Figure 1

(Color online) Bond arrangement for the computation of the free-energy difference in Eq. (8) (see main text). The crossover Hamiltonian $H_{\text{cr}}$ (c) belongs to a system which interpolates between those described by the Hamiltonians (a) $H_0$ and (b) $H_1$.Reuse & Permissions

Figure 2

(Color online) Monte Carlo data for $g\mathbf{\left(}y=\tau {\left(L-\frac{1}{2}\right)}^{1/\nu };L,2L,A={\left(L/\rho \right)}^2\mathbf{\right)}$ [see Eq. (12), $\tau =\left(T-T_c\right)/T_c$] in the three-dimensional $XY$ model for $L=10$, 15, 20, and fixed inverse aspect ratio ${\rho }^{-1}=6$. In (a) and (b) we present the result for $\left(O,O\right)$ and periodic BC, respectively. For $T\ge T_c$, i.e., $y=y_{+}>0$ one has $y_{+}={\left[\left(L-\frac{1}{2}\right){\xi }_0^{+}/{\xi }_{+}\right]}^{1/\nu }$. For the $XY$ model ${\xi }_{-}=\infty$ for all temperatures $T\le T_c$. The Kosterlitz-Thouless transition of the two-dimensional film occurs at $y=y_{c,OO}=-2.69\left(3\right)$ [41] and $y=y_{c,P}=-0.996\left(1\right)$ [42] in (a) and (b), respectively.Reuse & Permissions

Figure 3

(Color online) Monte Carlo data for $g\mathbf{\left(}y=\tau {\left(L-\frac{1}{2}\right)}^{1/\nu };L,2L,A={\left(L/\rho \right)}^2\mathbf{\right)}$ [see Eq. (12), $\tau =\left(T-T_c\right)/T_c$] within the three-dimensional $XY$ model with periodic BC for $L=10$ and different values of the inverse aspect ratio ${\rho }^{-1}$. For $y\ge 0$ one has $y=y_{+}={\left[\left(L-\frac{1}{2}\right){\xi }_0^{+}/{\xi }_{+}\right]}^{1/\nu }$. For the $XY$ model ${\xi }_{-}=\infty$ for all temperatures $T\le T_c$. The Kosterlitz-Thouless transition of the two-dimensional film occurs at $y=y_{c,P}=-0.996\left(1\right)$ [42]. Note the enlarged scales as compared with Fig. 2b.Reuse & Permissions

Figure 4

(Color online) Scaling function ${\vartheta }_{OO}$ of the Casimir force for the three-dimensional $XY$ model with $\left(O,O\right)$ BC. The MC data reported in this figure refer to lattices with $L=10$, 15, and 20, with fixed inverse aspect ratio $1/\rho =6$. Corrections to scaling have been accounted for according to two different ansätze, provided by Eqs. (20, 21); the corresponding numerical results are denoted by (i) and (ii), respectively. With corrections to scaling of the form (ii), the shape of the resulting scaling function is almost indistinguishable from the one obtained with corrections to scaling of the form (i), but its overall amplitude is reduced by a factor $R\simeq 0.9$. For (i) our MC data compare very well with the corresponding experimental data from Ref. [5] (solid line) and with the MC data of Ref. [17] (dashed). Due to the Goldstone modes ${\vartheta }_{OO}\left(x\to -\infty \right)=\text{const}\ne 0$. The dash-dotted line shows the mean-field scaling function [30, 31] normalized to the depth of the minimum of the MC data (i). The levelling off of the experimental data [5] for $x\to -\infty$ contains a component which is specific for ${}^4\text{H}\text{e}$ wetting films [29] and cannot be captured by an $XY$ lattice model. The gray bar indicates the position and uncertainty of the universal value $x_{O,O}^{\ast }=-7.64\left(15\right)$ of the scaling variable $x$ corresponding to the occurrence of the Kosterlitz-Thouless transition in the film, as inferred from MC simulations of lattice models in the $XY$ universality class presented in Ref. [41].Reuse & Permissions

Figure 5

(Color online) Critical Casimir amplitude ${\Delta }_P\left(L\right)$ for the three-dimensional $XY$ bulk universality class and periodic BC, estimated from lattices of several thicknesses $L$ and inverse aspect ratio $1/\rho =6$. Due to corrections to scaling, ${\Delta }_P$ depends on $L$. The solid line represents the best fit to the numerical database on Eq. (24) and allows one to extrapolate the value of ${\Delta }_P\left(L\right)$ to the scaling limit $L\to \infty$, resulting in ${\Delta }_P=-0.2993\left(7\right)$ (◼). With ● we indicate the numerical estimate ${\Delta }_P=-0.28$ provided in Ref. [16].Reuse & Permissions

Figure 6

(Color online) Scaling function ${\vartheta }_P$ of the Casimir force for the three-dimensional $XY$ model with periodic BC. The corrections to scaling are taken into account by Eq. (22) [case (iii)] and Eq. (18) with $r_1=0$. The shape of our MC data compares very well with the corresponding MC data (●) of Ref. [16]. For a discussion of the relative shift of the data sets see the main text. The solid line corresponds to the analytical prediction in Ref. [21]. Due to the Goldstone modes, in agreement with Eq. (25), ${\vartheta }_P\left(x\to -\infty \right)=-0.383\left(4\right)$, see horizontal dashed line. Contrary to $\left(O,O\right)$ BC in Fig. 4, for periodic BC MFT yields ${\vartheta }_P^{\left(\text{MFT}\right)}\left(x\right)\equiv 0$ for $x\lessgtr 0$. The gray vertical line indicates the position of the universal value $x_P^{\ast }=-2.82\left(2\right)$ of the scaling variable $x$ corresponding to the occurrence of the Kosterlitz-Thouless transition in the film, as inferred from MC simulations [42].Reuse & Permissions

Figure 7

(Color online) Plot of $g\mathbf{\left(}y;L,2L,A={\left(L/\rho \right)}^2\mathbf{\right)}$ [see Eq. (12)] for the three-dimensional Ising model with $L=10$ and $1/\rho =6$, 10, and 14. (a) and (b) refer to $\left(++\right)$ and $\left(+-\right)$ BC, respectively, and the coincidence of data points corresponding to different values of $\rho$ demonstrates that the geometry of the lattice does not affect the resulting finite-size critical behavior in the region $-4\lesssim y\lesssim 10$.Reuse & Permissions

Figure 8

(Color online) MC data for the critical Casimir amplitude $\Delta \left(L\right)$ of the three-dimensional Ising model with (a) $\left(++\right)$ and (b) $\left(+-\right)$ BC, as a function of the inverse lattice size $L$ (for lattices with fixed inverse aspect ratio $1/\rho =6$). $L$-dependent corrections to scaling give rise to the dependence $\Delta \left(L\right)$ such that ${\Delta }_{++/+-}\equiv {\Delta }_{++/+-}\left(L\to \infty \right)$. The solid line corresponds to the best fit obtained by using the fitting ansatz in Eq. (26) in the interval $0<1/L\le 0.1$. For comparison we present also the best fits using the ansätze (i) $\Delta \left(L\right)=\Delta {\left(1+g_1{L}^{-1}\right)}^{-1}$ and (ii) $\Delta \left(L\right)=\Delta \left(1+g_2{L}^{-1}\right)$. Our estimates (◼) for the asymptotic values of the Casimir amplitudes compare reasonably well with previous MC results (●) from Ref. [24] and with results (◆) obtained from the de Gennes–Fisher local-functional [18].Reuse & Permissions

Figure 9

(Color online) Scaling function ${\vartheta }_{++}$ of the critical Casimir force in the three-dimensional Ising model with $\left(++\right)$ BC and zero bulk field. Data points refer to lattices with fixed inverse aspect ratio $1/\rho =6$. The bottom and top data sets have been obtained by accounting for corrections to scaling according to Eq. (20) [case (i)] and Eq. (21) [case (ii)], respectively. The intermediate data set, instead, considers corrections of the rational form given by Eq. (23) [case (iv)]. In each case the data collapse turns out to be very good within the range of the scaling variable $x$ covered in the figure. The final estimate of the scaling function is biased by the functional form assumed for the corrections to scaling. The position $x_{\text{min}}\simeq 5.90\left(8\right)$ of the minimum is insensitive with respect to these choices for the form of the corrections. For comparison we provide the prediction of mean-field theory [24] (solid line), normalized such that ${\vartheta }_{++}^{\left(\text{MFT}\right)}\left(0\right)={\vartheta }_{++}^{\left(\text{MC}\right)}\left(0\right)$ [Fig. 8a], the exact result for the two-dimensional Ising model [23] (dashed line), and the result from the extended de Gennes–Fisher local-functional method [19] (dash-dotted line). Note that the actual phase transition of the film occurs at a nonzero value of the bulk field.Reuse & Permissions

Figure 10

(Color online) Scaling function ${\vartheta }_{+-}$ of the critical Casimir force in the three-dimensional Ising model with $\left(+-\right)$ BC and zero bulk field. Data points refer to lattices with fixed inverse aspect ratio $1/\rho =6$. For comparison we provide the mean-field prediction [24] (solid line), normalized such that ${\vartheta }_{+-}^{\left(\text{MFT}\right)}\left(0\right)={\vartheta }_{+-}^{\left(\text{MC}\right)}\left(0\right)$ [$=2{\Delta }_{+-}$, Fig. 8b], the exact result for the two-dimensional Ising model [23] (dashed line), and the set of experimental data points from Ref. [8]. The top and bottom data sets have been obtained by accounting for corrections to scaling according to Eq. (20) [case (i)] and Eq. (21) [case (ii)], respectively. The intermediate data set, instead, considers corrections of the rational form given by Eq. (23) [case (iv)]. In each case the data collapse turns out to be very good for $x\ge -20$. The final estimate of the scaling function is biased by the functional form assumed for the corrections to scaling. The position $x_{\text{max}}\simeq -5.4\left(1\right)$ of the maximum is insensitive with respect to these choices for the form of the corrections. In spite of this caveat the comparison with the experimental data is encouraging. Note that the actual phase transition of the film occurs at a nonzero value of the bulk field.Reuse & Permissions

Figure 11

(Color online) Monte Carlo data for $g_{OO}\left[y=\tau {\left(L-\frac{1}{2}\right)}^{1/\nu };L,2L,A={\left(L/\rho \right)}^2\right]$ [see Eq. (12), $\tau =\left(T-T_c\right)/T_c$] in the three-dimensional Ising model with $\left(O,O\right)$ BC for $L=8,12,16,20$, and for a fixed aspect ratio $\rho =1/6$. The 2D critical point of the film is located at $y=y_{c,OO}=-2.5\left(5\right)$, as inferred from extrapolating the data in Table II of Ref. [50] to $L\to \infty$.Reuse & Permissions

Figure 12

(Color online) Monte Carlo data for $g_{00}\mathbf{\left(}y=\tau {\left(L-\frac{1}{2}\right)}^{1/\nu };L,2L,A={\left(L/\rho \right)}^2\mathbf{\right)}$ [see Eq. (12), $\tau =\left(T-T_c\right)/T_c$] in the three-dimensional Ising model with $\left(O,O\right)$ BC for $L=12$ and various values of the inverse aspect ratio $1/\rho =\sqrt{A}/L$.Reuse & Permissions

Figure 13

(Color online) Scaling function ${\vartheta }_{OO}$ of the Casimir force for the three-dimensional Ising model with $\left(O,O\right)$ BC and zero bulk field. The MC data refer to lattices with $L=8,12,16,20$ and with a fixed inverse aspect ratio $1/\rho =6$. Corrections to scaling have been accounted for according to two different ansätze, provided by Eqs. (20, 21), and the corresponding numerical results are denoted by (i) and (ii), respectively. With corrections of the form (ii), the shape of the resulting scaling function is almost indistinguishable from the one obtained with corrections of the form (i), but its overall amplitude is reduced by a factor $R\simeq 0.866$. For comparison we show the exact result for the 2D Ising model [23] (dashed line) and the mean-field prediction [30] (dash-dotted line) normalized such that it yields the same depth of the minimum as the one of the MC data (i). In the inset we compare the MC data corresponding to the case (i) with the scaling function obtained from the $ϵ$ expansion [7]. The gray bar indicates the value $x_{OO}^{\ast }=-7.6\left(1.3\right)$ (and its uncertainty) of the scaling variable $x$ corresponding to the occurrence of the shifted critical point, inferred from extrapolating the data in Table II of Ref. [50] to $L\to \infty$.Reuse & Permissions

Figure 14

(Color online) MC data for the critical Casimir amplitude ${\Delta }_P\left(L\right)$ for the three-dimensional Ising model with periodic BC, as a function of the inverse film thickness $L$ (on lattices with fixed inverse aspect ratio $1/\rho =6$). Due to $L$-dependent corrections to scaling, ${\Delta }_P$ depends on $L$ and reaches its asymptotic value in the limit $L\to \infty$. The solid line corresponds to the best fit obtained by using the fitting ansatz given in Eq. (22) in the interval $0<1/L\le 0.25$. Our estimate (◼) for the asymptotic value of the Casimir amplitude ${\Delta }_P\left(\infty \right)$ compares very well with the previous MC result (●) from Ref. [24].Reuse & Permissions

Figure 15

(Color online) Scaling function ${\vartheta }_P\left(x\right)$ of the critical Casimir force in the three-dimensional Ising model with periodic BC and zero bulk field. The data points refer to lattices with the inverse aspect ratio $1/\rho =6$. The corrections to scaling are taken into account according to Eqs. (19, 22) [case (iii)], see the main text. For comparison we show also the data set corresponding to the lattice with thickness $L=20$ as investigated in Ref. [16], the analytical prediction of Ref. [21] (dash-dotted line) for $x\ge 0$, and results for 2D Ising model (dashed line) that we have obtained numerically by using the transfer matrix method. Due to the self-duality of the 2D Ising model one has ${\vartheta }_P\left(-x\right)={\vartheta }_P\left(x\right)$ for $d=2$ which allows for the occurrence of two symmetric minima [22]. We note that MFT yields ${\vartheta }_P\left(x\right)\equiv 0$ (solid line). The gray vertical line indicates the universal value $x_P^{\ast }=-1.60\left(2\right)$ of the scaling variable $x$ corresponding to the occurrence of the shifted critical point, inferred from extrapolating the data in Table I of Ref. [50] to $L\to \infty$.Reuse & Permissions

Figure 16

(Color online) Aspect-ratio dependence of the function $g_P\mathbf{\left(}y=\tau {\left(L-\frac{1}{2}\right)}^{1/\nu };L,2L,A={\left(L/\rho \right)}^2\mathbf{\right)}$ [see Eq. (12)] for the three-dimensional Ising model with periodic BC. For a fixed value of $\rho$, this function depends only weakly on $L$. By changing $\rho$, the function $g_P$ is affected mainly in the region $-0.6\lesssim y\lesssim 0$. The 2D critical point of the film is located at $y=y_{c,P}=-0.52\left(2\right)$, as inferred from extrapolating the data in Table I of Ref. [50] to $L\to \infty$.Reuse & Permissions

Figure 17

(Color online) Plot of the function $g_P\mathbf{\left(}y;L,2L,A={\left(L/\rho \right)}^2\right)$ [see Eq. (12)] and the associated scaling function $\widehat{\theta }\left(y\right)$ [i.e., the lattice estimate of $\theta \left(y\right)\equiv {\vartheta }_P\left(y/{\left({\xi }_0^{+}\right)}^{1/\nu }\right)$, which is calculated by solving Eq. (12)] iteratively for the three-dimensional Ising model with periodic BC. The data points refer to a lattice with $L=10$ and $1/\rho =6$. The data have not been corrected for the corrections to scaling. The pronounced shoulder originally present in $g_P$ is smoothed out upon calculating the associated scaling function.Reuse & Permissions