Quantum phase transition is interpreted as an evolution, at the end of which a parameter-dependent Hamiltonian $H(g)$ loses its observability. In the language of mathematics, such a quantum catastrophe occurs at an exceptional point of order $N$ (EPN). Although the Hamiltonian $H(g)$ itself becomes unphysical in the limit of $g \to g^{EPN}$, it is shown that it can play the role of an unperturbed operator in a perturbation-approximation analysis of the vicinity of the EPN singularity. Such an analysis is elementary at $N\leq 3$ and numerical at $N\geq 5$, so we pick up $N=4$. We demonstrate that the specific EP4 degeneracy becomes accessible via a unitary evolution process realizable inside a parametric domain ${\cal D}_{\rm physical}$, the boundaries of which are determined non-numerically. Possible relevance of such a mathematical result in the context of non-Hermitian photonics is emphasized.
At present, a deep and productive relevance of an originally purely mathematical concept of exceptional point (EP, [1]) in photonics is well known [2]. From a historical perspective (cf., e.g., the introductory chapter in dedicated monograph [3]), one could really localize several independent roots of the related successful theoretical as well as experimental developments.
During their first stage, for example, a key driving force originated from the relativistic quantum field theory, in the framework of which the challenges emerged due to the discoveries of divergences of many innocent-looking perturbation expansions. Soon, these phenomena found some of their very natural though not quite expected mathematical explanations precisely in the existence of the EP values of certain relevant phenomenological parameters (cf., e.g., [4,5]).
The currently popular idea of the use of the EPs in photonics only emerged when several authors realized that there must exist an intimate connection between the Schrödinger equation of quantum mechanics (in which the EPs already found their applications) and the classical Maxwell equations (especially when considered in the so-called paraxial approximation, cf., e.g., [6]). In any case, the subsequent progress was truly impressive (cf., e.g., the recent review [7]), based mainly on a wide range of mathematics shared by the quantum theory and photonics. Incidentally, this also motivated our present paper in which we are going to restrict our attention to several rather interesting interdisciplinary aspects of the shared mathematics using, in most cases, just the representative language of the so-called quasi-Hermitian quantum models in which the EPs manifest themselves as the instants of a specific class of the so-called quantum phase transitions.
The task of description of a quantum phase transition is, for several reasons, difficult. First of all, one has to leave the comparably comfortable formalism of classical mechanics. After the quantization, the numbers representing the classical observable quantities like, e.g., a (possibly, time-dependent) point-particle momentum p = p(t) must be replaced by operators. In particular, the quantized energy E, conserved or not, has to be treated as an eigenvalue of a preselected energy-representing Hamiltonian H, etc. Thus, a more or less purely geometric nature of the classical phase-transition theories (cf., e.g., [8]) is lost.
Secondly, all of the operators Λ(t) representing a relevant quantum observable have to be diagonalizable and, moreover, they are usually chosen or constructed as self-adjoint in a suitable (and, mostly, infinite-dimensional) Hilbert space of states H physical . In such a traditional quantum model-building setup, the very existence of a genuine “change of phase” (i.e., basically, of a loss of the reality of the spectrum of at least one of the observables) seems to contradict the well known tendency of eigenvalues of any self-adjoint operator to avoid any phase-transition-mimicking merger. Just an apparent “repulsion of levels” is encountered in many experiments.
On the level of abstract quantum theory, both of the latter obstacles can be circumvented when one resorts to the less usual formulation of quantum mechanics called quasi-Hermitian quantum mechanics (QHQM, cf. one of its oldest reviews [9] for a comprehensive introduction). The resolution of at least some of the quantum phase-transition puzzles is offered there by the possibility of the realization of the phase-transition-mimicking mergers of the eigenvalues as well as of the states at a singularity called, by mathematicians, exceptional point (EP, [1]). By definition, this means that in the singular, manifestly unphysical EP limit, the overall spectrum of an observable under consideration remains real but, at the same time, it degenerates and ceases to admit standard interpretation. Such a way of thinking about quantum phase-transition processes will be also accepted in our present paper.
We will start from the observation that in the majority of the existing constructions of models admitting an EP-related phase transition, the authors prefer the systems characterized by the observability of the mere parameter-dependent bound-state energy levels E n (g) with n = 0, 1, . . . . For the sake of definiteness, moreover, just the models admitting the most elementary EP mergers of order two (EP2) are usually considered. In the language of mathematics this means that the “first nontrivial” form of the dynamical scenario is only being described, in which the loss of the diagonalizability of the Hamiltonian H(g) in the limit of g → g (EP 2) is restricted to a two-dimensional subspace of the whole Hilbert space. In other words, the EP2-related merger of energies E n j (g) as well as of the related bound-state eigenvectors |ψ n j (g) only involves a single pair of states with subscripts j = 1 and j = 2.
Even the EP2-related studies need not be trivial. In the
This content is AI-processed based on open access ArXiv data.