The tensor renormalization group study of the general spin-S Blume-Capel model

TL;DR

This study uses tensor renormalization group methods to analyze phase transitions in the spin-S Blume-Capel model on a square lattice, revealing multiple first-order and continuous transitions depending on spin type and temperature.

Contribution

It applies a novel tensor renormalization group approach to explore phase transitions in the general spin-S Blume-Capel model, highlighting new insights into transition behaviors and symmetry breaking.

Findings

Integer spins undergo S first-order phase transitions with symmetry breaking.

Half-integer spins exhibit (S-1/2) first-order transitions and a continuous transition.

Critical temperature for the nth transition is independent of spin value.

Abstract

We focus on the special situation of $D=2J$ of the general spin-S Blume-Capel model on the square lattice. Under the infinitesimal external magnetic field, the phase transition behaviors due to the thermal fluctuations are discussed by the newly developed tensor renormalization group method. For the case of the integer spin-S, the system will undergo $S$ first-order phase transitions with the successive symmetry breaking with the magnetization $M=S,S-1,...0$. For the half-integer spin-S, there are similar $S-1/2$ first order phase transition with $M=S,S-1,...1/2$ stepwise structure, in addition, there is a continuous phase transition due to the spin-flip $Z_2$ symmetry breaking. In the low temperature regions, all first-order phase transitions are accompanied by the successive disappearance of the optional spin-component pairs($s,-s$), furthermore, the critical temperature for the nth first-order phase transition is the same, independent of the value of the spin-S. In the absence of the magnetic field, the visualization parameter characterizing the intrinsic degeneracy of the different phases clearly demonstrates the phase transition process.