[Paper Review] Data collapse in the critical region using finite-size scaling with subleading corrections

K. S. D. Beach, Ling Wang+1 more|arXiv (Cornell University)|May 8, 2005

Diverse Scientific and Engineering Research1 references3 citations

TL;DR

This paper proposes a finite-size scaling method that incorporates subleading corrections to improve data collapse in critical phenomena, enabling more accurate estimation of critical parameters like $T_c$ and $ u$. By introducing size-dependent renormalization and shift corrections to the scaling function, the approach enhances reliability and convergence speed, validated by high-accuracy results for 2D and 3D Ising models and a 2D quantum antiferromagnet.

ABSTRACT

We propose a treatment of the subleading corrections to finite-size scaling that preserves the notion of data collapse. This approach is used to extend and improve the usual Binder cumulant analysis. As a demonstration, we present results for the two- and three-dimensional classical Ising models and the two-dimensional, double-layer quantum antiferromagnet.

Motivation & Objective

• To address systematic errors in standard finite-size scaling (FSS) due to non-universal subleading corrections.
• To develop a data collapse method that remains robust and accurate even for finite system sizes.
• To provide a more reliable and convergent approach for estimating critical temperatures $T_c$ and critical exponents $ u$.
• To demonstrate that intermediate-size simulations can yield more accurate results than large-scale simulations without correction terms.

Proposed method

• Introduces a modified FSS ansatz: $ A(T,L)L^{-ar{\kappa}/\nu} = \mathcal{N}(L) g_A(tL^{1/\nu} - \epsilon(L)) $, where $\mathcal{N}(L)$ and $\epsilon(L)$ account for subleading corrections.
• Uses renormalization group theory to derive the asymptotic forms of $\mathcal{N}(L)$ and $\epsilon(L)$, ensuring consistency with universal scaling behavior.
• Applies the corrected scaling form to perform data collapse on thermodynamic quantities, minimizing scatter across different system sizes.
• Employs Binder ratio intersection points as a complementary method, fitting their $L$-dependence to extract $T_c$ and $\nu$ with improved convergence.
• Validates the method using Monte Carlo data for 2D and 3D classical Ising models and a 2D quantum bilayer antiferromagnet.
• Compares results with exact values and prior high-accuracy studies to confirm reliability and precision.

Experimental results

Research questions

• RQ1 Can subleading corrections in finite-size scaling be systematically incorporated to improve data collapse and parameter estimation?
• RQ2 Does the inclusion of size-dependent renormalization and shift terms lead to faster convergence to the true critical point $T_c$?
• RQ3 Can accurate estimates of $T_c$ and $\nu$ be obtained from intermediate-sized systems using this method, even without extremely large $L$?
• RQ4 How does the performance of this method compare to standard FSS and other advanced fitting techniques in terms of accuracy and error estimation?

Key findings

• For the 3D classical Ising model, the method yields $T_c = 4.5114(2)$ and $\nu = 0.625(1)$, consistent with recent high-accuracy Monte Carlo and Monte Carlo renormalization group results.
• The 2D Ising model analysis gives $T_c = 2.2692(8)$ and $\nu = 0.99(2)$, in good agreement with exact values and prior estimates.
• For the 2D quantum bilayer antiferromagnet, the method produces $\alpha_c = 2.52181(4)$, the most accurate value to date, and $\nu = 0.715(2)$, consistent with the 3D Heisenberg universality class.
• Data collapse using the corrected scaling form shows convincing collapse onto a single universal curve, while standard FSS fails to collapse the same data.
• The convergence of intersection points $T_i(L, L')$ to $T_c$ is faster than $L^{-1/\nu}$, confirming improved convergence properties.
• The method yields larger but more meaningful statistical errors, reducing the risk of overconfident, misleading fits common in standard FSS.