Tensor Renormalization Group Study of the General Spin-S Blume–Capel Model

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 Z2 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. where the spin variable S takes 2S + 1 values: (−S, −S + 1, ..., S − 1, S). J is the coupling constant, D is the strength of the single-ion anisotropy and h is the magnetic field. The sum of the first term runs over all the nearest neighbors. Firstly, we consider the case of h = 0. When D = 0, this model is reduced to the classical Ising model. When D goes to the positive infinity, the energy favorable state is the one with the full occupation of the smallest spin component. For the integer spin cases, the component is S = 0. For the half-integer cases, the component is S = ±1/2. If we look at the hamiltonian of any bond linking two * Electronic address: liping2012@cqu.edu.cn sites i, j, then we haveHere, q is the coordination number depending on the lattice structure. This formula can be rewritten asfrom which, it leads to the following conclusions for the ferromagnetic coupling(J > 0): when D > qJ/2, the configuration of the ground state is S i = S j = 0, S i = S j = 1/2(−1/2)(depending on the spontaneous breaking), respectively for the integer and half-odd spin-S cases; when D < qJ/2, the configuration of the ground state is S i = S j = max(S) for any spin-S cases. As a result, we have a special situation, i.e., D = qJ/2, where the ground state is S i = S j with (2S + 1)-fold degeneracy. Once the magnetic field h is turned on as a smallness, the ground state degeneracy is lifted in the case of D = qJ/2. The positive smallness h make the system go into the ground state S i = S j = max(S) as the case D < qJ/2 with h = 0. Without loss of the generality, we focus on the square lattice hereafter. Then, q = 4 and D = 2J is the special parameter. The phase boundaries of the square lattice [3,4,6,8] end at (T /J, D/J) = (0, 2), i.e., for D = 2J, and the critical temperature is T c = 0 for the general spin cases when h = 0.Motivated by the special parameter point and the though...

The Blume–Capel model for spins S=1 and 3∕2 in dimensions Butera1, Pernici2 2018Physica A: Statistical Mechanics and its Applications343360View full textAdd to dashboardCite

“…The BC model has been explored in a variety of analytical approximations (mean field (MF), effective field, Renormalization Group, etc.) 1,2,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] , by transfer-matrix methods [21][22][23] and by MonteCarlo (MC) simulation methods 14,20,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] , but only in a handful of papers [40][41][42][43][44][45] extrapolations of series-expansions were employed, in spite of the potential reliability and accuracy of this technique. The high-temperature (HT) and the low-temperature (LT) expansions have been jointly used 43 to map out the phase diagram, (the former being generally sufficient to locate the second-order part of the phase-boundary and to determine its universal parameters, the l...…”

The mixed spin-3/2 and spin-2 Blume-Capel model by renormalization group approach

“…Experimentally, the spins of Fe II ions are spin-2, and it is found that these ions have anisotropy (Mathonière et al , 1996). Theoretically, various magnetic properties of spin-2 BC systems have been studied by well-known methods in the equilibrium statistical physics such as the mean field approximation (Bahmad et al , 2007; Plascak and Moreira, 1993), the effective field theory with correlations (Ertaş et al , 2012; Jiang et al , 2000; Liang et al , 2004; Yigit and Albayrak, 2012), the Glauber-type stochastic dynamics (Temizer et al , 2016; Temizer and Korkmaz, 2012), the Monte Carlo simulation (Lara and Plascak, 1998; Iwashita et al , 2010), the exact recursion relations on the Bethe lattice (Erdinç et al , 2006), the four-spin model approximation (Iwashita et al , 2000) and the renormalization group method (Hachem et al , 2017; Yang and Xie, 2016).…”

Phase boundary location with information-theoretic entropy in tensor renormalization group flows

“…In contrast to the phase degeneracy method, the von Neumann entropy TRG method does not require coarse graining deep into the thermodynamic limit and does not require encoding of symmetries (numerical instability can blur the transition point in the phase degeneracy method if symmetries are not encoded, as seen in Ref. 29). In contrast to the approach of computing central charge, which locates only continuous transitions, the von Neumann entropy TRG method can locate both first-order and continuous transitions.…”