The matrix Riemann-Hilbert problem and differential equation for correlation function of the \(XXO\) Heisenberg chain.

*(English. Russian original)*Zbl 0836.35113
St. Petersbg. Math. J. 6, No. 2, 315-326 (1995); translation from Algebra Anal. 6, No. 2, 138-151 (1994).

The quantum inverse scattering method worked out by L. D. Faddeev and his co-workers at the end of the 70th and in the beginning of the 80th, has led to a considerable progress at studying the integrable systems of quantum field theory and statistical physics in one direction. In the paper [L. D. Faddeev, Sov. Sci. Rev., Sect. C, Math. Phys. Rev. 1, 107–155 (1980; Zbl 0569.35064)] basic principles of the method were formulated and the possible directions of its further development were also sketched out. A significant achievement was the creation of algebraic Bethe ansatz for the construction of eigenfunctions for Hamiltonians of some quantum systems.

One of the most difficult and important problems at studying the integrable systems is the problem of construction and description of their correlation functions. The basis for the correlation function description is the representation of these functions as the Fredholm determinant of some linear integral operator. For the first time such representation has been obtained for the model of unpenetrable bosons described by nonlinear SchrĂ¶dinger equation with infinite interaction constant in [A. Lenard, J. Math. Phys. 7, 1268–1272 (1966), J. Math. Phys. 5, 930–943 (1964)]. An algebraic Bethe ansatz provides the possibility to construct such representation for the case of finite interaction constant (see: [V. E. Korepin, Commun. Math. Phys. 113, 177–190 (1987; Zbl 0629.58039)] and [V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum inverse scattering method and correlation functions. Cambridge: Cambridge University Press (1993; Zbl 0787.47006)]).

In the paper under review the temperature dependent correlation functions of the \(XXO\) spin Heisenberg chain are considered. This model is related to the isotropic \(XY\) Lieb-Schultz-Mattis model [see E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16, 407–466 (1961; Zbl 0129.46401)]. Hamiltonian of the \(XXO\) chain describes the interaction of 1/2- spins at neighbouring sites of one-dimensional lattice. For the periodic lattice of the finite length with \(M\) sites in the external constant magnetic field \(h\) it is the \(2^M \times 2^M\) matrix:

\[ H = - \sum^M_{m = 1} \biggl( \sigma_x^{(m)} \sigma_x^{(m + 1)} + \sigma_y^{(m)} \sigma_y^{(m + 1)} + h \sigma_z^{(m)} \biggr), \] where \(\sigma_j^{(m)}\) \((j = x,y,z)\) are the Pauli matrices obeyed to the commutation rule

\[ \Bigl[ \sigma_j^{(m)}, \sigma_k^{(n)} \Bigr] = 2i \delta_{mn} \varepsilon_{kjl} \sigma_l^{(m)} \] and \[ \sigma_x = \left( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right),\;\sigma_y = \left( \begin{matrix} 0 & - i \\ i & 0 \end{matrix} \right),\;\sigma_z = \left( \begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix} \right). \] The authors consider some special linear integral operator \(V\), operating upon the functions on the unit circle \[ (Vf) (\lambda) = \oint V (\lambda, \mu) f(\mu)\,d \mu,\;|\lambda |= 1,\;|\mu |= 1. \tag{1} \] The symmetric kernel of the operator has the form \[ V (\lambda, \mu) = {e_+ (\lambda) e_- (\mu) - e_- (\lambda) e_+ (\mu) \over \pi (\lambda - \mu )} \] where the functions \(e_\pm\) are defined by the formulae

\[ e_- (\lambda) = \lambda^{- {m \over 2}} \exp \bigl[ - it (\lambda + \lambda^{-1}) \bigr] \sqrt {\theta (\lambda)},\;e_+ (\lambda) = e_- (\lambda) E(m,t, \lambda), \]

\[ \theta (\lambda) \equiv \theta (\lambda,h,T) = {1 \over 1 + \exp \bigl[ 2(h - \lambda - \lambda^{-1})/T \bigr]}, \]

\[ E(m,t, \lambda) = v.p. {1 \over \pi} \oint {\mu^m \exp \bigl[2it (\mu + \mu^{-1}) \bigr] \,d \mu \over \mu - \lambda}. \] Integral operator (1) is naturally connected to the matrix Riemann-Hilbert problem, the standard singular integral equations of which are equivalent to the nonsingular integral equations generated by the kernel \(V (\lambda, \mu)\).

The main result of the paper is deducing the system of integrable differential equations for the correlation functions of the \(XXO\) Heisenberg chain.

One of the most difficult and important problems at studying the integrable systems is the problem of construction and description of their correlation functions. The basis for the correlation function description is the representation of these functions as the Fredholm determinant of some linear integral operator. For the first time such representation has been obtained for the model of unpenetrable bosons described by nonlinear SchrĂ¶dinger equation with infinite interaction constant in [A. Lenard, J. Math. Phys. 7, 1268–1272 (1966), J. Math. Phys. 5, 930–943 (1964)]. An algebraic Bethe ansatz provides the possibility to construct such representation for the case of finite interaction constant (see: [V. E. Korepin, Commun. Math. Phys. 113, 177–190 (1987; Zbl 0629.58039)] and [V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum inverse scattering method and correlation functions. Cambridge: Cambridge University Press (1993; Zbl 0787.47006)]).

In the paper under review the temperature dependent correlation functions of the \(XXO\) spin Heisenberg chain are considered. This model is related to the isotropic \(XY\) Lieb-Schultz-Mattis model [see E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16, 407–466 (1961; Zbl 0129.46401)]. Hamiltonian of the \(XXO\) chain describes the interaction of 1/2- spins at neighbouring sites of one-dimensional lattice. For the periodic lattice of the finite length with \(M\) sites in the external constant magnetic field \(h\) it is the \(2^M \times 2^M\) matrix:

\[ H = - \sum^M_{m = 1} \biggl( \sigma_x^{(m)} \sigma_x^{(m + 1)} + \sigma_y^{(m)} \sigma_y^{(m + 1)} + h \sigma_z^{(m)} \biggr), \] where \(\sigma_j^{(m)}\) \((j = x,y,z)\) are the Pauli matrices obeyed to the commutation rule

\[ \Bigl[ \sigma_j^{(m)}, \sigma_k^{(n)} \Bigr] = 2i \delta_{mn} \varepsilon_{kjl} \sigma_l^{(m)} \] and \[ \sigma_x = \left( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right),\;\sigma_y = \left( \begin{matrix} 0 & - i \\ i & 0 \end{matrix} \right),\;\sigma_z = \left( \begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix} \right). \] The authors consider some special linear integral operator \(V\), operating upon the functions on the unit circle \[ (Vf) (\lambda) = \oint V (\lambda, \mu) f(\mu)\,d \mu,\;|\lambda |= 1,\;|\mu |= 1. \tag{1} \] The symmetric kernel of the operator has the form \[ V (\lambda, \mu) = {e_+ (\lambda) e_- (\mu) - e_- (\lambda) e_+ (\mu) \over \pi (\lambda - \mu )} \] where the functions \(e_\pm\) are defined by the formulae

\[ e_- (\lambda) = \lambda^{- {m \over 2}} \exp \bigl[ - it (\lambda + \lambda^{-1}) \bigr] \sqrt {\theta (\lambda)},\;e_+ (\lambda) = e_- (\lambda) E(m,t, \lambda), \]

\[ \theta (\lambda) \equiv \theta (\lambda,h,T) = {1 \over 1 + \exp \bigl[ 2(h - \lambda - \lambda^{-1})/T \bigr]}, \]

\[ E(m,t, \lambda) = v.p. {1 \over \pi} \oint {\mu^m \exp \bigl[2it (\mu + \mu^{-1}) \bigr] \,d \mu \over \mu - \lambda}. \] Integral operator (1) is naturally connected to the matrix Riemann-Hilbert problem, the standard singular integral equations of which are equivalent to the nonsingular integral equations generated by the kernel \(V (\lambda, \mu)\).

The main result of the paper is deducing the system of integrable differential equations for the correlation functions of the \(XXO\) Heisenberg chain.

Reviewer: Igor E. Tralle (Minsk)

##### MSC:

35Q15 | Riemann-Hilbert problems in context of PDEs |

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

35Q40 | PDEs in connection with quantum mechanics |

81U40 | Inverse scattering problems in quantum theory |

82B21 | Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics |