${ \mathcal P }{ \mathcal T }$-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.     

Topology of a parity-time symmetric non-Hermitian rhombic lattice

We investigate the topological properties of a trimerized parity–time(PT)symmetric non-Hermitian rhombic lattice.Although the system is PT-symmetric,the topology is not inherited from the Hermitian lattice;in contrast,the topology can be altered by the non-Hermiticity and depends on the couplings between the sublattices.The bulk–boundary correspondence is valid and the Bloch bulk captures the band topology.Topological edge states present in the two band gaps and are predicted from the global Zak phase obtained through the Wilson loop approach.In addition,the anomalous edge states compactly localize within two diamond plaquettes at the boundaries when all bands are flat at the exceptional point of the lattice.Our findings reveal the topological properties of the??PT-symmetric non-Hermitian rhombic lattice and shed light on the investigation of multi-band non-Hermitian topological phases.     

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.     

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.     

Hidden analytic relations for two-loop Higgs amplitudes in QCD

We compute the Higgs plus two-quark and one-gluon amplitudes(H→qqg) and Higgs plus three-gluon amplitudes(H→3 g) in the Higgs effective theory with a general class of operators.By changing the quadratic Casimir CF to CA,the maximally transcendental parts of the H→ qqg amplitudes turn out to be equivalent to that of the H→3 g amplitudes,which also coincide with the counterparts in N=4 SYM.This generalizes the so-called maximal transcendentality principle to the Higgs amplitudes with external quark states,thus the full QCD theory.We further verify that the correspondence applies also to two-loop form factors of more general operators,in both QCD and scalar-YM theory.Another interesting relation is also observed between the planar H→qqg amplitudes and the minimal density form factors in N=4 SYM.     

Phase sensitivity of light dynamics in PT-symmetric couplers

We study the sensitivity of light dynamics to the internal phase of propagating pulses in the two types of $\mathcal {PT}$ -symmetric models. The first is a waveguide array with an embedded pair of waveguides with gain and loss, called $\mathcal {PT}$ -coupler, and the second is a planar coupler which models a chain of $\mathcal {PT}$ -symmetric couplers. For the first model we investigate the soliton scattering on the mode localized on the coupler, while for second model we study the collision of two breathers. For both models we find that the light dynamics is sensitive to the internal phases of the interacting pulses. Particularly, the $\mathcal {PT}$ -symmetry breaking can take place or not, depending on the internal phases of two signals having identical other parameters.     

Data-driven parity-time-symmetric vector rogue wave solutions of multi-component nonlinear Schrödinger equation

Rogue waves are a class of nonlinear waves with extreme amplitudes, which usually appear suddenly and disappear without any trace. Recently, the parity-time ($\mathcal {PT}$)-symmetric vector rogue waves (RWs) of multi-component nonlinear Schrödinger equation ($n$-NLSE) are usually derived by the methods of integrable systems. In this paper, we utilize the multi-stage physics-informed neural networks (MS-PINNs) algorithm to derive the data-driven $\mathcal {PT}$ symmetric vector RWs solution of coupled NLS system in elliptic and X-shapes domains with nonzero boundary condition. The results of the experiment show that the multi-stage physics-informed neural networks are quite feasible and effective for multi-component nonlinear physical systems in the above domains and boundary conditions.     

Non-hermitian quantum mechanics in non-commutative space

A recent investigation of the possibility of having a -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a -symmetric deformation of this space. Specifically, a -symmetric harmonic oscillator together with an iC(x ₁+x ₂) interaction are discussed in this space, and solutions are obtained. We show that in the deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x ₁+x ₂) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not -symmetric. A complex interacting anisotropic oscillator system also is discussed.     

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.     

\mathcal{PT}-symmetric nonlinear metamaterials and zero-dimensional systems

A one dimensional, parity-time ( $\mathcal{PT}$ )-symmetric magnetic metamaterial comprising split-ring resonators having both gain and loss is investigated. In the linear regime, the transition from the exact to the broken $\mathcal{PT}$ -phase is determined through the calculation of the eigenfrequency spectrum for two different configurations; the one with equidistant split-rings and the other with the split-rings forming a binary pattern ( $\mathcal{PT}$ dimer chain). The latter system features a two-band, gapped spectrum with its shape determined by the gain/loss coefficient as well as the interelement coupling. In the presence of nonlinearity, the $\mathcal{PT}$ dimer chain configuration with balanced gain and loss supports nonlinear localized modes in the form of a novel type of discrete breathers below the lower branch of the linear spectrum. These breathers that can be excited from a weak applied magnetic field by frequency chirping, can be subsequently driven solely by the gain for very long times. The effect of a small imbalance between gain and loss is also considered. Fundamental gain-driven breathers occupy both sites of a dimer, while their energy is almost equally partitioned between the two split-rings, the one with gain and the other with loss. We also introduce a model equation for the investigation of classical $\mathcal{PT}$ symmetry in zero dimensions, realized by a simple harmonic oscillator with matched time-dependent gain and loss that exhibits a transition from oscillatory to diverging motion. This behavior is similar to a transition from the exact to the broken $\mathcal{PT}$ phase in higher-dimensional $\mathcal{PT}$ -symmetric systems. A stability condition relating the parameters of the problem is obtained in the case of a piece-wise constant gain/loss function that allows the construction of a phase diagram with alternating stable and unstable regions.     

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.     

Dark Energy Phenomenon from Backreaction Effect

In this paper, we interpret the dark energy phenomenon as an averaged effect caused by small scale inhomogeneities of the universe with the use of the spatial averaged approach of Buchert. Two models are considered here, one of which assumes that the backreaction term ${\cal Q}_\mathcal{D}$ and the averaged spatial Ricci scalar $\langle\mathcal{R}\rangle_\mathcal{D}$ obey the scaling laws of the volume scale factor $a_\mathcal{D}$ at adequately late times, and the other one adopts the ansatz that the backreaction term ${\cal Q}_\mathcal{D}$ is a constant in the recent universe. Thanks to the effective geometry introduced by Larena et al. in their previous work, we confront these two backreaction models with latest type Ia supernova and Hubble parameter observations, coming out with the results that the constant backreaction model is slightly favoured over the other model and the best fitting backreaction term in the scaling backreaction model behaves almost like a constant. Also, the numerical results show that the constant backreaction model predicts a smaller expansion rate and decelerated expansion rate than the other model does at redshifts higher than about 1, and both backreaction terms begin to accelerate the universe at a redshift around 0.5.     

A spin-less particle on a rotating curved surface in Minkowski space

In Minkowski space ${ \mathcal M }$, we derive the effective Schrödinger equation describing a spin-less particle confined to a rotating curved surface ${ \mathcal S }$. Using the thin-layer quantization formalism to constrain the particle on ${ \mathcal S }$, we obtain the relativity-corrected geometric potential ${V}_{g}^{{\prime} }$, and a novel effective potential ${\tilde{V}}_{g}$ related to both the Gaussian curvature and the geodesic curvature of the rotating surface. The Coriolis effect and the centrifugal potential also appear in the equation. Subsequently, we apply the surface Schrödinger equation to a rotating cylinder, sphere and torus surfaces, in which we find that the interplays between the rotation and surface geometry can contribute to the energy spectrum based on the potentials they offer.     

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.     

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.     

Spherical-separability of Non-Hermitian Hamiltonians and Pseudo- PT\mathcal{PT} -symmetry

Non-Hermitian but -symmetrized spherically-separable Dirac and Schr?dinger Hamiltonians are considered. It is observed that the descendant Hamiltonians H _r , H _θ , and H _φ play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a -symmetrized H _φ , we have shown that the conventional Dirac (relativistic) and Schr?dinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V(θ)≠0 in the descendant Hamiltonian H _θ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some -symmetrized H _φ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the -symmetric ones (here the non-Hermitian -symmetric Hamiltonians) are nicknamed as pseudo- -symmetric.     

Extensions of Ordering Sets of States from Effect Algebras onto Their MacNeille Completions

In (Rie?anová and Zajac in Rep. Math. Phys. 70(2):283–290, 2012) it was shown that an effect algebra E with an ordering set $\mathcal{M}$ of states can by embedded into a Hilbert space effect algebra $\mathcal{E}(l_{2}(\mathcal{M}))$ . We consider the problem when its effect algebraic MacNeille completion $\hat{E}$ can be also embedded into the same Hilbert space effect algebra $\mathcal {E}(l_{2}(\mathcal{M}))$ . That is when the ordering set $\mathcal{M}$ of states on E can be extended to an ordering set of states on $\hat{E}$ . We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.     

$\mathcal{PT}$ Symmetry in Statistical Mechanics and the Sign Problem

Generalized PT\mathcal{PT} symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized PT\mathcal{PT}-symmetric Hamiltonian. There is a corresponding class of PT\mathcal{PT}-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. We discuss a class of Z(N) spin models with explicit PT\mathcal{PT} symmetry and also the ANNNI model, which has a hidden PT\mathcal{PT} symmetry. For both quantum and classical models, the class of models with generalized PT\mathcal{PT} symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with PT\mathcal{PT} symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with PT\mathcal{PT} symmetry breaking of the ground state, and is generically a first-order transition. In the region where PT\mathcal{PT} symmetry is unbroken, the sign problem can always be solved in principle using the equivalence to a Hermitian theory in this region. The ANNNI model provides an example of a model with PT\mathcal{PT} symmetry which can be simulated for all parameter values, including cases where PT\mathcal{PT} symmetry is broken.     

Effects of Quantum Fluctuations on $\mathcal{PT}$-Symmetric Solitons of a Trapped Bose Gas *

Considering the quantum fluctuation effects, the existence and stabilityof solitons in a Bose-Einstein condensate subjected in a $\mathcal{PT}$-symmetric potentialare discussed. Using the variational approach, we investigate how the quantum fluctuationaffects the self-localization and stability of the condensate with attractivetwo-body interactions. The results show that the quantum fluctuation dramaticallyinfluences the shape, width, and chemical potential of the condensate.Analytical variational computation also predicts there exists a positive critical quantumfluctuation strength $q_{c}$ with each fixed attractive two-body interaction $g_{0}$, if thequantum fluctuation strength $q_{0}$ is bigger than $q_{c}$, there is no bright solitonsolution existence. We also study the effects of the quantum fluctuations on the stabilityof solitons using the Vakhitov-Kolokolov (VK) stability criterion. A robust stable brightsoliton will always exist when the quantum fluctuation strength $q_{0}$ belongs tothe parameter regimes $q_{c}\geq q_{0}＞0$.     

Searching for isotropic stochastic gravitational-wave background in the international pulsar timing array second data release

We search for isotropic stochastic gravitational-wave background (SGWB) in the International Pulsar Timing Array second data release. By modeling the SGWB as a power-law, we find very strong Bayesian evidence for a common-spectrum process, and further this process has scalar transverse (ST) correlations allowed in general metric theory of gravity as the Bayes factor in favor of the ST-correlated process versus the spatially uncorrelated common-spectrum process is 30 ± 2. The median and the 90% equal-tail amplitudes of ST mode are ${{ \mathcal A }}_{\mathrm{ST}}={1.29}_{-0.44}^{+0.51}\times {10}^{-15}$, or equivalently the energy density parameter per logarithm frequency is ${{\rm{\Omega }}}_{\mathrm{GW}}^{\mathrm{ST}}={2.31}_{-1.30}^{+2.19}\times {10}^{-9}$, at frequency of 1 year⁻¹. However, we do not find any statistically significant evidence for the tensor transverse (TT) mode and then place the 95% upper limits as ${{ \mathcal A }}_{\mathrm{TT}}\lt 3.95\times {10}^{-15}$, or equivalently ${{\rm{\Omega }}}_{\mathrm{GW}}^{\mathrm{TT}}\lt 2.16\times {10}^{-9}$, at frequency of 1 year⁻¹.