${ \mathcal P }{ \mathcal T }$-symmetry berry phases,topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking

We study the complex Berry phases in non-Hermitian systems with parity- and time-reversal $\left({ \mathcal P }{ \mathcal T }\right)$ symmetry. We investigate a kind of two-level system with ${ \mathcal P }{ \mathcal T }$ symmetry. We find that the real part of the the complex Berry phases have two quantized values and they are equal to either 0 or π, which originates from the topology of the Hermitian eigenstates. We also find that if we change the relative parameters of the Hamiltonian from the unbroken-${ \mathcal P }{ \mathcal T }$-symmetry phase to the broken-${ \mathcal P }{ \mathcal T }$-symmetry phase, the imaginary part of the complex Berry phases are divergent at the exceptional points. We exhibit two concrete examples in this work, one is a two-level toys model, which has nontrivial Berry phases; the other is the generalized Su–Schrieffer–Heeger (SSH) model that has physical loss and gain in every sublattice. Our results explicitly demonstrate the relation between complex Berry phases, topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking and enrich the field of the non-Hermitian physics.     

Impact of P T $\mathcal {P}\mathcal {T}$ -Symmetric Operation on Concurrence and the First-Order Coherence

International Journal of Theoretical Physics - In this paper, we lock the focus in effect of $\mathcal {P}\mathcal {T}$ -symmetric operation on the dynamics of concurrence and the first-order...     

To Quantify the Difference of η-Inner Products in P T $\mathcal {P}\mathcal {T}$ -Symmetric Theory

International Journal of Theoretical Physics - In this paper, we consider a typical continuous two dimensional $\mathcal {P}\mathcal {T}$ -symmetric Hamiltonian and propose two different approaches...     

Krein-space formulation of \mathcal{P}\mathcal{T} symmetry, \mathcal{C}\mathcal{P}\mathcal{T}-inner products, and pseudo-Hermiticity

Emphasizing the physical constraints on the formulation of the quantum theory, based on the standard measurement axiom and the Schrödinger equation, we comment on some conceptual issues arising in the formulation of the $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics. In particular, we elaborate on the requirements of the boundedness of the metric operator and the diagonalizability of the Hamiltonian. We also provide an accessible account of a Krein-space derivation of the $\mathcal{C}\mathcal{P}\mathcal{T}$ -inner product, that was widely known to mathematicians since 1950’s. We show how this derivation is linked with the pseudo-Hermitian formulation of the $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics.     

Understanding Partial P T $\mathcal {P}\mathcal {T}$ Symmetry as Weighted Composition Conjugation in Reproducing Kernel Hilbert Space: An application to Non-hermitian Bose-Hubbard Type Hamiltonian in Fock space

International Journal of Theoretical Physics - A new kind of symmetry behaviour introduced as partial $\mathcal {P}\mathcal {T}$ -symmetry(henceforth $\partial _{\mathcal {P}\mathcal {T}}$ ) is...     

Oscillatory stability of quantum droplets in $\text{}{ \mathcal P }{ \mathcal T }$-symmetric optical lattice

We study stability and collisions of quantum droplets (QDs) forming in a binary bosonic condensate trapped in parity-time (${ \mathcal P }{ \mathcal T }$)-symmetric optical lattices. It is found that the stability of QDs in the ${ \mathcal P }{ \mathcal T }$-symmetric system depends strongly on the values of the imaginary part W₀ of the ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices, self-repulsion strength g, and the condensate norm N. As expected, the ${ \mathcal P }{ \mathcal T }$-symmetric QDs are entirely unstable in the broken ${ \mathcal P }{ \mathcal T }$-symmetric phase. However, the ${ \mathcal P }{ \mathcal T }$-symmetric QDs exhibit oscillatory stability with the increase of N and g in the unbroken ${ \mathcal P }{ \mathcal T }$-symmetric phase. Finally, collisions between ${ \mathcal P }{ \mathcal T }$-symmetric QDs are considered. The collisions of droplets with unequal norms are completely different from that in free space. Besides, a stable ${ \mathcal P }{ \mathcal T }$-symmetric QDs collides with an unstable ones tend to merge into breathers after the collision.     

Pseudo-Hermitian Systems with $\mathcal {P}\mathcal {T}$-Symmetry: Degeneracy and Krein Space

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even \(\mathcal {P}\mathcal {T}\)-symmetry \((\mathcal {P}^{2}=1,\mathcal {T}^{2}=1)\) admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd \(\mathcal {P}\mathcal {T}\)-symmetric systems \((\mathcal {P}^{2}=1,\mathcal {T}^{2}=-1)\) which is appropriate to the fermions (Scolarici and Solombrino, Phys. Lett. A 303, 239 2002; Jones-Smith and Mathur, Phys. Rev. A 82, 042101 2010). We establish that the pseudo-Hermitian Hamiltonians with even \(\mathcal {P}\mathcal {T}\)-symmetry admit a degeneracy structure if the operator \(\mathcal {PT}\) anticommutes with the metric operator η _σ which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken \(\mathcal {P}\mathcal {T}\)-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even \(\mathcal {P}\mathcal {T}\) -symmetry.     

One-dimensional $\mathcal{PT}$-symmetric acoustic heterostructure

The explorations of parity-time ($\mathcal{PT}$)-symmetric acoustics have resided at the frontier in physics, and the pre-existing accessing of exceptional points typically depends on Fabry-Perot resonances of the coupling interlayer sandwiched between balanced gain and loss components. Nevertheless, the concise $\mathcal{PT}$-symmetric acoustic heterostructure, eliminating extra interactions caused by the interlayer, has not been researched in depth. Here we derive the generalized unitary relation for one-dimensional (1D) $\mathcal{PT}$-symmetric heterostructure of arbitrary complexity, and demonstrate four disparate patterns of anisotropic transmission resonances (ATRs) accompanied by corresponding spontaneous phase transitions. As a special case of ATR, the occasional bidirectional transmission resonance reconsolidates the ATR frequencies that split when waves incident from opposite directions, whose spatial profiles distinguish from a unitary structure. The derived theoretical relation can serve as a predominant signature for the presence of $\mathcal{PT}$ symmetry and $\mathcal{PT}$-symmetry-breaking transition, which may provide substantial support for the development of prototype devices with asymmetric acoustic responses.     

Analytic wave functions and energies for two-dimensional \mathcal{P}\mathcal{T} -symmetric quartic potentials

Analytic wave functions and the corresponding energies for a class of the $ \mathcal{P}\mathcal{T} $ -symmetric two-dimensional quartic potentials are found. The general form of the solutions is discussed.     

Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via PINN deep learning

In this paper, based on physics-informed neural networks (PINNs), a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations (PDEs) and other types of nonlinear physical models, we study the nonlinear Schrödinger equation (NLSE) with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential, which is an important physical model in many fields of nonlinear physics. Firstly, we choose three different initial values and the same Dirichlet boundary conditions to solve the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via the PINN deep learning method, and the obtained results are compared with those derived by the traditional numerical methods. Then, we investigate the effects of two factors (optimization steps and activation functions) on the performance of the PINN deep learning method in the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential. Ultimately, the data-driven coefficient discovery of the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential or the dispersion and nonlinear items of the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential can be approximately ascertained by using the PINN deep learning method. Our results may be meaningful for further investigation of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential in the deep learning.     

Effect of power-law nonlinearity on ${ \mathcal P }{ \mathcal T }$-symmetric optical system with fourth-order diffraction

Gaussian-type soliton solutions of the nonlinear Schrödinger (NLS) equation with fourth order dispersion, and power law nonlinearity in the novel parity-time (${ \mathcal P }{ \mathcal T }$)-symmetric quartic Gaussian potential are derived analytically and numerically. The exact analytical expressions of the solutions are obtained in the first two-dimensional (1D and 2D) power law NLS equations. By means of the linear stability analysis, the effect of power law nonlinearity on the stability of Gauss type solitons in different nonlinear media is carried out. Numerical investigations do confirm the stability of our soliton solutions in both focusing and defocusing cases, specially around the propagation parameters.     

Geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics

A new type of quantum theory known as time-dependent $\mathcal{PT}$-symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a $\mathcal{PT}$-symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a $\mathcal{PT}$-symmetric system, and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of $\mathcal{PT}$-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902 (2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.     

Spontaneous breakdown of $$ \mathcal{P}\mathcal{T} $$ symmetry in the complex Coulomb potential

The $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry of the Coulomb potential and its solutions are studied along trajectories satisfying the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry requirement. It is shown that with appropriate normalization constant the general solutions can be chosen $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric if the L parameter that corresponds to angular momentum in the Hermitian case is real. $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry is spontaneously broken, however, for complex L values of the form L = −1/2 + iλ. In this case the potential remains $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric, while the two independent solutions are transformed to each other by the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} operation and at the same time, the two series of discrete energy eigenvalues turn into each other’s complex conjugate.     

Superpotentials of $\mathcal{N}=1$ SUSY gauge theories

The European Physical Journal C - An introduction to recent advances in computing effective superpotentials of four dimensional $\mathcal{N}=1$ SUSY gauge theories coupled to matter is presented....     

New quasi-exactly solvable Hermitian as well as non-Hermitian $$ \mathcal{P}\mathcal{T} $$-invariant potentials

We start with quasi-exactly solvable (QES) Hermitian (and hence real) as well as complex $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -invariant, double sinh-Gordon potential and show that even after adding perturbation terms, the resulting potentials, in both cases, are still QES potentials. Further, by using anti-isospectral transformations, we obtain Hermitian as well as $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -invariant complex QES periodic potentials. We study in detail the various properties of the corresponding Bender-Dunne polynomials.     

Comparative analysis of real and \mathcal{P}\mathcal{T}-symmetric Scarf potentials

The Scarf I and Scarf II potentials are discussed within a common mathematical framework, which is then specified to handle the two potentials separately both in the conventional Hermitian and in the -symmetric setting. The physically admissible solutions are identified in each case together with the corresponding energy eigenvalues. Several main differences between the -symmetric Scarf I and II potentials are pointed out. These include the presence and absence of the quasi-parity quantum number, the sign of the pseudo-norm, the mechanism of the spontaneous breakdown of symmetry and the non- orthogonality of otherwise admissible solutions in the Scarf I potential. Similarities and differences with respect to the corresponding Hermitian systems are also pointed out.     

Scarcity of real discrete eigenvalues in non-analytic complex $$ \mathcal{P}\mathcal{T} $$-symmetric potentials

We find that a non-differentiability occurring whether in real or imaginary part of a complex $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric potential causes a scarcity of the real discrete eigenvalues despite the real part alone possessing an infinite spectrum. We demonstrate this by perturbing the real potentials x ² and |x| by imaginary $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric potentials ix/it|x| and ix, respectively.     

Gap solitons of spin-orbit-coupled Bose-Einstein condensates in $\mathcal{PT}$ periodic potential

We numerically investigate the gap solitons in Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) in the parity-time ($\mathcal{PT}$)-symmetric periodic potential. We find that the depths and periods of the imaginary lattice have an important influence on the shape and stability of these single-peak gap solitons and double-peak gap solitons in the first band gap. The dynamics of these gap solitons are checked by the split-time-step Crank-Nicolson method. It is proved that the depths of the imaginary part of the $\mathcal{PT}$-symmetric periodic potential gradually increase, and the gap solitons become unstable. But the different periods of imaginary part hardly affect the stability of the gap solitons in the corresponding parameter interval.     

On a few new quantization recipes using \mathcal{P}\mathcal{T}-symmetry

A review of a few recent developments in our analysis and applications of the coupled-channel version of the formalism of -symmetric quantum mechanics is given.