Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau–Lifschitz equation

Cen, Julia; Correa, Francisco; Fring, Andreas

doi:10.1088/1751-8121/ab81d9

Cited by 9 publications

(4 citation statements)

References 56 publications

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“…the more complicated system. Moreover, the iteration of Darboux transformations enables the construction of multi-soliton solutions in integrable systems, see [115][116][117][118][119]. Given their usefulness, let us now discuss their time-dependent versions.…”

Section: Quantum Fest 2021 Journal Of Physics: Conference Series 2448...mentioning

confidence: 99%

An Introduction to PT-Symmetric Quantum Mechanics-Time-Dependent Systems

Fring

2023

J. Phys.: Conf. Ser.

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I will provide a pedagogical introduction to non-Hermitian quantum systems that are PT-symmetric, that is they are left invariant under a simultaneous parity transformation (P) and time-reversal (T). I will explain how generalised versions of this antilinear symmetry can be utilised to explain that these type of systems possess real eigenvalue spectra in parts of their parameter spaces and how to set up a consistent quantum mechanical framework for them that enables a unitary time-evolution. In the second part I will explain how to extend this framework to explicitly time-dependent Hamiltonian systems and report in particular on recent progress made in this context. I will explain how to construct the essential key quantity in this framework, the time-dependent Dyson map and metric and solutions to the time-dependent Schrödinger equation, in an algebraic fashion, using time-dependent Darboux transformations, utilising Lewis-Riesenfeld invariants, point transformations and some approximation methods. I comment on the ambiguities of this metric and demonstrate that this can even lead to infinite series of metric operators. I conclude with some applications to PT-symmetrically coupled oscillators, demonstrate the equivalence of the time-dependent double wells and unstable anharmonic oscillators and show how the unphysical PT-symmetrically broken regions in the parameter space for the time-independent theory becomes physical in the explicitly time-dependent systems. I discuss how this leads to a prolongation of the otherwise rapidly decaying von Neumann entropy. The so-called sudden death of the entropy is stopped at a finite value.1

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Section: Quantum Fest 2021 Journal Of Physics: Conference Series 2448...mentioning

confidence: 99%

An Introduction to PT-Symmetric Quantum Mechanics-Time-Dependent Systems

Fring

2023

J. Phys.: Conf. Ser.

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“…In addition, it is often easier to solve the TDSE for one of the partner Hamiltonians than the other, Darboux transformation can facilitate the calculation of the eigenfunctions for the more complicated system. Moreover, the iteration of Darboux transformations enables the construction of multi-soliton solutions in integrable systems, see [115][116][117][118][119]. Given their usefulness, let us now discuss their time-dependent versions.…”

Section: Utilizing Lewis-riesenfeld Invariantsmentioning

confidence: 99%

An introduction to PT-symmetric quantum mechanics -- time-dependent systems

Fring¹

2022

Preprint

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I will provide a pedagogical introduction to non-Hermitian quantum systems that are PT -symmetric, that is they are left invariant under a simultaneous parity transformation (P) and time-reversal (T ). I will explain how generalised versions of this antilinear symmetry can be utilised to explain that these type of systems possess real eigenvalue spectra in parts of their parameter spaces and how to set up a consistent quantum mechanical framework for them that enables a unitary time-evolution. In the second part I will explain how to extend this framework to explicitly time-dependent Hamiltonian systems and report in particular on recent progress made in this context. I will explain how to construct the essential key quantity in this framework, the time-dependent Dyson map and metric and solutions to the time-dependent Schrödinger equation, in an algebraic fashion, using time-dependent Darboux transformations, utilising Lewis-Riesenfeld invariants, point transformations and some approximation methods. I comment on the ambiguities of this metric and demonstrate that this can even lead to infinite series of metric operators. I conclude with some applications to PT -symmetrically coupled oscillators, demonstrate the equivalence of the time-dependent double wells and unstable anharmonic oscillators and show how the unphysical PT -symmetrically broken regions in the parameter space for the time-independent theory becomes physical in the explicitly timedependent systems. I discuss how this leads to a prolongation of the otherwise rapidly decaying von Neumann entropy. The so-called sudden death of the entropy is stopped at a finite value. 1

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“…The focus on non-Hermitian PT -invariant systems has grown significantly following the pioneering studies of Bender and collaborators [1][2][3]. Such systems attract attention and find applications in various areas of physics, including optics [4][5][6][7][8], condensed matter physics [9][10][11], quantum field theory [12][13][14][15][16][17][18][19], gravity and cosmology [20,21], nonlinear waves [22], and theory of integrable models of finite [23][24][25][26][27] and infinite number of degrees of freedom [28][29][30][31][32][33][34][35][36][37][38][39][40].…”

Section: Introductionmentioning

confidence: 99%

Conformal bridge transformation, $$ \mathcal{PT} $$- and supersymmetry

Inzunza

Plyushchay

2022

J. High Energ. Phys.

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Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by i and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a $$ \mathcal{PT} $$ PT -invariant supersymmetric model with N subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic N = 2 supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the $$ \mathcal{PT} $$ PT -invariant supersymmetric systems that transform into the spin-1/2 Landau problem with and without an additional Aharonov-Bohm flux, where in the latter case, the well-defined integrals of motion appear only when the flux is quantized. We also build a 2D supersymmetric Hamiltonian related to the “exotic rotational invariant harmonic oscillator” system governed by a dynamical parameter γ. The bosonic and fermionic hidden symmetries for this model are shown to exist for rational values of γ.

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Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau–Lifschitz equation

Cited by 9 publications

References 56 publications

An Introduction to PT-Symmetric Quantum Mechanics-Time-Dependent Systems

An Introduction to PT-Symmetric Quantum Mechanics-Time-Dependent Systems

An introduction to PT-symmetric quantum mechanics -- time-dependent systems

Conformal bridge transformation, $$ \mathcal{PT} $$- and supersymmetry

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