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Three classes of finite-dimensional models of quantum systems exhibiting
spectral degeneracies called quantum catastrophes are described in detail.
Computer-assisted symbolic manipulation techniques are shown unexpectedly
efficient for the purpose.
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Context 1
... be summarized as follows. First of all, while starting from matrix dimension N = 4 it makes sense to pre-assist the algebraic manipulations by recalling some graphical software. A rough orientation can be obtained concerning both the topology of spectra and of the EP boundaries ∂D. A characteristic sample of the latter structure is provided by Fig. 1. Starting from N = 4 it makes also sense to search for the multiply and, in particular, maximally degenerate EPs (MEPs), i.e., for the vertices of the spikes as seen in our two-dimensional Fig. 1. The required algorithm may be entirely universal. Indeed, as long as all of the roots s n , n = 1, 2, . . . , N of our secular equation must ...
Context 2
... can be obtained concerning both the topology of spectra and of the EP boundaries ∂D. A characteristic sample of the latter structure is provided by Fig. 1. Starting from N = 4 it makes also sense to search for the multiply and, in particular, maximally degenerate EPs (MEPs), i.e., for the vertices of the spikes as seen in our two-dimensional Fig. 1. The required algorithm may be entirely universal. Indeed, as long as all of the roots s n , n = 1, 2, . . . , N of our secular equation must degenerate to zero at MEPs, each coefficient in the secular polynomial must vanish (up to the leading one of course). One obtains a multiplet of polynomial equations, perfectly fitting the ...
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Citations
... The intrigue surrounding EPs deepens when considering systems with more than two states, where higher-order EPs [18] appear, accompanied by the vanishing of more than one state, which provides new ideas for enhanced sensing [13,19] and emission [20]. In-depth studies of complex systems reveal the interaction can lead to the emergence, coalescence, and topological properties of multiple EPs [21][22][23][24][25][26]. These complex systems have been shown to exhibit a rich tapestry of physical phenomena, each with its own potential for groundbreaking applications that transcend those found in simpler two-state systems. ...
Exceptional points (EPs), which signify the singularity of eigenvalues and eigenstates in non-Hermitian systems, have garnered considerable attention in two-state systems, revealing a wealth of intriguing phenomena. However, the potential of EPs in multi-state systems, particularly their interaction and coalescence, has been underexplored, especially in the context of electromagnetic fields where far-field coupling can revolutionize spatial wave control. Here, we theoretically and computationally explore the coalescence of multiple EPs within a designer surface plasmonic quadrumer system. The coupled mode model shows that the multiple EPs can emerge and collide as the system parameters vary, leading to higher-order singularities. Numerically calculated results showcase that multiple EPs with different orders have special far-field responses. This pioneering strategy heralds a new era of wavefront engineering in non-Hermitian photonic structures, presenting a transformative class of radiative systems that transcend the conventional frequency spectrum from microwave to optical realms.
... Very recently we imagined that any quantization of a classical catastrophe must replace the usual schematic time-dependent trajectories x = x(t) in a classical space of states by certain selfadjoint operatorsx(t) in a suitable infinite-dimensional physical Hilbert space H (P) [31][32][33]. Thus, after quantization the concepts such as bifurcation (etc.) must necessarily be transferred from the domain of algebraic geometry to the area of functional analysis and spectral theory. ...
For the displaced harmonic double-well oscillator, the existence of exact polynomial bound states at certain displacements d is revealed. The N-plets of these quasi-exactly solvable (QES) states are constructed in closed form. For non-QES states, the Schrödinger equation can still be considered “non-polynomially exactly solvable” (NES) because the exact left and right parts of the wave function (proportional to confluent hypergeometric function) just have to be matched in the origin.
... Our choice of the AO model of [35] was motivated not only by its immediate phenomenological appeal (e.g. the details in [42][43][44]) but also by its most elementary real-matrix nature simplifying its non-numerical tractability at any finite matrix dimension N [45][46][47]. This merit contributed to the present project. ...
The description of unitary evolution using non-Hermitian but ‘hermitizable’ Hamiltonians H is feasible via an ad hoc metric Θ = Θ ( H ) and a (non-unique) amendment 〈 ψ 1 | ψ 2 〉 → 〈 ψ 1 | Θ | ψ 2 〉 of the inner product in Hilbert space. Via a proper fine-tuning of Θ ( H ) this opens the possibility of reaching the boundaries of stability (i.e. exceptional points) in many quantum systems sampled here by the fairly realistic Bose–Hubbard (BH) and discrete anharmonic oscillator (AO) models. In such a setting, it is conjectured that the EP singularity can play the role of a quantum phase-transition interface between different dynamical regimes. Three alternative ‘AO ↔ BH’ implementations of such an EP-mediated dynamical transmutation scenario are proposed and shown, at an arbitrary finite Hilbert-space dimension N , exact and non-numerical.
... The study of quantum systems in their quasi-Hermitian and PT −symmetric discrete representations usually originates from the needs of open-system studies [33] or of classical optics [34], especially in the light of the current quick developments of nanotechnologies [35]. Here we proceeded in a complementary direction of connecting these models with the simulations of the various forms of quantum phase transitions (cf., e.g., [36,37]). ...
At the lower edge of the energy continuum the birth of an isolated quantum bound state is studied as caused by an infinitesimally small change of the interaction. In our model a single, asymptotically free massive quantum particle is assumed moving along a discretized real line of coordinates, . The motion is assumed controlled by a weakly nonlocal 2J-parametric external potential which is non-Hermitian but PT-symmetric. Mathematically, the bound states are then reinterpreted as Sturmians, i.e., the bound-state energy is treated as a variable real parameter while the value of one of the couplings (responsible for the existence of the bound state) is determined via the standard secular equation. It is found that in such an arrangement the model is exactly solvable at all of the finite counts J of the couplings. For illustration, the explicit closed bound-state formulae are presented up to J=7.
... Such a point is defined by a certain ratio between the structure's parameters Γ : |τ 1 | : ... : |τ N −1 |, i.e. it is a specific point in the projective space of the parameters. Thus, one can treat an N -order EP as a quantum catastrophe 53,54 of the auxiliary Hamiltonian. At the very moment of the resonance coalescence the polynomial Q has only one degenerate root ω = ε 0 and so it has the form: Q CR = (ω − ε 0 ) N . ...
We study the phenomenon of spontaneous symmetry breaking in dissipationless resonant tunneling heterostructures (RTS). To describe the quantum transport in this system we apply both the nonequilibrium Green function formalism based on a tight-binding model and a numerical solution of the Schroedinger equation within the envelope wavefunction formalism. An auxiliary non-Hermitian Hamiltonian is introduced. Its eigenvalues determine exactly the transparency peak positions. We present a procedure how to construct a family of non-Hermitian Hamiltonians with real eigenvalues. In general these Hamiltonians do not have PT-symmetry. In spatially symmetric RTS the corresponding auxiliary non-Hermitian Hamiltonian becomes PT-symmetric and possesses real eigenvalues, which can coalesce at exceptional points (EP) of Hamiltonian. A coalescence of the auxiliary non-Hermitian Hamiltonian eigenvalues means a coalescence of resonances in RTS, which is accompanied be symmetry breaking of the electron wavefunction probability distribution (at a given direction of the particle flow). We construct a classification of different types of the peak coalescence in terms of the catastrophe theory and investigate the impact of dissipation and asymmetry on these phenomena. Possible applications include sensors and broad-band filters.
... For example, the PT symmetry recovery behaviors have been found in multiple optical waveguide system [30,31] and photonic crystals [32]. In general, multiple EPs can form [32][33][34][35][36] in a multistate system, and their interactions may lead to the coalescence of two or more EPs, which in turn gives rise to new physics including new singularities with different topological properties, which cannot be described by a 2 × 2 matrix [32][33][34][35][36]. ...
... For example, the PT symmetry recovery behaviors have been found in multiple optical waveguide system [30,31] and photonic crystals [32]. In general, multiple EPs can form [32][33][34][35][36] in a multistate system, and their interactions may lead to the coalescence of two or more EPs, which in turn gives rise to new physics including new singularities with different topological properties, which cannot be described by a 2 × 2 matrix [32][33][34][35][36]. ...
Non-Hermitian systems distinguish themselves from Hermitian systems by
exhibiting a phase transition point called an exceptional point (EP), which is
the point at which two eigenstates coalesce under a system parameter variation.
Many interesting EP phenomena such as level crossings/repulsions in
nuclear/molecular and condensed matter physics, and unusual phenomena in optics
such as loss-induced lasing and unidirectional transmission can be understood
by considering a simple 2x2 non-Hermitian matrix. At a higher dimension, more
complex EP physics not found in two-state systems arises. We consider the
emergence and interaction of multiple EPs in a four-state system theoretically
and realize the system experimentally using four coupled acoustic cavities with
asymmetric losses. We find that multiple EPs can emerge and as the system
parameters vary, these EPs can collide and merge, leading to higher order
singularities and topological characteristics much richer than those seen in
two-state systems.
... Finite-dimensional toy models, in addition to having diverse applications in quantum and solid-state physics per se [13,14,15,16], provide also vast possibilities for the description of various physical phenomena in simplified scenarios. Indeed, finite quasi-hermitian Hamiltonians have been succesfully used to model quantum phase transitions [17], quantum catastrophes [18] or even simplified big-bang scenarios [19]. The objects of interest in all these cases are the so-called exceptional points [20,21], which emerge inevitably on boundaries of observability domains. ...
We study a general class of -symmetric tridiagonal Hamiltonians
with purely imaginary interaction terms in the quasi-hermitian representation
of quantum mechanics. Our general Hamiltonian encompasses many previously
studied lattice models as special cases. We provide numerical results regarding
domains of observability and exceptional points, and discuss the possibility of
explicit construction of general metric operators (which in turn determine all
the physical Hilbert spaces). The condition of computational simplicity for the
metrics motivates the introduction of certain one-parametric special cases,
which consequently admit closed-form extrapolation patterns of the
low-dimensional results.
... A large subclass of tridiagonal quasi-hermitian matrices has a close connection to the theory of orthogonal polynomials [19,20]. Inspired by these models, we define the dual-SSH (or dSSH) model, which, in parallel with the operator-matrix correspondence of (7), has its two-site form ...
... SSH dSSH + k even, i odd + k odd, i odd − k even, i even − k odd, i even 1 k odd 1 k even (19) ...
A Su-Schrieffer-Heeger model with added PT-symmetric boundary
term is studied in the framework of pseudo-hermitian quantum mechanics. For two special cases, a complete set of pseudometrics is constructed in closed form. When complemented with a condition of positivity, the pseudometrics determine all the physical inner
products of the considered model.
... also Refs. [24,25]) it became clear that the transitions of quantum systems through their Jordan-block alias multiple-EP (or degenerate-EP) quantumphase-transition points may prove to be a phenomenologically relevant and interesting process. In fact, there emerged no true surprises in the context of mathematics where one simply observed differing patterns of behavior (and, in particular, of the complexification) of certain eigenvalues before and after the EP singularity. ...
The realization of a genuine phase transition in quantum mechanics requires
that at least one of the Kato's exceptional-point parameters becomes real. A
new family of finite-dimensional and time-parametrized quantum-lattice models
with such a property is proposed and studied. All of them exhibit, at a real
exceptional-point time t=0, the Jordan-block spectral degeneracy structure of
some of their observables sampled by Hamiltonian H(t) and site-position
Q(t). The passes through the critical instant t=0 are interpreted as
schematic simulations of non-equivalent versions of the Big-Bang-like quantum
catastrophes.
The phenomenon of degeneracy of an N-plet of bound states is studied in the framework of the quasi-Hermitian (a.k.a. PT-symmetric) formulation of quantum theory of closed systems. For a general non-Hermitian Hamiltonian H=H(λ) such a degeneracy may occur at a real Kato's exceptional point λ(EPN) of order N and of the geometric multiplicity alias clusterization index K. The corresponding unitary process of collapse (loss of observability) can be then interpreted as a generic quantum phase transition. The dedicated literature deals, predominantly, with the non-numerical benchmark models of the simplest processes where K=1. In our present paper it is shown that in the “anomalous” dynamical scenarios with 1<K≤N/2 an analogous approach is applicable. A multiparametric anharmonic-oscillator-type exemplification of such systems is constructed as a set of real-matrix N by N Hamiltonians which are exactly solvable, maximally non-Hermitian, and labeled by specific ad hoc partitionings R(N) of N.
