Skip to main content
SpringerLink
Log in

Higher Rank 1 + 1 Integrable Landau–Lifshitz Field Theories from the Associative Yang–Baxter Equation

  • METHODS OF THEORETICAL PHYSICS
  • Published:
JETP Letters Aims and scope Submit manuscript

We propose a construction of 1 + 1 integrable Heisenberg–Landau–Lifshitz type equations in the \({\text{g}}{{{\text{l}}}_{N}}\) case. The dynamical variables are matrix elements of an \(N \times N\) matrix S with the property \({{S}^{2}} = {\text{const}}{\kern 1pt} S\). The Lax pair with spectral parameter is constructed by means of a quantum R-matrix satisfying the associative Yang–Baxter equation. Equations of motion for \({\text{g}}{{{\text{l}}}_{N}}\) Landau–Lifshitz model are derived from the Zakharov–Shabat equations. The model is simplified when rank(S) = 1. In this case the Hamiltonian description is suggested. The described family of models includes the elliptic model coming from \({\text{G}}{{{\text{L}}}_{N}}\) Baxter–Belavin elliptic R-matrix. In \(N = 2\) case the widely known Sklyanin’s elliptic Lax pair for XYZ Landau–Lifshitz equation is reproduced. Our construction is also valid for trigonometric and rational degenerations of the elliptic R‑matrix.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. To derive ((7)) one should use the identities \(\varphi _{k}^{2}(z) = \wp (z) - \) \(\wp ({{\omega }_{k}})\), where \({{\omega }_{1}} = \tau {\text{/}}2\), \({{\omega }_{2}} = (\tau + 1){\text{/}}2\), \({{\omega }_{3}} = 1{\text{/}}2\) and \(\wp (z)\) is the Weierstrass \(\wp \)-function.

  2. For this purpose, one should use the identities \({{\partial }_{z}}{{\varphi }_{i}}(z) = \) \( - {{\varphi }_{j}}(z){{\varphi }_{k}}(z)\) valid for any set of distinct \(i,j,k \in \{ 1,2,3\} \).

  3. Here, \(\{ {{E}_{{ij}}}{\kern 1pt} ;{\kern 1pt} i,j = 1...N\} \) is the standard basis in \(N \times N\) matrices \({\text{Mat}}(N,\mathbb{C})\): \({{({{E}_{{ij}}})}_{{ab}}} = {{\delta }_{{ia}}}{{\delta }_{{jb}}}.\)

  4. The exchanging of indices from 12 to 21 means the exchanging of the tensor components. For example, \(R_{{21}}^{\hbar }(z) = {{P}_{{12}}}R_{{12}}^{\hbar }(z){{P}_{{12}}}\) = \(\sum\nolimits_{ijkl = 1}^N {{{R}_{{ij,kl}}}} (\hbar ,z){{E}_{{kl}}} \otimes {{E}_{{ij}}}\), where \({{P}_{{12}}}\) is the permutation operator. For any pair of N-dimensional vectors \(u,v\) the permutation operator acts as \({{P}_{{12}}}(u \otimes v) = v \otimes u\). For any pair of \(N \times N\) matrices \(A,B\): \({{P}_{{12}}}(A \otimes B) = (B \otimes A){{P}_{{12}}}\). Explicitly, \({{P}_{{12}}} = \sum\nolimits_{i,j = 1}^N {{{E}_{{ij}}}} \otimes {{E}_{{ji}}}\).

  5. On should use the expansion of \(\phi (z,u)\) near \(u = 0\): \(\phi (z,u) = {{u}^{{ - 1}}} + {{E}_{1}}(z) + u\rho (z) + O({{u}^{2}})\).

  6. For example, in this way one can deduce the properties \(SE(S) = 0\), \(SE(({{\partial }_{x}}S)S) = 0\), \(SE(S({{\partial }_{x}}S)) = - c({{\partial }_{x}}S)E(S)\).

REFERENCES

  1. L. D. Landau and E. M. Lifshitz, Phys. Zs. Sowjet. 8, 153 (1935).

    Google Scholar 

  2. E. K. Sklyanin, Preprint LOMI, E-3-79 (LOMI, Leningrad, 1979).

    Google Scholar 

  3. L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer, Berlin, 1987).

    Book  MATH  Google Scholar 

  4. V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl. 8, 226 (1974).

    Article  Google Scholar 

  5. V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl. 13, 166 (1979).

    Article  Google Scholar 

  6. What is Integrability?, Ed. by V. E. Zakharov, Springer Series in Nonlinear Dynamics (Springer, Berlin, 1991).

  7. R. J. Baxter, Ann. Phys. 70, 193 (1972).

    Article  ADS  Google Scholar 

  8. A. A. Belavin, Nucl. Phys. B 180, 189 (1981).

    Article  ADS  Google Scholar 

  9. L. Takhtajan and L. Faddeev, Russ. Math. Surv. 34 (5), 11 (1979).

    Google Scholar 

  10. E. K. Sklyanin, J. Sov. Math. 46, 1664 (1989).

    Article  Google Scholar 

  11. S. Fomin and A. N. Kirillov, in Advances in Geometry, Prog. Math. 172, 147 (1999).

    MathSciNet  Google Scholar 

  12. A. Polishchuk, Adv. Math. 168, 56 (2002).

    Article  MathSciNet  Google Scholar 

  13. A. Levin, M. Olshanetsky, and A. Zotov, J. High Energy Phys. 1407, 012 (2014); arXiv: 1405.7523 [hep-th].

  14. A. Levin, M. Olshanetsky, and A. Zotov, Nucl. Phys. B 887, 400 (2014); arXiv: 1406.2995 [math-ph].

    Article  ADS  Google Scholar 

  15. A. Levin, M. Olshanetsky, and A. Zotov, J. High Energy Phys. 10, 109 (2014); arXiv: 1408.6246 [hep-th].

    Article  ADS  Google Scholar 

  16. A. Levin, M. Olshanetsky, and A. Zotov, Theor. Math. Phys. 184, 924 (2015); arXiv: 1501.07351.

    Article  Google Scholar 

  17. A. Levin, M. Olshanetsky, and A. Zotov, J. Phys. A: Math. Theor. 49, 014003 (2016); arXiv: 1507.02617.

  18. A. Levin, M. Olshanetsky, and A. Zotov, J. Phys. A: Math. Theor. 49, 395202 (2016); arXiv: 1603.06101.

  19. A. Zabrodin and A. Zotov, arXiv: 2107.01697 [math-ph].

  20. A. Zotov, Mod. Phys. Lett. A 32, 1750169 (2017); arXiv: 1706.05601 [math-ph].

  21. I. V. Cherednik, Theor. Math. Phys. 47, 422 (1981).

    Article  Google Scholar 

  22. I. V. Cherednik, J. Sov. Math. 38, 1989 (1987).

    Article  Google Scholar 

  23. A. Zotov, SIGMA 7, 067 (2011); arXiv: 1012.1072 [math-ph].

  24. A. Levin, M. Olshanetsky, and A. Zotov, arXiv: 2202.10106 [hep-th].

  25. G. Aminov, S. Arthamonov, A. Smirnov, and A. Zotov, J. Phys. A: Math. Theor. 47, 305207 (2014); arXiv: 1402.3189 [math-ph].

  26. A. Grekov, I. Sechin, and A. Zotov, J. High Energy Phys. 10, 081 (2019); arXiv: 1905.07820 [math-ph].

  27. T. Krasnov and A. Zotov, Ann. Henri Poincaré 20, 2671 (2019); arXiv: 1812.04209 [math-ph].

    Article  ADS  MathSciNet  Google Scholar 

  28. K. Atalikov and A. Zotov, J. Geom. Phys. 164, 104161 (2021); arXiv: 2010.14297 [math-ph].

  29. I. Z. Golubchik and V. V. Sokolov, Theor. Math. Phys. 124, 909 (2000).

    Article  Google Scholar 

  30. T. V. Skrypnyk, Theor. Math. Phys. 142, 275 (2005).

    Article  Google Scholar 

  31. T. Skrypnyk, J. Math. Phys. 54, 103507 (2013).

Download references

Funding

A. Zotov acknowledges the support of the Russian Science Foundation, project no. 21-41-09011.

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, 119991, Moscow, Russia

    K. Atalikov & A. Zotov

  2. Institute for Theoretical and Experimental Physics, National Research Center Kurchatov Institute, 117218, Moscow, Russia

    K. Atalikov

  3. National Research University Higher School of Economics, 119048, Moscow, Russia

    A. Zotov

Authors
  1. K. Atalikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Zotov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to K. Atalikov or A. Zotov.

Ethics declarations

The authors declare that they have no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Atalikov, K., Zotov, A. Higher Rank 1 + 1 Integrable Landau–Lifshitz Field Theories from the Associative Yang–Baxter Equation. Jetp Lett. 115, 757–762 (2022). https://doi.org/10.1134/S0021364022600811

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0021364022600811

Search

Navigation