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Thermodynamic model of social influence on two-dimensional square lattice: Case for two features

We propose a thermodynamic multi-state spin model in order to describe equilibrial behavior of a society. Our model is inspired by the Axelrod model used in social network studies. In the framework of the statistical mechanics language, we analyze phase transitions of our model, in which the spin interaction $J$ is interpreted as a mutual communication among individuals forming a society. The thermal fluctuations introduce a noise $T$ into the communication, which suppresses long-range correlations. Below a certain phase transition point $T_t$, large-scale clusters of the individuals, who share a specific dominant property, are formed. The measure of the cluster sizes is an order parameter after spontaneous symmetry breaking. By means of the Corner transfer matrix renormalization group algorithm, we treat our model in the thermodynamic limit and classify the phase transitions with respect to inherent degrees of freedom. Each individual is chosen to possess two independent features $f=2$ and each feature can assume one of $q$ traits (e.g. interests). Hence, each individual is described by $q^2$ degrees of freedom. A single first order phase transition is detected in our model if $q>2$, whereas two distinct continuous phase transitions are found if $q=2$ only. Evaluating the free energy, order parameters, specific heat, and the entanglement von Neumann entropy, we classify the phase transitions $T_t(q)$ in detail. The permanent existence of the ordered phase (the large-scale cluster formation with a non-zero order parameter) is conjectured below a non-zero transition point $T_t(q)\approx0.5$ in the asymptotic regime $q\to\infty$.