Calogero–Sutherland system at a free fermion point

Abstract

We present two ways to obtain precise expressions for the commuting Hamiltonians of the integrable system regarded as a fermionic limit of the quantum Calogero–Sutherland system as the number of particles tends to infinity with some special values of the coupling constant \(\beta\). The construction is realized in the Fock space.

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References

  1. 1

    D. Bernard, M. Gaudin, F. D. M. Haldane, and V. Pasquier, “Yang–Baxter equation in spin chains with long range interactions,” J. Phys. A: Math. Gen., 26, 5219–5236 (1993).

    ADS  Article  Google Scholar 

  2. 2

    Y. Kato and Y. Kuramoto, “Exact solution of the Sutherland model with arbitrary internal symmetry,” Phys. Rev. Lett., 74, 1222–1225 (1995); arXiv:cond-mat/9409031v2 (1994).

    ADS  Article  Google Scholar 

  3. 3

    C. F. Dunkl, “Differential-difference operators associated to reflection groups,” Trans. Amer. Math. Soc., 311, 167–183 (1989).

    MathSciNet  Article  Google Scholar 

  4. 4

    G. J. Heckman, “An elementary approach to the hypergeometric shift operators of Opdam,” Invent. Math., 103, 341–350 (1991).

    ADS  MathSciNet  Article  Google Scholar 

  5. 5

    A. P. Polychronakos, “Exchange operator formalism for integrable systems of particles,” Phys. Rev. Lett., 69, 703–705 (1992).

    ADS  MathSciNet  Article  Google Scholar 

  6. 6

    I. Andrić, A. Jevicki, and H. Levine, “On the large-\(N\) limit in symplectic matrix models,” Nucl. Phys. B, 215, 307–315 (1983).

    ADS  MathSciNet  Article  Google Scholar 

  7. 7

    H. Awata, Y. Matsuo, S. Odake, and J. Shiraishi, “Collective field theory, Calogero–Sutherland model, and generalized matrix models,” Phys. Lett. B, 347, 49–55 (1995); arXiv:hep-th/9411053v3 (1994).

    ADS  MathSciNet  Article  Google Scholar 

  8. 8

    H. Awata, Y. Matsuo, and T. Yamamoto, “Collective field description of spin Calogero–Sutherland models,” J. Phys. A, 29, 3089–3098 (1996); arXiv:hep-th/9512065v3 (1995).

    ADS  MathSciNet  Article  Google Scholar 

  9. 9

    I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, Oxford (1998).

    MATH  Google Scholar 

  10. 10

    M. L. Nazarov and E. K. Sklyanin, “Integrable hierarchy of the quantum Benjamin–Ono equation,” SIGMA, 9, 078 (2013); arXiv:1309.6464v2 [nlin.SI] (2013).

    MathSciNet  MATH  Google Scholar 

  11. 11

    A. N. Sergeev and A. P. Veselov, “Dunkl operators at infinity and Calogero–Moser systems,” Internat. Math. Res. Notices, 2015, 10959–10986 (2015).

    MathSciNet  Article  Google Scholar 

  12. 12

    A. G. Abanov and P. B. Wiegmann, “Quantum hydrodynamics, the quantum Benjamin–Ono equation, and the Calogero model,” Phys. Rev. Lett., 95, 076402 (2005); arXiv:cond-mat/0504041v1 (2005).

    ADS  Article  Google Scholar 

  13. 13

    A. P. Polychronakos, “Waves and solitons in the continuum limit of the Calogero–Sutherland model,” Phys. Rev. Lett., 74, 5153–5157 (1995).

    ADS  MathSciNet  Article  Google Scholar 

  14. 14

    S. M. Khoroshkin and M. G. Matushko, “Fermionic limit of the Calogero–Sutherland system,” J. Math. Phys., 60, 071706 (2019).

    ADS  MathSciNet  Article  Google Scholar 

  15. 15

    A. K. Pogrebkov, “Boson–fermion correspondence and quantum integrable and dispersionless models,” Russian Math. Surveys, 58, 1003–1037 (2003).

    ADS  MathSciNet  Article  Google Scholar 

  16. 16

    P. Rossi, “Gromov–Witten invariants of target curves via symplectic field theory,” J. Geom. Phys., 58, 931–941 (2008).

    ADS  MathSciNet  Article  Google Scholar 

  17. 17

    A. Alexandrov and A. Zabrodin, “Free fermions and tau-functions,” J. Geom. Phys., 67, 37–80 (2013).

    ADS  MathSciNet  Article  Google Scholar 

  18. 18

    V. G. Kac, A. K. Raina, and N. Rozhkovskaya, Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras (Adv. Ser. Math. Phys., Vol. 29), World Scientific, Singapore (2013).

    Book  Google Scholar 

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Acknowledgments

The author thanks S. M. Khoroshkin and A. K. Pogrebkov for the valuable remarks and fruitful discussions on the subject of the paper.

Funding

This research was supported by a grant from the Russian Science Foundation (Project No. 20-41-09009) and by the Simons Foundation.

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Correspondence to M. G. Matushko.

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Appendix

It was proved in Proposition 2 that the densities \( \mathcal{W} _k(z)\) are linearly expressed in terms of \(w_n(z)\) given by (3.13). The first densities \(w_n(z)\) are given in Sec. 3.3. Here, we present the expressions for the first densities \( \mathcal{W} _k(z)\),

$$\begin{aligned} \, & \mathcal{W} _0(z)=w_1(z),\qquad \mathcal{W} _1(z)=\frac{1}{2}w_2(z)-\frac{1}{2}w_1(z), \\ & \mathcal{W} _2(z)=\frac{1}{3}w_3(z)-\frac{1}{2}w_2(z)+\frac{1}{6}w_1(z), \\ & \mathcal{W} _3(z)=\frac{1}{4}w_4(z)-\frac{1}{2}w_3(z)+\frac{1}{4}w_2(z), \end{aligned}$$
and the expressions for the corresponding Hamiltonians,
$$\begin{aligned} \, \mathcal{H} _0={}&p_0,\qquad \mathcal{H} _1=\sum_{n>0}np_n\frac{ \partial }{ \partial p_n}+\frac{1}{2}(p_0^2-p_0), \\ \mathcal{H} _2={}&\sum_{n,k>0}nkp_{n+k}\frac{ \partial }{ \partial p_n}\frac{ \partial }{ \partial p_k}+ \sum_{n,k>0}(n+k)p_np_k\frac{ \partial }{ \partial p_{n+k}}+{} \\ &{}+(2p_0-1)\sum_{n>0}np_n\frac{ \partial }{ \partial p_n}+ \frac{1}{6}(2p_0^3-3p_0^2+p_0), \\ \mathcal{H} _3={}&\sum_{n,k,m>0}nkmp_{n+k+m}\frac{ \partial }{ \partial p_n} \frac{ \partial }{ \partial p_k} \frac{ \partial }{ \partial p_m} +\sum_{n,k,m>0}(n+k+m)p_np_kp_m \frac{ \partial }{ \partial p_{n+k+m}}+{} \\ &{}+\frac{3}{2}\sum_{k,m>0}\sum_{n=1}^{m+k-1}kmp_np_{m+k-n} \frac{ \partial }{ \partial p_k}\frac{ \partial }{ \partial p_m}+ \frac{1}{2}\sum_{n>0}n^3p_n\frac{ \partial }{ \partial p_n}+{} \\ &{}+\biggl(3p_0-\frac{3}{2}\biggr)\sum_{n,k>0}nkp_{n+k} \frac{ \partial }{ \partial p_n}\frac{ \partial }{ \partial p_k}+ \biggl(3p_0-\frac{3}{2}\biggr)\sum_{n,k>0}(n+k) p_np_k\frac{ \partial }{ \partial p_{n+k}}+{} \\ &{}+\biggl(3p_0^2-3p_0+\frac{1}{2}\biggr)\sum_{n>0}np_n \frac{ \partial }{ \partial p_n}+\frac{1}{4}(p_0^4-2p_0^3+p_0^2). \end{aligned}$$
The first line in the formula for the Hamiltonian \( \mathcal{H} _2\) is called a cut-and-join operator, which has applications in areas such as Hurwitz theory and knot theory.

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Matushko, M.G. Calogero–Sutherland system at a free fermion point. Theor Math Phys 205, 1593–1610 (2020). https://doi.org/10.1134/S0040577920120041

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Keywords

  • Calogero–Sutherland system
  • free fermion
  • boson–fermion correspondence