Browsing by Subject "Renormalization Group Method"

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    A Dynamical Approach to the Potts Model on Cayley Tree
    (2024-12) Pannipitiya, Diyath Nelaka; Kitchens, Bruce P.; Roeder, Roland K. W.; Geller, William; Perez, Rodrigo A.
    The Ising model is one of the most important theoretical models in statistical physics, which was originally developed to describe ferromagnetism. A system of magnetic particles, for example, can be modeled as a linear chain in one dimension or a lattice in two dimensions, with one particle at each lattice point. Then each particle is assigned a spin σi ∈ {±1}. The q-state Potts model is a generalization of the Ising model, where each spin σi may take on q ≥3 number of states {0,··· ,q−1}. Both models have temperature T and an externally applied magnetic field h as parameters. Many statistical and physical properties of the q- state Potts model can be derived by studying its partition function. This includes phase transitions as T and/or h are varied. The celebrated Lee-Yang Theorem characterizes such phase transitions of the 2-state Potts model (the Ising model). This theorem does not hold for q > 2. Thus, phase transitions for the Potts model as h is varied are more complicated and mysterious. We give some results that characterize the phase transitions of the 3-state Potts model as h is varied for constant T on the binary rooted Cayley tree. Similarly to the Ising model, we show that for fixed T >0the 3-state Potts model for the ferromagnetic case exhibits a phase transition at one critical value of h or not at all, depending on T. However, an interesting new phenomenon occurs for the 3-state Potts model because the critical value of h can be non-zero for some range of temperatures. The 3-state Potts model for the antiferromagnetic case exhibits a phase transition at up to two critical values of h. The recursive constructions of the (n + 1)st level Cayley tree from two copies of the nth level Cayley tree allows one to write a relatively simple rational function relating the Lee-Yang zeros at one level to the next. This allows us to use techniques from dynamical systems.