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In this paper we study effective actions of the compact torus on smooth compact manifolds of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to , the complement to the union of disjoint open subsets of the -sphere. The results obtained are applied to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices of step type. Bibliography: 23 titles.

We investigate novel features of three-dimensional non-Hermitian Weyl semimetals, paying special attention to the unconventional bulk–boundary correspondence. We use the non-Bloch Chern numbers as the tool to obtain the topological phase diagram, which is also confirmed by the energy spectra from our numerical results. It is shown that, in sharp contrast to Hermitian systems, the conventional (Bloch) bulk–boundary correspondence breaks down in non-Hermitian topological semimetals, which is caused by the non-Hermitian skin effect. We establish the non-Bloch bulk–boundary correspondence for non-Hermitian Weyl semimetals: the topological edge modes are determined by the non-Bloch Chern number of the bulk bands. Moreover, these topological edge modes can manifest as the unidirectional edge motion, and their signatures are consistent with the non-Bloch bulk–boundary correspondence. Our work establishes the non-Bloch bulk–boundary correspondence for non-Hermitian topological semimetals.

The bulk boundary correspondence, which connects the topological invariant, the continuum band and energies under different boundary conditions, is the core concept in the non-Bloch band theory, in which the generalized Brillouin zone (GBZ), appearing as a closed loop generally, is a fundamental tool to rebuild it. In this work, it can be shown that the recovery of the open boundary energy spectrum by the continuum band remains unchanged even if the GBZ itself shrinks into a point. Contrastively, if the bizarreness of the GBZ occurs, the winding number will become illness. Namely, we find that the bulk boundary correspondence can still be established whereas the GBZ has singularities from the perspective of the energy, but not from the topological invariant. Meanwhile, regardless of the fact that the GBZ comes out with the closed loop, the bulk boundary correspondence cannot be well characterized yet because of the ill-definition of the topological number. Here, the results obtained may be useful for improving the existing non-Bloch band theory.

In open quantum systems, it is known that if the system and environment are in a product state, the evolution of the system is given by a linear completely positive (CP) Hermitian map. CP maps are a subset of general linear Hermitian maps, which also include non completely positive (NCP) maps. NCP maps can arise in evolutions such as non-Markovian evolution, where the CP divisibility of the map (writing the overall evolution as a composition of CP maps) usually fails. Positive but NCP maps are also useful as entanglement witnesses. In this paper, we focus on transforming an initial NCP map to a CP map through composition with the asymmetric depolarizing map. We use separate asymmetric depolarizing maps acting on the individual subsystems. Previous work have looked at structural physical approximation (SPA), which is a CP approximation of an NCP map using a mixture of the NCP map with a completely depolarizing map. We prove that the composition can always be made CP without completely depolarizing in any direction. It is possible to depolarize less in some directions. We give the general proof by using the Choi matrix and an isomorphism from a maximally entangled two qudit state to a set of qubits. We also give measures that describe the amount of disturbance the depolarization introduces to the original map. Given our measures, we show that asymmetric depolarization has many advantages over SPA in preserving the structure of the original NCP map. Finally, we give some examples. For some measures and examples, completely depolarizing (while not necessary) in some directions can give a better approximation than keeping the depolarizing parameters bounded by the required depolarization if symmetric depolarization is used.

$L$-ensembles are a class of determinantal point processes which can be viewed as a statistical mechanical systems in the grand canonical ensemble. Circulant $L$-ensembles are the subclass which are locally translationally invariant and furthermore subject to periodic boundary conditions. Existing theory can very simply be specialised to this setting, allowing for the derivation of formulas for the system pressure, and the correlation kernel, in the thermodynamic limit. For a one-dimensional domain, this is possible when the circulant matrix is both real symmetric, or complex Hermitian. The special case of the former having a Gaussian functional form for the entries is shown to correspond to free fermions at finite temperature, and be generalisable to higher dimensions. A special case of the latter is shown to be the statistical mechanical model introduced by Gaudin to interpolate between Poisson and unitary symmetry statistics in random matrix theory. It is shown in all cases that the compressibility sum rule for the two-point correlation is obeyed, and the small and large distance asymptotics of the latter are considered. Also, a conjecture relating the asymptotic form of the hole probability to the pressure is verified., 20 pages

The conformal bridge transformation (CBT) is reviewed in the light of the $\mathcal{PT}$ symmetry. Originally, the CBT was presented as a non-unitary transformation (a complex canonical transformation in the classical case) that relates two different forms of dynamics in the sense of Dirac. Namely, it maps the asymptotically free form into the harmonically confined form of dynamics associated with the $\mathfrak{so}(2,1)\cong \mathfrak{sl}(2,{\mathbb R})$ conformal symmetry. However, as the transformation relates the non-Hermitian operator $i\hat{D}$, where $\hat{D}$ is the generator of dilations, with the compact Hermitian generator $\hat{\mathcal{J}}_0$ of the $\mathfrak{sl}(2,{\mathbb R})$ algebra, the CBT generator can be associated with a $\mathcal{PT}$-symmetric metric. In this work we review the applications of this transformation for one- and two-dimensional systems, as well as for systems on a cosmic string background, and for a conformally extended charged particle in the field of Dirac monopole. We also compare and unify the CBT with the Darboux transformation. The latter is used to construct $\mathcal{PT}$-symmetric solutions of the equations of the KdV hierarchy with the properties of extreme waves. As a new result, by using a modified CBT we relate the one-dimensional $\mathcal{PT}$-regularized asymptotically free conformal mechanics model with the $\mathcal{PT}$-regularized version of the de Alfaro, Fubini and Furlan system., Comment: 42 pages, review article

We review recent work on asymmetric scattering by Non-Hermitian (NH) Hamiltonians. Quantum devices with an asymmetric scattering response to particles incident from right or left in effective ID waveguides will be important to develop quantum technologies. They act as microscopic equivalents of familiar macroscopic devices such as diodes, rectifiers, or valves. The symmetry of the underlying NH Hamiltonian leads to selection rules which restrict or allow asymmetric response. NH-symmetry operations may be organized into group structures that determine equivalences among operations once a symmetry is satisfied. The NH Hamiltonian posseses a particular symmetry if it is invariant with respect to the corresponding symmetry operation, which can be conveniently expressed by a unitary or antiunitary superoperator. A simple group is formed by eight symmetry operations, which include the ones for Parity-Time symmetry and Hermiticity as specific cases. The symmetries also determine the structure of poles and zeros of the S matrix. The ground-state potentials for two-level atoms crossing properly designed laser beams realize different NH symmetries to achieve transmission or reflection asymmetries.

We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of this new type of random matrices, we focus on two specific models of matrices which are pseudo-hermitian with respect to a given indefinite metric B. Eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. The diagrammatic method is applied to deriving explicit analytical expressions for the density of eigenvalues in the complex plane and on the real axis, in the large-N, planar limit. In one of the models we discuss, the metric B depends on a certain real parameter t. As t varies, the model exhibits various "phase transitions" associated with eigenvalues flowing from the complex plane onto the real axis, causing disjoint eigenvalue support intervals to merge. Our analytical results agree well with presented numerical simulations., Comment: 18 pages, 9 figures. Minor typos corrected

We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might exhibit Poisson or Poisson-like statistics. The Poisson statistics have been already observed for heavy-tailed Wigner matrices. We conclude with two conjectures on this peculiar behaviour., Comment: 30 pages, 7 figures

Corresponding to a hypergraph $G$ with $d$ vertices, a quantum hypergraph state is defined by $|G\rangle = \frac{1}{\sqrt{2^d}}\sum_{n = 0}^{2^d - 1} (-1)^{f(n)} |n \rangle$, where $f$ is a $d$-variable Boolean function depending on the hypergraph $G$, and $|n \rangle$ denotes a binary vector of length $2^d$ with $1$ at $n$-th position for $n = 0, 1, \dots (2^d - 1)$. The non-classical properties of these states are studied. We consider annihilation and creation operator on the Hilbert space of dimension $2^d$ acting on the number states $\{|n \rangle: n = 0, 1, \dots (2^d - 1)\}$. The Hermitian number and phase operators, in finite dimensions, are constructed. The number-phase uncertainty for these states leads to the idea of phase squeezing. We establish that these states are squeezed in the phase quadrature only and satisfy the Agarwal-Tara criterion for non-classicality, which only depends on the number of vertices of the hypergraphs. We also point out that coherence is observed in the phase quadrature.

Non-Hermitian Hamiltonians with anti- PT symmetry show distinct discrepancies compared to PT -symmetric systems. Here we investigate the non-Hermitian Su–Schrieffer–Heeger (SSH) model and reveal the interplay between anti- PT symmetry and different topological phases, in which the anti- PT symmetry is triggered by introducing real chemical potential and imaginary nearest-neighbor hopping. Three types of anti- PT -symmetric SSH models with different on-site configurations are investigated, and the effect of each on-site configuration on the energy spectrum of system is analyzed and discussed. We find that the type of on-site configuration affects the energy spectrum of the same topological region significantly and each on-site configuration has distinguishable impacts on the energy spectra of different topological regions. Moreover, the spontaneous anti- PT -symmetry-breaking transitions undergone in both of the topologically trivial and nontrivial regions are also clarified. Furthermore, we address the corresponding eigenstates of some specific eigenvalues and some interesting localized phenomena are observed. Our results enrich the study of non-Hermitian topological phases and broaden the understanding of anti- PT -symmetric systems.

We study the non-Hermitian extension of the Lagrangian of the Standard Model extended by singlet right-handed heavy neutrinos. The Yukawa coupling matrices comprise of hermitian and non-hermitian components and the neutrino mass eigenvalues are calculated for three generation case. The increased number of unknown parameters in the theory due to non-hermitian nature of coupling matrices impose problem for its productiveness. Hence we apply four zero texture for the Yukawa matrices. We consider the normal hierarchy of neutrino mass ordering and find the mixing angles and Dirac CP violating phase by diagonalizing the lepton mass matrices. The parameters of the mixing matrix are obtained in terms of the parameters of the non-hermitian Yukawa coupling matrix. The values of the angles and phase are compatible with experimental data available for neutrino mass observables and using this the non-hermitian parameters are constrained. Also, we study the constraints imposed by $CP$ violation in the lepton sector and leptogenesis on the model parameters., Comment: 29 pages, matches with the published version

We report an unusual buildup of the quantum coherence in a qubit subjected to non-Hermitian evolution generated by a Parity-Time ($\mathcal{PT}$) symmetric Hamiltonian, which is reinterpreted as a Hermitian system in a higher dimensional space using Naimark dilation. The coherence is found to be maximum about the exceptional points (EPs), i.e., the points of coalescence of the eigenvalues as well as the eigenvectors. The nontrivial physics about EPs has been observed in various systems, particularly in photonic systems. As a consequence of enhancement in coherence, the various formulations of Leggett-Garg inequality tests show maximal violation about the EPs.

The dynamical and topological properties of non-Hermitian systems have attracted great attention in recent years. In this work, we establish an intrinsic connection between two classes of intriguing phenomena -- topological phases and dynamical quantum phase transitions (DQPTs) -- in non-Hermitian systems. Focusing on one-dimensional models with chiral symmetry, we find DQPTs following the quench from a trivial to a non-Hermitian topological phase. Moreover, the number of critical momenta and critical time periods of the DQPTs are found to be directly related to the topological invariants of the non-Hermitian system. We further demonstrate our theory in three prototypical non-Hermitian lattice models, the lossy Kitaev chain (LKC), the LKC with next-nearest-neighbor hoppings, and the nonreciprocal Su-Schrieffer-Heeger model. Finally, we present a proposal to experimentally verify the found connection by a nitrogen-vacancy center in diamond., 33 pages, 9 figures, 3 tables

Stimulated Raman adiabatic passage allows robust transfer between two ends of a three-state quantum system and has been employed to transfer weakly bound Feshbach molecules into their deeply bound rovibrational ground state. However, the efficient transfer remains to be explored. Here we propose a possible alternative route, based on a recently developed non-Hermitian shortcut to adiabaticity method. It is able to realize single-step transfer efficiencies up to 100% even in the presence of a decaying excited level, surpassing all the previous methods. We also prove that our scheme is robust against the external field parameter fluctuations and is expected to be applicable for abundant molecular species.

We propose time-dependent Darboux (supersymmetric) transformations that provide a scheme for the calculation of explicitly time-dependent solvable non-Hermitian partner Hamiltonians. Together with two Hermitian Hamilitonians the latter form a quadruple of Hamiltonians that are related by two time-dependent Dyson equations and two intertwining relations in form of a commutative diagram. Our construction is extended to the entire hierarchy of Hamiltonians obtained from time dependent Darboux-Crum transformations. As an alternative approach we also discuss the intertwining relations for Lewis-Riesenfeld invariants for Hermitian as well as non-Hermitian Hamiltonians that reduce the time-dependent equations to auxiliary eigenvalue equations. The working of our proposals is discussed for a hierarchy of explicitly time-dependent rational, hyperbolic, Airy function and nonlocal potentials., 22 pages, 4 figures

We propose a scheme to realize various non-Hermitian topological phases in a topolectrical (TE) circuit network consisting of resistors, inductors, and capacitors. These phases are characterized by topologically protected exceptional points and lines. The positive and negative resistive couplings R g in the circuit provide loss and gain factors which break the Hermiticity of the circuit Laplacian. By controlling R g, the exceptional lines of the circuit can be modulated, e.g. from open curves to closed ellipses in the Brillouin zone. In practice, the topology of the exceptional lines can be detected by the impedance spectra of the circuit. We also considered finite TE systems with open boundary conditions, the admittance spectra of which exhibit highly tunable zero-admittance states demarcated by boundary points (BPs). The phase diagram of the system shows topological phases that are characterized by the number of their BPs. The transition between different phases can be controlled by varying the circuit parameters and tracked via the impedance readout between the terminal nodes. Our TE model offers an accessible and tunable means of realizing different topological phases in a non-Hermitian framework and characterizing them based on their boundary point and exceptional line configurations.

We study the dynamics of a non-Hermitian PT- symmetric Hamiltonian of a two level system (TLS) which has real eigenvalues. Within the framework of Hermitian quantum mechanics, it is known that maximal violation of Leggett–Garg inequality (LGI) is bounded by K 3 = 3/2 (Luder’s bound). We show that this absolute bound can be evaded when dynamics is governed by non-Hermitian PT- symmetric Hamiltonians with real eigenvalues. Moreover, the extent of violation can be optimized to asymptotically approach the algebraic maximum of K 3 max = 3 , which is otherwise observed for Hermitian Hamiltonian with infinite dimensional Hilbert space. The extreme violation of LGI is shown to be directly related to the two basic ingredients: (i) the Bloch equation for the TLS having a non-linear terms which allow for accelerated dynamics of states on the Bloch sphere exceeding all known quantum speed limits of state evolution; and (ii) quantum trajectory of states lie on a great circle (geodesic path) on the Bloch sphere at all times. We demonstrate that such extreme temporal correlation of TLS can be simulated in realistic system by embedding the TLS into a higher dimensional Hilbert space such that the composite system obeys unitary dynamics. Specifically we show that a four dimensional embedding of non-Hermitian PT- symmetric TLS is enough to host K 3 → 3 limit. We also discuss the effect of random noise on our results. Finally we conclude with a comparative study of our results with existing experimental realization of embedding.

The non hermiticity of the Hamiltonian is shown to be generated from the q-deformation of the algebra. As an example the q-deformed harmonic oscillator is investigated within the leading order approximation in the pseudo Hermitian perturbation theory. A generalized form of the metric is constructed. The spectrum as well as eigenstates of the system are derived explicitly.. Some applications and illustrations in q-deformed quantum statistical mechanics are also discussed.

The underwater target tracking environment usually has the characteristics of strong nonlinearity and non-Gaussian. The target tracking problem usually uses the nonlinear filtering algorithm, which combines the nonlinear measurement model with the linear system dynamics. The primary goal of target tracking is to extract accurate information about the real-time state of the target from the noise nonlinear observation obtained by the sensor. A new Gaussian Hermitt Kalman particle filter algorithm (GHKF-PF) is used to improve the tracking accuracy of the particle filter algorithm (PF). GHKF-PF uses the GHKF sampling method to sample particles from the posterior distribution of the target, calculates the mean and covariance for each particle, and uses the mean and variance to guide sampling. In the simulation experiment, the uniform linear motion model of underwater 3D target in the environment of Gaussian mixture noise (GMN) is established, and the application of GHKF-PF algorithm in 3D space is realized, and the accuracy is higher than that of PF. In order to further verify the effectiveness of the algorithm, a six-dimensional uniform acceleration motion model is established and compared with the extended particle filter (EPF) algorithm and the unscented particle filter (UPF) algorithm. The simulation results show that the performance of the GHKF-PF algorithm is similar to that of the UPF algorithm and better than the EPF algorithm.

Sorting and assigning phonon branches (e.g., longitudinal acoustic) of phonon modes is important for characterizing the phonon bands of a crystal and the determination of phonon properties such as the Gr\"uneisan parameter and group velocity. To do this, the phonon band indices (including the longitudinal and transverse acoustic) have to be assigned correctly to all phonon modes across a path or paths in the Brillouin zone. As our solution to this challenging problem, we propose a computationally efficient and robust two-stage hybrid method that combines two approaches with their own merits. The first is the perturbative approach in which we connect the modes using degenerate perturbation theory. In the second approach, we use numerical fitting based on least-squares fits to circumvent local connectivity errors at or near exact degenerate modes. The method can be easily generalized to other condensed matter problems involving Hermitian matrix operators such as electronic bands in tight-binding Hamiltonians or in a standard density-functional calculation, and photonic bands in photonic crystals., Comment: 6 figures, 7 pages

This article is dedicated to the following class of problems. Start with an $N\times N$ Hermitian matrix randomly picked from a matrix ensemble - the reference matrix. Applying a rank-$t$ perturbation to it, with $t$ taking the values $1\le t \le N$, we study the difference between the spectra of the perturbed and the reference matrices as a function of $t$ and its dependence on the underlying universality class of the random matrix ensemble. We consider both, the weaker kind of perturbation which either permutes or randomizes $t$ diagonal elements and a stronger perturbation randomizing successively $t$ rows and columns. In the first case we derive universal expressions in the scaled parameter $\tau=t/N$ for the expectation of the variance of the spectral shift functions, choosing as random-matrix ensembles Dyson's three Gaussian ensembles. In the second case we find an additional dependence on the matrix size $N$., Comment: 23 pages, 6 figures. Slightly revised version following referee comments

We provide the first solution of a time-dependent metric operator for the non-Hermitian Jaynes-Cummings Hamiltonian. We use this solution to calculate the entanglement between two identical isolated such Hamiltonians. The presence of a non-Hermitian interaction term leads to a spontaneously broken $\mathcal{PT}$-symmetric regime which manifests itself in the exotic time-evolution of entanglement. When the symmetry is broken, oscillatory modes transition into decay. As such that there is a drastic difference in behaviour between the broken and unbroken regimes., 12 pages, 1 figure

We propose complex Berry curvatures associated with the non-Hermitian Hamiltonian and its Hermitian adjoint and use these to reveal new physics in non-Hermitian systems. We give the complex Berry curvature and Berry phase for the two-dimensional non-Hermitian Dirac model. The imaginary part of the complex Berry phase induces susceptance so that the quantum Hall conductance is generalized to admittance for non-Hermitian systems. This implies that the non-Hermiticity of physical systems can induce intrinsic capacitive or inductive properties, depending on the non-Hermitian parameters. We analyze the complex energy band structures of the two-dimensional non-Hermitian Dirac model, determine the point and line gaps, and identify the conditions for their closure. We find that closure is associated with the exceptional degeneracy of the energy bands in the parameter space, which, in turn, is associated with topological phase transitions. In the continuum limit, we obtain the complex Berry phase in the parameter space.

Breakdown of bulk-boundary correspondence in non-Hermitian (NH) topological systems with generalized inversion symmetries is a controversial issue. The non-Bloch topological invariants determine the existence of edge states, but fail to describe the number and distribution of defective edge states in non-Hermitian topological systems. The state-dependent topological invariants, instead of a global topological invariant, are developed to accurately characterize the bulk-boundary correspondence of the NH systems, which is very different from their Hermitian counterparts. At the same time, we obtain the accurate phase diagram of the one-dimensional non-Hermitian Su–Schrieffer–Heeger model with a generalized inversion symmetry from the state-dependent topological invariants. Therefore, these results will be helpful for understanding the exotic topological properties of various non-Hermitian systems.

In this paper, we study how to apply a periodic driving field to control stable spin tunneling in a non-Hermitian spin–orbit (SO) coupled bosonic double-well system. By means of a high-frequency approximation, we obtain the analytical Floquet solutions and their associated quasienergies and thus construct the general non-Floquet solutions of the dissipative SO coupled bosonic system. Based on detailed analysis of the Floquet quasienergy spectrum, the profound effect of system parameters and the periodic driving field on the stability of spin-dependent tunneling is investigated analytically and numerically for both balanced and unbalanced gain–loss between two wells. Under balanced gain and loss, we find that the stable spin-flipping tunneling is preferentially suppressed with the increase of gain–loss strength. When the ratio of Zeeman field strength to periodic driving frequency Ω/ω is even, there is a possibility that continuous stable parameter regions will exist. When Ω/ω is odd, nevertheless, only discrete stable parameter regions are found. Under unbalanced gain and loss, whether Ω/ω is even or odd, we can get parametric equilibrium conditions for the existence of stable spin tunneling. The results could be useful for the experiments of controlling stable spin transportation in a non-Hermitian SO coupled system.

We develop a graphic approach for characterizing one-dimensional non-Hermitian topological phases. The eigenstates of energy bands are mapped to a graph on the torus, where a nontrivial topology exhibits as links. The topology of band touching exceptional points is a crucial aspect of a non-Hermitian system; the existence of exceptional point results in networks. We discuss the parity-time (   ) symmetric two-band models. The pseudo-anti-Hermiticity protects the band topology, and the eigenstate graphs in the exact   -symmetric phase locate on the torus surface under the   symmetry protection. For the Su-Schrieffer-Heeger ladder, the eigenstate graph is a Hopf link in the gapped nontrivial phase; chiral-time symmetry protects that the movable exceptional points appear in pairs in the real-energy gapless phase, and each exceptional point splits into a pair of exceptional points when the   symmetry breaks. The proposed graphic approach is applicable in one-dimensional N-band models. Our findings provide insight into one-dimensional non-Hermitian topology phases through visualizing the eigenstates.

We describe recent progress in understanding the continuous symmetry properties of non-Hermitian, PT-symmetric quantum field theories. Focussing on a simple non-Hermitian theory composed of one complex scalar and one complex pseudoscalar, we revisit the derivation of Noether's theorem to show that the conserved currents of non-Hermitian theories correspond to transformations that do not leave the Lagrangian invariant. We illustrate the impact that this has on the consistent formulation of (Abelian) gauge theories by studying a non-Hermitian extension of scalar quantum electrodynamics. We consider the spontaneous breakdown of both global and local symmetries, and describe how the Goldstone theorem and the Englert-Brout-Higgs mechanism are borne out for non-Hermitian, PT-symmetric theories., Comment: 7 pages, JPCS format. Prepared for the proceedings of DISCRETE2018: the Sixth Symposium on Prospects in the Physics of Discrete Symmetries, 26 - 30 November 2018, Austrian Academy of Sciences, Vienna, to appear in the Journal of Physics: Conference Series

In this work we address the study of topological phase protection of open quantum systems. Using the self-energy formalism, we investigate the paradigmatic case of an extended Kitaev model. The results show how the topological order can be affected by coupling the system to two external leads, giving rise to Non-Hermitian topological phases. Our results could be useful in spectroscopic measurements made on nanowire-based mesoscopic devices.

The Eigen problem is a problem that exists when there is an operator that is affected by the function, then that function does not change except being multipled by the constant in the form of an eigenvalue. In general, the eigenvalue problem uses the Hermitian operators by using observable variable in the form of mathematical operations. Hermitian operator is an operator that happens when the Hermit conjugation is performed, then the operator will return to its original condition. This research aims to determine the base matrix of Hermitian operators with order n ≤ 4 (size 2 × 2, 3 × 3 and 4 × 4). Entry element of the hermitian matrix operator uses complex numbers of which each order consists of few Hermitian operators. The method used is an analytical method by inserting problems into the eigen equation then normalized to obtain a diagonal matrix, so as to obtain a complete solution of the eigen problem on form of a matrix (including eigenvalues, normalized eigenvectors, and base matrix). Base matrix that has been obtained will be a diagonal matrix with diagonal elements of the eigenvalue. The results of this research indicate that the base matrix obtained in each order forms a square matrix that has entry elements in the form of complex numbers.

In this paper, based on a one-dimensional non-Hermitian spin model with $\mathcal{RT}$-invariant term, we study the non-Hermitian physics for the two (nearly) degenerate ground states. By using the high-order perturbation method, an effective pseudo-spin model is obtained to describe non-Hermitian physics for the two (nearly) degenerate ground states, which are precisely consistent with the numerical calculations. We found that there may exist effective (anti) $\mathcal{PT}$ symmetry for the effective pseudo-spin model of the two (nearly) degenerate ground states. In particular, there exists spontaneous (anti) $\mathcal{PT}$ -symmetry breaking for the topological degenerate ground states with tunable parameters in external fields. We also found that even a very tiny imaginary external field applied will drive $\mathcal{PT}$ phase transition.

The monopole for the geometric curvature is studied for non-Hermitian systems. We find that the monopole contains not only the exceptional points but also branch cuts. As the mathematical choice of branch cut in the complex plane is rather arbitrary, the monopole changes with the branch-cut choice. Despite this branch-cut dependence, our monopole is invariant under the $GL(l,\mathbb{C})$ gauge transformation that is inherent in non-Hermitian systems. Although our results are generic, they are presented in the context of a two-mode non-Hermitian Dirac model. A corresponding two-mode Hermitian system is also discussed to illustrate the essential difference between monopoles in Hermitian systems and non-Hermitian systems., Comment: 6 pages, 3 figures

Extrapolated new Hermitian and skew-Hermitian splitting method is presented for solving non-Hermitian and normal positive definite linear systems. We theoretically prove that this method is convergent in three different ranges of the parameters α and ω. Specially, the convergence results of the NHSS method are also obtained, which generalize the conclusions in [2]. The numerical example further verifies the results.

The geometry of double field theory (DFT) is described by a para-Hermitian manifold M. A tangent bundle of the para-Hermitian manifold T M is decomposed into two eigenbundles L and ∼L associated with the eigenvalues of the para-complex structure K. We define a Lie algebroid structure on the eigenbundles L, ∼L. The gauge symmetry algebra of DFT is governed by the C-bracket. The algebraic structure based on the C-bracket is not a Courant algebroid, but a metric algebroid proposed by Vaisman (the Vaisman algebroid). We show that the Vaisman algebroid in DFT is naturally defined on T M by an analogue of the Drinfel’d double of L, ∼L. We also find that an algebraic origin of the strong constraint is the condition for (L, ∼L) to become a Lie bialgebroid.

In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum $\bsA+\bsB$ of two independent random Hermitian matrices $\bsA$ and $\bsB$. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of $n$ random density matrices chosen from a random state ensemble (specified in the text) increases with the number $n$. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components $n$ in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels., 40 pages, 10 figures, LaTeX, the final version accepted for publication in J. Phys. A

The search of topological states in non-Hermitian systems has gained a strong momentum over the last two years climbing to the level of an emergent research front. In this Perspective we give an overview with a focus in connecting this topic to others like Floquet systems. Furthermore, using a simple scattering picture we discuss an interpretation of concepts like the Hamiltonian's defectiveness, i.e. the lack of a full basis of eigenstates, crucial in many discussions of topological phases of non-Hermitian Hamiltonians., 22 pages, 4 figures, invited article to appear in Journal of Physics: Materials, special issue

We study state conversion in parity-time (PT) symmetry broken non-Hermitian two level system. We construct a theory and explain underlying mechanism for state conversion and define adiabatic evolutions in non-Hermitian systems. The adiabatic theorem can be used if initial state is an instantaneous eigenstate in Hermitian systems. We show that the system can adiabatically follow the modulationally stable instantaneous eigenstate sooner or later, regardless of initial state if the system parameters are varied slowly in non-Hermitian systems. We discuss the topological feature of dynamical circling in the parameter space., Comment: To Appear in Journal of Physics A: Mathematical and Theoretical

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples --- adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors., Comment: 60 pages, 10 figures

Working within the context of PT-symmetric quantum mechanics, we begin by describing a non-Hermitian extension of QED that is both Lorentz invariant and consistent with unitarity. We show that the non-Hermitian Dirac mass matrix of this theory exhibits an exceptional point, corresponding to an effectively massless theory whose conserved current is either right- or left-chiral dominated. With this inspiration, we are able to construct a non-Hermitian model of light Dirac neutrino masses from Hermitian and anti-Hermitian Yukawa couplings that are both of order unity. We finish by highlighting potential phenomenological implications of this model., Comment: 10 pages, 1 figure. Prepared for the proceedings of DISCRETE2016: the Fifth Symposium on Prospects in the Physics of Discrete Symmetries, 28 November-3 December 2016, University of Warsaw, Poland, to appear in the Journal of Physics: Conference Series (JPCS). Presented by P. Millington

The Nearest Neighbour Spacing (NNS) distribution can be computed for generalized symmetric 2x2 matrices having different variances in the diagonal and in the off-diagonal elements. Tuning the relative value of the variances we show that the distributions of the level spacings exhibit a crossover from clustering to repulsion as in GOE. The analysis is extended to 3x3 matrices where distributions of NNS as well as Ratio of Nearest Neighbour Spacing (RNNS) show similar crossovers. We show that it is possible to calculate NNS distributions for Hermitian matrices (N=2, 3) where also crossovers take place between clustering and repulsion as in GUE. For large symmetric and Hermitian matrices we use interpolation between clustered and repulsive regimes and identify phase diagrams with respect to the variances., 15 pages, 7 figures

In this study, we introduce a two dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time independent Schr\"odinger equation yields real eigenvalues with complex eigenfunctions. We also construct the coherent state of the system by using a superposition of 12 eigenfunctions. Using the complex correspondence principle for the probability density we investigate the possible modifications in the probability densities due to the non-Hermitian aspect of the Hamiltonian., Comment: 11 pages and 5 figures

We consider wave dynamics for a Schrodinger equation with a non-Hermitian Hamiltonian satisfying the generalized (anyonic) parity-time symmetry , where and are the parity and time-reversal operators. For a stationary potential, the anyonic phase φ just rotates the energy spectrum of in a complex plane, however, for a drifting potential the energy spectrum is deformed and the scattering and localization properties of the potential show intriguing behaviors arising from the breakdown of the Galilean invariance when . In particular, in the unbroken phase the drift makes a scattering potential barrier reflectionless, whereas for a potential well the number of bound states decreases as the drift velocity increases because of a non-Hermitian delocalization transition.

It has been suggested that star products in phase-space quantization may be augmented to describe additional, classical effects. That proposal is examined critically here. Two known star products that introduce classical effects are: the generalized Husimi product of coarse-grained quantization, and a non-Hermitian damped star product for the harmonic oscillator. Following these examples, we consider products related by transition differential operators to the classic Moyal star product. We restrict to Hermitian star products, avoiding problems already pointed out for the original damped product. It is shown, however, that with such star products, augmented quantization is impossible, since an appropriate classical limit does not result. For a more complete study, we then also consider generalized, or local, transition operators, that depend on the local phase-space coordinates, as well as their derivatives. In this framework, one example of possible physical interest is constructed. Because of its limited validity and complicated form, however, it cannot be concluded that augmented quantization with local transition operators is practical., 17 pages, 1 figure

Canonical quantum mechanics postulates Hermitian Hamiltonians to ensure real eigenvalues. Counterintuitively, a non-Hermitian Hamiltonian, satisfying combined parity-time (PT) symmetry, could display entirely real spectra above some phase-transition threshold. Such a counterintuitive discovery has aroused extensive theoretical interest in extending canonical quantum theory by including non-Hermitian but PT-symmetric operators in the last two decades. Despite much fundamental theoretical success in the development of PT-symmetric quantum mechanics, an experimental observation of pseudo-Hermiticity remains elusive as these systems with a complex potential seem absent in Nature. But nevertheless, the notion of PT symmetry has highly survived in many other branches of physics including optics, photonics, AMO physics, acoustics, electronic circuits, material science over the past ten years, and others, where a judicious balance of gain and loss constitutes a PT-symmetric system. Here, although we concentrate upon reviewing recent progress on PT symmetry in optical microcavity systems, we also wish to present some new results that may help to accelerate the research in the area. Such compound photonic structures with gain and loss provide a powerful platform for testing various theoretical proposals on PT symmetry, and initiate new possibilities for shaping optical beams and pulses beyond conservative structures. Throughout this article there is an effort to clearly present the physical aspects of PT-symmetry in optical microcavity systems, but mathematical formulations are reduced to the indispensable ones. Readers who prefer strict mathematical treatments should resort to the extensive list of references. Despite the rapid progress on the subject, new ideas and applications of PT symmetry using optical microcavities are still expected in the future.

Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on Cd, it is proved that H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.

Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it can be constructed, as an advantage, directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out., 13 pages, no figures; v2: 14 pages, typos corrected, clarifications and references added, this version accepted for publication in Commun. Theor. Phys

We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. Our main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic Ensemble (GSE), respectively, in the limit $N\to \infty$. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. As a main input, we establish that the density of eigenvalues converges to the Wigner semicircle law and this holds even down to the smallest possible scale, and, moreover, we show that eigenvectors are fully delocalized. These results hold even without the condition that the matrix elements are identically distributed, only independence is used. In fact, we give strong estimates on the matrix elements of the Green function as well that imply that the local statistics of any two ensembles in the bulk are identical if the first four moments of the matrix elements match. Universality at the spectral edges requires matching only two moments. We also prove a Wegner type estimate and that the eigenvalues repel each other on arbitrarily small scales., Comment: 111 pages