QKZ-Ruijsenaars correspondence revisited

TL;DR

This paper revisits the correspondence between quantum Knizhnik-Zamolodchikov equations and Ruijsenaars models, exploring their quantum and classical relationships and extending the understanding of integrable systems.

Contribution

It provides a detailed analysis of the Matsuo-Cherednik correspondence for $GL(N)$ and extends the quantum-classical correspondence to broader settings.

Findings

Established a link between quantum KZ equations and Ruijsenaars models.

Extended the quantum-classical correspondence to non-equal particle and spin chain sizes.

Clarified the role of the quasiclassical limit in integrable systems.

Abstract

We discuss the Matsuo-Cherednik type correspondence between the quantum Knizhnik-Zamolodchikov equations associated with $GL(N)$ and the $n$-particle quantum Ruijsenaars model, with $n$ being not necessarily equal to $N$. The quasiclassical limit of this construction yields the quantum-classical correspondence between the quantum spin chains and the classical Ruijsenaars models.

Equations

$$e^{\eta\hbar\partial_{x_{i}}}\Bigl{|}\Phi\Bigr{>}={\bf K}_{i}^{(\hbar)}\Bigl{|}\Phi\Bigr{>},\qquad i=1,\ldots,n$$

$${\mathbf{K}}_i^{(\hslash )}={\mathbf{R}}_{ii-1}(x_i-x_{i-1}+\eta \hslash )\dots {\mathbf{R}}_{i1}(x_i-x_1+\eta \hslash ){\mathbf{g}}^{(i)}{\mathbf{R}}_{in}(x_i-x_n)\dots {\mathbf{R}}_{ii+1}(x_i-x_{i+1}).$$

$${\mathbf{R}}_{ij}(x)=\frac{x\mathbf{I}+\eta {\mathbf{P}}_{ij}}{x+\eta },$$

$$\hat{\mathcal{H}}=\sum _{i=1}^n\left(\prod _{j\ne i}^n\frac{x_i-x_j+\eta }{x_i-x_j}\right)e^{\eta \hslash {\partial }_{x_i}}$$

$$\Bigl{|}\Phi\Bigr{>}=\sum_{\sigma\in S_{n}}\Phi_{\sigma}\Bigl{|}e_{\sigma}\Bigr{>},\qquad\Bigl{|}e_{\sigma}\Bigr{>}=e_{\sigma(1)}\otimes e_{\sigma(2)}\otimes\ldots\otimes e_{\sigma(n)},$$

$$\sum _{i=1}^n\prod _{j\ne i}^n\frac{x_i-x_j+\eta }{x_i-x_j}\Psi (x_1,\dots ,x_i+\eta \hslash ,\dots ,x_n)=E\Psi (x_1,\dots ,x_n),E=\sum _{a=1}^N{g}_a.$$

$$\Bigl{|}\Phi\Bigr{>}=\left(\Bigl{|}\phi_{0}\Bigr{>}+\hbar\Bigl{|}\phi_{1}\Bigr{>}+\ldots\right)e^{S/\hbar},$$

$${\bf K}_{i}^{(0)}\Bigl{|}\phi_{0}\Bigr{>}=e^{\eta p_{i}}\Bigl{|}\phi_{0}\Bigr{>},\qquad p_{i}=\frac{\partial S}{\partial x_{i}},\qquad i=1,\ldots,n,$$

$$\tilde{\mathbf{R}}(x)=\frac{x+\eta }{x}\mathbf{R}(x)=\mathbf{I}+\frac{\eta }{x}\mathbf{P}.$$

$${\bf T}(x)=\mbox{tr}_{0}\Bigl{(}\widetilde{\bf R}_{0n}(x-x_{n})\ldots\widetilde{\bf R}_{02}(x-x_{2})\widetilde{\bf R}_{01}(x-x_{1})({\bf g}\otimes{\bf I})\Bigr{)},$$

$$\mathbf{T}(x)=\text{mbox}tr\mathbf{g}\cdot \mathbf{I}+\sum _{j=1}^n\frac{\eta {\mathbf{H}}_j}{x-x_j}.$$

$${\mathbf{H}}_i={\tilde{\mathbf{R}}}_{ii-1}(x_i-x_{i-1})\dots {\tilde{\mathbf{R}}}_{i1}(x_i-x_1){\mathbf{g}}^{(i)}{\tilde{\mathbf{R}}}_{in}(x_i-x_n)\dots {\tilde{\mathbf{R}}}_{ii+1}(x_i-x_{i+1}).$$

$${\mathbf{H}}_i={\mathbf{K}}_i^{(0)}\prod _{j\ne i}^n\frac{x_i-x_j+\eta }{x_i-x_j},$$

$${\mathbf{M}}_a=\sum _{l=1}^n{e}_{aa}^{(l)}.$$

$$\begin{array}[]{lll}{\bf T}(x)&=&\displaystyle{\mbox{tr}_{0}\left[\Bigl{(}{\bf I}+\frac{\eta{\bf P}_{0n}}{x-x_{n}}\Bigr{)}\ldots\Bigl{(}{\bf I}+\frac{\eta{\bf P}_{01}}{x-x_{1}}\Bigr{)}{\bf g}^{(0)}\right]}\\ &&\\ &=&\displaystyle{\mbox{tr}\,{\bf g}\cdot{\bf I}+\frac{\eta}{x}\sum_{i=1}^{n}\mbox{tr}_{0}\Bigl{(}{\bf P}_{0i}{\bf g}^{(0)}\Bigr{)}+\ldots}\\ &&\\ &=&\displaystyle{\mbox{tr}\,{\bf g}\cdot{\bf I}+\frac{\eta}{x}\sum_{i=1}^{n}{\bf g}^{(i)}+\ldots}\,,\end{array}$$

$$\sum _{i=1}^n{\mathbf{H}}_i=\sum _{i=1}^n{\mathbf{g}}^{(i)}=\sum _{a=1}^N{g}_a{\mathbf{M}}_a,$$

$$\left\{\begin{array}[]{l}{\bf H}_{i}\Bigl{|}\phi\Bigr{>}=H_{i}\Bigl{|}\phi\Bigr{>}\\ {\bf M}_{a}\Bigl{|}\phi\Bigr{>}=M_{a}\Bigl{|}\phi\Bigr{>}\end{array}\right.$$

$$\mathcal{V}=V^{\otimes n}=\underset{M_1,\dots ,M_N}{⨁}\mathcal{V}(\{M_a\})$$

$$\text{mbox}dim\mathcal{V}(\{M_a\})=\frac{n!}{M_1!\dots M_N!}.$$

$$\Bigl{(}e^{\eta\hbar\partial_{x_{i}}}{\bf K}_{j}^{(\hbar)}\Bigr{)}{\bf K}_{i}^{(\hbar)}=\Bigl{(}e^{\eta\hbar\partial_{x_{j}}}{\bf K}_{i}^{(\hbar)}\Bigr{)}{\bf K}_{j}^{(\hbar)}$$

$$\Psi =\sum _J{\Phi }_J$$

$$\sum _{i=1}^n\prod _{j\ne i}^n\frac{x_i-x_j+\eta }{x_i-x_j}\Psi (x_1,\dots ,x_i+\eta \hslash ,\dots ,x_n)=E\Psi (x_1,\dots ,x_n).$$

$$\Bigl{<}\Omega\Bigr{|}=\sum_{J}\Bigl{<}J\Bigr{|},$$

$$e^{\eta\hbar\partial_{x_{i}}}\Bigl{<}\Omega\Bigr{|}\Phi\Bigr{>}=e^{\eta\hbar\partial_{x_{i}}}\Psi=\Bigl{<}\Omega\Bigr{|}{\bf K}_{i}^{(\hbar)}\Bigl{|}\Phi\Bigr{>}=\Bigl{<}\Omega\Bigr{|}{\bf K}_{i}^{(0)}\Bigl{|}\Phi\Bigr{>}.$$

$$\sum_{i=1}^{n}\left(\prod_{j\neq i}^{n}\frac{x_{i}-x_{j}+\eta}{x_{i}-x_{j}}\right)e^{\eta\hbar\partial_{x_{i}}}\Psi\,\,=\,\,\sum_{i=1}^{n}\prod_{j\neq i}^{n}\frac{x_{i}-x_{j}+\eta}{x_{i}-x_{j}}\,\Bigl{<}\Omega\Bigr{|}{\bf K}_{i}^{(0)}\Bigl{|}\Phi\Bigr{>}$$

$$=\,\sum_{i=1}^{n}\Bigl{<}\Omega\Bigr{|}{\bf H}_{i}\Bigl{|}\Phi\Bigr{>}\,\,=\,\,\sum_{i=1}^{n}\Bigl{<}\Omega\Bigr{|}{\bf g}^{(i)}\Bigl{|}\Phi\Bigr{>}\,\,=\,\,\sum_{a=1}^{N}g_{a}\Bigl{<}\Omega\Bigr{|}{\bf M}_{a}\Bigl{|}\Phi\Bigr{>}\,\,=\,\,\left(\sum_{a=1}^{N}g_{a}M_{a}\right)\Psi,$$

$$\displaystyle{\hat{\cal H}_{d}=\sum\limits_{I\subset\{1,\ldots,n\},|I|=d}\Bigl{(}\prod\limits_{s\in I,r\not\in I}\frac{x_{s}-x_{r}+\eta}{x_{s}-x_{r}}\Bigr{)}\prod\limits_{i\in I}e^{\eta\hbar\partial_{x_{i}}}}$$

$${\hat{\mathcal{H}}}_d=\sum _{1\le i_1<\dots <i_d\le n}\prod _{k=1}^d\prod _{r\ne i_k}^n\frac{x_{i_k}-x_r+\eta }{x_{i_k}-x_r}\prod _{1\le \alpha <\beta \le d}{\left(1-\frac{{\eta }^2}{(x_{i_{\alpha }}-x_{i_{\beta }}{)}^2}\right)}^{-1}\prod _{k=1}^d{e}^{\eta \hslash \frac{\partial }{\partial x_{i_k}}}.$$

$$\begin{array}[]{c}\displaystyle{\hat{\cal H}_{2}=\sum\limits_{i<j}^{n}\Bigl{(}\prod\limits_{k\neq i,j}^{n}\frac{x_{i}-x_{k}+\eta}{x_{i}-x_{k}}\prod\limits_{l\neq i,j}^{n}\frac{x_{j}-x_{l}+\eta}{x_{j}-x_{l}}\Bigr{)}\,e^{\eta\hbar\partial_{x_{i}}}e^{\eta\hbar\partial_{x_{j}}}=}\\ \\ \displaystyle{=\sum\limits_{i<j}^{n}\Bigl{(}\prod\limits_{k\neq i}^{n}\frac{x_{i}-x_{k}+\eta}{x_{i}-x_{k}}\prod\limits_{l\neq j}^{n}\frac{x_{j}-x_{l}+\eta}{x_{j}-x_{l}}\Bigr{)}\left(1-\frac{\eta^{2}}{(x_{i}-x_{j})^{2}}\right)^{-1}e^{\eta\hbar\partial_{x_{i}}}e^{\eta\hbar\partial_{x_{j}}}\,.}\end{array}$$

$$\displaystyle{\prod\limits_{s=1}^{d}e^{\eta\hbar\frac{\partial\ }{\partial x_{i_{s}}}}\Psi=\Bigl{<}\Omega\Bigr{|}{\bf K}_{i_{1}}^{(0)}\ldots{\bf K}_{i_{d}}^{(0)}\Bigr{|}\Phi\Bigr{>}\quad\hbox{for}\quad i_{k}\neq i_{m}\,.}$$

Full text

QKZ-Ruijsenaars correspondence revisited

A. Zabrodin National Research University Higher School of Economics, Russian Federation; Skolkovo Institute of Science and Technology, 143026 Moscow, Russian Federation; e-mail: zabrodin@itep.ru   

A. Zotov Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, 119991, Moscow, Russia; ITEP, B.Cheremushkinskaya 25, Moscow 117218, Russia; Moscow Institute of Physics and Technology, Inststitutskii per. 9, Dolgoprudny, Moscow region, 141700, Russia; e-mail: zotov@mi.ras.ru

(April 2017)

ITEP-TH-04/17

We discuss the Matsuo-Cherednik type correspondence between the quantum Knizhnik-Zamolodchikov equations associated with $GL(N)$ and the $n$-particle quantum Ruijsenaars model, with $n$ being not necessarily equal to $N$. The quasiclassical limit of this construction yields the quantum-classical correspondence between the quantum spin chains and the classical Ruijsenaars models.

1 Introduction

The quantum Knizhnik-Zamolodchikov (qKZ) equations [8] is a system of holonomic difference equations

[TABLE]

for the vector \Bigl{|}\Phi\Bigr{>}=\Bigl{|}\Phi\Bigr{>}(x_{1},\ldots,x_{n}) belonging to the tensor product ${\cal V}=V\otimes V\otimes\ldots\otimes V=V^{\otimes n}$ of the vector spaces $V=\mbox{\Bbb C}^{N}$. The operator ${\bf K}_{i}^{(\hbar)}$ in the right hand side is constructed as a chain product of quantum $R$-matrices:

[TABLE]

Here ${\bf R}_{ij}(x)$ is the $R$-matrix acting in the $i$-th and $j$-th tensor factors (it has to satisfy the unitarity condition ${\bf R}_{ij}(x){\bf R}_{ji}(-x)=\mbox{id}$), ${\bf g}=\mbox{diag}(g_{1},\ldots,g_{N})$ is a diagonal $N\!\times\!N$ matrix and ${\bf g}^{(i)}$ is the operator in ${\cal V}$ acting as ${\bf g}$ on the $i$-th factor (and identically on all other factors). For example, the rational $R$-matrix is of the form

[TABLE]

where $x$ is the spectral parameter, ${\bf I}$ is the identity operator and ${\bf P}_{ij}$ is the permutation of the $i$-th and $j$-th tensor factors. Compatibility of the qKZ equations follows from the Yang-Baxter equation for the $R$-matrix and from the commutativity $[{\bf g}\otimes{\bf g},\,{\bf R}(x)]=0$.

The remarkable correspondence of the qKZ equations with the Macdonald type difference operator

[TABLE]

was discussed in [5, 11, 14, 17] in the case $N=n$. In this case solutions to (1) can be found in the form

[TABLE]

where $e_{a}$ are standard basis vectors in $\displaystyle{V=\mbox{\Bbb C}^{N}=\bigoplus_{a=1}^{N}\mbox{\Bbb C}e_{a}}$ and $S_{n}$ is the symmetric group. If such \Bigl{|}\Phi\Bigr{>} is a solution to the qKZ equations, then the function $\displaystyle{\Psi=\sum_{\sigma\in S_{n}}\Phi_{\sigma}}$ solves the spectral problem for the operator $\hat{\cal H}$:

[TABLE]

This is referred to as the Matsuo-Cherednik type correspondence [13, 4]. A similar correspondence holds true for qKZ equations with trigonometric $R$-matrices.

The operator $\hat{\cal H}$ is essentially the Hamiltonian of the quantum Ruijsenaars model [15] which is a relativistic version of the Calogero model (more precisely, the operator $\hat{\cal H}$ and the Ruijsenaars Hamiltonian are connected by a similarity transformation, see [10]). The parameter $\eta$ plays the role of the inverse velocity of light.

We will give a simple proof of the correspondence valid also in the case when $n$ is not necessarily equal to $N=\mbox{dim}\,V$. In this form it looks like a quantum deformation of the quantum-classical correspondence [1, 2, 9, 21, 3] between the quantum XXX or XXZ spin chains and the classical Ruijsenaars models.

The spectral problem for the spin chain appears as a “quasiclassical” limit of qKZ as $\hbar\to 0$. Indeed, as $\hbar\to 0$ the qKZ solutions have the asymptotic form [18, 19]

[TABLE]

where $S$ is some scalar function. Upon substitution to the qKZ equations (1), this leads, in the leading order, to the joint eigenvalue problems

[TABLE]

for the commuting operators ${\bf K}_{i}^{(0)}$. They are Hamiltonians of the inhomogeneous quantum spin chain. They can be diagonalized using the algebraic Bethe ansatz. In the quantum-classical correspondence, the $p_{i}$’s are identified with momenta of the Ruijsenaars particles. The parameter $\hbar$ becomes the true Planck constant after quantization of the corresponding Ruijsenaars model.

The paper is organized as follows. In section 2, we review the inhomogeneous XXX spin chain and the associated qKZ equations. In section 3 the qKZ-Ruijsenaars correspondence is established by a simple direct calculation. In section 4 we extend the result to higher Ruijsenaars Hamiltonians. Section 5 is devoted to the qKZ equations with trigonometric $R$-matrices. Finally, in section 6 we discuss the interpretation of the results as a “quantum” deformation of the quantum-classical correspondence. The link to spin chains and their solutions by means of the algebraic Bethe ansatz appears naturally because of the usage of $R$-matrices in fundamental representation. It is alternative to the approach based on the affine Hecke algebras [4, 5, 6], where similar results were originally presented. Explicit relationship between these two approaches deserves further consideration.

2 The XXX spin chain and qKZ equations

Let $e_{ab}$ be the standard basis in the space of $N\!\times\!N$ matrices: the matrix $e_{ab}$ has only one non-zero element (equal to 1) at the place $ab$: $(e_{ab})_{a^{\prime}b^{\prime}}=\delta_{aa^{\prime}}\delta_{bb^{\prime}}$. Note that ${\bf I}=\sum_{a}e_{aa}$ is the unity operator and ${\bf P}=\sum_{ab}e_{ab}\otimes e_{ba}$ is the permutation operator in the space $\mbox{\Bbb C}^{N}\otimes\mbox{\Bbb C}^{N}$. We embed $e_{ab}$ into $\mbox{End}\,(V^{\otimes n})$ in the usual way: $e^{(i)}_{ab}:={\bf I}^{\otimes(i-1)}\otimes e_{ab}\otimes{\bf I}^{\otimes(n-i)}$. It is clear that $e^{(i)}_{ab}$, $e^{(j)}_{a^{\prime}b^{\prime}}$ commute for any $i\neq j$ because they act non-trivially in different spaces. Similarly, for any matrix ${\bf g}\in\mbox{End}(\mbox{\Bbb C}^{N})$ we define ${\bf g}^{(i)}$ acting in the tensor product $V^{\otimes n}$: ${\bf g}^{(i)}={\bf I}^{\otimes(i-1)}\otimes{\bf g}\otimes{\bf I}^{\otimes(n-i)}\in\mbox{End}({\cal V})$. In this notation, the permutation operator of the $i$-th and $j$-th tensor factors in ${\cal V}=\mbox{\Bbb C}^{N}\otimes\ldots\otimes\mbox{\Bbb C}^{N}$ is $\displaystyle{{\bf P}_{ij}=\sum_{a,b}e^{(i)}_{ab}e^{(j)}_{ba}}$.

Let ${\bf g}\in GL(N)$ be a diagonal matrix $\displaystyle{{\bf g}=\mbox{diag}\,(g_{1},g_{2},\ldots,g_{N})=\sum_{a=1}^{N}g_{a}e_{aa}}$. We call it the twist matrix with twist parameters $g_{a}$. It is used for the construction of an integrable spin chain with twisted boundary conditions. Together with the $R$-matrix (3), we introduce another rational $R$-matrix which differs from the ${\bf R}(x)$ by a scalar factor:

[TABLE]

It obeys the same Yang-Baxter equation and commutes with ${\bf g}\otimes{\bf g}$. The transfer matrix of the inhomogeneous spin chain (or, equivalently, of the associated statistical vertex model on the 2D lattice) is defined in the standard way as a trace of the chain product of $R$-matrices in the auxiliary space $V=\mbox{\Bbb C}^{N}$ with index [math]:

[TABLE]

where $x_{1},x_{2},\ldots,x_{n}$ are inhomogeneity parameters. (We assume that they are in general position meaning that $x_{i}\neq x_{j}$ and $x_{i}\neq x_{j}\pm\eta$ for all $i\neq j$.) As is known, the Yang-Baxter equation for the $R$-matrix implies that the transfer matrices with fixed inhomogeneity and twist parameters commute: $[{\bf T}(x),\,{\bf T}(x^{\prime})]=0$.

The dynamical variables of the model (we call them “spins” in analogy with the rank 1 case) are vectors in the vector representation of $GL(N)$ realized in the spaces $V=\mbox{\Bbb C}^{N}$ attached to each site. Non-local commuting Hamiltonians ${\bf H}_{j}$ are defined as residues of ${\bf T}(x)$ at $x=x_{j}$:

[TABLE]

From (8) it follows that their explicit form is

[TABLE]

We obviously have

[TABLE]

where ${\bf K}_{i}^{(0)}$ is the operator (2) at $\hbar=0$.

Let us introduce the operators

[TABLE]

They commute among themselves and with the Hamiltonians: $[{\bf M}_{a},{\bf M}_{b}]=[{\bf H}_{i},{\bf M}_{a}]=0$. We call them weight operators. Clearly, $\sum_{a}{\bf M}_{a}=n{\bf I}$. Comparing the expansion of (8) as $x\to\infty$,

[TABLE]

with that of (9), we conclude that

[TABLE]

so the system has $n+N-2$ independent integrals of motion. The joint spectral problem is

[TABLE]

The common eigenstates of the Hamiltonians can be classified according to eigenvalues of the operators ${\bf M}_{a}$.

Let

[TABLE]

be the weight decomposition of the Hilbert space of the spin chain, ${\cal V}$, into the direct sum of eigenspaces for the operators ${\bf M}_{a}$ with the eigenvalues $M_{a}\in\mbox{\Bbb Z}_{\geq 0}$, $a=1,\ldots,N$ (recall that $M_{1}+\ldots+M_{N}=n$). The common eigenstates of the operators ${\bf M}_{a}$ and ${\bf H}_{i}$ belong to the spaces ${\cal V}(\{M_{a}\})$. The dimension of ${\cal V}(\{M_{a}\})$ is

[TABLE]

The basis vectors in ${\cal V}(\{M_{a}\})$ are \Bigl{|}J\Bigr{>}=e_{j_{1}}\otimes e_{j_{2}}\otimes\ldots\otimes e_{j_{n}}, where the number of indices $j_{k}$ such that $j_{k}=a$ is equal to $M_{a}$ for all $a=1,\ldots,N$. We also introduce dual vectors \Bigl{<}J\Bigr{|} such that \Bigl{<}J\Bigr{|}J^{\prime}\Bigr{>}=\delta_{J,J^{\prime}}. Matrix elements of an operator ${\bf O}$ are \Bigl{<}J\Bigr{|}{\bf O}\Bigl{|}J^{\prime}\Bigr{>}.

Associated with the inhomogeneous XXX spin chain is the system of qKZ equations e^{\eta\hbar\partial_{x_{i}}}\Bigl{|}\Phi\Bigr{>}={\bf K}_{i}^{(\hbar)}\Bigl{|}\Phi\Bigr{>} (1) with the operators ${\bf K}_{i}^{(\hbar)}$ given by (2). The compatibility condition

[TABLE]

follows from the Yang-Baxter equation for the $R$-matrix. The operators ${\bf K}_{i}^{(\hbar)}$ respect the weight decomposition (14), hence the solutions to the qKZ system belong to the weight subspaces ${\cal V}(\{M_{a}\})$.

3 The qKZ-Ruijsenaars correspondence in the rational case

Let \displaystyle{\Bigl{|}\Phi\Bigr{>}=\sum_{J}\Phi_{J}\Bigl{|}J\Bigr{>}} be any solution of the qKZ equations belonging to the weight subspace ${\cal V}(\{M_{a}\})$. We claim that the function

[TABLE]

is an eigenfunction of the Macdonald operator $\hat{\cal H}$ with the eigenvalue $\displaystyle{E=\sum_{a=1}^{N}M_{a}g_{a}}$:

[TABLE]

For the proof we consider the covector \Bigl{<}\Omega\Bigr{|} equal to the sum of all basis (dual) vectors:

[TABLE]

then \Psi=\Bigl{<}\Omega\Bigr{|}\Phi\Bigr{>}. It is important to note that \Bigl{<}\Omega\Bigr{|}{\bf P}_{ij}=\Bigl{<}\Omega\Bigr{|} and, therefore, \Bigl{<}\Omega\Bigr{|}{\bf R}_{ij}(x)=\Bigl{<}\Omega\Bigr{|}. It then follows that \Bigl{<}\Omega\Bigr{|}{\bf K}_{i}^{(\hbar)}=\Bigl{<}\Omega\Bigr{|}{\bf K}_{i}^{(0)}, so the projection of the $i$-th qKZ equation onto the covector \Bigl{<}\Omega\Bigr{|} reads

[TABLE]

Therefore, multiplying by $\displaystyle{\prod_{j\neq i}^{n}\frac{x_{i}-x_{j}+\eta}{x_{i}-x_{j}}}$ and summing over $i$, we get:

[TABLE]

[TABLE]

where we have used (11) and (13).

In the next section we show that $\Psi$ is the common eigenfunction for all higher Ruijsenaars Hamiltonians $\hat{\cal H}_{d}$ with the eigenvalues $e_{d}(\underbrace{g_{1},\ldots,g_{1}}_{M_{1}},\,\ldots\,\underbrace{g_{N},\ldots,g_{N}}_{M_{N}})$, where $e_{d}$ is the elementary symmetric polynomial of $n$ variables.

4 Higher Hamiltonians

Here we show that $\Psi$ (15) is an eigenfunction of the higher rational Macdonald-Ruijsenaars Hamiltonians $\hat{\cal H}_{d}$ defined by

[TABLE]

or

[TABLE]

For example, for $d=2$

[TABLE]

Proposition 4.1

The operators ${\bf K}_{j}^{(\hbar)}$ from (1) and the wave function (15) \Psi=\Bigl{<}\Omega|\Phi\Bigr{>} satisfy

[TABLE]

For the proof we introduce the notation ${\bf R}_{ab}={\bf R}_{ab}(x_{a}-x_{b})$ and ${\bf R}^{+}_{ab}={\bf R}_{ab}(x_{a}-x_{b}+\eta\hbar)$, so that

[TABLE]

Consider first the case $d=2$. For $i<j$ we have

[TABLE]

The “underbraced” expressions in (21) consist of commuting products of $R$-matrices thanks to $i<j$. Therefore, these expressions can be permuted. Using then the property \Bigl{<}\Omega\Bigr{|}\,{\bf R}_{kl}(x)=\Bigl{<}\Omega\Bigr{|} we conclude that all shifts by $\eta\hbar$ in the $R$-matrix arguments can be removed. Finally, we can permute the products of $R$-matrices coming back to the initial order (but with non-shifted arguments).

For $d=3$ and $i<j<k$ we have

[TABLE]

Again, since $i<j<k$ we see that “underbraced” expressions commute as well as the “overbraced” expressions. Permute first the “underbraced” expressions. Then we get

[TABLE]

Now the “overbraced” expression from the second line of (23) commutes with the whole third line. By permuting them we get the product where all $R$-matrices with shifted arguments ($\bf R^{+}$) are to the left from twist matrices and act on \Bigl{<}\Omega\Bigr{|} (to the left). Then we can apply the previously used reasoning and remove all the shifts of arguments.

It is easy to see that the same proof holds true for arbitrary $d$. The choice of ordering $i_{1}<...<i_{d}$ is convenient but the final answer is independent of it since $[{\bf K}_{i}^{(0)},{\bf K}_{j}^{(0)}]=0$ (because ${\bf K}_{i}^{(0)}\propto\bf H_{i}$).

Multiplying both parts of (20) by the products transforming ${\bf R}(x)$ to $\widetilde{\bf R}(x)$ (7), (11), we get

[TABLE]

A comparison between (18) and (24) shows that

[TABLE]

We are now in a position to use the following operator relation (see [20, eq. (4.2)]):

[TABLE]

The both sides are polynomials in $z$. Equating the coefficients in front of the powers of $z$, we have

[TABLE]

where $e_{d}(\{{\bf P}_{j}\})$ are elementary symmetric functions defined by the generating function as

[TABLE]

and ${\bf P}_{k}=\sum_{a}{\bf M}_{a}g_{a}^{k}$. In particular, for $k=1,2,3$

[TABLE]

The vector \Bigl{|}\Phi\Bigr{>} is an eigenvector for these operators with the eigenvalues given by the same formulas with ${\bf M}_{a}\to M_{a}$. Therefore, plugging (27) into (25) we get $\hat{\cal H}_{d}\Psi=E_{d}\Psi$ with the eigenvalue

[TABLE]

It is easy to see that the right hand side is the elementary symmetric polynomial of $n$ variables $e_{d}(\underbrace{g_{1},\ldots,g_{1}}_{M_{1}},\,\ldots\,\underbrace{g_{N},\ldots,g_{N}}_{M_{N}})$.

5 The qKZ-Ruijsenaars correspondence in the trigonometric case

The trigonometric (hyperbolic) analog of the $R$-matrix (3) which participates in the qKZ equations (1) reads

[TABLE]

(In this section we use the same notation for analogous objects in the trigonometric and rational cases.) After some algebra it can be represented in the form

[TABLE]

where

[TABLE]

is the $q$-permutation operator acting as follows:

[TABLE]

Note that ${\bf P}^{q}_{ij}={\bf P}^{1/q}_{j\,i}$. The unitarity condition can be easily checked.

We also introduce the $R$-matrix

[TABLE]

and the transfer matrix {\bf T}(x)=\mbox{tr}_{0}\Bigl{(}\widetilde{\bf R}_{0n}(x-x_{n})\ldots\widetilde{\bf R}_{01}(x-x_{1})\,{\bf g}^{(0)}\Bigr{)} of the inhomogeneous XXZ spin chain with twisted boundary conditions. Similarly to the rational case, the commuting Hamiltonians are defined by the pole expansion

[TABLE]

They are expressed through the $R$-matrices by the same formula (10). We have (see [2]): $\displaystyle{{\bf T}(\pm\infty)={\bf C}\pm\sinh\eta\sum_{k}{\bf H}_{k}=\sum_{a=1}^{N}g_{a}e^{\pm\eta{\bf M}_{a}},}$ where ${\bf C}$ is some operator and the weight operators ${\bf M}_{a}$ are defined in the same way as before. Hence

[TABLE]

Similarly to (11), we have:

[TABLE]

As in the rational case, the trigonometric operators ${\bf K}_{i}^{(\hbar)}$ respect the weight decomposition (14). Let \displaystyle{\Bigl{|}\Phi\Bigr{>}=\sum_{J}\Phi_{J}\Bigl{|}J\Bigr{>}} be any solution of the trigonometric qKZ equations belonging to the weight subspace ${\cal V}(\{M_{a}\})$. Let $\ell(J)$ be the minimal number of elementary permutations $\sigma_{i\,i\!+\!1}\in S_{n}$ which are required to get the multi-index $J=(j_{1},j_{2},\ldots,j_{n})$ from the “minimal” one111Put it differently, any \Bigl{|}J\Bigr{>} can be constructed from the minimal one \displaystyle{\Bigl{|}J_{min}\Bigr{>}=e_{1}^{\otimes M_{1}}\otimes...\otimes e_{N}^{\otimes M_{N}}} in $\ell(J)$ steps by applying elementary permutations of neighboring tensor components $\sigma_{k_{i}\,k_{i}\!+\!1}$:

\displaystyle{\Bigl{|}J_{min}\Bigr{>}=\Bigl{|}J^{(0)}\Bigr{>}\stackrel{{\scriptstyle\sigma_{k_{1}\,k_{1}\!+\!1}}}{{\longrightarrow}}\Bigl{|}J^{(1)}\Bigr{>}\stackrel{{\scriptstyle\sigma_{k_{2}\,k_{2}\!+\!1}}}{{\longrightarrow}}\Bigl{|}J^{(2)}\Bigr{>}\stackrel{{\scriptstyle\sigma_{k_{3}\,k_{3}\!+\!1}}}{{\longrightarrow}}...\stackrel{{\scriptstyle\sigma_{k_{\ell(J)}\,k_{\ell(J)}\!+\!1}}}{{\longrightarrow}}\Bigl{|}J^{({\ell(J)})}\Bigr{>}=\Bigl{|}J\Bigr{>}\,,}

where \displaystyle{\Bigl{|}J^{(i)}\Bigr{>}=e_{j_{1}^{(i)}}\otimes...\otimes e_{j_{n}^{(i)}}} and each time $j_{k_{i}}^{(i-1)}<j_{k_{i}+1}^{(i-1)}$., where the $j_{k}$’s are ordered as $1\leq j_{1}\leq j_{2}\leq\ldots\leq j_{n}\leq N$. In the case $n=N$, $M_{1}=M_{2}=\ldots=M_{N}=1$ the $\ell(J)$ is what is called length of the permutation $(12\ldots N)\to(j_{1}j_{2}\ldots j_{N})$. We claim that the function

[TABLE]

is an eigenfunction of the trigonometric Macdonald operator $\hat{\cal H}$ with the eigenvalue $\displaystyle{E=\sum_{a=1}^{N}g_{a}}\frac{\sinh(\eta M_{a})}{\sinh\eta}\,$:

[TABLE]

The idea of the proof is the same as in the rational case. We consider the covector

[TABLE]

then \Psi=\Bigl{<}\Omega_{q}\Bigr{|}\Phi\Bigr{>}. It is not difficult to see that \Bigl{<}\Omega_{q}\Bigr{|}{\bf P}^{q}_{i,i\!-\!1}=\Bigl{<}\Omega_{q}\Bigr{|}.222Indeed, from (32) and construction of \Bigl{|}J\Bigr{>} from \Bigl{|}J_{min}\Bigr{>} one gets {\bf P}^{q}_{i\!-\!1,i}\Bigl{|}\Omega_{q}\Bigr{>}=\Bigl{|}\Omega_{q}\Bigr{>}. The matrix transposition of this equality gives \Bigl{<}\Omega_{q}\Bigr{|}=\Bigl{<}\Omega_{q}\Bigr{|}\Bigl{(}{\bf P}^{q}_{i\!-\!1,i}\Bigr{)}^{\!T}=\Bigl{<}\Omega_{q}\Bigr{|}\,{\bf P}^{1/q}_{i\!-\!1,i}=\Bigl{<}\Omega_{q}\Bigr{|}\,{\bf P}^{q}_{i,i\!-\!1}. This implies the important relation

[TABLE]

(the second term in (31) disappears). Using the relation ${\bf P}_{i\,i\!-\!1}{\bf P}^{q}_{i\,i\!-\!2}={\bf P}^{q}_{i\!-\!1\,i\!-\!2}{\bf P}_{i\,i\!-\!1}$, one can show by induction that

[TABLE]

for all $i=2,\ldots,n$. Since the right hand side does not depend on the spectral parameters, we can substitute each $R$-matrix in the left hand side by the same one with $\hbar=0$ and conclude that \Bigl{<}\Omega_{q}\Bigr{|}{\bf K}_{i}^{(\hbar)}=\Bigl{<}\Omega_{q}\Bigr{|}{\bf K}_{i}^{(0)}. The projection of the $i$-th qKZ equation onto the covector \Bigl{<}\Omega_{q}\Bigr{|} reads

[TABLE]

In the same way as in the rational case, we multiply this by $\displaystyle{\prod_{j\neq i}^{n}\frac{\sinh(x_{i}-x_{j}+\eta)}{\sinh(x_{i}-x_{j})}}$ and sum over $i$ to get:

[TABLE]

[TABLE]

Here we have used (34) and (35).

Similarly to the rational case, the correspondence can be extended to the higher trigonometric Macdonald-Ruijsenaars operators.

6 Conclusion and discussion

We have established the correspondence between solutions to the qKZ equations (1) in different weight subspaces of $V^{\otimes n}$ and solutions to the spectral problem for the $n$-body Ruijsenaars model with Planck’s constant $\hbar$, with $V$ being the space of $N$-dimensional vector representation of $GL(N)$. The proof appears to be even simpler than in the case of the correspondence between the differential KZ equations and the quantum Calogero model (see [13, 4, 7] and [22]). In the limit $\hbar\to 0$ we obtain the quantum-classsical correspondence [1, 9, 20] between the quantum spin chain (XXX or XXZ) and the classical Ruijsenaars system of particles [16] (rational or trigonometric).

The Hamiltonian of the classical Ruijsenaars system has the form

[TABLE]

with the usual Poisson brackets $\{p_{i},x_{j}\}=\delta_{ij}$ (for simplicity we consider the rational case). The model is integrable, with the Lax matrix

[TABLE]

where

[TABLE]

is the velocity of the $i$-th particle. The higher Hamiltonians in involution are coefficients of the characteristic polynomial of the Lax matrix:

[TABLE]

The correspondence with the quantum spin chain goes as follows. Let the eigenvalues of the $n\!\times\!n$ Lax matrix be the twist parameters $g_{a}$ with multiplicities $M_{a}$ (recall that $M_{1}+\ldots+M_{a}=n$). This means that we consider the level set of all the classical Hamiltonians

[TABLE]

with fixed coordinates $x_{i}$. Then the admissible values of velocities, $\dot{x}_{i}$, are equal to $\eta H_{i}$, where the $H_{i}$’s are eigenvalues of the spin chain Hamiltonians ${\bf H}_{i}$ in the weight subspace ${\cal V}(\{M_{a}\})$ for the model with the inhomogeneity parameters $x_{i}$ and the twist matrix ${\bf g}=\mbox{diag}\,(g_{1},\ldots,g_{N})$. In fact the admissible values of $\dot{x}_{i}$’s obey a system of algebraic equations (see [20]). Different solutions of this system correspond to different eigenstates of the spin chain Hamiltonians.

In the trigonometric case eigenvalues of the Lax matrix $\displaystyle{L^{\rm trig}_{ij}=\frac{\dot{x}_{j}}{\sinh(x_{i}-x_{j}+\eta)}}$ should form “multiplicative strings” of lengths $M_{a}$ centered at $g_{a}$:

[TABLE]

Then $\dot{x}_{i}/\eta$ are eigenvalues of the XXZ spin chain Hamiltonians. The eigenvalue $E=\displaystyle{\sum_{a=1}^{N}g_{a}\frac{\sinh(\eta M_{a})}{\sinh\eta}}$ of the Ruijsenaars operator agrees with this since it is clear that

[TABLE]

Comparing this with (6) and taking into account (11), (40), we see that in the limit $\hbar\to 0$ of the qKZ system (which is the spectral problem for ${\bf H}_{i}$’s or ${\bf K}_{i}^{(0)}$’s) the other side of the correspondence becomes the classical Ruijsenaars model. In other words, we can say that the quantization of the classical Ruijsenaars system of particles with the Planck constant $\hbar$ ($p_{i}\to\hbar\partial_{x_{i}}$) corresponds to the passage from the spectral problem for the spin chain to the system of qKZ equations with the step parameter $\eta\hbar$.

Acknowledgments

We thank A.Liashyk for discussions. The work of A. Zabrodin has been funded by the Russian Academic Excellence Project ‘5-100’. It was also supported in part by RFBR grant 15-01-05990 and by joint RFBR grant 15-52-50041 YaFa. The work of A. Zotov was supported by RFBR grant 15-01-04217 and by joint RFBR project 15-51-52031 HHCa.