Abstract
We present and solve a soliton equation which we call the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation. This equation, which depends on a parameter δ > 0, describes the time evolution of two coupled spin densities propagating on the real line, and in the limit δ → ∞ it reduces to two decoupled half-wave maps (HWM) equations of opposite chirality. We show that the ncIHF equation is related to the A-type hyperbolic spin Calogero-Moser (CM) system in two distinct ways: (i) it is obtained as a particular continuum limit of an Inozemtsev-type spin chain related to this CM system, (ii) it has multi-soliton solutions obtained by a spin-pole ansatz and with parameters satisfying the equations of motion of a complexified version of this CM system. The integrability of the ncIHF equation is shown by constructing a Lax pair. We also propose a periodic variant of the ncIHF equation related to the A-type elliptic spin CM system.
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B. Estienne, V. Pasquier, R. Santachiara and D. Serban, Conformal blocks in Virasoro and W theories: Duality and the Calogero-Sutherland model, Nucl. Phys. B 860 (2012) 377 [arXiv:1110.1101] [INSPIRE].
M. Isachenkov and V. Schomerus, Superintegrability of d-dimensional Conformal Blocks, Phys. Rev. Lett. 117 (2016) 071602 [arXiv:1602.01858] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in 16th International Congress on Mathematical Physics, (2009), pp. 265–289, DOI [arXiv:0908.4052] [INSPIRE].
G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics, JHEP 07 (2014) 141 [arXiv:1403.6454] [INSPIRE].
P. Koroteev and A. Sciarappa, On Elliptic Algebras and Large-N Supersymmetric Gauge Theories, J. Math. Phys. 57 (2016) 112302 [arXiv:1601.08238] [INSPIRE].
P. Koroteev and A. Sciarappa, Quantum Hydrodynamics from Large-N Supersymmetric Gauge Theories, Lett. Math. Phys. 108 (2018) 45 [arXiv:1510.00972] [INSPIRE].
A. Gorsky, O. Koroteeva, P. Koroteev and A. Vainshtein, On dimensional transmutation in 1 + 1D quantum hydrodynamics, J. Math. Phys. 61 (2020) 082302 [arXiv:1910.02606] [INSPIRE].
A.P. Polychronakos, Waves and solitons in the continuum limit of the Calogero-Sutherland model, Phys. Rev. Lett. 74 (1995) 5153 [hep-th/9411054] [INSPIRE].
A.G. Abanov and P.B. Wiegmann, Quantum hydrodynamics, quantum Benjamin-Ono equation, and Calogero model, Phys. Rev. Lett. 95 (2005) 076402 [cond-mat/0504041] [INSPIRE].
T. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967) 559.
H. Ono, Algebraic Solitary Waves in Stratified Fluids, J. Phys. Soc. Jap. 39 (1975) 1082.
A. Jevicki, Nonperturbative collective field theory, Nucl. Phys. B 376 (1992) 75 [INSPIRE].
H. Azuma and S. Iso, Explicit relation of quantum hall effect and Calogero-Sutherland model, Phys. Lett. B 331 (1994) 107 [hep-th/9312001] [INSPIRE].
S. Iso, Anyon basis of c = 1 conformal field theory, Nucl. Phys. B 443 (1995) 581 [hep-th/9411051] [INSPIRE].
H. Awata, Y. Matsuo, S. Odake and J. Shiraishi, Excited states of Calogero-Sutherland model and singular vectors of the W(N) algebra, Nucl. Phys. B 449 (1995) 347 [hep-th/9503043] [INSPIRE].
A.L. Carey and E. Langmann, Loop groups, anyons and the Calogero-Sutherland model, Commun. Math. Phys. 201 (1999) 1 [math-ph/9805010] [INSPIRE].
A.G. Abanov, E. Bettelheim and P. Wiegmann, Integrable hydrodynamics of Calogero-Sutherland model: Bidirectional Benjamin-Ono equation, J. Phys. A 42 (2009) 135201 [arXiv:0810.5327] [INSPIRE].
B.K. Berntson, E. Langmann and J. Lenells, Nonchiral intermediate long-wave equation and interedge effects in narrow quantum Hall systems, Phys. Rev. B 102 (2020) 155308 [arXiv:2001.04462] [INSPIRE].
E. Langmann, Solution algorithm for the elliptic Calogero-Sutherland model, Lett. Math. Phys. 54 (2000) 279 [math-ph/0007036] [INSPIRE].
T. Zhou and M. Stone, Solitons in a continuous classical Haldane-Shastry spin chain, Phys. Lett. A 379 (2015) 2817 [arXiv:1504.00873].
E. Lenzmann and A. Schikorra, On energy-critical half-wave maps into 𝕊2, Invent. Math. 213 (2018) 1 [arXiv:1702.05995].
E. Lenzmann and J. Sok, Derivation of the Half-Wave Maps Equation from Calogero-Moser Spin Systems, arXiv:2007.15323.
P. Gérard and E. Lenzmann, A Lax pair structure for the half-wave maps equation, Lett. Math. Phys. 108 (2018) 1635 [arXiv:1707.05028].
B.K. Berntson, R. Klabbers and E. Langmann, Multi-solitons of the half-wave maps equation and Calogero-Moser spin-pole dynamics, J. Phys. A 53 (2020) 505702 [arXiv:2006.16826] [INSPIRE].
V. Inozemtsev, On the connection between the one-dimensional S = 1/2 Heisenberg chain and Haldane-Shastry model, J. Stat. Phys. 59 (1990) 1143.
B. Berntson, E. Langmann and J. Lenells, On the non-chiral intermediate long wave equation II: periodic case, arXiv:2103.02572.
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.26 (15-03-2020).
F. Calogero, Classical Many-Body Problems Amenable to Exact Treatments, vol. 66 of Lecture Notes in Physics Monographs, Spring-Verlag Berlin Heidelberg (2001).
A.P. Polychronakos, Lattice integrable systems of Haldane-Shastry type, Phys. Rev. Lett. 70 (1993) 2329 [hep-th/9210109] [INSPIRE].
F. Calogero, A Sequence of Lax Matrices for Certain Integrable Hamiltonian Systems, Lett. Nuovo Cim. 16 (1976) 22 [INSPIRE].
F. Finkel and A. González-López, A new perspective on the integrability of Inozemtsev’s elliptic spin chain, Annals Phys. 351 (2014) 797 [arXiv:1405.7855] [INSPIRE].
F.D.M. Haldane, Exact Jastrow-Gutzwiller resonating valence bond ground state of the spin 1/2 antiferromagnetic Heisenberg chain with 1/R2 exchange, Phys. Rev. Lett. 60 (1988) 635 [INSPIRE].
B. Sriram Shastry, Exact solution of an S = 1/2 Heisenberg antiferromagnetic chain with long ranged interactions, Phys. Rev. Lett. 60 (1988) 639 [INSPIRE].
F. Calogero, Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable?, in What Is Integrability?, (Berlin, Heidelberg), pp. 1–62, Springer Berlin Heidelberg (1991), [DOI].
K. Tamizhmani, J. Satsuma, B. Grammaticos and A. Ramani, Nonlinear integrodifferential equations as discrete systems, Inverse Probl. 15 (1999) 787.
M. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, U.K. (1991).
Y. Kodama, J. Satsuma and M. Ablowitz, Nonlinear Intermediate Long-Wave Equation: Analysis and Method of Solution, Phys. Rev. Lett. 46 (1981) 687.
B. Berntson, E. Langmann and J. Lenells, On the non-chiral intermediate long wave equation, arXiv:2005.10781.
H. Chen, Y. Lee and N. Pereira, Algebraic internal wave solitons and the integrable Calogero-Moser-Sutherland N-body problem, Phys. Fluids 22 (1979) 187.
D. Uglov, The Trigonometric counterpart of the Haldane-Shastry model, hep-th/9508145 [INSPIRE].
J. Lamers, Resurrecting the partially isotropic Haldane-Shastry model, Phys. Rev. B 97 (2018) 214416 [arXiv:1801.05728] [INSPIRE].
I. Krichever and A. Zabrodin, Spin generalization of the Ruijsenaars-Schneider model, nonAbelian 2-D Toda chain and representations of Sklyanin algebra, Russ. Math. Surveys 50 (1995) 1101 [hep-th/9505039] [INSPIRE].
G.E. Arutyunov and S.A. Frolov, On Hamiltonian structure of the spin Ruijsenaars-Schneider model, J. Phys. A 31 (1998) 4203 [hep-th/9703119] [INSPIRE].
M. Kulkarni, F. Franchini and A.G. Abanov, Nonlinear dynamics of spin and charge in spin-Calogero model, Phys. Rev. B 80 (2009) 165105 [arXiv:0904.3762] [INSPIRE].
L. Xing, Classical hydrodynamics of Calogero-Sutherland models, Ph.D. Thesis, University of Illinois at Urbana-Champaign, U.S.A. (2015).
F. Calogero, Exactly Solvable One-Dimensional Many Body Problems, Lett. Nuovo Cim. 13 (1975) 411 [INSPIRE].
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Berntson, B.K., Klabbers, R. & Langmann, E. The non-chiral intermediate Heisenberg ferromagnet equation. J. High Energ. Phys. 2022, 46 (2022). https://doi.org/10.1007/JHEP03(2022)046
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DOI: https://doi.org/10.1007/JHEP03(2022)046