Abstract
We study a non-Hermitian (NH) sl(2) affine Toda model coupled to fermions through soliton theory techniques and the realizations of the pseudo-chiral and pseudo- Hermitian symmetries. The interplay of non-Hermiticity, integrability, nonlinearity, and topology significantly influence the formation and behavior of a continuum of bound state modes (CBM) and extended waves in the localized continuum (ELC). The non-Hermitian soliton-fermion duality, the complex scalar field topological charges and winding numbers in the spectral topology are uncovered. The biorthogonal Majorana zero modes, dual to the NH Toda solitons with topological charges \( \frac{2}{\pi}\arg \left(z=\pm i\right)=\pm 1 \), appear at the complex-energy point gap and are pinned at zero energy. The zero eigenvalue λ(z = ± i) = 0, besides being a zero mode, plays the role of exceptional points (EPs), and each EP separates a real eigenvalue \( \mathcal{A} \)-symmetric and \( \mathcal{A} \)-symmetry broken regimes for an antilinear symmetry \( \mathcal{A}\in \left\{\mathcal{PT},{\gamma}_5\mathcal{PT}\right\} \). Our findings improve the understanding of exotic quantum states, but also paves the way for future research in harnessing non-Hermitian phenomena for topological quantum computation, as well as the exploration of integrability and NH solitons in the theory of topological phases of matter.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Y. Ashida, Z. Gong and M. Ueda, Non-Hermitian physics, Adv. Phys. 69 (2021) 249 [arXiv:2006.01837] [INSPIRE].
E.J. Bergholtz, J.C. Budich and F.K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93 (2021) 015005 [arXiv:1912.10048] [INSPIRE].
K. Kawabata, K. Shiozaki, M. Ueda and M. Sato, Symmetry and Topology in Non-Hermitian Physics, Phys. Rev. X 9 (2019) 041015 [arXiv:1812.09133] [INSPIRE].
E.A. Ivanov and A.V. Smilga, Cryptoreality of nonanticommutative Hamiltonians, JHEP 07 (2007) 036 [hep-th/0703038] [INSPIRE].
J.D.H. Rivero and L. Ge, Pseudochirality: A Manifestation of Noether’s Theorem in Non-Hermitian Systems, Phys. Rev. Lett. 125 (2020) 083902.
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].
A. Mostafazadeh, PseudoHermiticity versus PT symmetry 3: Equivalence of pseudoHermiticity and the presence of antilinear symmetries, J. Math. Phys. 43 (2002) 3944 [math-ph/0203005] [INSPIRE].
A. Mostafazadeh, PseudoHermiticity versus PT symmetry 2. A complete characterization of nonHermitian Hamiltonians with a real spectrum, J. Math. Phys. 43 (2002) 2814 [math-ph/0110016] [INSPIRE].
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, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70 (2007) 947 [hep-th/0703096] [INSPIRE].
C. Bender et al., PT-Symmetry in Quantum And Classical Physics, World Scientific Publishing Company (2018).
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) 74 [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].
N. Okuma and M. Sato, Non-Hermitian Topological Phenomena: A Review, Ann. Rev. Condensed Matter Phys. 14 (2023) 83 [arXiv:2205.10379] [INSPIRE].
P. Delplace, J.B. Marston and A. Venaille, Topological origin of equatorial waves, Science 358 (2017) 1075.
H. Blas et al., Zero mode-soliton duality and pKdV kinks in Boussinesq system for non-linear shallow water waves, arXiv:2305.04037 [INSPIRE].
L. Jezequel and P. Delplace, Nonlinear edge modes from topological one-dimensional lattices, Phys. Rev. B 105 (2022) 035410.
R. Jackiw and C. Rebbi, Solitons with Fermion Number 1/2, Phys. Rev. D 13 (1976) 3398 [INSPIRE].
J. Goldstone and F. Wilczek, Fractional Quantum Numbers on Solitons, Phys. Rev. Lett. 47 (1981) 986 [INSPIRE].
R. MacKenzie and F. Wilczek, Illustrations of Vacuum Polarization by Solitons, Phys. Rev. D 30 (1984) 2194 [INSPIRE].
H. Blas, J.J. Monsalve, R. Quicaño and J.R.V. Pereira, Majorana zero mode-soliton duality and in-gap and BIC bound states in modified Toda model coupled to fermion, JHEP 09 (2022) 082 [arXiv:2207.01161] [INSPIRE].
H.S. Blas Achic and L.A. Ferreira, Confinement, solitons and the equivalence between the sine-Gordon and massive Thirring models, Nucl. Phys. B 571 (2000) 607 [hep-th/9909118] [INSPIRE].
H. Blas, Noether and topological currents equivalence and soliton / particle correspondence in affine sl(2)(1) Toda theory coupled to matter, Nucl. Phys. B 596 (2001) 471 [hep-th/0011243] [INSPIRE].
F.K. Kunst, E. Edvardsson, J.C. Budich and E.J. Bergholtz, Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems, Phys. Rev. Lett. 121 (2018) 026808 [INSPIRE].
A. Udupa, A. Banerjee, K. Sengupta and D. Sen, One-dimensional spin-orbit coupled Dirac system with extended s-wave superconductivity: Majorana modes and Josephson effects, J. Phys. Condens. Matter 33 (2021) 145301 [INSPIRE].
C. Spånslätt, E. Ardonne, J.C. Budich and T.H. Hansson, Topological aspects of π phase winding junctions in superconducting wires, J. Phys. Condens. Matter 27 (2015) 405701 [arXiv:1501.03413] [INSPIRE].
R. Eneias and A. Ferraz, BCS coupling in a 1D Luttinger liquid, New J. Phys. 17 (2015) 123006.
H. Blas, Affine toda model coupled to matter and the string tension in QCD, Phys. Rev. D 66 (2002) 127701 [hep-th/0209037] [INSPIRE].
Q. Wang et al., Continuum of Bound States in a Non-Hermitian Model, Phys. Rev. Lett. 130 (2023) 103602 [arXiv:2210.03738] [INSPIRE].
W. Wang, X. Wang and G. Ma, Extended State in a Localized Continuum, Phys. Rev. Lett. 129 (2022) 264301 [INSPIRE].
A.M. Jazayeri, Fixed points on band structures of non-Hermitian models: Extended states in the bandgap and ideal superluminal tunneling, Phys. Rev. B 107 (2023) 144302.
T.J. Hollowood, Solitons in affine Toda field theories, Nucl. Phys. B 384 (1992) 523 [hep-th/9110010] [INSPIRE].
J. Cen and A. Fring, Complex solitons with real energies, J. Phys. A 49 (2016) 365202 [arXiv:1602.05465] [INSPIRE].
Z.-Y. Ge et al., Topological band theory for non-Hermitian systems from the Dirac equation, Phys. Rev. B 100 (2019) 054105 [arXiv:1903.09985] [INSPIRE].
C.-G. Oh, S.-H. Han and S. Cheon, Symmetry-protected solitons and bulk-boundary correspondence in generalized Jackiw-Rebbi models, Sci. Rep. 11 (2021) 21652 [arXiv:2108.00240] [INSPIRE].
D. Sticlet, B. Dóra and C.P. Moca, Kubo Formula for Non-Hermitian Systems and Tachyon Optical Conductivity, Phys. Rev. Lett. 128 (2022) 016802 [arXiv:2104.02428] [INSPIRE].
L.A. Ferreira, J.-L. Gervais, J. Sánchez Guillen and M.V. Savelev, Affine Toda systems coupled to matter fields, Nucl. Phys. B 470 (1996) 236 [hep-th/9512105] [INSPIRE].
H. Blas and B.M. Pimentel, The Faddeev-Jackiw approach and the affine sl(2) Toda model coupled to matter field, Annals Phys. 282 (2000) 67 [hep-th/9905026] [INSPIRE].
S. Franca et al., Non-Hermitian Physics without Gain or Loss: The Skin Effect of Reflected Waves, Phys. Rev. Lett. 129 (2022) 086601 [arXiv:2111.02263] [INSPIRE].
A. Fring, An Introduction to PT-Symmetric Quantum Mechanics-Time-Dependent Systems, J. Phys. Conf. Ser. 2448 (2023) 012002 [arXiv:2201.05140] [INSPIRE].
S.-J. Chang, S.D. Ellis and B.W. Lee, Chiral Confinement: An Exact Solution of the Massive Thirring Model, Phys. Rev. D 11 (1975) 3572 [INSPIRE].
E. Witten, Chiral Symmetry, the 1/n Expansion, and the SU(N) Thirring Model, Nucl. Phys. B 145 (1978) 110 [INSPIRE].
D.A. Takahashi and M. Nitta, Self-Consistent Multiple Complex-Kink Solutions in Bogoliubov-de Gennes and Chiral Gross-Neveu Systems, Phys. Rev. Lett. 110 (2013) 131601 [arXiv:1209.6206] [INSPIRE].
M. Stone, Bosonization, World Scientific (1994).
J. Wiersig, Distance between exceptional points and diabolic points and its implication for the response strength of non-Hermitian systems, Phys. Rev. Res. 4 (2022) 033179 [arXiv:2205.15685] [INSPIRE].
M. Sato, K. Hasebe, K. Esaki and M. Kohmoto, Time-Reversal Symmetry in Non-Hermitian Systems, Prog. Theor. Phys. 127 (2012) 937 [arXiv:1106.1806] [INSPIRE].
A. Ghatak and T. Das, New topological invariants in non-Hermitian systems, J. Phys. Condens. Matter 31 (2019) 263001 [arXiv:1902.07972] [INSPIRE].
Z.-H. Wang et al., Majorana polarization in non-Hermitian topological superconductors, Phys. Rev. B 103 (2021) 134507.
A. Mostafazadeh, PseudoHermiticity and generalized PT and CPT symmetries, J. Math. Phys. 44 (2003) 974 [math-ph/0209018] [INSPIRE].
A. Beygi, S.P. Klevansky and C.M. Bender, Coupled Oscillator Systems Having Partial PT Symmetry, Phys. Rev. A 91 (2015) 062101 [arXiv:1503.05725] [INSPIRE].
Y. Long, H. Xue and B. Zhang, Always-Real-Eigenvalued Non-Hermitian Topological Systems, Phys. Rev. B 105 (2022) L100102 [arXiv:2111.02701] [INSPIRE].
A. Melkani, Degeneracies and symmetry breaking in pseudo-Hermitian matrices, Phys. Rev. Res. 5 (2023) 023035 [arXiv:2209.06887] [INSPIRE].
C.M. Bender, H.F. Jones and R.J. Rivers, Dual PT-symmetric quantum field theories, Phys. Lett. B 625 (2005) 333 [hep-th/0508105] [INSPIRE].
B. Basu-Mallick and D. Sinha, Integrable coupled bosonic massive Thirring model and its nonlocal reductions, JHEP 03 (2024) 071 [arXiv:2307.00351] [INSPIRE].
B. Basu-Mallick, F. Finkel, A. González-López and D. Sinha, Integrable coupled massive Thirring model with field values in a Grassmann algebra, JHEP 11 (2023) 018 [arXiv:2307.03626] [INSPIRE].
H. Blas, Higher grading conformal affine Toda theory and (generalized) sine-Gordon/massive Thirring duality, JHEP 11 (2003) 054 [hep-th/0306171] [INSPIRE].
H. Blas and H.L. Carrion, Solitons, kinks and extended hadron model based on the generalized sine-Gordon theory, JHEP 01 (2007) 027 [hep-th/0610107] [INSPIRE].
B. Liégeois, C. Ramasubramanian and N. Defenu, Tunable tachyon mass in the PT-broken massive Thirring model, Phys. Rev. D 108 (2023) 116014 [arXiv:2212.08110] [INSPIRE].
Y. Ashida, S. Furukawa and M. Ueda, Parity-time-symmetric quantum critical phenomena, Nature Commun. 8 (2017) 15791 [INSPIRE].
R. Arouca, J. Cayao and A.M. Black-Schaffer, Topological superconductivity enhanced by exceptional points, Phys. Rev. B 108 (2023) L060506 [arXiv:2206.15324] [INSPIRE].
H. Blas, H.F. Callisaya and J.P.R. Campos, Riccati-type pseudo-potentials, conservation laws and solitons of deformed sine-Gordon models, Nucl. Phys. B 950 (2020) 114852 [arXiv:1801.00866] [INSPIRE].
Acknowledgments
The author is grateful to J.J. Monsalve and J.R.V. Pereira for useful discussions, and Dr. A.M. Jazayeri for calling his attention to reference [31].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2310.03215
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Blas, H. Biorthogonal Majorana zero modes, ELC waves and soliton-fermion duality in non-Hermitian sl(2) affine Toda coupled to fermions. J. High Energ. Phys. 2024, 7 (2024). https://doi.org/10.1007/JHEP06(2024)007
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2024)007