Skip to main content
SpringerLink
Log in

Integrability of Limit Shapes of the Inhomogeneous Six Vertex Model

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper we prove that the Euler–Lagrange equations for the limit shape for the inhomogeneous six vertex model on a cylinder have infinitely many conserved quantities.

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

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. The idea of using a superposition of plane waves as an eigenfunction for the Heisenberg spin Hamiltonian (which is the the logarithmic derivative of the transfer-matrix of the homogeneous 6-vertex model at \(u=0\)) goes back to H. Bethe [B]. It was first applied to the 6-vertex model by E. Lieb in [L] for zero magnetic fields. Shorty after C.P. Yang [Y] applied it to the asymmetric 6-vertex model (with magnetic fields). The algebraic form we use is due to L. Faddeev and L. Takhtajan [FT2].

References

  1. Allison, D., Reshetikhin, N.: The 6-vertex model with fixed boundary conditions. Annales de l’institut Fourier. 55(6), (2005). arXiv:cond-mat/0502314

  2. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Dover Publications, London (1982)

    MATH  Google Scholar 

  3. Bazhanov, V., Mangazeev, V.: Analytic theory of the eight vertex model. Nucl. Phys. B 775(FS), 225–282 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  4. Bethe, H.: Zur Theorie der Metalle I. Eigenwerte und Eigenfunktionen der Hnearen Atom- kette, Zeitschrift fur Physik 71, 205–226 (1931)

    ADS  MATH  Google Scholar 

  5. Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. Am. Math. Soc. 14, 297–346 (2001)

    Article  MathSciNet  Google Scholar 

  6. Colomo, F., Pronko, A.G.: The arctic curve of the domain-wall six-vertex model. J. Stat. Phys. 138, 662–700 (2010). https://doi.org/10.1007/s10955-009-9902-2. arXiv:0907.1264

  7. Duminil-Copin, H., Kozlowski, K., Krachum, D., Manolescu, I., Tikhonovskaia, T.: On the six-vertex model’s free energy, arXiv:2012.11675 [math-ph]

  8. de Gier, J., Kenyon, R., Watson, S.: Limit shapes for the asymmetric five vertex model. arXiv: 1812.11934v1

  9. Faddeev, L., Reshetikhin, N.: Integrability of the principal chiral field model in \(1+1\) dimension. Ann. Phys. 167(2), 227–256 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  10. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer (2007)

  11. Faddeev, L.D., Takhtajan, L.A.: The quantum method for the inverse problem and the XY Z Heisenberg model. Uspekhi Mat. Nauk 34 (1979); (5)(209), 13–63 (in Russian)

  12. Izergin, A., Korepin, V., Reshetikhin, N.: Conformal dimension in Bethe ansatz solvable models. J. Phys. A Math. Gen. 22, 2615–2620 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  13. Korepin, V.E., Bogolyubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press (1993)

  14. Lieb, E.: Exact solution of the problem of the entropy of two-dimensional ice. Phys. Rev. Lett. 18(n17), 692–694 (1967)

    Article  ADS  Google Scholar 

  15. Lieb, E., Wu, F.Y.: Two-dimensional ferroelectric models. In: Domb, C., Green, M.S. (Eds.) Phase Transitions and Critical Phenomena, Vol. 1, pp. 331–490. Academic Press (1972)

  16. Noh, J.D., Kim, D.: Finite size scaling and the toroidal partition function of the critical asymmetric six-vertex model. Phys. Rev. E 53(4), 3225 (1996). arXiv:cond-mat/9511001

  17. Nolden, I.M.: The asymmetric six-vertex model. J. Stat. Phys. 67(1/2), 155–201 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  18. Plamarchuk, K., Reshetikhin, N.: The 6-vertex model with fixed boundary conditions, arXiv:1010.5011

  19. Reshetikhin, N.: Lectures on integrable models in statistical mechanics. In: Exact methods in low-dimensional statistical physics and quantum computing. In: Proceedings of Les Houches School in Theoretical Physics, Oxford University Press (2010) arXiv:1010.5031

  20. Reshetikhin, N., Sridhar, A.: Integrability of Limit Shapes in the Six Vertex Model, arXiv:1510.01053 [math-ph]

  21. Sridhar, A., Suris, Y.: Commutativity in Lagrangian and Hamiltonian Mechanics, arXiv:1801.06076 [math-ph]

  22. Sutherland, B., Yang, C.N., Yang, C.P.: Exact solution of a model of two-dimensional ferroelectrics in an arbitrary external electric field. Phys. Rev. Lett. 19(10), 588–591 (1967)

    Article  ADS  Google Scholar 

  23. Yang, C.P.: Exact solution of a model of two-dimensional ferroelectric in an arbitrary external electric field. Phys. Rev. Lett. 16(n10), 586–588 (1967)

    Article  ADS  Google Scholar 

  24. Yang, C.N., Yang, C.P.: One-dimensional chain of anisotropic spin-spin interactions, I and II. Phys. Rev. 150(n1), 321–339 (1967)

    Article  ADS  Google Scholar 

  25. Zinn-Justin, P.: The Influence of Boundary Conditions in the Six-Vertex Model, arXiv:cond-mat/0205192

Download references

Acknowledgements

The work of N.R. was partly supported by Grants NSF FRG DMS-1664521 and RSF-18-11-00-297. The work of D.K. was partly supported by the Grant NSF DMS-1902226. N.R. and D.K. are grateful for the hospitality at ITS ETH where he was visiting when the work was completed. N.R. would like to thank A. Borodin, I. Corwin and A. Pronko for helpful discussions on various aspects of the six vertex model.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    David Keating, Nicolai Reshetikhin & Ananth Sridhar

  2. Saint Petersburg University, Saint Petersburg, Russia

    Nicolai Reshetikhin

  3. KdV Institute for Mathematics, University of Amsterdam, 1098 XH, Amsterdam, The Netherlands

    Nicolai Reshetikhin

Authors
  1. David Keating
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolai Reshetikhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ananth Sridhar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Keating.

Additional information

Communicated by J.-d. Gier.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Keating, D., Reshetikhin, N. & Sridhar, A. Integrability of Limit Shapes of the Inhomogeneous Six Vertex Model. Commun. Math. Phys. 391, 1181–1207 (2022). https://doi.org/10.1007/s00220-022-04334-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-022-04334-9

Search

Navigation