Binder parameter

The Binder parameter or Binder cumulant[1][2] in statistical physics, also known as the fourth-order cumulant ${\displaystyle U_{L}=1-{\frac {{\langle s^{4}\rangle }_{L}}{3{\langle s^{2}\rangle }_{L}^{2}}}}$ is defined as the kurtosis of the order parameter, s introduced by Austrian Theoretical Physicist Kurt Binder. It is frequently used to determine accurately phase transition points in numerical simulations of various models. [3]

The phase transition point is usually identified comparing the behavior of ${\displaystyle U}$ as a function of the temperature for different values of the system size ${\displaystyle L}$. The transition temperature is the unique point where the different curves cross in the thermodynamic limit. This behavior is based on the fact that in the critical region, ${\displaystyle T\approx T_{c}}$, the Binder parameter behaves as ${\displaystyle U(T,L)=b(\epsilon L^{1/\nu })}$, where ${\displaystyle \epsilon ={\frac {T-T_{c}}{T}}}$.

Accordingly, the cumulant may also be used to identify the universality class of the transition by determining the value of the critical exponent ${\displaystyle \nu }$ of the correlation length. [1]

In the thermodynamic limit, at the critical point, the value of the Binder parameter depends on boundary conditions, the shape of the system, and anisotropy of correlations. [1][4][5][6]

References

1. Binder, K. (1981). "Finite size scaling analysis of ising model block distribution functions". Zeitschrift für Physik B: Condensed Matter. 43 (2): 119–140. Bibcode:1981ZPhyB..43..119B. doi:10.1007/bf01293604. ISSN 0340-224X. S2CID 121873477.
2. Binder, K. (1981-08-31). "Critical Properties from Monte Carlo Coarse Graining and Renormalization". Physical Review Letters. 47 (9): 693–696. Bibcode:1981PhRvL..47..693B. doi:10.1103/physrevlett.47.693. ISSN 0031-9007.
3. K. Binder, D. W. Heermann, Monte Carlo Simulation in Statistical Physics: An Introduction (2010) Springer
4. Kamieniarz, G; Blote, H W J (1993-01-21). "Universal ratio of magnetization moments in two-dimensional Ising models". Journal of Physics A: Mathematical and General. 26 (2): 201–212. Bibcode:1993JPhA...26..201K. doi:10.1088/0305-4470/26/2/009. ISSN 0305-4470.
5. Chen, X. S.; Dohm, V. (2004-11-30). "Nonuniversal finite-size scaling in anisotropic systems". Physical Review E. 70 (5): 056136. arXiv:cond-mat/0408511. Bibcode:2004PhRvE..70e6136C. doi:10.1103/physreve.70.056136. ISSN 1539-3755. PMID 15600721. S2CID 44785145.
6. Selke, W; Shchur, L N (2005-10-19). "Critical Binder cumulant in two-dimensional anisotropic Ising models". Journal of Physics A: Mathematical and General. 38 (44): L739–L744. arXiv:cond-mat/0509369. doi:10.1088/0305-4470/38/44/l03. ISSN 0305-4470. S2CID 14774533.