Abstract
We refine the notion of eclectic spin chains introduced in [1] by including a maximal number of deformation parameters. These models are integrable, nearest-neighbor n-state spin chains with exceedingly simple non-hermitian Hamiltonians. They turn out to be non-diagonalizable in the multiparticle sector (n > 2), where their “spectrum” consists of an intricate collection of Jordan blocks of arbitrary size and multiplicity. We show how and why the quantum inverse scattering method, sought to be universally applicable to integrable nearest-neighbor spin chains, essentially fails to reproduce the details of this spectrum. We then provide, for n=3, detailed evidence by a variety of analytical and numerical techniques that the spectrum is not “random”, but instead shows surprisingly subtle and regular patterns that moreover exhibit universality for generic deformation parameters. We also introduce a new model, the hypereclectic spin chain, where all parameters are zero except for one. Despite the extreme simplicity of its Hamiltonian, it still seems to reproduce the above “generic” spectra as a subset of an even more intricate overall spectrum. Our models are inspired by parts of the one-loop dilatation operator of a strongly twisted, double-scaled deformation of \( \mathcal{N} \) = 4 Super Yang-Mills Theory.
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References
A.C. Ipsen, M. Staudacher and L. Zippelius, The one-loop spectral problem of strongly twisted N = 4 super Yang-Mills theory, JHEP 04 (2019) 044 [arXiv:1812.08794] [INSPIRE].
H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain, Z. Phys. 71 (1931) 205 [INSPIRE].
Ö. Gürdoğan and V. Kazakov, New integrable 4D quantum field theories from strongly deformed planar N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 117 (2016) 201602 [Addendum ibid. 117 (2016) 259903] [arXiv:1512.06704] [INSPIRE].
C. Sieg and M. Wilhelm, On a CFT limit of planar γi-deformed N = 4 SYM theory, Phys. Lett. B 756 (2016) 118 [arXiv:1602.05817] [INSPIRE].
J. Caetano, Ö. Gürdoğan and V. Kazakov, Chiral limit of N = 4 SYM and ABJM and integrable Feynman graphs, JHEP 03 (2018) 077 [arXiv:1612.05895] [INSPIRE].
D. Chicherin, V. Kazakov, F. Loebbert, D. Müller and D.-L. Zhong, Yangian symmetry for bi-scalar loop amplitudes, JHEP 05 (2018) 003 [arXiv:1704.01967] [INSPIRE].
N. Gromov, V. Kazakov, G. Korchemsky, S. Negro and G. Sizov, Integrability of conformal fishnet theory, JHEP 01 (2018) 095 [arXiv:1706.04167] [INSPIRE].
D. Chicherin, V. Kazakov, F. Loebbert, D. Müller and D.-L. Zhong, Yangian symmetry for fishnet Feynman graphs, Phys. Rev. D 96 (2017) 121901 [arXiv:1708.00007] [INSPIRE].
D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, Strongly γ-deformed N = 4 supersymmetric Yang-Mills theory as an integrable conformal field theory, Phys. Rev. Lett. 120 (2018) 111601 [arXiv:1711.04786] [INSPIRE].
V. Kazakov, Quantum spectral curve of γ-twisted N = 4 SYM theory and fishnet CFT, Rev. Math. Phys. 30 (2018) 1840010 [arXiv:1802.02160] [INSPIRE].
N. Gromov, V. Kazakov and G. Korchemsky, Exact correlation functions in conformal fishnet theory, JHEP 08 (2019) 123 [arXiv:1808.02688] [INSPIRE].
L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Les Houches school of physics: astrophysical sources of gravitational radiation, (1996), pg. 149 [hep-th/9605187] [INSPIRE].
R.I. Nepomechie, A spin chain primer, Int. J. Mod. Phys. B 13 (1999) 2973 [hep-th/9810032] [INSPIRE].
F. Levkovich-Maslyuk, The Bethe ansatz, J. Phys. A 49 (2016) 323004 [arXiv:1606.02950] [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
K. Zoubos, Review of AdS/CFT integrability, chapter IV.2: deformations, orbifolds and open boundaries, Lett. Math. Phys. 99 (2012) 375 [arXiv:1012.3998] [INSPIRE].
A.M. Gainutdinov and R.I. Nepomechie, Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity, Nucl. Phys. B 909 (2016) 796 [arXiv:1603.09249] [INSPIRE].
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Ahn, C., Staudacher, M. The integrable (hyper)eclectic spin chain. J. High Energ. Phys. 2021, 19 (2021). https://doi.org/10.1007/JHEP02(2021)019
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DOI: https://doi.org/10.1007/JHEP02(2021)019