QES solutions of a two-dimensional system with quadratic nonlinearities

Mandal, Bhabani Prasad; Mourya, Brijesh Kumar; Singh, Aman

doi:10.1140/epjp/s13360-020-00335-6

Cited by 3 publications

(2 citation statements)

References 71 publications

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“…In these latter two papers the Englert-Brout-Higgs mechanism was explored in a non-Hermitian context. Subsequent follow up may be found in [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

Section: So What Happens To the Goldstone And Englert-brout-higgs Mec...mentioning

confidence: 99%

Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories

Mannheim

2019

Phys. Rev. D

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In recent work Alexandre, Ellis, Millington and Seynaeve have extended the Goldstone theorem to non-Hermitian Hamiltonians that possess a discrete antilinear symmetry such as P T and possess a continuous global symmetry. They restricted their discussion to those realizations of antilinear symmetry in which all the energy eigenvalues of the Hamiltonian are real. Here we extend the discussion to the two other realizations possible with antilinear symmetry, namely energies in complex conjugate pairs or Jordan-block Hamiltonians that are not diagonalizable at all. In particular, we show that under certain circumstances it is possible for the Goldstone boson mode itself to be one of the zero-norm states that are characteristic of Jordan-block Hamiltonians. While we discuss the same model as Alexandre, Ellis, Millington and Seynaeve our treatment is quite different, though their main conclusion that one can have Goldstone bosons in the non-Hermitian case remains intact. We extend our analysis to a continuous local symmetry and find that the gauge boson acquires a non-zero mass by the Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the fact that the continuous local symmetry has been spontaneously broken, the gauge boson remains massless. E * . Thus as originally noted by Wigner in his study of time reversal invariance, energies can thus be real or appear in complex conjugate pairs with complex conjugate eigenfunctions. It is often the case that one can move between these two realizations by a change in the parameters in H. There will thus be a transition point (known as an exceptional point) at which the switch over occurs. However, at this transition point the two complex conjugate wave functions (|ψ and A|ψ ) have to collapse into a single common wave function as there are no complex conjugate pairs on the real energy side. Since this collapse to a single common wave function reduces the number of energy eigenfunctions, at the transition point the eigenspectrum of the Hamiltonian becomes incomplete, with the Hamiltonian then being of non-diagonalizable Jordan-block form, the thus third possible realization of antilinear symmetry.While the above analysis would in principle apply to any antilinear symmetry, because of its H = p 2 +ix 3 progenitor, the antilinear symmetry program is conventionally referred to as the P T -symmetry program. However, P T symmetry can actually be selected out for a different reason, namely it has a connection to spacetime. Specifically, it was noted in [11] and emphasized in [3] that for the spacetime coordinates the linear part of a P T transformation is the same as a particular complex Lorentz transformation, while in [7,12] it was noted that for spinors the linear part of a CP T transformation is the same as that very same particular complex Lorentz transformation, where C denotes charge conjugation. 2 Then in [7,12] it was shown that if one imposes only two requirement...

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Section: So What Happens To the Goldstone And Englert-brout-higgs Mec...mentioning

confidence: 99%

Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories

Mannheim

2019

Phys. Rev. D

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“…In these latter two papers the Englert-Brout-Higgs mechanism was explored in a non-Hermitian context. Subsequent follow up may be found in [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25].…”

Section: So What Happens To the Goldstone And Englert-brout-higgs Mec...mentioning

confidence: 99%

Extension of the Goldstone and the Englert-Brout-Higgs mechanisms to non-Hermitian theories

Mannheim

2023

J. Phys.: Conf. Ser.

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We discuss the extension of the Goldstone and Englert-Brout-Higgs mechanisms to non-Hermitian Hamiltonians that possess an antilinear PT symmetry. We study a model due to Alexandre, Ellis, Millington and Seynaeve and show that for the spontaneous breakdown of a continuous global symmetry we obtain a massless Goldstone boson in all three of the antilinear symmetry realizations: eigenvalues real, eigenvalues in complex conjugate pairs, and eigenvalues real but eigenvectors incomplete. In this last case we show that it is possible for the Goldstone boson mode to be a zero-norm state. For the breakdown of a continuous local symmetry the gauge boson acquires a non-zero mass by the Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the fact that the continuous local symmetry has been spontaneously broken, the gauge boson remains massless.

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Examples of PT Phase Transition : QM to QFT

Mandal

2021

J. Phys.: Conf. Ser.

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Parity Time Reversal (PT) phase transition is a typical characteristic of most of the PT symmetric non-Hermitian (NH) systems. Depending on the theory, a particular system and spacetime dimensionality PT phase transition has various interesting features. In this article we review some of our works on PT phase transitions in quantum mechanics (QM) as well as in Quantum Field theory (QFT). We demonstrate typical characteristics of PT phase transition with the help of several analytically solved examples. In one dimensional QM, we consider examples with exactly as well as quasi exactly solvable (QES) models to capture essential features of PT phase transition. The discrete symmetries have rich structures in higher dimensions which are used to explore the PT phase transition in higher dimensional systems. We consider anisotropic SHOs in two and three dimensions to realize some connection between the symmetry of original hermitian Hamiltonian and the unbroken phase of the NH system. We consider the 2+1 dimensional massless Dirac particle in the external magnetic field with PT symmetric non-Hermitian spin-orbit interaction in the background of the Dirac oscillator potential to show the PT phase transition in a relativistic system. A small mass gap, consistent with the other approaches and experimental observations is generated only in the unbroken phase of the system. Finally we develop the NH formulation in an SU(N) gauge field theoretic model by using the natural but unconventional Hermiticity properties of the ghost fields. Deconfinement to confinement phase transition has been realized as PT phase transition in such a non-hermitian model.

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QES solutions of a two-dimensional system with quadratic nonlinearities

Cited by 3 publications

References 71 publications

Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories

Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories

Extension of the Goldstone and the Englert-Brout-Higgs mechanisms to non-Hermitian theories

Examples of PT Phase Transition : QM to QFT

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