Abstract
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 \( \mathcal{PT} \)-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 \( \mathcal{PT} \)-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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References
C.M. Bender and S. Boettcher, Real spectra in nonHermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243 [physics/9712001] [INSPIRE].
C.M. Bender, S. Boettcher and P. Meisinger, PT symmetric quantum mechanics, J. Math. Phys. 40 (1999) 2201 [quant-ph/9809072] [INSPIRE].
C.M. Bender, D.C. Brody and H.F. Jones, Complex extension of quantum mechanics, Phys. Rev. Lett. 89 (2002) 270401 [Erratum ibid. 92 (2004) 119902] [quant-ph/0208076] [INSPIRE].
C.E. Rüter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev and D. Kip, Observation of parity-time symmetry in optics, Nature Phys. 6 (2010) 192.
A. Guo et al., Observation of PT-Symmetry Breaking in Complex Optical Potentials, Phys. Rev. Lett. 103 (2009) 093902 [INSPIRE].
A.A. Zyablovsky, A.P. Vinogradov A.A. Pukhov, A.V. Dorofeenko and A.A. Lisyansky, PT-symmetry in optics, Phys.-Usp. 57 (2014) 1063.
Z. Zhang et al., Observation of parity-time symmetry in optically induced atomic lattices, Phys. Rev. Lett. 117 (2016) 123601 [arXiv:1604.04025] [INSPIRE].
F. Correa, V. Jakubsky and M.S. Plyushchay, PT-symmetric invisible defects and confluent Darboux-Crum transformations, Phys. Rev. A 92 (2015) 023839 [arXiv:1506.00991] [INSPIRE].
K. Kawabata, Y. Ashida, H. Katsura and M. Ueda, Parity-time-symmetric topological superconductor, Phys. Rev. B 98 (2018) 085116 [arXiv:1801.00499] [INSPIRE].
A. Ghatak and T. Das, Theory of superconductivity with non-Hermitian and parity-time reversal symmetric Cooper pairing symmetry, Phys. Rev. B 97 (2018) 014512 [arXiv:1708.09108].
K. Yamamoto, M. Nakagawa, K. Adachi, K. Takasan, M. Ueda and N. Kawakami, Theory of non-Hermitian fermionic superfluidity with a complex-valued interaction, Phys. Rev. Lett. 123 (2019) 123601 [arXiv:1903.04720].
C.M. Bender, D.C. Brody and H.F. Jones, Extension of PT symmetric quantum mechanics to quantum field theory with cubic interaction, Phys. Rev. D 70 (2004) 025001 [Erratum ibid. 71 (2005) 049901] [hep-th/0402183] [INSPIRE].
C.M. Bender, N. Hassanpour, S.P. Klevansky and S. Sarkar, PT-symmetric quantum field theory in D dimensions, Phys. Rev. D 98 (2018) 125003 [arXiv:1810.12479] [INSPIRE].
A. Felski, C.M. Bender, S.P. Klevansky and S. Sarkar, Towards perturbative renormalization of ϕ2(iϕ)ϵ quantum field theory, Phys. Rev. D 104 (2021) 085011 [arXiv:2103.07577] [INSPIRE].
A. Felski, A. Beygi and S.P. Klevansky, Non-Hermitian extension of the Nambu-Jona-Lasinio model in 3 + 1 and 1 + 1 dimensions, Phys. Rev. D 101 (2020) 116001 [arXiv:2004.04011] [INSPIRE].
A. Fring and T. Taira, Pseudo-Hermitian approach to Goldstone’s theorem in non-Abelian non-Hermitian quantum field theories, Phys. Rev. D 101 (2020) 045014 [arXiv:1911.01405] [INSPIRE].
J. Alexandre, J. Ellis, P. Millington and D. Seynaeve, Spontaneously breaking non-Abelian gauge symmetry in non-Hermitian field theories, Phys. Rev. D 101 (2020) 035008 [arXiv:1910.03985] [INSPIRE].
J. Alexandre, J. Ellis and P. Millington, \( \mathcal{PT} \)-symmetric non-Hermitian quantum field theories with supersymmetry, Phys. Rev. D 101 (2020) 085015 [arXiv:2001.11996] [INSPIRE].
A. Fring and T. Taira, ‘t Hooft-Polyakov monopoles in non-Hermitian quantum field theory, Phys. Lett. B 807 (2020) 135583 [arXiv:2006.02718] [INSPIRE].
P.D. Mannheim, Making the case for conformal gravity, Found. Phys. 42 (2012) 388 [arXiv:1101.2186] [INSPIRE].
P.D. Mannheim, Astrophysical evidence for the non-Hermitian but PT-symmetric Hamiltonian of conformal gravity, Fortsch. Phys. 61 (2013) 140 [arXiv:1205.5717] [INSPIRE].
V.V. Konotop, J. Yang and D.A. Zezyulin, Nonlinear waves in PT-symmetric systems, Rev. Mod. Phys. 88 (2016) 035002 [arXiv:1603.06826] [INSPIRE].
A. Fring, PT-symmetric deformations of integrable models, Phil. Trans. Roy. Soc. Lond. A 371 (2013) 20120046 [arXiv:1204.2291] [INSPIRE].
D. Sinha and P.K. Ghosh, \( \mathcal{PT} \)-symmetric rational Calogero model with balanced loss and gain, Eur. Phys. J. Plus 132 (2017) 460 [arXiv:1705.03426] [INSPIRE].
F. Correa and O. Lechtenfeld, \( \mathcal{PT} \) deformation of angular Calogero models, JHEP 11 (2017) 122 [arXiv:1705.05425] [INSPIRE].
F. Correa and O. Lechtenfeld, \( \mathcal{PT} \) deformation of Calogero-Sutherland models, JHEP 05 (2019) 166 [arXiv:1903.06481] [INSPIRE].
F. Correa and O. Lechtenfeld, Algebraic integrability of \( \mathcal{PT} \)-deformed Calogero models, (2021) [arXiv:2106.05428] [INSPIRE].
M.J. Ablowitz and Z.H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett. 110 (2013) 064105.
M.J. Ablowitz and Z.H. Musslimani, Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity 29 (2016) 915.
S.Y. Lou, Alice-Bob systems, Ps-Td-C principles and multi-soliton solutions, arXiv:1603.03975.
S.Y. Lou and F. Huang, Alice-Bob physics: coherent solutions of nonlocal KdV systems, Sci. Rep. 7 (2017) 1.
X. Zhang, Y. Chen and Y. Zhang, Breather, lump and X soliton solutions to nonlocal KP equation, Comput. & Math. Appl. 74 (2017) 2341.
J. Cen, F. Correa and A. Fring, Integrable nonlocal Hirota equations, J. Math. Phys. 60 (2019) 081508 [arXiv:1710.11560] [INSPIRE].
J. Cen, F. Correa and A. Fring, Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau-Lifschitz equation, J. Phys. A 53 (2020) 195201 [arXiv:1910.07272] [INSPIRE].
J-G Liu, M.S. Osman, W-H Zhu, L. Zhou and G-P Ai, Different complex wave structures described by the Hirota equation with variable coefficients in inhomogeneous optical fibers, Appl. Phys. B 125 (2019) 1.
J. Cen and A. Fring, Complex solitons with real energies, J. Phys. A 49 (2016) 365202 [arXiv:1602.05465] [INSPIRE].
F. Correa and A. Fring, Regularized degenerate multi-solitons, JHEP 09 (2016) 008 [arXiv:1605.06371] [INSPIRE].
J. Cen, F. Correa and A. Fring, Time-delay and reality conditions for complex solitons, J. Math. Phys. 58 (2017) 032901 [arXiv:1608.01691] [INSPIRE].
J. Mateos Guilarte and M.S. Plyushchay, Perfectly invisible \( \mathcal{PT} \)-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry, JHEP 12 (2017) 061 [arXiv:1710.00356] [INSPIRE].
J. Cen, F. Correa and A. Fring, Degenerate multi-solitons in the sine-Gordon equation, J. Phys. A 50 (2017) 435201 [arXiv:1705.04749] [INSPIRE].
F.G. Scholtz, H.B. Geyer and F.J.W. Hahne, Quasi-Hermitian operators in quantum mechanics and the variational principle, Annals Phys. 213 (1992).
A. Mostafazadeh, PseudoHermiticity versus PT symmetry. The necessary condition for the reality of the spectrum, J. Math. Phys. 43 (2002) 205 [math-ph/0107001] [INSPIRE].
C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70 (2007) 947 [hep-th/0703096] [INSPIRE].
C.M. Bender, PT-symmetric quantum theory, J. Phys. Conf. Ser. 631 (2015) 012002 [INSPIRE].
C.M. Bender et al., PT Symmetry, WSP (2019), [DOI] [INSPIRE].
M.S. Swanson, Transition elements for a non-Hermitian quadratic Hamiltonian, J. Math. Phys. 45 (2004) 585.
C.F. de Morisson Faria and A. Fring, Time evolution of non-Hermitian Hamiltonian systems, J. Phys. A 39 (2006) 9269 [quant-ph/0604014].
D.P. Musumbu, H.B. Geyer and W.D. Heiss, Choice of a metric for the non-Hermitian oscillator, J. Phys. A 40 (2006) F75 [quant-ph/0611150].
P.E. Assis and A. Fring, Non-Hermitian Hamiltonians of Lie algebraic type, J. Phys. A 42 (2008) 015203 [arXiv:0804.4677].
A. Mostafazadeh, Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 1191 [arXiv:0810.5643] [INSPIRE].
F.J. Dyson, Thermodynamic behavior of an ideal ferromagnet, Phys. Rev. 102 (1956) 1230 [INSPIRE].
J.M. Leinaas and J. Myrheim, On the theory of identical particles, Nuovo Cim. B 37 (1977) 1 [INSPIRE].
R. Mackenzie and F. Wilczek, Peculiar spin and statistics in two space dimensions, Int. J. Mod. Phys. A 03 (1988) 2827.
D. Yoshioka, The Quantum Hall Effect, Springer Berlin, Heidelberg, Germany (2002) [DOI].
A. Khare, Fractional Statistics and Quantum Theory, World Scientific, Singapore (2005) [DOI].
N.R. Cooper, Rapidly rotating atomic gases, Adv. Phys. 57 (2008) 539 [arXiv:0810.4398].
L. Inzunza and M.S. Plyushchay, Conformal generation of an exotic rotationally invariant harmonic oscillator, Phys. Rev. D 103 (2021) 106004 [arXiv:2103.07752] [INSPIRE].
L. Inzunza and M.S. Plyushchay, Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background, JHEP 01 (2022) 179 [arXiv:2109.05161] [INSPIRE].
V.B. Matveev and M.A. Salle, Darboux Transformations and Solitons, Springer Berlin, Heidelberg, Germany (1991).
F. Cooper, A. Khare and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rept. 251 (1995) 267 [hep-th/9405029] [INSPIRE].
L. Inzunza and M.S. Plyushchay, Klein four-group and Darboux duality in conformal mechanics, Phys. Rev. D 99 (2019) 125016 [arXiv:1902.00538] [INSPIRE].
A. Sinha and P. Roy, Pseudo supersymmetric partners for the generalized Swanson model, J. Phys. A (2008) 41 335306 [arXiv:0806.4490].
Özlem Yesiltas, Quantum isotonic nonlinear oscillator as a Hermitian counterpart of Swanson Hamiltonian and pseudo-supersymmetry, J. Phys. A 44 (2011) 305305 [arXiv:1108.0106].
L. Inzunza, M.S. Plyushchay and A. Wipf, Conformal bridge between asymptotic freedom and confinement, Phys. Rev. D 101 (2020) 105019 [arXiv:1912.11752] [INSPIRE].
L. Inzunza, M.S. Plyushchay and A. Wipf, Hidden symmetry and (super)conformal mechanics in a monopole background, JHEP 04 (2020) 028 [arXiv:2002.04341] [INSPIRE].
L. Inzunza and M.S. Plyushchay, Conformal bridge in a cosmic string background, JHEP 21 (2020) 165 [arXiv:2012.04613] [INSPIRE].
J.B. Achour and E.R. Livine, Symmetries and conformal bridge in Schwarschild-(A)dS black hole mechanics, JHEP 12 (2021) 152 [arXiv:2110.01455] [INSPIRE].
L. Inzunza and M.S. Plyushchay, Conformal bridge transformation and PT symmetry, J. Phys. Conf. Ser. 2038 (2021) 012014 [arXiv:2104.08351] [INSPIRE].
J.F. Cariñena and M.S. Plyushchay, Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator, Phys. Rev. D 94 (2016) 105022 [arXiv:1611.08051] [INSPIRE].
M.V. Berry and J.P. Keating, H = xp and the Riemann zeros, in Supersymmetry and Trace Formulae: Chaos and Disorder, I.V. Lerner, J.P. Keating and D.E. Khmelnitskii eds., Springer, Boston, U.S.A. (1999), pp. 355–367 [DOI].
V. M Berry and J.P. Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Review 41 (1999) 236.
A. Connes, Trace formula in noncommutative geometry and the zeros of the Riemann zeta function, Selecta Math. 5 (1999) 29 [math/9811068].
G. Sierra and P.K. Townsend, Landau levels and Riemann zeros, Phys. Rev. Lett. 101 (2008) 110201 [arXiv:0805.4079] [INSPIRE].
C.M. Bender, D.C. Brody and M.P. Müller, Hamiltonian for the zeros of the Riemann zeta function, Phys. Rev. Lett. 118 (2017) 130201 [arXiv:1608.03679] [INSPIRE].
C.M. Bender and D.C. Brody, Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian, J. Phys. A 51 (2018) 135203 [arXiv:1710.04411].
G. Sierra, The Riemann zeros as spectrum and the Riemann hypothesis, Symmetry 11 (2019) 494 [arXiv:1601.01797] [INSPIRE].
L. Inzunza and M.S. Plyushchay, Hidden superconformal symmetry: Where does it come from?, Phys. Rev. D 97 (2018) 045002 [arXiv:1711.00616] [INSPIRE].
L. Inzunza and M.S. Plyushchay, Hidden symmetries of rationally deformed superconformal mechanics, Phys. Rev. D 99 (2019) 025001 [arXiv:1809.08527] [INSPIRE].
S.M. Klishevich and M.S. Plyushchay, Nonlinear supersymmetry, quantum anomaly and quasiexactly solvable systems, Nucl. Phys. B 606 (2001) 583 [hep-th/0012023] [INSPIRE].
L. Frappat, P. Sorba and A. Sciarrino, Dictionary on Lie superalgebras, hep-th/9607161 [INSPIRE].
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956) 47.
M.V. Berry, Riemann’s Zeta function: A model for quantum chaos?, Lect. Notes Phys. 263 (1986) 1 [INSPIRE].
H.P. McKean, Selberg’s trace formula as applied to a compact Riemann surface Comm. Pure and Applied Math 25 (1972) 225.
J. Marklof, Selberg’s trace formula: an introduction, in: Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology, J. Bolte and F. Steiner eds., Cambridge University Press, Cambridge, U.K. (2011), pp. 83–119 [math/0407288].
B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. 174 (1968) 1559 [INSPIRE].
G.W. Gibbons, R.H. Rietdijk and J.W. van Holten, SUSY in the sky, Nucl. Phys. B 404 (1993) 42 [hep-th/9303112] [INSPIRE].
M. Cariglia, V.P. Frolov, P. Krtous and D. Kubiznak, Geometry of Lax pairs: particle motion and Killing-Yano tensors, Phys. Rev. D 87 (2013) 024002 [arXiv:1210.3079] [INSPIRE].
M. Cariglia, Hidden Symmetries of Dynamics in Classical and Quantum Physics, Rev. Mod. Phys. 86 (2014) 1283 [arXiv:1411.1262] [INSPIRE].
V. Frolov, P. Krtous and D. Kubiznak, Black holes, hidden symmetries, and complete integrability, Living Rev. Rel. 20 (2017) 6 [arXiv:1705.05482] [INSPIRE].
F. Correa, H. Falomir, V. Jakubsky and M.S. Plyushchay, Hidden superconformal symmetry of spinless Aharonov-Bohm system, J. Phys. A 43 (2010) 075202 [arXiv:0906.4055] [INSPIRE].
F. Correa and M.S. Plyushchay, Hidden supersymmetry in quantum bosonic systems, Annals Phys. 322 (2007) 2493 [hep-th/0605104] [INSPIRE].
V. Jakubsky, L.-M. Nieto and M.S. Plyushchay, The origin of the hidden supersymmetry, Phys. Lett. B 692 (2010) 51 [arXiv:1004.5489] [INSPIRE].
F. Correa, V. Jakubsky, L.-M. Nieto and M.S. Plyushchay, Self-isospectrality, special supersymmetry, and their effect on the band structure, Phys. Rev. Lett. 101 (2008) 030403 [arXiv:0801.1671] [INSPIRE].
M.S. Plyushchay and L.-M. Nieto, Self-isospectrality, mirror symmetry, and exotic nonlinear supersymmetry, Phys. Rev. D 82 (2010) 065022 [arXiv:1007.1962] [INSPIRE].
F. Correa, H. Falomir, V. Jakubsky and M.S. Plyushchay, Supersymmetries of the spin-1/2 particle in the field of magnetic vortex, and anyons, Annals Phys. 325 (2010) 2653 [arXiv:1003.1434] [INSPIRE].
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Inzunza, L., Plyushchay, M.S. Conformal bridge transformation, \( \mathcal{PT} \)- and supersymmetry. J. High Energ. Phys. 2022, 228 (2022). https://doi.org/10.1007/JHEP08(2022)228
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DOI: https://doi.org/10.1007/JHEP08(2022)228