Finite-size scaling analysis

The finite-size scaling ansatz

Consider a system with some parameter \(\varrho\), which undergoes a phase transition at a critical value \(\varrho_c\). Divergences in the correlation length \(\xi\) and of fluctuations (e.g. the susceptibility \(\chi\)) characterize the critical point in infinite systems. Typically, in the critical region a diverging quantity \(A_\infty(\varrho)\) scales as \(|\varrho - \varrho_c|^{-\zeta}\) with some critical exponent \(\zeta\). In fact, this critical behavior should also hold in systems of finite size \(L\) at scales much larger than the characteristic length scale \(\xi\). Here and in the following, the characteristic length scale \(\xi\) is the correlation length in the infinite system (\(L \to \infty\)). As the correlation length diverges as \(\xi \sim |\varrho - \varrho_c|^{-\nu}\) for \(\varrho \to \varrho_c\), we have

\[A_L(\varrho) \sim |\varrho - \varrho_c|^{-\zeta} \sim \xi^{\zeta / \nu}, \qquad (L \gg \xi, \varrho \to \varrho_c).\]

For \(L \ll \xi\), the system size \(L\) takes over the role as the cutoff, such that we expect

\[A_L(\varrho) \sim L^{\zeta/\nu}, \qquad (L \ll \xi, \varrho \to \varrho_c).\]

These considerations constitute the finite-size scaling ansatz [NB99][BH10][FB72]

\[A_L(\varrho) = \xi^{\zeta/\nu} f(L / \xi), \qquad (L \to \infty, \varrho \to \varrho_c),\]


\[\begin{split}f(x) \begin{cases} = \text{const.} & \text{for } |x| \gg 1, \\ \sim x^{\zeta/\nu} & \text{for } x \to 0. \end{cases}\end{split}\]

The scaling function \(f(x)\) is a dimensionless function of the dimensionless ratio \(L/\xi\) of the finite system size and the infinite-system correlation length. This ratio controls the finite-size effects. The conventional scaling function is \(\tilde{f}(x) = x^{-\zeta} f(x^\nu)\) [NB99][BH10] such that

\[A_L(\varrho) = L^{\zeta/\nu} \tilde{f}\left(L^{1/\nu} (\varrho - \varrho_c)\right), \qquad (L \to \infty, \varrho \to \varrho_c),\]


\[\begin{split}\tilde{f}(x) \begin{cases} = \text{const.} & \text{for } x \to 0 \quad (L \ll \xi), \\ \sim L^{-\zeta/\nu} (\varrho - \varrho_c)^{-\zeta} & \text{for } |x| \gg 1 \quad (L \gg \xi). \end{cases}\end{split}\]

The data collapse method

A simulation experiment yields the quantity \(a_{L, \varrho}\) at system size \(L\) and parameter \(\varrho\), with standard error \(da_{L, \varrho}\). The scaling function is

\[\tilde{f}\left(L^{1/\nu} (\varrho - \varrho_c) \right) = L^{-\zeta/\nu} A_L(\varrho).\]

Thus, plotting \(L^{-\zeta/\nu} a_{L, \varrho}\) against \(L^{1/\nu}(\varrho-\varrho_c)\) should let the experimental data collapse onto the single curve \(\tilde{f}(x)\). For this to happen, the critical parameter value \(\varrho_c\) and the critical exponents \(\zeta, \nu\) need to be correct. These assumptions hold for \(L \to \infty\), with systematic errors at finite sizes [NB99][BH10].

The intersection method

At a first-order transition where the order parameter \(P\) remains constant on both sides of the transition, its critical exponent is \(\beta = 0\).

So we have

\[P_L(\varrho) = \tilde{P}\left(L^{1/\nu}(\varrho-\varrho_c)\right),\]

and hence, \(P_L(\varrho_c) = \tilde{P}(0)\) independent of the system size \(L\). Thus, the common intersection point of the measured curves \(p_{L, \varrho}\) yields an estimate of the threshold \(\varrho_c\). This estimate is unbiased with regards to the critical exponents, and “should be free” from systematic errors due to finite system size [BH10].

Implementation in the fssa package


fssa.fssa.scaledata Scale experimental data according to critical exponents


fssa.fssa.ScaledData A namedtuple for scaledata()