Abstract
We investigate a special class of the \( \mathcal{P}\mathcal{T} \) -symmetric quantum models being perfectly invisible zero-gap systems with a unique bound state at the very edge of continuous spectrum of scattering states. The family includes the \( \mathcal{P}\mathcal{T} \) -regularized two particle Calogero systems (conformal quantum mechanics models of de Alfaro-Fubini-Furlan) and their rational extensions whose potentials satisfy equations of the KdV hierarchy and exhibit, particularly, a behaviour typical for extreme waves. We show that the two simplest Hamiltonians from the Calogero subfamily determine the fluctuation spectra around the \( \mathcal{P}\mathcal{T} \)-regularized kinks arising as traveling waves in the field-theoretical Liouville and SU(3) conformal Toda systems. Peculiar properties of the quantum systems are reflected in the associated exotic nonlinear supersymmetry in the unbroken or partially broken phases. The conventional \( \mathcal{N}=2 \) supersymmetry is extended here to the \( \mathcal{N}=4 \) nonlinear supersymmetry that involves two bosonic generators composed from Lax-Novikov integrals of the subsystems, one of which is the central charge of the superalgebra. Jordan states are shown to play an essential role in the construction.
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C.M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243 [physics/9712001] [INSPIRE].
C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70 (2007) 947 [hep-th/0703096] [INSPIRE].
A. Mostafazadeh, Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 1191 [arXiv:0810.5643] [INSPIRE].
P. Dorey, C. Dunning and R. Tateo, Spectral equivalences, Bethe ansatz equations and reality properties in PT-symmetric quantum mechanics, J. Phys. A 34 (2001) 5679 [hep-th/0103051] [INSPIRE].
P. Dorey, C. Dunning and R. Tateo, The ODE/IM correspondence, J. Phys. A 40 (2007) R205 [hep-th/0703066] [INSPIRE].
F. Calogero, Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419 [INSPIRE].
M. Znojil and M. Tater, Complex Calogero model with real energies, J. Phys. A 34 (2001) 1793 [quant-ph/0010087].
P.K. Ghosh and K.S. Gupta, On the real spectra of Calogero model with complex coupling, Phys. Lett. A 323 (2004) 29 [hep-th/0310276] [INSPIRE].
B. Basu-Mallick, T. Bhattacharyya, A. Kundu and B.P. Mandal, Bound and scattering states of extended Calogero model with an additional PT invariant interaction, Czech. J. Phys. 54 (2004) 5 [hep-th/0309136] [INSPIRE].
A. Fring and M. Znojil, PT-symmetric deformations of Calogero models, J. Phys. A 41 (2008) 194010 [arXiv:0802.0624] [INSPIRE].
A. Fring and M. Smith, Non-Hermitian multi-particle systems from complex root spaces, J. Phys. A 45 (2012) 085203 [arXiv:1108.1719] [INSPIRE].
A. Fring, PT-symmetric deformations of integrable models, Phil. Trans. Roy. Soc. Lond. A 371 (2013) 20120046 [arXiv:1204.2291] [INSPIRE].
F. Correa and O. Lechtenfeld, PT deformation of angular Calogero models, JHEP 11 (2017) 122 [arXiv:1705.05425] [INSPIRE].
H. Airault, H.P. McKean and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977) 95.
M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Commun. Math. Phys. 61 (1978) 1 [INSPIRE].
A. Gorsky and N. Nekrasov, Hamiltonian systems of Calogero type and two-dimensional Yang-Mills theory, Nucl. Phys. B 414 (1994) 213 [hep-th/9304047] [INSPIRE].
M. Znojil, F. Cannata, B. Bagchi and R. Roychoudhury, Supersymmetry without hermiticity within PT symmetric quantum mechanics, Phys. Lett. B 483 (2000) 284 [hep-th/0003277] [INSPIRE].
M. Znojil, PT symmetrized supersymmetric quantum mechanics, Czech. J. Phys. 51 (2001) 420 [hep-ph/0101038] [INSPIRE].
P. Dorey, C. Dunning and R. Tateo, Supersymmetry and the spontaneous breakdown of PT symmetry, J. Phys. A 34 (2001) L391 [hep-th/0104119] [INSPIRE].
B. Bagchi, S. Mallik and C. Quesne, Complexified PSUSY and SSUSY interpretations of some PT symmetric Hamiltonians possessing two series of real energy eigenvalues, Int. J. Mod. Phys. A 17 (2002) 51 [quant-ph/0106021] [INSPIRE].
F. Correa and M.S. Plyushchay, Self-isospectral tri-supersymmetry in PT-symmetric quantum systems with pure imaginary periodicity, Annals Phys. 327 (2012) 1761 [arXiv:1201.2750] [INSPIRE].
F. Correa and M.S. Plyushchay, Spectral singularities in PT-symmetric periodic finite-gap systems, Phys. Rev. D 86 (2012) 085028 [arXiv:1208.4448] [INSPIRE].
S.P. Novikov, S.V. Manakov, L.P. Pitaevskii and V.E. Zakharov, Theory of solitons, Plenum, New York U.S.A., (1984).
V.B. Matveev and M.A. Salle, Darboux transformations and solitons, Springer, Berlin Germany, (1991).
L.A. Bordag and V.B. Matveev, Self-similar solutions of the Korteweg-de Vries equation and potentials with a trivial S-matrix, Theor. Math. Phys. 34 (1978) 272 [Teor. Mat. Fiz. 34 (1978) 426].
V.B. Matveev, Positons: slowly decreasing analogues of solitons, Theor. Math. Phys. 131 (2002) 483 [Teor. Mat. Fiz. 131 (2002) 44] [INSPIRE].
A. Arancibia and M.S. Plyushchay, Chiral asymmetry in propagation of soliton defects in crystalline backgrounds, Phys. Rev. D 92 (2015) 105009 [arXiv:1507.07060] [INSPIRE].
F. Correa, M.A. del Olmo and M.S. Plyushchay, On hidden broken nonlinear superconformal symmetry of conformal mechanics and nature of double nonlinear superconformal symmetry, Phys. Lett. B 628 (2005) 157 [hep-th/0508223] [INSPIRE].
V. de Alfaro, S. Fubini and G. Furlan, Conformal invariance in quantum mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].
C.M. Bender, D.C. Brody, J. Chen and E. Furlan, PT-symmetric extension of the Korteweg-de Vries equation, J. Phys. A 40 (2007) F153 [math-ph/0610003] [INSPIRE].
A. Fring, PT-symmetric deformations of the Korteweg-de Vries equation, J. Phys. A 40 (2007) 4215 [math-ph/0701036] [INSPIRE].
C.M. Bender and J. Feinberg, Does the complex deformation of the Riemann equation exhibit shocks?, J. Phys. A 41 (2008) 244004 [arXiv:0709.2727] [INSPIRE].
A. Cavaglia and A. Fring, PT-symmetrically deformed shock waves, J. Phys. A 45 (2012) 444010 [arXiv:1201.5809] [INSPIRE].
S. Longhi and G. Della Valle, Invisible defects in complex crystals, Annals Phys. 334 (2013) 35 [arXiv:1306.0667].
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].
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].
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].
F. Correa and M.S. Plyushchay, Hidden supersymmetry in quantum bosonic systems, Annals Phys. 322 (2007) 2493 [hep-th/0605104] [INSPIRE].
J.L. Burchnall and T.W. Chaundy, Commutative ordinary differential operators, Proc. London Math. Soc. 21 (1923) 420.
J.L. Burchnall and T.W. Chaundy, Commutative ordinary differential operators, Proc. Roy. Soc. London A 118 (1928) 557.
I.M. Krichever, Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl. 11 (1977) 12.
I.M. Krichever, Baker-Akhiezer functions and integrable systems, in Integrability. The Seiberg-Witten and Whitham equations, H.W. Braden and I.M. Krichever eds., Gordon and Breach Science Publishers, Amsterdam The Netherlands, (2000), pg. 1.
A. Schulze-Halberg, Wronskian representation for confluent supersymmetric transformation chains of arbitrary order, Eur. Phys. J. Plus 128 (2013) 68.
A. Contreras-Astorga and A. Schulze-Halberg, Recursive representation of Wronskians in confluent supersymmetric quantum mechanics, J. Phys. A 50 (2017) 105301 [arXiv:1702.00843] [INSPIRE].
P. Drazin and R. Johnson, Solitons: an introduction, Cambridge University Press, Cambridge U.K., (1996).
J. Kumar, Conformal mechanics and the Virasoro algebra, JHEP 04 (1999) 006 [hep-th/9901139] [INSPIRE].
S. Cacciatori, D. Klemm and D. Zanon, W ∞ algebras, conformal mechanics and black holes, Class. Quant. Grav. 17 (2000) 1731 [hep-th/9910065] [INSPIRE].
E. D’Hoker and R. Jackiw, Liouville field theory, Phys. Rev. D 26 (1982) 3517 [INSPIRE].
R. Jackiw, Liouville field theory: a two-dimensional model for gravity?, in Quantum theory of gravity, S. Christensen ed, Adam Hilger, Bristol U.K., (1984), pg. 403.
A. Bilal and J.-L. Gervais, Extended C = ∞ conformal systems from classical Toda field theories, Nucl. Phys. B 314 (1989) 646 [INSPIRE].
K.E. Cahill, A. Comtet and R.J. Glauber, Mass formulas for static solitons, Phys. Lett. B 64 (1976) 283 [INSPIRE].
J. Mateos Guilarte, A. Alonso-Izquierdo, W. Garcia Fuertes, M. de la Torre Mayado and M.J. Senosiain, Quantum fluctuations around low-dimensional topological defects, PoS(ISFTG)013 [arXiv:0909.2107] [INSPIRE].
A. Alonso-Izquierdo and J. Mateos Guilarte, One-loop kink mass shifts: a computational approach, Nucl. Phys. B 852 (2011) 696 [arXiv:1107.2216] [INSPIRE].
A. Arancibia, J. Mateos Guilarte and M.S. Plyushchay, Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems, Phys. Rev. D 87 (2013) 045009 [arXiv:1210.3666] [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].
A. Arancibia and M.S. Plyushchay, Transmutations of supersymmetry through soliton scattering and self-consistent condensates, Phys. Rev. D 90 (2014) 025008 [arXiv:1401.6709] [INSPIRE].
A. Arancibia, J. Mateos Guilarte and M.S. Plyushchay, Fermion in a multi-kink-antikink soliton background and exotic supersymmetry, Phys. Rev. D 88 (2013) 085034 [arXiv:1309.1816] [INSPIRE].
S. Fedoruk, E. Ivanov and O. Lechtenfeld, OSp(4|2) superconformal mechanics, JHEP 08 (2009) 081 [arXiv:0905.4951] [INSPIRE].
E. Ivanov, S. Krivonos and O. Lechtenfeld, New variant of N = 4 superconformal mechanics, JHEP 03 (2003) 014 [hep-th/0212303] [INSPIRE].
C. Leiva and M.S. Plyushchay, Superconformal mechanics and nonlinear supersymmetry, JHEP 10 (2003) 069 [hep-th/0304257] [INSPIRE].
J. Mateos Guilarte and M.S. Plyushchay, Extended nonlinear super-Schrödinger symmetry of PT-symmetric perfectly invisible zero-gap quantum systems, in preparation.
A. Cavaglia, A. Fring and B. Bagchi, PT-symmetry breaking in complex nonlinear wave equations and their deformations, J. Phys. A 44 (2011) 325201 [arXiv:1103.1832] [INSPIRE].
J. Cen and A. Fring, Complex solitons with real energies, J. Phys. A 49 (2016) 365202 [arXiv:1602.05465] [INSPIRE].
J.M. Leinaas and J. Myrheim, On the theory of identical particles, Nuovo Cim. B 37 (1977) 1 [INSPIRE].
A.P. Polychronakos, Physics and mathematics of Calogero particles, J. Phys. A 39 (2006) 12793 [hep-th/0607033] [INSPIRE].
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Guilarte, J.M., Plyushchay, M.S. Perfectly invisible \( \mathcal{P}\mathcal{T} \) -symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry. J. High Energ. Phys. 2017, 61 (2017). https://doi.org/10.1007/JHEP12(2017)061
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DOI: https://doi.org/10.1007/JHEP12(2017)061