Hermitian-to-quasi-Hermitian quantum phase transitions

Znojil, Miloslav

doi:10.1103/physreva.97.042117

Cited by 7 publications

(4 citation statements)

References 68 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As long as the latter claims (of a top mathematical quality) are accompanied by certain fairly vague quantum-theoretical considerations (which could certainly prove misleading), we feel forced to point out that recently, the study of the parametric domains of unitarity near EPs [38] clarified the point (cf. also the less formal explanation in [9]).…”

Section: False Instabilities and Open Systems In Disguisementioning

confidence: 99%

See 1 more Smart Citation

Paths of unitary access to exceptional points

Znojil

2021

Preprint

Self Cite

View full text Add to dashboard Cite

With an innovative ideas of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson's papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato's exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states L into a triplet (viz., in our notation, spaces K and H besides the conventional L). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.

show abstract

Section: False Instabilities and Open Systems In Disguisementioning

confidence: 99%

“…Interested readers are recommended to consult Refs. [38] and [45] for the further details of the solution of such an equation. For our present purposes the essence of the latter technicalities may be explained using the elementary unperturbed real-matrix Hamiltonians of Ref.…”

Section: Perturbation Theory Near Eps Using Nonstandard Unperturbed H...mentioning

confidence: 99%

Paths of unitary access to exceptional points

Znojil

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the boundary of both regions of the model space, two or more eigenvalues and their corresponding eigenstates can be coalescent. This set of parameters are called exceptional points [8][9][10][11][12][13][14][15][16][17][18][19][20]. The time evolution of a given initial state under the action of these hamiltonians strongly depends on the characteristics of the spectrum, particularly at EPs [21][22][23][24][25][26][27][28][29][30], where the exponential law is generally not valid.…”

Section: Introductionmentioning

confidence: 99%

Swanson Hamiltonian: non-PT-symmetry phase

Fernández¹,

Ramírez²,

Reboiro³

2021

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

In this work, we study the non-hermitian Swanson hamiltonian, particularly the non-PT symmetry phase. We use the formalism of Gel’fand triplet to construct the generalized eigenfunctions and the corresponding spectrum. Depending on the region of the parameter model space, we show that the Swanson hamiltonian represents diﬀerent physical systems, i.e. parabolic barrier, negative mass oscillators. We also discussed the presence of Exceptional Points of inﬁnite order.

show abstract

“…The recent growth of interest in non-Hermitian and, in particular, parity-times-time-reversal-symmetric (PTsymmetric) quantum Hamiltonians H with real spectra [1][2][3] upgraded also the status of Kato's exceptional points (EPs, [4]) from an abstract, purely mathematical concept to an interesting, experimentally accessible singularity [5][6][7][8][9][10][11][12]. As a sample of this tendency one can recall paper [13], the authors of which managed to throw new light upon the traditional phenomenological problem of Bose-Einstein condensation as well as upon its EP alias 'non-Hermitian degeneracy' [14] mathematical background.…”

Section: Introductionmentioning

confidence: 99%

Passage through exceptional point: case study

Znojil

2020

Proc. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Hermitian-to-quasi-Hermitian quantum phase transitions

Cited by 7 publications

References 68 publications

Paths of unitary access to exceptional points

Paths of unitary access to exceptional points

Swanson Hamiltonian: non-PT-symmetry phase

Passage through exceptional point: case study

Contact Info

Product

Resources

About