Infinite cascades of phase transitions in the classical Ising chain

We report the new exact results on one of the best studied models in statistical physics: the classical antiferromagnetic Ising chain in a magnetic field. We show that the model possesses an infinite cascade of thermal phase transitions (also known as "disorder lines" or geometric phase transitions). The phase transition is signalled by a change of asymptotic behavior of the nonlocal string-string correlation functions when their monotonous decay becomes modulated by incommensurate oscillations. The transitions occur for rarefied ($m$-periodic) strings with arbitrary odd $m$. We propose a duality transformation which maps the Ising chain onto the $m$-leg Ising tube with nearest-neighbor couplings along the legs and the plaquette four-spin interactions of adjacent legs. Then the $m$-string correlation functions of the Ising chain are mapped onto the two-point spin-spin correlation functions along the legs of the $m$-leg tube. We trace the origin of these cascades of phase transitions to the lines of the Lee-Yang zeros of the Ising chain in $m$-periodic complex magnetic field, allowing us to relate these zeros to the observable (and potentially measurable) quantities.