Supersymmetry and the spontaneous breakdown of 𝒫𝒯 symmetry

Dorey, Patrick; Dunning, Clare; Tateo, Roberto

doi:10.1088/0305-4470/34/28/102

Cited by 305 publications

(392 citation statements)

References 26 publications

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“…The first complete proof of spectral reality and positivity for H in (2) was given by Dorey et al in Ref. [27,28].…”

Section: P T Quantum Mechanicsmentioning

confidence: 99%

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Bender¹,

Brody²

2009

Lecture Notes in Physics

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The shortest path between two truths in the real domain passes through the complex domain. -Jacques Hadamard, The Mathematical Intelligencer 13 (1991) Consider the set of all Hamiltonians whose largest and smallest energy eigenvalues, E max and E min , differ by a fixed energy ω. Given two quantum states, an initial state |ψ I and a final state |ψ F , there exist many Hamiltonians H belonging to this set under which |ψ I evolves in time into |ψ F . Which Hamiltonian transforms the initial state to the final state in the least possible time τ ? For Hermitian Hamiltonians, τ has a nonzero lower bound. However, among complex non-Hermitian P T -symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of τ can be made arbitrarily small because for P T -symmetric Hamiltonians the evolution path from the vector |ψ I to the vector |ψ F , as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here resembles the effect in general relativity in which two space-time points can be made arbitrarily close if they are connected by a wormhole. This result may have applications in quantum computing.

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“…The first complete proof of spectral reality and positivity for H in (2) was given by Dorey et al in Ref. [27,28].…”

Section: P T Quantum Mechanicsmentioning

confidence: 99%

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Bender¹,

Brody²

2009

Lecture Notes in Physics

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“…Firstly, our study [2] revealed that the manifest non-Hermiticity of the models of the type (1) leads to the reliable leading order approximation only after we select our harmonic oscillator approximant as lying very far from the real axis (i.e., from the Hermitian regime). Such a recipe is, apparently, deeply incompatible with a smooth modification of the traditional zero-order approximants occurring in current Hermitian 1/ℓ recipes (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Within this class, the strong-coupling version of DDT oscillators (1) with α ≫ 1 forms a particularly suitable testing ground as it combines the necessary reality of its spectrum with the smallness of the inverse quantity 1/ℓ. Moreover, the phenomenologically appealing non-Hermitian models like (1) are rarely solvable in closed form so that the presence of a "universal" small parameter 1/ℓ ≪ 1 offers one of not too many ways towards their systematic approximate solution.…”

Section: Introductionmentioning

confidence: 99%

PT-symmetric pseudo-perturbation recipe: an imaginary cubic oscillator with spikes

Mustafa

Znojil

2002

J. Phys. A: Math. Gen.

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“…When ≥ 0, the PT symmetry is unbroken, but when < 0, the PT symmetry is broken ( figure 1). Moreover, some extensions of non-Hermitian PT -symmetric Hamiltonians H given by equation (2.1) have also been considered [15][16][17][18][19][20]. The notion of pseudo-Hermiticity has also been introduced [21][22][23][24] and it has been shown that every Hamiltonian with a real spectrum is pseudo-Hermitian [25].…”

Section: Non-hermitian Pt -Symmetric Hamiltoniansmentioning

confidence: 99%

Complex PT -symmetric nonlinear Schrödinger equation and Burgers equation

Yan

2013

Phil. Trans. R. Soc. A.

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One contribution of 17 to a Theme Issue 'PT quantum mechanics' . The complex PT -symmetric nonlinear wave models have drawn much attention in recent years since the complex PT -symmetric extensions of the Kortewegde Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex PT -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex PT symmetry. We then report on exact solutions of one-and twodimensional nonlinear Schrödinger equations (known as the Gross-Pitaevskii equation in Bose-Einstein condensates) with several complex PT -symmetric potentials. Finally, some complex PT -symmetric extension principles are used to generate some complex PT -symmetric nonlinear wave equations starting from both PT -symmetric (e.g. the KdV equation) and non-PT -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex PT -symmetric Burgers equation in detail.

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Supersymmetry and the spontaneous breakdown of 𝒫𝒯 symmetry

Cited by 305 publications

References 26 publications

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

PT-symmetric pseudo-perturbation recipe: an imaginary cubic oscillator with spikes

Complex PT -symmetric nonlinear Schrödinger equation and Burgers equation

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