The evaluation of the entropy  of entanglement of the ground state in a wide family of one-dimensional quantum spin can be reduced to the Wiener-Hopf factorization of certain 2x2 algebraic  matrix valued functions. We show how this factorization can be performed using the apparatus of the Riemann-Hilbert method  and algebra-geometricintegration  borrowed from the theory of integrable systems. We would like to thinkabout these calculations as a basis for a conjecture that  the Wiener-Hopf factorization of a general  algebraic matrix   can be  performed in terms of the Riemann theta functions associated with a certain  algebraic curve. The talk is based on the speaker works with  V. Korepin and B. Q. Jin and on his works with  F. Mezzadri and M. Y. Mo.