Abstract
The Skyrme-Faddeev model has planar soliton solutions with the target space ℂPN. An Abelian Chern-Simons term (the Hopf term) in the Lagrangian of the model plays a crucial role for the statistical properties of the solutions. Because П3(ℂP1) = ℤ, the term becomes an integer for N = 1. On the other hand, for N > 1, it becomes perturbative because П3(ℂPN) is trivial. The prefactor Θ of the Hopf term is not quantized, and its value depends on the physical system. We study the spectral flow of normalizable fermions coupled with the baby-Skyrme model (ℂPN Skyrme-Faddeev model). We discuss whether the statistical nature of solitons can be explained using their constituents, i.e., quarks.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. A. Ferreira and P. Klimas, “Exact vortex solutions in a CP N Skyrme-Faddeev type model,” JHEP, 1010, 008 (2010); arXiv:1007.1667v1 [hep-th] (2010).
Y. Amari, P. Klimas, N. Sawado, and Y. Tamaki, “Potentials and the vortex solutions in the CP N Skyrme-Faddeev model,” Phys. Rev. D, 92, 045007 (2015); arXiv:1504.02848v1 [hep-th] (2015).
B. M. A. G. Piette, B. J. Schroers, and W. J. Zakrzewski, “Multi-solitons in a two-dimensional Skyrme model,” Z. Phys. C, 65, 165–174 (1995); arXiv:hep-th/9406160v1 (1994).
F. Wilczek and A. Zee, “Linking numbers, spin, and statistics of solitons,” Phys. Rev. Lett., 51, 2250–2252 (1983).
A. G. Abanov, “Hopf term induced by fermions,” Phys. Lett. B, 492, 321–323 (2000); arXiv:hep-th/0005150v2 (2000).
A. G. Abanov and P. B. Wiegmann, “On the correspondence between fermionic number and statistics of solitons,” JHEP, 0110, 030 (2001); arXiv:hep-th/0105213v1 (2001).
O. Bar, M. Imboden, and U. J. Wiese, “Pions versus magnons: From QCD to antiferromagnets and quantum Hall ferromagnets,” Nucl. Phys. B, 686, 347–376 (2004); arXiv:cond-mat/0310353v1 (2003).
T. Jaroszewicz, “Induced topological terms, spin, and statistics in (2+1)-dimensions,” Phys. Lett. B, 159, 299–302 (1985).
M. F. Atiyah and I. M. Singer, “The index of elliptic operators: 1,” Ann. Math.Ser.2, 87, 484–530 (1968).
S. Kahana and G. Ripka, “Baryon density of quarks coupled to a chiral field,” Nucl. Phys. A, 429, 462–476 (1984).
A. Acus, E. Norvaisas, and Ya. Shnir, “Baby Skyrmions stabilized by canonical quantization,” Phys. Lett. B, 682, 155–162 (2009); arXiv:0909.5281v1 [hep-th] (2009).
C.-C. Liu, P. Goswami, and Q. Si, “Skyrmion defects of antiferromagnet and competing singlet orders of a Kondo-Heisenberg model on honeycomb lattice,” Phys. Rev. B, 96, 125101 (2017); arXiv:1704.07818v2 [condmat.str-el] (2017).
E. Witten, “Global aspects of current algebra,” Nucl. Phys. B, 223, 422–432 (1983).
R. Jackiw and P. Rossi, “Zero modes of the vortex-fermion system,” Nucl. Phys. B, 190, 681–691 (1981).
D. I. Diakonov, V. Yu. Petrov, and P. V. Pobylitsa, “A chiral theory of nucleons,” Nucl. Phys. B, 306, 809–848 (1988).
R. Alkofer and H. Reinhardt, Chiral Quark Dynamics (Lect. Notes Phys. Monogr., Vol. 33), Springer, Berlin (1995).
A. D’Adda and A. C. Davis, “Chiral symmetry restoration in the CP(n−1) in two-dimensions model with fermions,” Phys. Lett. B, 101, 85–88 (1981).
A. D’Adda, M. Luscher, and P. Di Vecchia, “A 1/n expandable series of nonlinear σ models with instantons,” Nucl. Phys. B, 146, 63–76 (1978).
A. E. Kudryavtsev, B. M. A. G. Piette, and W. J. Zakrzewski, “Skyrmions and domain walls in (2+1)-dimensions,” Nonlinearity, 11, 783–795 (1998); arXiv:hep-th/9709187v1 (1997).
Y. Amari, P. Klimas, and N. Sawado, “Collective coordinate quantization, spin statistics of the solitons in the ℂP N Skyrme-Faddeev model,” Phys. Rev. D, 94, 025032 (2016); arXiv:1604.06125v2 [hep-th] (2016).
Acknowledgments
One of the authors (N. S.) thanks the conference organizers of MQFT-2018 for the kind accommodation and hospitality.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Conflicts of interest
The authors declare no conflicts of interest.
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 200, No. 3, pp. 381–398, September, 2019.
Rights and permissions
About this article
Cite this article
Amari, Y., Iida, M. & Sawado, N. Statistical Nature of Skyrme-Faddeev Models in 2+1 Dimensions and Normalizable Fermions. Theor Math Phys 200, 1253–1268 (2019). https://doi.org/10.1134/S0040577919090010
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577919090010