Transfer Operators and Dynamical Zeta Functions for a Class of Lattice Spin Models
Abstract
We investigate the location of zeros and poles of a dynamical zeta function for a family of subshifts of finite type with an interaction function depending on the parameters . The system corresponds to the well known KacBaker lattice spin model in statistical mechanics. Its dynamical zeta function can be expressed in terms of the Fredholm determinants of two transfer operators and with the Ruelle operator acting in a Banach space of holomorphic functions, and an integral operator introduced originally by Kac, which acts in the space with a kernel which is symmetric and positive definite for positive β. By relating via the SegalBargmann transform to an operator closely related to the Kac operator we can prove equality of their spectra and hence reality, respectively positivity, for the eigenvalues of the operator for real, respectively positive, β. For a restricted range of parameters we can determine the asymptotic behavior of the eigenvalues of for large positive and negative values of β and deduce from this the existence of infinitely many nontrivial zeros and poles of the dynamical zeta functions on the real β line at least for generic . For the special choice , we find a family of eigenfunctions and eigenvalues of leading to an infinite sequence of equally spaced ``trivial'' zeros and poles of the zeta function on a line parallel to the imaginary βaxis. Hence there seems to hold some generalized Riemann hypothesis also for this kind of dynamical zeta functions.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2002
 DOI:
 10.1007/s0022000207468
 arXiv:
 arXiv:math/0203191
 Bibcode:
 2002CMaPh.232...19H
 Keywords:

 Mathematics  Dynamical Systems;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 Mathematics  Number Theory;
 Mathematics  Spectral Theory;
 37C30;
 37A30;
 37C25;
 11M26
 EPrint:
 39 pages