Quantum phase transitions in nonhermitian harmonic oscillator

Abstract

The Stone theorem requires that in a physical Hilbert space $${{{\mathcal {H}}}}$$ H the time-evolution of a stable quantum system is unitary if and only if the corresponding Hamiltonian H is self-adjoint. Sometimes, a simpler picture of the evolution may be constructed in a manifestly unphysical Hilbert space $${{{\mathcal {K}}}}$$ K in which H is nonhermitian but $${{\mathcal {PT}}}$$ PT -symmetric. In applications, unfortunately, one only rarely succeeds in circumventing the key technical obstacle which lies in the necessary reconstruction of the physical Hilbert space $${{{\mathcal {H}}}}$$ H . For a $${{\mathcal {PT}}}$$ PT -symmetric version of the spiked harmonic oscillator we show that in the dynamical regime of the unavoided level crossings such a reconstruction of $${{{\mathcal {H}}}}$$ H becomes feasible and, moreover, obtainable by non-numerical means. The general form of such a reconstruction of $${{{\mathcal {H}}}}$$ H enables one to render every exceptional unavoided-crossing point tractable as a genuine, phenomenologically most appealing quantum-phase-transition instant.

Confluences of exceptional points and a systematic classification of quantum catastrophes

“…( 7 ) one obtains, for them, an alternative, infinite-dimensional but partitioned Jordan-block limit of the form of Eq. ( 15 ) with infinite sequence of the energy mergers available in closed form, 46 .…”

“…In our recent follow-up paper 46 a closer connection has been established between the harmonic-oscillator physics of collapse and the mathematics of its exceptional points. Near , in particular, explicit form has been found of all of the admissible duality maps defining all of the available physical Hilbert spaces and metrics .…”

Paths of unitary access to exceptional points

“…Fig. 6) enabled us to pay more attention, in paper [56], to one of the key challenges connected with the theory, viz., to the constructive analysis of the practical consequences of the nontriviality and of the ambiguity of the related angular-momentum-dependent metrics Θ = Θ(L).…”

“…The basic technical ingredient in the construction of the metrics (see its details as well as the rather long explicit formulae in [56]) was twofold. Firstly, the availability of the closed-form diagonalization of H (HO) (L) enabled us to replace the Hamiltonian, at any one of its EP limits, by an equivalent matrix called canonical or Jordan-block representation.…”

Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities

“…We should only add that an extreme care must be paid to the Stone theorem [52] requiring the Hermiticity of H(λ) in the related physical Hilbert space H. This means that a hermitization of the Hamiltonian is needed [9]. Such a process usually involves a reconstruction of an appropriate amended inner product in the conventional but manifestly unphysical Hilbert space K. Interested readers may find a sample of such a reconstruction of H, say, in [53].…”