protection by a symmetry. A number of methods was introduced to construct non-Hermitian flatbands relying either on a presence of a symmetry, or specific frustrated geometries, often inspired by Hermitian models. We discuss a systematic method of construction of non-Hermitian flatbands using 1D two band tight-binding networks as an example, extending the methods used to construct systematically Hermitian flatbands. We show that the non-Hermitian case admits fine-tuned, non-symmetry protected flatbands and provides more types of flatbands than the Hermitian case., Comment: 14 pagers, 4 figures

We propose, design, and experimentally test a non-Hermitian acoustic superlattice that acts as a tunable precise filter. The superlattice is composed of two concatenated sublattices. The first sublattice is Hermitian, while the other can be adjusted to be Hermitian or non-Hermitian. The existence of non-Hermiticity, in terms of an induced loss in the second sublattice, results in the generation of absorption resonances that appear in the reflected spectrum. This provides us with a powerful knob to absorb or reflect several frequencies at will with high accuracy. The number of filtered frequencies can be controlled by designing the resonances in the first sublattice. Our proposed tunable acoustic filter can be extended to higher-frequency ranges, such as ultrasound, and other areas, such as photonics.

In this paper, the interplay of the non-Herimiticity and the cascade of delocalization transition in a quasiperiodic chain are studied. The study is applied in the non-Hermitian interpolating Aubry-Andr\'e-Fibonacci (IAAF) model, which combines the non-Hermitian Aubry-Andr\'e (AA) model and the non-Hermitian Fibonacci model through a varying parameter, and the non-Hermiticity in this model is introduced by nonreciprocal hopping. In the non-Hermitian AA limit, the system undergoes a delocalization transition by tuning the potential strength. At the critical point, the spatial distribution of the critical state shows a self-similar structure with the relative distance between the peaks being the Fibonacci sequence, and the finite-size scaling of the inverse participation ratios $(\mathrm{IPRs})$ of the critical ground state with lattice size $L$ shows that ${\mathrm{IPR}}_{g}\ensuremath{\propto}{L}^{\ensuremath{-}0.1189}$. In the non-Hermitian Fibonacci limit, we find that the system is always in the extended phase. Along the continuous deformation from the non-Hermitian AA model into the non-Hermitian Fibonacci model in the IAAF model, the cascade of the delocalization transition is found, but only a few plateaux appear. Moreover, the self-similar structure of spatial distribution for the critical modes along the cascade transition is also found. In addition, we find that the delocalization transition and the real-complex transition for the excited states happen at almost the same parameter. Our results show that the non-Hermiticity provides an additional knob to control the cascade of the delocalization transition besides the on-site potential.

Line waves are recently discovered wave entities that are localized along two directions and therefore can be viewed as the one-dimensional counterpart of surface waves. These waves can be supported at discontinuities of the surface reactance and/or resistance of low-dimensional materials such as metasurfaces or graphene. Here, a broader class of non-Hermitian surface-impedance junctions is studied that can support coupled line waves and that allows the investigation of different one-dimensional waveguiding mechanisms in a unified framework. It is theoretically demonstrated that, under parity--time-symmetry conditions, exceptional points can occur in a truly flat-optics scenario, hence endowing these waveguiding systems with the attractive features of both line-wave and exceptional-point physics and shedding further light on the phase transitions existing in these systems. It is also shown that the required surface-impedance parameters are compatible with those attainable with typical models of photoexcited graphene metasurfaces at terahertz frequencies. Besides providing additional understanding in the physics of line waves, which is still in its infancy, these results pave the way toward intriguing developments in the largely unexplored field of non-Hermitian flat optics, with possible applications ranging from sensing to lasing and on-chip optical signal processing.

The introduction of non-Hermiticity has greatly enriched the research field of traditional condensed matter physics, and eventually led to a series of discoveries of exotic phenomena. We investigate the effect of non-Hermitian imaginary hoppings on the attractive Hubbard model. The exact bound-pair solution shows that the electron-electron correlation suppresses the non-Hermiticity, resulting in off-diagonal long-range order (ODLRO) ground state. In a large negative $U $ limit, the ODLRO ground state corresponds to $\eta $-spin ferromagnetic states. We also study the system with mixed hopping configuration. The numerical result indicates the existence of the transition from normal to $\eta $-pairing ground states by increasing the imaginary hopping strength. Our results provide a promising approach for the non-Hermitian strongly correlated system., Comment: 10 pages, 4 figures

We theoretically study the role of the exceptional point (EP) in a non-Hermitian cavity magnomechanical (CMM) system, with a tunable dissipative magnon-photon coupling. We find that the EP emerging in such a system can radically change the properties of photons and phonons. As a result, flexible on-off optical transmission, coherent switching of slow and fast lights, and enhanced mechanical cooling deep into the ground state can be achievable by approaching the EP. Detailed comparisons between this system and its Hermitian counterpart are given. Our results show that EP-assisted CMM devices can serve as tools for applications in optical communications and electromechanical switching or sensing.

This paper shows that non-Hermiticity by loss can induce topological phase transition of a Floquet system .

Recently, topological phases in non-Hermitian systems have attracted great attention. In most of these works, disorders are added to the chemical potential or hopping term and the topological transition driven by disordered loss and gain has not been found. In this paper, we investigate the one-dimensional dimerized Kitaev superconductor chain with balanced loss and gain. In the clean limit, the phase diagram is given by calculating the Bloch winding number and real-space energy spectrum. Furthermore, by calculating the disorder-averaged real-space winding number and disorder-averaged real-space energy spectrum, we identify a topological transition driven by the disordered balanced loss and gain. The disorder-averaged inverse participate ratio of each eigenstate, mean inverse participate ratio, and density distribution of the edge mode are also calculated to support the existence of the topological Anderson phase. Finally, we compare our model with the Hermitian disordered Kitaev chain and discuss the possible experimental scheme of our model. Our work offers a minimal start point to explore the interplay between the dissipation and Majorana zero modes in the topological superconductor.

We introduce a powerful analytic method to study the statistics of the number $\mathcal{N}_{\textbf{A}}(\gamma)$ of eigenvalues inside any contour $\gamma \in \mathbb{C}$ for infinitely large non-Hermitian random matrices ${\textbf A}$. Our generic approach can be applied to different random matrix ensembles, even when the analytic expression for the joint distribution of eigenvalues is not known. We illustrate the method on the adjacency matrices of weighted random graphs with asymmetric couplings, for which standard random-matrix tools are inapplicable. The main outcome is an effective theory that determines the cumulant generating function of $\mathcal{N}_{\textbf{A}}$ via a path integral along $\gamma$, with the path probability distribution following from the solution of a self-consistent equation. We derive the expressions for the mean and the variance of $\mathcal{N}_{\textbf{A}}$ as well as for the rate function governing rare fluctuations of ${\mathcal{N}}_{\textbf{A}}{(\gamma)}$. All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement., Comment: 6 pages, 2 figures. SI as ancillary file

In this study, the level statistics of the three-dimensional tight-binding model with non-Hermitian disorder, introduced by Tzortzakakis, Makris, and Economou (TME), and its variants were analyzed. The shift-and-invert Arnoldi method was used to analyze the eigenvalues of the TME model up to a $48\ifmmode\times\else\texttimes\fi{}48\ifmmode\times\else\texttimes\fi{}48$ cubic lattice, which was larger than those used in previous studies. The results showed that the magnitude of critical disorders in the isotropic TME model was larger than that reported in previous numerical studies for smaller systems. The localization-delocalization transition in an anisotropic model was also investigated. The computational method adopted in this study was shown to be quite effective for large-scale analysis of the calculation of all eigenvalues of sparse non-Hermitian matrices.

We employ electric circuit networks to study topological states of matter in non-Hermitian systems enriched by parity-time symmetry PT and chiral symmetry anti-PT (APT). The topological structure manifests itself in the complex admittance bands which yields excellent measurability and signal to noise ratio. We analyze the impact of PT-symmetric gain and loss on localized edge and defect states in a non-Hermitian Su-Schrieffer-Heeger (SSH) circuit. We realize all three symmetry phases of the system, including the APT-symmetric regime that occurs at large gain and loss. We measure the admittance spectrum and eigenstates for arbitrary boundary conditions, which allows us to resolve not only topological edge states, but also a novel PT-symmetric Z_{2} invariant of the bulk. We discover the distinct properties of topological edge states and defect states in the phase diagram. In the regime that is not PT symmetric, the topological defect state disappears and only reemerges when APT symmetry is reached, while the topological edge states always prevail and only experience a shift in eigenvalue. Our findings unveil a future route for topological defect engineering and tuning in non-Hermitian systems of arbitrary dimension.

We study the breaking of reciprocity in non-Hermitian coupled photonic waveguides by the simultaneous action of the nonlinear effect of saturable absorption, and the presence of exceptional points. The nonlinear response of such a system is studied in depth, giving physical insight into the operational principles and fundamental limitations, while producing analytical and semi-analytical results for the transmission and nonreciprocal intensity range. We numerically validate the theoretical results and assess important performance metrics for the nonreciprocal operation of such a device. This novel class of passive non-resonant devices can offer on-chip broadband nonreciprocal operation, making them potentially very important for high-speed applications., 10 pages, 6 figures, submitted to Physical Review A on 15-Feb-2021

We propose an ultracold-atom setting where a fermionic superfluidity with attractive s-wave interaction is uploaded in a non-Hermitian Lieb optical lattice. The existence of a real-energy flat band solution is revealed. We show that the interplay between the skin effect and flat-band localization leads to exotic localization properties. We develop a multiband mean-field description of this system and use both order parameters and superfluid weight to describe the phase transition. A relation between the superfluid weight and non-Hermitian quantum metric of the quantum states manifold is built. We find non-monotone criticality depending on the non-Hermiticity, and the non-reciprocity prominently enhances the phase coherence of the pairing field, suggesting ubiquitous critical behavior of the non-Hermitian fermionic superfluidity., 10 pages, 4 figures

Non-Hermitian dynamics has been widely studied to enhance the precision of quantum sensing; and non-reciprocity can be a powerful resource for non-Hermitian quantum sensing, as non-reciprocity allows to arbitrarily exceed the fundamental bound on the measurement rate of any reciprocal sensors. Here we establish fundamental limits on signal-to-noise ratio for reciprocal and non-reciprocal non-Hermitian quantum sensing. In particular, for two-mode linear systems with two coherent drives, an approximately attainable uniform bound on the best possible measurement rate per photon is derived for both reciprocal and non-reciprocal sensors. This bound is only related to the coupling coefficients and, in principle, can be made arbitrarily large. Our results thus demonstrate that a conventional reciprocal sensor with two drives can simulate any non-reciprocal sensor. This work also demonstrates a clear signature on how the excitation signals affect the signal-to-noise ratio in non-Hermitian quantum sensing., 14 pages, 3 figures

In recent research, it has been shown that non-Hermitian systems exhibit sensitivity to boundaries, and it is caused by the non-Hermitian skin effect. In this work, we construct the non-Bloch band theory in bosonic Bogoliubov--de Gennes (BdG) systems. From our theory, we can calculate the generalized Brillouin zone and the energy spectrum in such systems with open boundary conditions in the thermodynamic limit, and we can thus discuss its non-Hermitian nature, despite Hermiticity of an original Hamiltonian. In fact, we find that the bosonic Kitaev-Majorana chain exhibits rich aspects of the non-Hermitian skin effect, such as instability against infinitesimal perturbations and reentrant behavior, in terms of the non-Bloch band theory. This result indicates that our theory is powerful tool for studying non-Hermitian nature in bosonic BdG systems., 9 pages, 3 figures

Non-Hermitian Hamiltonians are relevant to describe the features of a broad class of physical phenomena, ranging from photonics and atomic and molecular systems to nuclear physics and mesoscopic electronic systems. An important question relies on the understanding of the influence of curved background on the static and dynamical properties of non-Hermitian systems. In this work, we study the interplay of geometry and non-Hermitian dynamics by unveiling the existence of curvature-dependent non-Hermitian phase transitions. We investigate a prototypical model of Dirac fermions on a sphere with an imaginary mass term. This exactly-solvable model admits an infinite set of curvature-dependent pseudo-Landau levels. We characterize these phases by computing an order parameter given by the pseudo-magnetization and, independently, the non-Hermitian fidelity susceptibility. Finally, we probe the non-Hermitian phase transitions by computing the (generalized) Loschmidt echo and the dynamical fidelity after a quantum quench of the imaginary mass and find singularities in correspondence of exceptional radii of the sphere., Comment: 5+5 pages, 6 figures

We proposed a framework for the topological characterization of non-Hermitian band structures. Different from previous $K$-theoretical approaches, our approach is homotopical, which enables us to see more topological invariants. Specifically, we considered the classification of non-Hermitian systems with separable band structures. We found that the whole classification set is decomposed into several sectors, based on the braiding of energy levels. Each sector can be further classified based on the topology of eigenstates (wave functions). Due to the interplay between energy levels braiding and eigenstates topology, we found some torsion invariants, which only appear in the non-Hermitian world via homotopical approach. We further proved that these new topological invariants are unstable, in the sense that adding more bands will trivialize these invariants.

Topological invariants play an important role in characterizing topological phases. However, the topological invariants of Hermitian systems usually fail to characterize non-Hermitian topological systems due to non-Hermiticity. In this work, we generalize the Majorana polarization, which is initially defined to describe Hermitian topological superconductors, as a topological edge invariant to characterize the non-Hermitian topological superconductors. The definition of the generalized Majorana polarization depends upon two inequivalent particle-hole symmetries in non-Hermitian systems. The spinless Kitaev chain model and topological superconductor model on the honeycomb lattice are considered to examine the reliability and validity of the generalized Majorana polarization. We find that the phase transitions obtained by using Majorana polarization are consistent with the commonly used complex-energy point gap descriptions, which indicates the Majorana polarization is a reliable topological invariant to characterize the topological phase transition in non-Hermitian topological superconductors. In addition, the non-Hermitian skin effect on Majorana bound states in non-Hermitian topological superconductors is also discussed.

Unlike their Hermitian counterparts, non-Hermitian (NH) systems may display an exponential sensitivity to boundary conditions and an extensive number of edge-localized states in systems with open boundaries, a phenomena dubbed the "non-Hermitian skin effect." The NH skin effect is one of the primary challenges to defining a topological theory of NH Hamiltonians, as the sensitivity to boundary conditions invalidates the traditional bulk-boundary correspondence. The NH skin effect has recently been connected to the winding number, a topological invariant unique to NH systems. In this paper, we extend the definition of the winding number to disordered NH systems by generalizing established results on disordered Hermitian topological insulators. Our real-space winding number is self-averaging, continuous as a function of the parameters in the problem, and remains quantized even in the presence of strong disorder. We verify that our real-space formula still predicts the NH skin effect, allowing for the possibility of predicting and observing the NH skin effect in strongly disordered NH systems. As an application we apply our results to predict a NH Anderson skin effect where a skin effect is developed as disorder is added to a clean system, and to explain recent results in optical funnels., Comment: 6 pages, 3 pages supplement

The problem of the time of arrival of a quantum system in a specified state is considered in the framework of the repeated measurement protocol and in particular the limit of continuous measurements is discussed. It is shown that for a particular choice of system-detector coupling, the Zeno effect is avoided and the system can be described effectively by a non-Hermitian effective Hamiltonian. As a specific example we consider the evolution of a quantum particle on a one-dimensional lattice that is subjected to position measurements at a specific site. By solving the corresponding non-Hermitian wave function evolution equation, we present analytic closed-form results on the survival probability and the first arrival time distribution. Finally we discuss the limit of vanishing lattice spacing and show that this leads to a continuum description where the particle evolves via the free Schrodinger equation with complex Robin boundary conditions at the detector site. Several interesting physical results for this dynamics are presented., Comment: 30 pages, 5 figures

We analyze a correlated system in equilibrium with special emphasis on non-Hermitian topology inducing a skin effect. The pseudospectrum, computed by the real-space dynamical mean field theory, elucidates that additional pseudoeigenstates emerge for the open boundary condition in contrast to the dependence of the density of states on the boundary condition. We further discuss how the line-gap topology, another type of non-Hermitian topology, affects the pseudospectrum. Our numerical simulation clarifies that while the damping of the quasiparticles induces the nontrivial point-gap topology, it destroys the nontrivial line-gap topology. The above two effects are also reflected in the temperature dependence of the local pseudospectral weight.

We study a one-dimensional $p$-wave superconductor subject to non-Hermitian quasiperiodic potentials. Although non-Hermiticity exists, the Majorana zero mode is still robust against the disorder perturbation. The analytic topological phase boundary is verified by calculating the energy gap closing point and the topological invariant. Furthermore, we investigate the localized properties of this model, quantitatively revealing that the topological phase transition is accompanied by the Anderson localization phase transition, and a wide critical phase emerges with amplitude increments of the non-Hermitian quasiperiodic potentials. Finally, we numerically uncover a unconventional real-complex transition of the energy spectrum, which is different from the conventional $\mathcal{PT}$ symmetric transition.

A broad family of phase transitions in the closed as well as open quantum systems is known to be mediated by a non-Hermitian degeneracy (a.k.a. exceptional point, EP) of the Hamiltonian. In the EP limit, in general, the merger of an $N-$plet of the energy eigenvalues is accompanied by a parallel (though not necessarily complete) degeneracy of eigenstates (forming an EP-asociated $K-$plet; in mathematics, $K$ is called the geometric multiplicity of the EP). In the literature, unfortunately, only the benchmark matrix models with $K=1$ can be found. In our paper the gap is filled: the EP-mediated quantum phase transitions with $K>1$ are called "clustered", and a family of benchmark models admitting such a clustering phenomenon is proposed and described. For the sake of maximal simplicity our attention is restricted to the real perturbed-harmonic-oscillator-type N by N matrix Hamiltonians which are exactly solvable and in which the perturbation is multiparametric (i.e., maximally variable) and antisymmetric (i.e., maximally non-Hermitian). A labeling (i.e., an exhaustive classification) of these models is provided by a specific partitioning of N., 26 pages, supersedes unpublished arXiv:2010.15014

Recently, the topological phase in non-Hermitian systems has been a rapidly expanding field. The iconic features of non-Hermitian systems are exceptional points at which the eigenmodes coalesce and the non-Hermitian skin effect. We study the non-Hermitian Su-Schrieffer-Heeger model with long-range nonreciprocal hopping and find the model exhibiting topologically nontrivial phases which can be characterized by the non-Bloch winding number. With specific parameter values, the skin effect can be eliminated. As long-range nonreciprocal hopping is not easy for experimental implementations, we furthermore propose a feasible electrical-circuit simulation with operational amplifiers to implement the non-Hermitian term to realize these interesting states.

I determine the intrinsic Hamiltonian of the relative motion of the constituent fermions of bosonic composites in superfluid fermion systems, assuming the effective fermion-fermion potential to be a sum of separable terms. The derivation is based on an expansion that treats composites, quasiparticles, and their interactions on the same footing. The intrinsic Hamiltonian of the composites is expressed in terms of the solution of the gap equation and of the form factors of the potential. It has the Brillouin-Wigner form, namely it is Hermitian and energy-dependent. In such a context, a solid justification is given for discarding negative energies of the composites. As a check, I rederive the dispersion law of the Bogoliubov-Anderson mode and the spectrum of the Bardasis and Schrieffer Hamiltonian. The possible occurrence of gapped modes depending on the form of the fermion-fermion interaction is discussed. The intrinsic wave functions are derived explicitly in the long wave limit, while their full determination requires numerical calculations.

Non-Hermitian systems can host topological states with novel topological invariants and bulk-edge correspondences that are distinct from conventional Hermitian systems. Here we show that two unique classes of non-Hermitian 2D topological phases, a 2$\mathbb{Z}$ non-Hermitian Chern insulator and a $\mathbb{Z}_{2}$ topological semimetal, can be realized by tuning staggered asymmetric hopping strengths in a 1D superlattice. These non-Hermitian topological phases support real edge modes due to robust $\mathcal{PT}$-symmetric-like spectra and can coexist in certain parameter regime. The proposed phases can be experimentally realized in photonic or atomic systems and may open an avenue for exploring novel classes of non-Hermitian topological phases with 1D superlattices., 8 pages; 6 figures

We study the transport and spectral properties of a non-Hermitian one-dimensional disordered lattice, the diagonal matrix elements of which are random complex variables taking both positive (loss) and negative (gain) imaginary values: Their distribution is either the usual rectangular one or a binary pair-correlated one possessing, in its Hermitian version, delocalized states, and unusual transport properties. Contrary to the Hermitian case, all states in our non-Hermitian system are localized. In addition, the eigenvalue spectrum, for the binary pair-correlated case, exhibits an unexpected intricate fractallike structure on the complex plane and with increasing non-Hermitian disorder, the eigenvalues tend to coalesce in particular small areas of the complex plane, a feature termed "eigenvalue condensation". Despite the strong Anderson localization of all eigenstates, the system appears to exhibit transport not by diffusion but by a new mechanism through sudden jumps between states located even at distant sites. This seems to be a general feature of open non-Hermitian random systems. The relation of our findings to recent experimental results is also discussed., 10 pages, 10 figures

When designing a metamaterial, to obtain excellent resonance characteristics the dissipative loss of subcomponents usually should be minimized. However, this study takes the opposite approach and harnesses the dissipation. In such a non-Hermitian metamaterial, a bound state in the continuum (BIC) with an infinite $Q$-factor is observed, and an extremely steep transition from complete extinction to nearly perfect transmission is achieved in a narrow band. By repurposing dissipation as a coupling mechanism, non-Hermitian metamaterials open avenues for microwave sensing, switching, and the realization of slow microwave light.

In the presence of Lindblad decoherence, i.e. dissipative effects in an open quantum system due to interaction with an environment, we examine the transition probabilities between the eigenstates in the two-level quantum system described by non-Hermitian Hamiltonians with the Lindblad equation, for which the parity-time-reversal (PT) symmetry is conserved. First, the density matrix formalism for PT-symmetric non-Hermitian Hamiltonian systems is developed. It is shown that the Lindblad operators $L^{}_j$ are pseudo-Hermitian, namely, $\eta L^{}_j \eta^{-1} = L^\dagger_j$ with $\eta$ being a linear and positive-definite metric, and respect the PT symmetry as well. We demonstrate that the generalized density matrix $\rho^{}_{\rm G}(t) \equiv \rho(t) \eta$, instead of the normalized density matrix $\rho^{}_{\rm N}(t) \equiv \rho(t)/{\rm tr}\left[\rho(t)\right]$, should be implemented for the calculation of the transition probabilities in accordance with the linearity requirement. Second, the density matrix formalism is used to derive the transition probabilities in general cases of PT-symmetric non-Hermitian Hamiltonians. In some concrete examples, we calculate compact analytical formulas for the transition probabilities and explore their main features with numerical illustrations. We also make a comparison between our present results and our previous ones using state vectors in the absence of Lindblad decoherence., Comment: 35 pages, 10 figures. Final version published in Phys. Rev. A