Critical behavior of Gaussian model on diamond-type hierarchical lattices

Abstract

It is proposed that the Gaussian type distribution constantb _qi in the Gaussian model depends on the coordination numberq _i of sitei, and that the relation${{b_{q_i } } \mathord{\left/ {\vphantom {{b_{q_i } } {b_{q_j } }}} \right. \kern-\nulldelimiterspace} {b_{q_j } }} = {{q_q } \mathord{\left/ {\vphantom {{q_q } {q_q }}} \right. \kern-\nulldelimiterspace} {q_q }}$ holds amongb _qi ’s. The Gaussian model is then studied on a family of the diamond-type hierarchical (or DH) lattices, by the decimation real-space renormalization group following spin-rescaling method. It is found that the magnetic property of the Gaussian model belongs to the same universal class, and that the critical pointK* and the critical exponentv are given by${{K^ * = b_{q_i } } \mathord{\left/ {\vphantom {{K^ * = b_{q_i } } {q_i }}} \right. \kern-\nulldelimiterspace} {q_i }}$ and$\nu = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}$, respectively.