Thermodynamic Casimir effect: Universality and corrections to scaling

We study the thermodynamic Casimir force for films in the three-dimensional Ising universality class with symmetry breaking boundary conditions. We focus on the effect of corrections to scaling and probe numerically the universality of our results. In particular, we check the hypothesis that corrections are well described by an effective thickness $L_{0,\mathrm{eff}}=L_0+c{(L_0+L_s)}^{1-\omega }+L_s$, where $c$ and $L_s$ are system specific parameters and $\omega \approx 0.8$ is the exponent of the leading bulk correction. We simulate the improved Blume-Capel model and the spin-1/2 Ising model on the simple cubic lattice. First, we analyze the behavior of various quantities at the critical point. Taking into account corrections $\propto L_0^{-\omega }$ in the case of the Ising model, we find good consistency of results obtained from these two different models. In particular, we get from the analysis of our data for the Ising model for the difference of Casimir amplitudes ${\Delta }_{+-}-{\Delta }_{++}=3.200\left(5\right)$, which nicely compares with ${\Delta }_{+-}-{\Delta }_{++}=3.208\left(5\right)$ obtained by studying the improved Blume-Capel model. Next, we study the behavior of the thermodynamic Casimir force for large values of the scaling variable $x=t{(L_0/{\xi }_0)}^{1/\nu }$. It can be obtained up to an overall amplitude by expressing the partition function of the film in terms of eigenvalues and eigenstates of the transfer matrix and boundary states. Here, we demonstrate how this amplitude can be computed with high accuracy. Finally, we discuss our results for the scaling functions ${\theta }_{+-}$ and ${\theta }_{++}$ of the thermodynamic Casimir force for the whole range of the scaling variable. We conclude that our numerical results are in accordance with universality. Corrections to scaling are well approximated by an effective thickness.

Figure 1

The amplitude $C\left(\xi \right)$ for the Ising and the Blume-Capel models at $D=0.655$ as a function of $1/\xi$.

Figure 2

We plot $\tilde{q}\left(x\right)x^{-\nu }$ as a function of the scaling variable $x$ in the range that is relevant for our problem. For the definition of $\tilde{q}\left(x\right)$ and a discussion, see the text.

Figure 3

We plot ${\theta }_{+-}$, $-{\theta }_{++}$ and the approximation (84). The data are taken for the Blume-Capel model at $D=0.655$ and the finite difference is computed from $L_0=32$ and 34.

Figure 4

We plot $-\Delta f{L}_{0,\mathrm{eff}}^3$ as a function of $t{(L_{0,\mathrm{eff}}/{\xi }_0)}^{1/\nu }$ for $++$ boundary conditions. The thick lines give the result obtained for the Blume-Capel model at $D=0.655$ and the two thicknesses $L_0=16.5$ and 33. In the case of the Blume-Capel model, we used $L_{0,\mathrm{eff}}=L_0+1.91$ as the effective thickness of the film. Our results for the Ising model are given by thin lines. In the case of the Ising model, we used the effective thicknesses $L_{0,\mathrm{eff}}=19.712$, $36.509$, and $69.936$ for $L_0=16.5$, 33, and 66, respectively. These effective thicknesses are chosen such that at the minima the curves fall on top of the one for the Blume-Capel model and $L_0=33$. At the resolution of the plot, all five curves fall on top of each other almost everywhere. Only for $20\stackrel{<}{{}_{\sim }}x\stackrel{<}{{}_{\sim }}40$, the curve for the Ising model and $L_0=16.5$ can be distinguished from the other four.

Figure 5

We plot $-\Delta f{L}_{0,\mathrm{eff}}^3$ as a function of $t{[L_{0,\mathrm{eff}}/{\xi }_0]}^{1/\nu }$ for $+-$ boundary conditions. The thick lines give the result obtained for the Blume-Capel model at $D=0.655$ and the two thicknesses $L_0=16.5$ and 33. In the case of the Blume-Capel model, we used $L_{0,\mathrm{eff}}=L_0+1.91$ as the effective thickness of the film. The data for $L_0=33$ are attached as Supplemental Material.[43] Our results for the Ising model are given by thin lines. In the case of the Ising model, we used the effective thicknesses $L_{0,\mathrm{eff}}=19.712$, $36.509$, and $69.936$, for $L_0=16.5$, 33, and 66, respectively. These values are taken from the analysis of $++$ boundary conditions above. At the resolution of the plot, all five curves fall on top of each other almost everywhere. Near the maximum, the curve for the Ising model and $L_0=16.5$ stays slightly below the other ones. For $x\stackrel{<}{{}_{\sim }}-30$ the curves slightly fork. Note that in this range, the difference between the Blume-Capel results for $L_0=16.5$ and 33 is of a similar size as the one between the Ising results for $L_0=16.5$ and 33 and between Blume-Capel and Ising models.