non-Hermitian Problems Research Articles

Exceptional ring of the buoyancy instability in stars

We reveal properties of global modes of linear buoyancy instability in stars, characterized by the celebrated Schwarzschild criterion, using non-Hermitian topology. We identify a ring of exceptional points of order 4 that originates from the pseudo-Hermitian and pseudochiral symmetries of the system. The ring results from the merging of a dipole of degeneracy points in the Hermitian stably-stratified counterpart of the problem. Its existence is related to spherically symmetric unstable modes. We obtain the conditions for which convection grows over such radial modes. Those are met at early stages of low-mass stars formation. We finally show that a topological wave is robust to the presence of convective regions by reporting the presence of a mode transiting between the wavebands in the non-Hermitian problem, strengthening their relevance for asteroseismology. Published by the American Physical Society 2024

Dispersive hydrodynamics in non-Hermitian nonlinear Schrödinger equation with complex external potential

In this paper dispersive hydrodynamics associated with the non-Hermitian nonlinear Schrödinger (NLS) equation with generic complex external potential is studied. In particular, a set of dispersive hydrodynamic equations are obtained. They differ from their classical counterparts (without an external potential), by the presence of additional source terms that alter the density and momentum equations. When restricted to a class of Wadati-type complex potentials, the resulting hydrodynamic system conserves a modified momentum and admits constant intensity/density solutions. This motivates the construction and study of an initial value problem (IVP) comprised of a centred (or non-centred) step-like initial condition that connects two constant intensity/density states. Interestingly, this IVP is shown to be related to a Riemann problem posed for the hydrodynamic system in an appropriate traveling reference frame. The study of such IVPs allows one to interpret the underlying non-Hermitian Riemann problem in terms of an ‘optical flow’ over an obstacle. A broad class of non-Hermitian potentials that lead to modulationally stable constant intensity states are identified. They are subsequently used to numerically solve the associated Riemann problem for various initial conditions. Due to the lack of translation symmetry, the resulting long-time dynamics show a dependence on the location of the step relative to the potential. This is in sharp contrast to the NLS case without potential, where the dynamics are independent of the step location. This fact leads to the formation of diverse nonlinear wave patterns that are otherwise absent. In particular, various gain-loss generated near-field features are present, which in turn drive the optical flow in the far-field which could be comprised of various rich nonlinear wave structures, including DSW-DSW, DSW-rarefaction, and soliton-DSW interactions.

Simulation of Linear Non-Hermitian Boundary-Value Problems with Quantum Singular-Value Transformation

We propose a quantum algorithm for simulating dissipative waves in inhomogeneous linear media as a boundary-value problem. Using the so-called quantum singular value transformation (QSVT), we construct a quantum circuit that models the propagation of electromagnetic waves in a one-dimensional system with outgoing boundary conditions. The corresponding measurement procedure is also discussed. Limitations of the QSVT algorithm are identified in connection with the large condition numbers that the dispersion matrices exhibit at weak dissipation.

Entanglement and correlation spreading in non-Hermitian spin chains

Non-Hermitian quantum many-body systems are attracting widespread interest for their exotic properties, including unconventional quantum criticality and topology. Here we study how quantum information and correlations spread under a quantum quench generated by a prototypical non-Hermitian spin chain. Using the mapping to fermions we solve exactly the problem and compute the entanglement entropy and the correlation dynamics in the thermodynamic limit. Depending on the quench parameters, we identify two dynamical phases. One is characterized by rapidly saturating entanglement and correlations. The other instead presents a logarithmic growth in time, and correlations spreading faster than the Lieb-Robinson bound, with collapses and revivals giving rise to a modulated light-cone structure. Here, in the long-time limit, we compute analytically the entanglement entropy that we show to scale logarithmically with the size of the cut, with an effective central charge that we obtain in closed form. Our results provide an example of an exactly solvable non-Hermitian many-body problem that shows rich physics including entanglement and spectral transitions.

Real-complex quantum phase transition in non-Hermitian disorder-free systems

Localization phenomena and the quantum phase transition are two core concepts of low-dimensional condensed-matter physics. The localization transition problem related to vortex line pinning in superconductors can be studied by mapping it to a real-complex non-Hermitian problem. Its relation to the quantum phase transition has to be discerned. We explore these two phase transitions induced by interactions rather than by the disorder in non-Hermitian systems. It is shown that the phase diagram is divided into the classical and quantum regimes by a characteristic temperature. The classical regime contains topological localization transitions and a tricritical point which connects the first-order phase transition line to the second-order transition line. The quantum regime is a nonchaotic and first-order phase transition. In such a quantum regime, the relaxation time does not always satisfy the bound on chaos. We show that the oscillation phase transition line due to quantized Matsubara frequencies can give an index similar in structure to the quantum oscillation in an imaginary magnetic field, which makes the first-order quantum phase diagram behave as a quantum critical phase diagram.

Multi-Resolution Localized Orthogonal Decomposition for Helmholtz Problems

We introduce a novel multi-resolution localized orthogonal decomposition (LOD) for time-harmonic acoustic scattering problems that can be modeled by the Helmholtz equation. The method merges the concepts of LOD and operator-adapted wavelets (gamblets) and proves its applicability for a class of complex-valued, non-hermitian, and indefinite problems. It computes hierarchical bases that block-diagonalize the Helmholtz operator and thereby decouples the discretization scales. Sparsity is preserved by a novel localization strategy that improves stability properties even in the elliptic case. We present a rigorous stability and a priori error analysis of the proposed method for homogeneous media. In addition, we investigate the fast solvability of the blocks by a standard iterative method. A sequence of numerical experiments illustrates the sharpness of the theoretical findings and demonstrates the applicability to scattering problems in heterogeneous media.

Modal approximation for plasmonic resonators in the time domain: the scalar case

We study the electromagnetic field scattered by a metallic nanoparticle with dispersive material parameters in a resonant regime. We consider the particle placed in a homogeneous medium in a low-frequency regime. We define modes for the non-Hermitian problem as perturbations of electro-static modes, and obtain a modal approximation of the scattered field in the frequency domain. The poles of the expansion correspond to the eigenvalues of a singular boundary integral operator and are shown to lie in a bounded region near the origin of the lower-half complex plane. Finally, we show that this modal representation gives a very good approximation of the field in the time domain. We present numerical simulations in two dimensions to corroborate our results.

Generalized SOR-Like Iteration Method for Linear Complementarity Problem

In this paper, we present a generalized SOR-like iteration method to solve the non-Hermitian positive definite linear complementarity problem (LCP), which is obtained by reformulating equivalently the implicit fixed-point equation of the LCP as a two-by-two block nonlinear equation. The convergence properties of the generalized SOR-like iteration method are discussed under certain conditions. Numerical experiments show that the generalized SOR-like method is efficient, compared with the SOR-like method and the modulus-based SOR method.

On the regularization matrix of the regularized DPSS preconditioner for non-Hermitian saddle-point problems

In this work, with respect to the regularization matrix of the new regularized deteriorated positive and skew-Hermitian splitting (RDPSS) preconditioner for non-Hermitian saddle-point problems, we provide a class of Hermitian positive semidefinite matrices, which depend on certain parameters, for practical computations. A precise description about the eigenvalue distribution of the corresponding preconditioned matrix is given. The condition number of the eigenvector matrix, which partly determines the convergence rate of the related preconditioned Krylov subspace method, is also discussed in this work. Finally, some numerical experiments are carried out to identify the effectiveness of the presented special choices for the regularization matrix to solve the non-Hermitian saddle-point problems.

A Contour-Integral Based Method with Schur–Rayleigh–Ritz Procedure for Generalized Eigenvalue Problems

Recently, a class of eigensolvers based on contour integrals has been developed for computing the eigenvalues inside a given region in the complex plane. The CIRR method (a Rayleigh–Ritz type method with contour integrals) is a classic example among this kind of methods. It first constructs a subspace to contain the eigenspace of interest via a set of contour integrals, and then uses the standard Rayleigh–Ritz procedure to extract desired eigenpairs. However, it was shown that the CIRR method may fail to find the desired eigenpairs when the considered eigenproblem is non-Hermitian. This fact motivates us to develop a non-Hermitian scheme for the CIRR method. To this end, we formulate a Schur–Rayleigh–Ritz procedure to extract the desired eigenpairs. The theoretical analysis shows that our new extraction scheme can make the CIRR method also applicable for the non-Hermitian problems. Some implementation issues arising in practical applications are also studied. Numerical experiments are reported to illustrate the numerical performance of our new method.

Interacting non-Hermitian ultracold atoms in a harmonic trap: Two-body exact solution and a high-order exceptional point

We study interacting ultracold atoms in a three-dimensional (3D) harmonic trap with spin-selective dissipations, which can be effectively described by non-Hermitian parity-time ($\mathcal{PT}$) symmetric Hamiltonians. By exactly solving the non-Hermitian two-body problem of spin-1/2 (spin-1) bosons in a 3D harmonic trap, we find that the system can exhibit third-order (fifth-order) exceptional points (EPs) with ultrasensitive cube-root (fifth-root) spectral response due to interaction anisotropies in spin channels. We also present the general principle for the creation of high-order EPs and their spectral sensitivities with arbitrary particle number $N$ and arbitrary spin $s$. Generally, with spin-independent interactions, the EP order of bosons can be as high as $2Ns+1$, and the spectral response around EP can be as sensitive as $\ensuremath{\sim}{\ensuremath{\epsilon}}^{1/(2ks+1)}$ under a $k$-body interaction anisotropy. Moreover, we propose to detect the ultrasensitive spectral response through the probability dynamics of certain state. These results suggest a convenient route towards more powerful sensor devices in spinor cold atomic systems.

A harmonic FEAST algorithm for non-Hermitian generalized eigenvalue problems

The FEAST algorithm, a contour-integral based eigensolver, was developed for computing the eigenvalues inside a given interval, along with their eigenvectors, of a Hermitian generalized eigenproblem. The FEAST algorithm is a fast and stable technique, and is easily parallelizable. However, it was shown that this algorithm may fail to find the desired eigenpairs when applied to non-Hermitian problems. Efforts have been made to adapt FEAST to non-Hermitian problems. In this work, we aim at formulating a new non-Hermitian scheme for the FEAST algorithm. Our new scheme is based on the partial generalized Schur decomposition. Unlike the existing non-Hermitian FEAST methods, our method seeks approximation to the generalized Schur vectors instead of the potentially ill-conditioned eigenvectors. Our new method can compute not only the eigenvectors but also the right and left generalized Schur vectors corresponding to target eigenvalues. We present standard benchmark numerical experiments in which our method achieves better stability and efficiency comparing with the existing non-Hermitian FEAST algorithms.

Compact numerical solutions to the two-dimensional repulsive Hubbard model obtained via nonunitary similarity transformations

Similarity transformation of the Hubbard Hamiltonian using a Gutzwiller correlator leads to a non-Hermitian effective Hamiltonian, which can be expressed exactly in momentum-space representation, and contains three-body interactions. We apply this methodology to study the two-dimensional Hubbard model with repulsive interactions near half-filling in the intermediate interaction strength regime ($U/t=4$). We show that at optimal or near optimal strength of the Gutzwiller correlator, the similarity transformed Hamiltonian has extremely compact right eigenvectors, which can be sampled to high accuracy using the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method, and its initiator approximation. Near-optimal correlators can be obtained using a simple projective equation, thus obviating the need for a numerical optimisation of the correlator. The FCIQMC method, as a projective technique, is well-suited for such non-Hermitian problems, and its stochastic nature can handle the 3-body interactions exactly without undue increase in computational cost. The highly compact nature of the right eigenvectors means that the initiator approximation in FCIQMC is not severe, and that large lattices can be simulated, well beyond the reach of the method applied to the original Hubbard Hamiltonian. Results are provided in lattice sizes upto 50 sites and compared to auxiliary-field QMC. New benchmark results are provided in the off half-filling regime, with no severe sign-problem being encountered. In addition, we show that methodology can be used to calculate excited states of the Hubbard model and lay the groundwork for the calculation of observables other than the energy.

On the non-Hermitian FEAST algorithms with oblique projection for eigenvalue problems

The FEAST algorithm, due to Polizzi, is a typical contour-integral based eigensolver for computing the eigenvalues, along with their eigenvectors, inside a given region in the complex plane. It was formulated under the circumstance that the considered eigenproblem is Hermitian. The FEAST algorithm is stable and accurate, and has attracted much attention in recent years. However, Yin et al. (2017) observed that the FEAST algorithm may fail to find the target eigenpairs when applied to non-Hermitian problems. This observation motivated them to formulate a non-Hermitian version for the FEAST algorithm. The key to the success of their method is that they use the oblique projection, rather than the orthogonal projection used in the FEAST algorithm, to extract desired eigenpairs. The objective of the present work is two-folded. Firstly, we would like to present a new non-Hermitian scheme for the FEAST algorithm, which also uses the oblique projection technique to extract the desired eigenpairs. Secondly, we would like to study the convergence properties of the non-Hermitian FEAST algorithms based on oblique projection to illustrate the effectiveness. Numerical experiments are reported to demonstrate the numerical performance of the non-Hermitian FEAST algorithms based on oblique projection and to validate the convergence properties studied.

High-order exceptional points in ultracold Bose gases

We show that arbitrarily high-order exceptional points (EPs) can be achieved in a repulsively interacting two-species Bose gas in one dimension. By exactly solving the non-Hermitian two-boson problem, we demonstrate the existence of third-order EPs when the system is driven across the parity-time symmetry breaking transition. We further address the fourth-order EPs with three bosons and generalize the results to $N$-body system, where the EP order can be as high as $N+1$. Physically, such high order originates from the intrinsic ferromagnetic correlation in spinor bosons, which renders the entire system collectively behave as a single huge spin. Moreover, we show how to create ultra-sensitive spectral response around EPs via an interaction anisotropy in different spin channels. Our work puts forward the possibility of atomic sensors made from highly controllable ultracold gases.

On the Spectrum of Deflated Matrices with Applications to the Deflated Shifted Laplace Preconditioner for the Helmholtz Equation

The deflation technique for accelerating Krylov subspace iterative methods for the solution of linear systems has long been well established. The first landmark papers of Nicolaides and Dostál date back to the late 1980s, where deflation is used for Hermitian positive definite (HPD) linear systems. In the last decade deflation has been used and analyzed in combination with domain decomposition and multigrid methods, leading to very efficient algorithms. Examples are the multilevel Krylov methods introduced by Erlangga and Nabben, and the works of Sheikh et al. in which multilevel deflation techniques are presented. Although these algorithms work very well in practice for non-Hermitian problems, not many theoretical results are yet known for this class of problems. We study here the general case of deflation operators in arbitrary inner products, and give inclusion regions for the spectrum of an arbitrary deflated matrix based on the field of values. The new inclusion regions generalize previous results for HPD matrices. Moreover, we give a bound based on the field of values for the norms of the residuals of deflated GMRES. We apply our results to linear systems arising from the Helmholtz equation, focusing on the combination of the complex shifted Laplace (CSL) preconditioner with a two-level deflation preconditioner, which is the basis of a multilevel method introduced by Erlangga and Nabben. The analysis here only covers the case of exact inversion of the CSL preconditioner. It has been observed in numerical experiments that the eigenvalues of the deflated CSL-preconditioned system lie on the exact same disks as the CSL-preconditioned linear systems and are shifted away from zero. Here we are able to prove these surprising results for a class of problems with Dirichlet and first-order Sommerfeld boundary conditions. Our results help to explain the high performance of multilevel Krylov methods based on deflation for the Helmholtz equation.

Self-hybridization within non-Hermitian localized plasmonic systems

Common intuition in physics is based on the concept of orthogonal eigenmodes. Those are well de- fined solutions of Hermitian equations used to describe many physical situations, from quantum mechanics to acoustics. A large variety of non-Hermitian problems, including gravitational waves close to black holes or leaky electromagnetic cavities require the use of bi-orthogonal eigenbasis. Physical consequences of non-Hermiticity challenge our physical understanding. However, the usual need to compensate for energy losses made the few successful attempts to probe non-Hermiticity extremely complicated. We show that this issue can be overcome considering localized plasmonic systems. Indeed, since the non-Hermiticity in these systems does not stem from from temporal invariance breaking but from spatial symmetry breaking, its consequences can be observed more easily. We report on the theoretical and experimental evidence of non-Hermiticity induced strong coupling between surface plasmon modes of different orders within silver nano-crosses. The sym- metry conditions for triggering this counter-intuitive self-hybridization phenomenon are provided. Similarly, observable effects are expected to be observed in any system subtending bi-orthogonal eigenmodes.

Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems

A detailed exposition of highly efficient and accurate method for the propagation of the time-dependent Schrödinger equation [50] is presented. The method is readily generalized to solve an arbitrary set of ODE's. The propagation is based on a global approach, in which large time-intervals are treated as a whole, replacing the local considerations of the common propagators. The new method is suitable for various classes of problems, including problems with a time-dependent Hamiltonian, nonlinear problems, non-Hermitian problems and problems with an inhomogeneous source term. In this paper, a thorough presentation of the basic principles of the propagator is given. We give also a special emphasis on the details of the numerical implementation of the method. For the first time, we present the application for a non-Hermitian problem by a numerical example of a one-dimensional atom under the influence of an intense laser field. The efficiency of the method is demonstrated by a comparison with the common Runge–Kutta approach.

A FEAST algorithm with oblique projection for generalized eigenvalue problems

The contour integral-based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best-known members are the Sakurai–Sugiura method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non-Hermitian problems. Recently, a dual subspace FEAST algorithm was proposed to extend the FEAST algorithm to non-Hermitian problems. In this paper, we instead use the oblique projection technique to extend FEAST to the non-Hermitian problems. Our approach can be summarized as follows: (a) construct a particular contour integral to form a search subspace containing the desired eigenspace and (b) use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. Comparing to the dual subspace FEAST algorithm, we can save the computational cost roughly by a half if only the eigenvalues or the eigenvalues together with their right eigenvectors are needed. We also address some implementation issues such as how to choose a suitable starting matrix and design-efficient stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.

A New Uzawa-Type Iteration Method for Non-Hermitian Saddle-Point Problems

AbstractBased on a preconditioned shift-splitting of the (1,1)-block of non-Hermitian saddle-point matrix and the Uzawa iteration method, we establish a new Uzawa-type iteration method. The convergence properties of this iteration method are analyzed. In addition, based on this iteration method, a preconditioner is proposed. The spectral properties of the preconditioned saddle-point matrix are also analyzed. Numerical results are presented to verify the robustness and the efficiency of the new iteration method and the preconditioner.