Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians

Abstract

A phenomenological Hamiltonian of a closed (i.e., unitary) quantum system is assumed to have an $N$ by $N$ real-matrix form composed of a unperturbed diagonal-matrix part $H_0^{(N)}$ and of a tridiagonal-matrix perturbation $\lambda W^{(N)}(\lambda )$. The requirement of the unitarity of the evolution of the system (i.e., of the diagonalizability and of the reality of the spectrum) restricts, naturally, the variability of the matrix elements to a "physical" domain ${\mathcal{D}}^{[N]}\subset {\mathbb{R}}^d$. We fix the unperturbed matrix (simulating a non-equidistant, square-well-type unperturbed spectrum) and we only admit the maximally non-Hermitian antisymmetric-matrix perturbations. This yields the hiddenly Hermitian model with the measure of perturbation $\lambda$ and with the $d=N$ matrix elements which are, inside ${\mathcal{D}}^{[N]}$, freely variable. Our aim is to describe the quantum phase-transition boundary $\partial {\mathcal{D}}^{[N]}$ (alias exceptional-point boundary) at which the unitarity of the system is lost. Our main attention is paid to the strong-coupling extremes of stability, i.e., to the Kato's exceptional points of order $N$ (EPN) and to the (sharply spiked) shape of the boundary $\partial {\mathcal{D}}^{[N]}$ in their vicinity. The feasibility of our constructions is based on the use of the high-precision arithmetics in combination with the computer-assisted symbolic manipulations (including, in particular, the Gröbner basis elimination technique).

arXiv comment27 pp

References in corpus (24)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermitian Representation of Quantum Mechanics
• An Atom Laser with a cw Output Coupler
• The emergence, coalescence and topological properties of multiple exceptional points and their experimental realization
• A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points
• Pseudospectra in non-Hermitian quantum mechanics
• On the metric operator for the imaginary cubic oscillator
• Operator (quasi-)similarity, quasi-Hermitian operators and all that
• Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime
• Bifurcations in Resonance Widths of an Open Bose-Hubbard Dimer
• Maximal couplings in PT-symmetric chain-models with the real spectrum of energies
• Fragile PT-symmetry in a solvable model
• Admissible perturbations and false instabilities in PT-symmetric quantum systems
• Complex symmetric Hamiltonians and exceptional points of order four and five
• Quantum inner-product metrics via recurrent solution of Dieudonne equation
• Determination of the domain of the admissible matrix elements in the four-dimensional PT-symmetric anharmonic model
• Two patterns of PT-symmetry breakdown in a non-numerical six-state simulation
• Generalized Bose-Hubbard Hamiltonians exhibiting a complete non-Hermitian degeneracy
• Anomalous mechanisms of the loss of observability in non-Hermitian quantum models
• IR-truncated $\mathcal{P}\mathcal{T}-$symmetric $i{x}^3$ model and its asymptotic spectral scaling graph
• Unitary unfoldings of Bose-Hubbard exceptional point with and without particle number conservation
• Quantum phase transitions mediated by clustered non-Hermitian degeneracies
• Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities

Cited by in corpus (8)

• Confluences of exceptional points and a systematic classification of quantum catastrophes
• Exceptional point in a coupled Swanson system
• Exceptional points and quantum phase transition in a fermionic extension of the Swanson oscillator
• Analogue Hawking radiation as a tunneling in a two-level $\mathcal{P}\mathcal{T}$-symmetric system
• Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via $\mathcal{P}\mathcal{T}-$symmetric and exactly solvable linear-Bose-Hubbard building blocks
• A reappraisal of Lagrangians with non-quadratic velocity dependence and branched Hamiltonians
• Interference of non-Hermiticity with Hermiticity at exceptional points
• Swanson Hamiltonian: non-PT-symmetry phase