PT-symmetric Model Research Articles

Electric conductivity in non-Hermitian holography

We study the phase structure and charge transport at finite temperature and chemical potential in the non-Hermitian \mathcal{PT}𝒫𝒯-symmetric holographic model of [SciPost Phys. 9, 032 (2020)]. The non-Hermitian \mathcal{PT}𝒫𝒯-symmetric deformation is realized by promoting the parameter of a global U(1) symmetry to a complex number. Depending on the strength of the deformation, we find three phases: stable \mathcal{PT}𝒫𝒯-symmetric phase, unstable \mathcal{PT}𝒫𝒯-symmetric phase, and an unstable \mathcal{PT}𝒫𝒯-symmetry broken phase. In the three phases, the square of the condensate and also the spectral weight of the AC conductivity at zero frequency are, respectively, positive, negative, and complex. We check that the Ferrell-Glover-Tinkham sum rule for the AC conductivity holds in all the three phases. We also investigate a complexified U(1) rotor model with \mathcal{PT}𝒫𝒯-symmetric deformation, derive its phase structure and condensation pattern, and find a zero frequency spectral weight analogous to the holographic model.

Exceptional points in the Baxter-Fendley free parafermion model

Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled two-level fermionic systems. Free parafermions are a generalisation of this idea to \mathbb{Z}_NℤN-symmetric clock models. In 1989 Baxter discovered a non-Hermitian but \mathcal{PT}𝒫𝒯-symmetric model directly generalising the Ising chain, which was later described by Fendley as a free parafermion spectrum. By extending the model’s magnetic field parameter to the complex plane, it is shown that a series of exceptional points emerges, where the quasienergies defining the free spectrum become degenerate. An analytic expression for the locations of these points is derived, and various numerical investigations are performed. These exceptional points also exist in the Ising chain with a complex transverse field. Although the model is not in general \mathcal{PT}𝒫𝒯-symmetric at these exceptional points, their proximity can have a profound impact on the model on the \mathcal{PT}𝒫𝒯-symmetric real line. Furthermore, in certain cases an exceptional point may appear on the real line (with negative field).

Entanglement spectrum and entropy in topological non-Hermitian systems and nonunitary conformal field theory

We propose a method of computing and studying entanglement quantities in non-Hermitian systems by use of a biorthogonal basis. We find that the entanglement spectrum characterizes the topological properties in terms of the existence of mid-gap states in the non-Hermitian Su-Schrieffer-Heeger (SSH) model with parity and time-reversal symmetry (PT symmetry) and the non-Hermitian Chern insulators. In addition, we find that at a critical point in the PT symmetric SSH model, the entanglement entropy has a logarithmic scaling with corresponding central charge $c=-2$. This critical point then is a free-fermion lattice realization of the non-unitary conformal field theory.

PT -symmetric non-Hermitian quantum field theories with supersymmetry

We formulate supersymmetric non-Hermitian quantum field theories with PT symmetry, starting with free chiral boson/fermion models and then including trilinear superpotential interactions. We consider models with both Dirac and Majorana fermions, analyzing them in terms of superfields and at the component level. We also discuss the relation between the equations of motion, the (non-)invariance of the Lagrangian and the (non-)conservation of the supercurrents in the two models. We exhibit a similarity transformation that maps the free-field supersymmetric PT-symmetric Dirac model to a supersymmetric Hermitian theory, but there is, in general, no corresponding similarity transformation for the Majorana model. In this model, we find generically a mass splitting between bosons and fermions, even though its construction is explicitly supersymmetric, offering a novel non-Hermitian mechanism for soft supersymmetry breaking.

Nonlinear Parity-Time-Symmetric Model for Constant Efficiency Wireless Power Transfer: Application to a Drone-in-Flight Wireless Charging Platform

A major challenge for practical wireless power transfer (WPT) applications is to attain stable power transfer with high and constant transfer efficiency under a dynamic change of coupling condition. In order to address the issue, this paper proposes a novel nonlinear parity-time (PT) symmetric model, wherein the nonlinear saturable gain is provided by a self-oscillating controlled inverter. In this paper, the transfer performance and stability criterion of the nonlinear PT-based WPT system are analyzed based on the coupled-mode theory. The theoretical analysis shows that the proposed system automatically achieves constant output power with constant transfer efficiency against the variation of coupling coefficient. Moreover, based on the gain saturation mechanism, the control strategy for the inverter needs to detect only the current in the transmitter, which eliminates auxiliary circuits of wireless communication for feedback control from the receiver. As a case study of dynamic charging, a drone-in-flight wireless charging platform is improved by applying the nonlinear PT-symmetric model. Experimental results show that when the flying drone hovers in a confined three-dimensional volume of space above the WPT platform, a stable output power is maintained with approximately constant transfer efficiency of 93.6%.

Tunable photon statistics in a non-Hermitian system

In the present study, the antibunching effect is considered in a parity time (PT)-symmetric system, using the recently proposed non-Hermitian model for resonant cavities. The model consists of a cavity coupled to an optomechanical system by a chiral mirror, with both cavities being driven coherently. The nonreciprocal coupling strength between the two cavities leads to the generation of the non-Hermitian but PT-symmetric model. The optimal conditions of strong antibunching are calculated in this system, and then the antibunching effect is considered using the two-photon correlation function and the Wigner function. The results showed that this model presented the best control over the strong antibunching effect by tuning the nonreciprocal coupling strength between the cavities, the strength ratio between the two fields, and nonlinearity strength—an outcome inaccessible by the Hermitian system.

Nonreciprocal behaviour between photon and phonon in coupled optomechanical systems

It is shown that the asymmetry coupling between two coupled optomechanical cavities leads to special class of PT-symmetric model for optomechanical structure. Under these conditions, Hamiltonian is considered in blue and red sideband regime. In these cases, the asymmetric coupling between two cavities has been transferred such that the asymmetric beam-splitter or squeezing interaction is generated between optical and mechanical modes. Then, the amount of entanglement between the different optical and mechanical modes is calculated. The results define that PT-symmetry can improve the entanglement in special conditions. The proposed system provides good condition to investigate the nonreciprocal interaction between photon and phonon.

Solitary Waves of a <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}$</tex-math> </inline-formula><inline-formula> <tex-math notation="LaTeX">$\mathcal {T}$</tex-math> </inline-formula>-Symmetric Nonlinear Dirac Equation

In the present work, we consider a prototypical example of a PT-symmetric Dirac model. We discuss the underlying linear limit of the model and identify the threshold of the PT-phase transition in an analytical form. We then focus on the examination of the nonlinear model. We consider the continuation in the PT-symmetric model of the solutions of the corresponding Hamiltonian model and find that the solutions can be continued robustly as stable ones all the way up to the PT-transition threshold. In the latter, they degenerate into linear waves. We also examine the dynamics of the model. Given the stability of the waveforms in the PT-exact phase we consider them as initial conditions for parameters outside of that phase. We find that both oscillatory dynamics and exponential growth may arise, depending on the size of the corresponding "quench". The former can be characterized by an interesting form of bi-frequency solutions that have been predicted on the basis of the SU(1,1) symmetry. Finally, we explore some special, analytically tractable, but not PT-symmetric solutions in the massless limit of the model.

Phase transitions in Wick-rotated [formula omitted]-symmetric optics

PT-symmetric models with a Wick rotation of time (t→±it) show spectral phase transitions that are similar to those of dissipative systems driven out of equilibrium. Optics can provide an accessible test bed to explore spectral phase transitions of Wick-rotated PT-symmetric models. This is shown by considering the transverse dynamics of laser light in optical cavities with variable reflectivity and tilted mirrors. Two specific examples are discussed: the optical analogue of the hydrodynamic Squire model of vorticity, and the Wick-rotated PT-symmetric nonlinear dimer model. In the latter case the spectral phase transition is associated with the universal phase locking–unlocking transition in Adler’s theory of coupled oscillators.

Spectral and transport properties of the PT-symmetric dimer model

We study the scattering properties of the PT-symmetric tight-binding model with balanced gain and loss parameters. Our main interest is to establish the link between the spectral properties of scattering states and transport characteristics for the case of non-equal couplings between gain/loss sites. The analytical approach we have used allows one to reveal a quite unexpected role of this set-up in comparison with that of equal couplings. In particular, we demonstrate that for the exceptional points characterized by equal eigenvalues of the transfer matrix, the transmission coefficient can be different from one in contrast with the model with equal couplings. The analytical results are complemented by the numerical data.

Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator

The non-Hermitian quadratic oscillator known as the Swanson oscillator is one of the popular PT-symmetric model systems. Here a full classical description of its dynamics is derived using recently developed metriplectic flow equations, which combine the classical symplectic flow for Hermitian systems with a dissipative metric flow for the anti-Hermitian part. Closed form expressions for the metric and phase-space trajectories are presented which are found to be periodic in time. Since the Hamiltonian is only quadratic the classical dynamics exactly describe the quantum dynamics of Gaussian wave packets. It is shown that the classical metric and trajectories as well as the quantum wave functions can diverge in finite time even though the PT-symmetry is unbroken, i.e., the eigenvalues are purely real.

PT symmetric Aubry–Andre model

PT symmetric Aubry-Andre model describes an array of N coupled optical waveguides with position dependent gain and loss. We show that the reality of the spectrum depends sensitively on the degree of disorder for small number of lattice sites. We obtain the Hofstadter Butterfly spectrum and discuss the existence of the phase transition from extended to localized states. We show that rapidly changing periodical gain/loss materials almost conserves the total intensity.

PT symmetry and large-N models

Recently developed methods for PT-symmetric models can be applied to quantum-mechanical matrix and vector models. In matrix models, the calculation of all singlet wavefunctions can be reduced to the solution of a one-dimensional PT-symmetric model. The large-N limit of a wide class of matrix models exists, and properties of the lowest-lying singlet state can be computed using WKB. For models with cubic and quartic interactions, the ground-state energy appears to show rapid convergence to the large-N limit. For the special case of a quartic model, we find explicitly an isospectral Hermitian matrix model. The Hermitian form for a vector model with O(N) symmetry can also be found, and shows many unusual features. The effective potential obtained in the large-N limit of the Hermitian form is shown to be identical to the form obtained from the original PT-symmetric model using familiar constraint field methods. The analogous constraint field prescription in four dimensions suggests that PT-symmetric scalar field theories are asymptotically free.

Novel phase in the phase structure of the(gϕ4+hϕ6)1+1field theoretic model

In view of the newly discovered and physically acceptable PT symmetric and non-Hermitian models, we reinvestigated the phase structure of the (g{phi}{sup 4}+h{phi}{sup 6}){sub 1+1} Hermitian model. The reinvestigation concerns the possibility of a phase transition from the original Hermitian and PT symmetric phase to a non-Hermitian and PT symmetric one. This kind of phase transition, if verified experimentally, will lead to the first proof that non-Hermitian and PT symmetric models are not just a mathematical research framework but are a nature desire. To do the investigation, we calculated the effective potential up to second order in the couplings and found a Hermitian to non-Hermitian phase transition. This leads us to introduce, for the first time, Hermiticity as a symmetry which can be broken due to quantum corrections, i.e., when starting with a model which is Hermitian in the classical level, quantum corrections can break Hermiticity while the theory stays physically acceptable. In fact, ignoring this phase will lead to violation of universality when comparing these model predictions with other models in the same class of universality. For instance, in a previous work we obtained a second order phase transition for the PT symmetric and non-Hermitian (-g{phi}{sup 4}) model andmore » according to universality, this phase should exist in the phase structure of the (g{phi}{sup 4}+h{phi}{sup 6}) model for negative g. Finally, among the novelties in this paper, in our calculation for the effective potential, we introduced a new renormalization group equation which describes the invariance of the bare vacuum energy under the change of the scale. We showed that without this invariance, the original theory and the effective one are inequivalent.« less

Closed formula for the metric in the Hilbert space of a -symmetric model

We introduce a very simple, exactly solvable PT-symmetric non-Hermitian model with real spectrum, and derive a closed formula for the metric operator which relates the problem to a Hermitian one.

Solvable PT-symmetric model with a tunable interspersion of nonmerging levels

We study the spectrum in such a PT-symmetric square well (of a diameter L⩽∞) where the “strength of the non-Hermiticity” is controlled by the two parameters, viz., by an imaginary coupling ig and by the distance ℓ<L of its onset from the origin. We solve this problem and confirm that the spectrum is discrete and real in a nonempty interval of g⩽g0(ℓ,L). Surprisingly, a specific distinction between the bound states is found in their asymptotic stability∕instability with respect to an unlimited growth of g beyond g0(ℓ,L). In our model, all of the low-lying levels remain asymptotically unstable at the small ℓ⪡L and finite L while only the stable levels survive near ℓ≈L<∞ or in the purely imaginary force limit with 0<ℓ<L=∞. In between these two extremes, an unusual and tunable, variable pattern of the interspersed “robust” and “fragile” subspectra of the real levels is obtained.

MHD α2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator

It is shown that the α2-dynamo of magnetohydrodynamics, the hydrodynamic Squire equation as well as an interpolation model of PT-symmetric quantum mechanics are closely related as spectral problems in Krein spaces. For the α2-dynamo and the PT-symmetric model the strong similarities are demonstrated with the help of a 2×2 operator matrix representation, whereas the Squire equation is reinterpreted as a rescaled and Wick-rotated PT-symmetric problem. Based on recent results on the Squire equation the spectrum of the PT-symmetric interpolation model is analyzed in detail and the Herbst limit is described as spectral singularity.