Behavior of eigenvalues in a region of broken 
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 symmetry

Bender, Carl M.; Hassanpour, Nima; Hook, Daniel W.; Klevansky, S. P.; Sünderhauf, Christoph; Wen, Zichao

doi:10.1103/physreva.95.052113

Cited by 19 publications

(20 citation statements)

References 29 publications

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“…Numerical evidence of what happens when ε is below these lower limits is presented in the recent paper [3]. The method of our paper breaks down when m = 2M + ε ≤ 1.…”

Section: Introductionmentioning

confidence: 86%

PT-symmetric eigenvalues for homogeneous potentials

Erëmenko

Gabrielov

2018

Journal of Mathematical Physics

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We consider one-dimensional Schrödinger equations with potential x 2M (ix) ε , where M ≥ 1 is an integer, and ε is real, under appropriate PT-symmetric boundary conditions. We prove the phenomenon which was discovered by Bender and Boettcher by numerical computation: as ε changes, the real spectrum suddenly becomes non-real in the sense that all but finitely many eigenvalues become non-real. We find the limit arguments of these non-real eigenvalues E as E → ∞.MSC: 81Q05, 34M60, 34A05.

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“…Numerical evidence of what happens when ε is below these lower limits is presented in the recent paper [3]. The method of our paper breaks down when m = 2M + ε ≤ 1.…”

Section: Introductionmentioning

confidence: 86%

PT-symmetric eigenvalues for homogeneous potentials

Erëmenko

Gabrielov

2018

Journal of Mathematical Physics

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“…32 e textbook example of a P T -symmetric Hamiltonian is 44Ĥ = p 2 + ix 3 (1) which has been extensively studied by Bender and coworkers. [24][25][26][27][28][29][30][31][32][33][37][38][39] It is clear that the application of the combined space-time re ection P T , where…”

Section: A Bosonsmentioning

confidence: 99%

PT-Symmetry in Hartree–Fock Theory

Burton¹,

Thom²,

Loos³

2019

Preprint

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<div> <div> <p>P T -symmetry — invariance with respect to combined space reflection P and time reversal T — provides a weaker condition than (Dirac) Hermiticity for ensuring a real energy spectrum of a general non-Hermitian Hamiltonian. PT -symmetric Hamiltonians therefore form an intermediate class between Hermitian and non-Hermitian Hamiltonians. In this work, we derive the conditions for PT-symmetry in the context of electronic structure theory, and specifically, within the Hartree–Fock (HF) approximation. We show that the HF orbitals are symmetric with respect to the P T operator if and only if the effective Fock Hamiltonian is PT -symmetric, and vice versa. By extension, if an optimal self-consistent solution is invariant under PT , then its eigenvalues and corresponding HF energy must be real. Moreover, we demonstrate how one can construct explicitly PT -symmetric Slater determinants by forming PT doublets (i.e. pairing each occupied orbital with its PT -transformed analogue), allowing PT -symmetry to be conserved throughout the self-consistent process. Finally, considering the H2 molecule as an illustrative example, we observe PT-symmetry in the HF energy landscape and find that the symmetry-broken unrestricted HF wave functions (i.e. diradical configurations) are P T -symmetric, while the symmetry-broken restricted HF wave functions (i.e. ionic configurations) break PT -symmetry.</p> </div> </div>

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“…which has been extensively studied by Bender and coworkers. [24][25][26][27][28][29][30][31][32][33][37][38][39] From the standard action of P and T , where…”

Section: A Spinless P T -Symmetrymentioning

confidence: 99%

$\mathcal{PT}$-Symmetry in Hartree-Fock Theory

Burton,

Thom,

Loos

2019

Preprint

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P T -symmetry -invariance with respect to combined space re ection P and time reversal T -provides a weaker condition than (Dirac) Hermiticity for ensuring a real energy spectrum of a general non-Hermitian Hamiltonian. P T -symmetric Hamiltonians therefore form an intermediate class between Hermitian and non-Hermitian Hamiltonians. In this work, we derive the conditions for P T -symmetry in the context of electronic structure theory, and speci cally, within the Hartree-Fock (HF) approximation. We show that the HF orbitals are symmetric with respect to the P T operator if and only if the e ective Fock Hamiltonian is P T -symmetric, and vice versa. By extension, if an optimal self-consistent solution is invariant under P T , then its eigenvalues and corresponding HF energy must be real. Moreover, we demonstrate how one can construct explicitly P T -symmetric Slater determinants by forming P T doublets (i.e. pairing each occupied orbital with its P T -transformed analogue), allowing P T -symmetry to be conserved throughout the self-consistent process. Finally, considering the H 2 molecule as an illustrative example, we observe P T -symmetry in the HF energy landscape and nd that the spatially symmetry-broken unrestricted HF wave functions (i.e. diradical con gurations) are P T -symmetric, while the spatially symmetry-broken restricted HF wave functions (i.e. ionic con gurations) break P T -symmetry.

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Behavior of eigenvalues in a region of broken PT symmetry

Cited by 19 publications

References 29 publications

PT-symmetric eigenvalues for homogeneous potentials

PT-symmetric eigenvalues for homogeneous potentials

PT-Symmetry in Hartree–Fock Theory

$\mathcal{PT}$-Symmetry in Hartree-Fock Theory

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