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.

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.