On spin and matrix models in the complex plane

Tadashi Kadowaki, Yoshihiko Nonomura, 西森 秀稔 +1 more

We propose a new method for exact analytical calculation of the ground-state energy of the Ising spin glass on strips. An outstanding advantage of this method over the numerical transfer matrix technique is that the energy is obtained for complex values of the probability describing quenched randomness. We study the $\pm J$ and the site-random models using this method for strips of various sizes up to $5\times\infty$. The ground-state energy of these models is found to have singular points in the complex-probability plane, reminiscent of Lee-Yang zeros in the complex-field plane for the Ising ferromagnet. The $\pm J$ Ising model has a series of singularities which may approach a limiting point around $p \sim 0.9$ on the real axis in the limit of infinite width.

Marek Czachor

Andreas Wipf

Tadashi Kadowaki, Yoshihiko Nonomura, Hidetoshi Nishimori

We propose a new method for exact analytical calculation of the ground-state energy of the Ising spin glass on strips. An outstanding advantage of this method over the numerical transfer matrix technique is that the energy is obtained for complex values of the probability describing quenched randomness. We study the $\pm J$ and the site-random models using this method for strips of various sizes up to $5\times\infty$. The ground-state energy of these models is found to have singular points in the complex-probability plane, reminiscent of Lee-Yang zeros in the complex-field plane for the Ising ferromagnet. The $\pm J$ Ising model has a series of singularities which may approach a limiting point around $p \sim 0.9$ on the real axis in the limit of infinite width.

N. Angelescu, M. Bundaru, G. Costache +1 more

Giuseppe Albertini

The eigenvalues of the Fateev-Zamolodchikov Z N invariant model transfer matrix are found for N odd. Their zeros in the complex plane of the rapidity variable are shown to satisfy a set of Bethe-ansatz type equations similar to those obtained for the integrable XXZ chains. The eigenvalue for a filled sea of (N-1)-strings gives the free energy found by the matrix inversion method.