Stimulation of the fluctuation superconductivity by PT symmetry

Experimentally Realizable PT Phase Transitions in Reflectionless Quantum Scattering

Above-barrier quantum scattering with truncated real potentials V(x)=-|x|^{p} provides an experimentally accessible platform that exhibits spontaneous parity-time symmetry breaking as p is varied. The unbroken phase has reflectionless states that correspond to bound states in the continuum of the nontruncated potentials at arbitrarily high discrete real energies. In the fully broken phase there are no bound states. There is a mixed phase in which exceptional points occur at specific energies and values of p. These effects should be observable in cold-atom scattering experiments.

Vortices in 𝒫𝒯-symmetric non-Hermitian superfluid

We discuss the properties of the non-Hermitian 𝒫𝒯-symmetric two–scalar fields model. We investigate stability areas of this system and properties of vortices that emerge in the system of two interacting scalar fields. The phase diagram of the model contains stable and unstable regions depending on 𝒫𝒯-symmetry breaking, which intercross the regions of U(1)-symmetric and U(1)-broken phases in a nontrivial way. At non-zero quartic couplings, the non-Hermitian model possesses classical vortex solutions in the 𝒫𝒯-symmetric regions. We also consider a close Hermitian analog of the theory and compare the results with the non-Hermitian model.

Magnetic Suppression of Non-Hermitian Skin Effects

Skin effect, where macroscopically many bulk states are aggregated toward the system boundary, is one of the most important and distinguishing phenomena in non-Hermitian quantum systems. We discuss a new aspect of this effect whereby, despite its topological origin, applying a magnetic field can largely suppress it. Skin states are pushed back into the bulk, and the skin topological area, which we define, is sharply reduced. As seen from exact solutions of representative models, this is fundamentally rooted in the fact that the applied magnetic field restores the validity of the low-energy description that is rendered inapplicable in the presence of non-Bloch skin states. We further study this phenomenon using rational gauge fluxes, which reveals a unique irrelevance of the generalized Brillouin zone in the standard non-Bloch band theory of non-Hermitian systems.

Quantum Steering in Two- and Three-Mode ??-Symmetric Systems

We consider two PT-symmetric models, consisting of two or three single-mode cavities. In both models, the cavities are coupled to each other by linear interactions, forming a linear chain. Additionally, the first and last of such cavities interact with an environment. Since the models are PT-symmetric, they are described by non-Hermitian Hamiltonians that, for a specific range of system parameters, possess real eigenvalues. We show that in the models considered in the article, the steering generation process strongly depends on the coupling strengths and rates of the gains/losses in energy. Moreover, we find the values of parameters describing the system for which the steering appears.

Perturbation theory in the complex plane: exceptional points and where to find them

We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree-Fock approximation and Rayleigh-Schrödinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Møller-Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller-Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

We explore the eigenvalue statistics of a non-Hermitian version of the Su-Schrieffer-Heeger model, with imaginary on-site potentials and randomly distributed hopping terms. We find that owing to the structure of the Hamiltonian, eigenvalues can be purely real in a certain range of parameters, even in the absence of parity and time-reversal symmetry. As it turns out, in this case of purely real spectrum, the level statistics is that of the Gaussian orthogonal ensemble. This demonstrates a general feature which we clarify that a non-Hermitian Hamiltonian whose eigenvalues are purely real can be mapped to a Hermitian Hamiltonian which inherits the symmetries of the original Hamiltonian. When the spectrum contains imaginary eigenvalues, we show that the density of states (DOS) vanishes at the origin and diverges at the spectral edges on the imaginary axis. We show that the divergence of the DOS originates from the Dyson singularity in chiral-symmetric one-dimensional Hermitian systems and derive analytically the asymptotes of the DOS which is different from that in Hermitian systems.

Theory of Non-Hermitian Fermionic Superfluidity with a Complex-Valued Interaction

Motivated by recent experimental advances in ultracold atoms, we analyze a non-Hermitian (NH) BCS Hamiltonian with a complex-valued interaction arising from inelastic scattering between fermions. We develop a mean-field theory to obtain a NH gap equation for order parameters, which are different from the standard BCS ones due to the inequivalence of left and right eigenstates in the NH physics. We find unconventional phase transitions unique to NH systems: superfluidity shows reentrant behavior with increasing dissipation, as a consequence of nondiagonalizable exceptional points, lines, and surfaces in the quasiparticle Hamiltonian for weak attractive interactions. For strong attractive interactions, the superfluid gap never collapses but is enhanced by dissipation due to an interplay between the BCS-BEC crossover and the quantum Zeno effect. Our results lay the groundwork for studies of fermionic superfluidity subject to inelastic collisions.

Demonstration of a two-dimensional [Formula: see text]-symmetric crystal

With the discovery of [Formula: see text]-symmetric quantum mechanics, it was shown that even non-Hermitian systems may exhibit entirely real eigenvalue spectra. This finding did not only change the perception of quantum mechanics itself, it also significantly influenced the field of photonics. By appropriately designing one-dimensional distributions of gain and loss, it was possible to experimentally verify some of the hallmark features of [Formula: see text]-symmetry using electromagnetic waves. Nevertheless, an experimental platform to study the impact of [Formula: see text] -symmetry in two spatial dimensions has so far remained elusive. We break new grounds by devising a two-dimensional [Formula: see text]-symmetric system based on photonic waveguide lattices with judiciously designed refractive index landscape and alternating loss. With this system at hand, we demonstrate a non-Hermitian two-dimensional topological phase transition that is closely linked to the emergence of topological mid-gap edge states.

Optical solitons in media with focusing and defocusing saturable nonlinearity and a parity-time-symmetric external potential

We report results for solitons in models of waveguides with focusing or defocusing saturable nonlinearity and a parity-time ([Formula: see text])-symmetric complex-valued external potential of the Scarf-II type. The model applies to the nonlinear wave propagation in graded-index optical waveguides with balanced gain and loss. We find both fundamental and multipole solitons for both focusing and defocusing signs of the saturable nonlinearity in such [Formula: see text]-symmetric waveguides. The dependence of the propagation constant on the soliton's power is presented for different strengths of the nonlinearity saturation, S The stability of fundamental, dipole, tripole and quadrupole solitons is investigated by means of the linear-stability analysis and direct numerical simulations of the corresponding (1+1)-dimensional nonlinear Schrödinger-type equation. The results show that the instability of the stationary solutions can be mitigated or completely suppressed, increasing the value of SThis article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Symmetric and asymmetric solitons supported by a 𝒫𝒯-symmetric potential with saturable nonlinearity: bifurcation, stability and dynamics

The symmetry breaking bifurcation of solitons in an optical waveguide with focusing saturable nonlinearity and parity-time (𝒫𝒯)-symmetric complex-valued external potentials is investigated. As the soliton power increases, it is found that the branches of asymmetric solitons split off from the base branches of 𝒫𝒯-symmetric fundamental soliton. The bifurcation diagrams, consisting essentially of the propagation constants of optical solitons, indicate that symmetric fundamental and multipole solitons, as well as asymmetric solitons can exist. The stabilities and the dynamics characteristics of solitons are comprehensively investigated. We find the different instability scenarios of the symmetric solitons, but the symmetry breaking bifurcation is caused only by the onset of instability of the symmetric fundamental solitons. This result is further confirmed by the numerical examples with the different saturable nonlinearity parameters. In particular, we find that the soliton power and the stability of soliton at the bifurcation points are significantly changed by varying the strength of the saturable nonlinearities. These results provide additional way to control symmetry breaking bifurcations in 𝒫𝒯-symmetric optical waveguide.

Information Retrieval and Criticality in Parity-Time-Symmetric Systems

By investigating information flow between a general parity-time (PT-)symmetric non-Hermitian system and an environment, we find that the complete information retrieval from the environment can be achieved in the PT-unbroken phase, whereas no information can be retrieved in the PT-broken phase. The PT-transition point thus marks the reversible-irreversible criticality of information flow, around which many physical quantities such as the recurrence time and the distinguishability between quantum states exhibit power-law behavior. Moreover, by embedding a PT-symmetric system into a larger Hilbert space so that the entire system obeys unitary dynamics, we reveal that behind the information retrieval lies a hidden entangled partner protected by PT symmetry. Possible experimental situations are also discussed.

Parity-time-symmetric quantum critical phenomena

Synthetic non-conservative systems with parity-time (PT) symmetric gain–loss structures can exhibit unusual spontaneous symmetry breaking that accompanies spectral singularity. Recent studies on PT symmetry in optics and weakly interacting open quantum systems have revealed intriguing physical properties, yet many-body correlations still play no role. Here by extending the idea of PT symmetry to strongly correlated many-body systems, we report that a combination of spectral singularity and quantum criticality yields an exotic universality class which has no counterpart in known critical phenomena. Moreover, we find unconventional low-dimensional quantum criticality, where superfluid correlation is anomalously enhanced owing to non-monotonic renormalization group flows in a PT-symmetry-broken quantum critical phase, in stark contrast to the Berezinskii–Kosterlitz–Thouless paradigm. Our findings can be experimentally tested in ultracold atoms and predict critical phenomena beyond the Hermitian paradigm of quantum many-body physics.

Parity-time (PT) symmetry has been mainly studied in optical and weakly interacting open quantum systems without many-body correlations. Here the authors show that in a strongly correlated many-body system the interplay between correlations and PT symmetry leads to the emergence of new critical phenomena.

Parametric instability of optical non-Hermitian systems near the exceptional point

In contrast to Hermitian systems, the modes of non-Hermitian systems are generally nonorthogonal. As a result, the power of the system signal depends not only on the mode amplitudes but also on the phase shift between them. In this work, we show that it is possible to increase the mode amplitudes without increasing the power of the signal. Moreover, we demonstrate that when the system is at the exceptional point, any infinitesimally small change in the system parameters increases the mode amplitudes. As a result, the system becomes unstable with respect to such perturbation. We show such instability by using the example of two coupled waveguides in which loss prevails over gain and all modes are decaying. This phenomenon enables compensation for losses in dissipative systems and opens a wide range of applications in optics, plasmonics, and optoelectronics, in which loss is an inevitable problem and plays a crucial role.

Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance

Scattering experiments with microwave cavities were performed and the effects of broken time-reversal invariance (TRI), induced by means of a magnetized ferrite placed inside the cavity, on an isolated doublet of nearly degenerate resonances were investigated. All elements of the effective Hamiltonian of this two-level system were extracted. As a function of two experimental parameters, the doublet and the associated eigenvectors could be tuned to coalesce at a so-called exceptional point (EP). The behavior of the eigenvalues and eigenvectors when encircling the EP in parameter space was studied, including the geometric amplitude that builds up in the case of broken TRI. A one-dimensional subspace of parameters was found where the differences of the eigenvalues are either real or purely imaginary. There, the Hamiltonians were found to be PT invariant under the combined operation of parity (P) and time reversal (T) in a generalized sense. The EP is the point of transition between both regions. There a spontaneous breaking of PT occurs.

PT quantum mechanics

PT-symmetric quantum mechanics (PTQM) has become a hot area of research and investigation. Since its beginnings in 1998, there have been over 1000 published papers and more than 15 international conferences entirely devoted to this research topic. Originally, PTQM was studied at a highly mathematical level and the techniques of complex variables, asymptotics, differential equations and perturbation theory were used to understand the subtleties associated with the analytic continuation of eigenvalue problems. However, as experiments on PT-symmetric physical systems have been performed, a simple and beautiful physical picture has emerged, and a PT-symmetric system can be understood as one that has a balanced loss and gain. Furthermore, the PT phase transition can now be understood intuitively without resorting to sophisticated mathematics. Research on PTQM is following two different paths: at a fundamental level, physicists are attempting to understand the underlying mathematical structure of these theories with the long-range objective of applying the techniques of PTQM to understanding some of the outstanding problems in physics today, such as the nature of the Higgs particle, the properties of dark matter, the matter-antimatter asymmetry in the universe, neutrino oscillations and the cosmological constant; at an applied level, new kinds of PT-synthetic materials are being developed, and the PT phase transition is being observed in many physical contexts, such as lasers, optical wave guides, microwave cavities, superconducting wires and electronic circuits. The purpose of this Theme Issue is to acquaint the reader with the latest developments in PTQM. The articles in this volume are written in the style of mini-reviews and address diverse areas of the emerging and exciting new area of PT-symmetric quantum mechanics.