Calogero–Sutherland system at a free fermion point


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


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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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).

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