Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding

Abstract

The practical use of non-Hermitian (i.e., typically, PT-symmetric) phenomenological quantum Hamiltonians is discussed as requiring an explicit reconstruction of the {\em ad hoc} Hilbert-space metrics which would render the time-evolution unitary. Just the N-dimensional matrix toy models Hamiltonians are considered, therefore. For them, the matrix elements of alternative metrics are constructed via solution of a coupled set of polynomial equations, using the computer-assisted symbolic manipulations for the purpose. The feasibility and some consequences of such a model-construction strategy are illustrated via a discrete square well model endowed with multi-parametric close-to-the-boundary real bidiagonal-matrix interaction. The degenerate exceptional points marking the phase transitions are then studied numerically. A way towards classification of their unfoldings in topologically non-equivalent dynamical scenarios is outlined.

19 pp, 5 figures

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Cited by in corpus (14)

• Integrable nonlocal vector nonlinear Schrödinger equation with self-induced parity-time-symmetric Potential
• On Symmetries and Exact Solutions of a Class of Non-local Non-linear Schrodinger Equations with Self-induced PT-symmetric Potential
• Multiply Degenerate Exceptional Points and Quantum Phase Transitions
• Three solvable matrix models of a quantum catastrophe
• Passage through exceptional point: Case study
• Complex symmetric Hamiltonians and exceptional points of order four and five
• Two patterns of PT-symmetry breakdown in a non-numerical six-state simulation
• Hamiltonian formulation of systems with balanced loss-gain and exactly solvable models
• Bound states emerging from below the continuum in a solvable PT-symmetric discrete Schroedinger equation
• Broken Hermiticity phase transition in Bose-Hubbard model
• Hermitian - non-Hermitian interfaces in quantum theory
• Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via $\mathcal{P}\mathcal{T}-$symmetric and exactly solvable linear-Bose-Hubbard building blocks
• Hermitian--to--quasi-Hermitian quantum phase transitions
• Nonlocality of Observables in Quasi-Hermitian Quantum Theory