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arXiv · 1305.4822

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

Miloslav Znojil
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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.

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Miloslav Znojil. 2013-05-21. Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding. https://doi.org/10.1016/j.aop.2013.05.016

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