Monte Carlo renormalization group study of the XY model

Gaoke Hu, S. C. Ying

Vieri Mastropietro

Paweł Jakubczyk, Andreas Eberlein

We solve the nonperturbative renormalization-group flow equations for the two-dimensional XY model at the truncation level of the (complete) second-order derivative expansion. We compute the thermodynamic properties in the high-temperature phase and compare the nonuniversal features specific to the XY model with results from Monte Carlo simulations. In particular, we study the position and magnitude of the specific-heat peak as a function of temperature. The obtained results compare well with Monte Carlo simulations. We additionally gauge the accuracy of simplified nonperturbative renormalization-group treatments relying on ϕ^{4}-type truncations. Our computation indicates that such an approximation is insufficient in the high-T phase and a correct analysis of the specific-heat profile requires account of an infinite number of interaction vertices.

Ji-Feng Yu, Z. Y. Xie, Yannick Meurice +6 more

Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the thermodynamic properties of the continuous XY model on the square lattice. The temperature dependence of the free energy, the internal energy, and the specific heat agree with the Monte Carlo calculations. From the field dependence of the magnetic susceptibility, we find the Kosterlitz-Thouless transition temperature to be 0.8921(19), consistent with the Monte Carlo as well as the high temperature series expansion results. At the transition temperature, the critical exponent δ is estimated as 14.5, close to the analytic value by Kosterlitz.

Guy Segall, Snir Gazit, Daniel K. Podolsky

We introduce a numerical method to study the critical properties of the XY model close to the ferromagnet to paramagnet phase transition. Our method applies ideas of the tensor renormalization group to obtain an improved action. This improved action allows the extraction of universal critical properties through Monte Carlo simulations on relatively small system sizes. We demonstrate the effectiveness of this method on the XY model in three dimensions.

Robert H. Swendsen