Critical free energy and Casimir forces in rectangular geometries

We study the critical behavior of the free energy and the thermodynamic Casimir force in a $L_{\parallel }^{d-1}\times L$ block geometry in $2<d<4$ dimensions with aspect ratio $\rho =L/L_{\parallel }$ on the basis of the $O\left(n\right)$ symmetric ${\varphi }^4$ lattice model with periodic boundary conditions and with isotropic short-range interactions. Exact results are derived in the large-$n$ limit describing the geometric crossover from film ($\rho =0$) over cubic ($\rho =1$) to cylindrical ($\rho =\infty$) geometries. For $n=1$, three perturbation approaches in the minimal renormalization scheme at fixed $d$ are presented that cover both the central finite-size regime near $T_c$ for $1/4\lesssim \rho \lesssim 3$ and the region well above and below $T_c$. At bulk $T_c$, we predict the critical Casimir force in the vertical $\left(L\right)$ direction to be negative (attractive) for a slab ($\rho <1$), positive (repulsive) for a rod ($\rho >1$), and zero for a cube $(\rho =1)$. Our results for finite-size scaling functions agree well with Monte Carlo data for the three-dimensional Ising model by Hasenbusch for $\rho =1$ and by Vasilyev et al. for $\rho =1/6$ above, at, and below $T_c$.

Figure 1

Three-dimensional $L_{\parallel }^2\times L$ block geometries with aspect ratio $\rho =L/L_{\parallel }$: (a) slab ($\rho <1$), (b) rod ($\rho >1$). Arrows: critical Casimir force (2.15). Our theory (in Secs. 3, 4, 5) predicts that, for isotropic systems with periodic BC and short-range interactions, $F_{\text{Casimir},s}$ at bulk $T_c$ is negative (attractive) for a slab, positive (repulsive) for a rod, and zero for a cube $(\rho =1)$.

Figure 2

Schematic plot of the asymptotic part of the $L^{-1/\nu }-t$ plane, with $t\equiv (T-T_c)/T_c$, for the ${\varphi }^4$ model with $n=1$ and with isotropic interactions on a simple-cubic lattice with lattice spacing $\tilde{a}$ in a $L_{\parallel }^{d-1}\times L$ block geometry with periodic boundary conditions and finite aspect ratio $\rho =L/L_{\parallel }$. In the central finite-size region (above the dashed lines), the lowest mode must be separated, whereas, outside this region (below the dashed lines), ordinary perturbation theory is applicable. Above the shaded region, universal finite-size scaling is valid for isotropic systems. In the large-$L$ regime at $t\ne 0$ (shaded region), the exponential form of the size dependence violates both finite-size scaling and universality because of a non-negligible dependence on the lattice spacing $\tilde{a}$. An analogous plot applies to the $L_{\parallel }^{-1/\nu }-t$ plane.

Figure 3

Scaling functions (a) $F^{ex}(\tilde{x},\rho )$ [Eqs. (2.27) and (3.6)–(3.8)] and (b) $X(\tilde{x},\rho )$ [Eq. (2.29)] as a function of $\tilde{x}=t{(L/{\xi }_{0+})}^{1/\nu }$ in the large-$n$ limit in three dimensions for film geometry ($\rho =0$, dotted lines), for slab geometry (solid lines: $\rho =1/4$, dashed lines: $\rho =1/2$, dotted-dashed lines: $\rho =3/4$), and for cubic geometry ($\rho =1$, double-dotted-dashed lines). For $\tilde{x}\to -\infty$, the curves in (a) with $\rho >0$ diverge logarithmically toward $-\infty$, whereas the $\rho =0$ curve in (a) has a finite low-temperature limit $-0.191$. All curves in (b) have finite limits for $\tilde{x}\to -\infty$ as given by (3.21) and (3.23).

Figure 4

Scaling functions (a) ${\Phi }^{ex}({\tilde{x}}_{\parallel },\rho )$ [Eq. (2.40)] and (b) $\Xi ({\tilde{x}}_{\parallel },\rho )$ [Eq. (2.41)] as a function of ${\tilde{x}}_{\parallel }=t{(L_{\parallel }/{\xi }_{0+})}^{1/\nu }$ in the large-$n$ limit in three dimensions for cylinder geometry ($1/\rho =0$, dotted lines), for rod geometry (solid lines: $1/\rho =1/4$, dashed lines: $1/\rho =1/2$), and for cubic geometry ($1/\rho =1$, double-dotted-dashed lines). For ${\tilde{x}}_{\parallel }\to -\infty$, the curves in (a) with $1/\rho >0$ diverge logarithmically toward $-\infty$, whereas the $1/\rho =0$ curve in (a) has a finite low-temperature limit $-0.719$. All curves in (b) have finite limits for ${\tilde{x}}_{\parallel }\to -\infty$ as given by (3.22) and (3.24).

Figure 5

Critical amplitudes (a) $F(0,\rho )$ [Eq. (2.27)] and (b) $X(0,\rho )$ [Eq. (5.10)] at $T_c$ in three dimensions as a function of the aspect ratio $\rho$ in the large-$n$ limit [thin lines, from (3.6)] and for $n=1$ [thick lines, from (5.1)]. For $n=\infty$, $F_{\text{film}}\left(0\right)\equiv F(0,0)=-0.153$ and $X_{\text{film}}\left(0\right)\equiv X(0,0)=2F_{\text{film}}\left(0\right)=-0.306$. At $\rho =1$, $X(0,1)$ vanishes for both $n=\infty$ and 1. Monte Carlo data of the $d=3$ Ising model for $\rho =1$ by Mon [41] [full circle in (a)] and for $\rho =0$ by Vasilyev et al. [16] [triangle in (b)]. See also Fig. 7.

Figure 6

Critical amplitudes (a) $\Phi (0,\rho )$ [Eq. (2.39)] and (b) $\Xi (0,\rho )$ [Eq. (5.11)] at $T_c$ in three dimensions as a function of the inverse aspect ratio $1/\rho$ in the large-$n$ limit [thin lines, from (3.6)] and for $n=1$ [thick lines, from (5.1)]. For $n=\infty$, ${\Phi }_{\mathrm{cyl}}\left(0\right)\equiv \Phi (0,\infty )=-0.329=-{\Xi }_{\mathrm{cyl}}\left(0\right)\equiv -\Xi (0,\infty )$. At $1/\rho =1$, $\Xi (0,1)$ vanishes for both $n=\infty$ and 1. Monte Carlo data of the $d=3$ Ising model for $\rho =1$ by Mon [41] [full circle in (a)]. See also Fig. 8.

Figure 7

Critical amplitudes (a) $F(0,\rho )$ [Eq. (2.27)] and (b) $X(0,\rho )$ [Eq. (5.10)] at $T_c$ in three dimensions as a function of the aspect ratio $\rho$ for $n=1$ [thick lines, from (5.1)] and in the large-$n$ limit [thin lines, from (3.6)]. The maximum $-0.1636$ of the $n=1$ line in (a) is at ${\rho }_{\max }=0.2470$. The dashed lines are the extrapolations of the $n=1$ lines from $\rho ={\rho }_{\max }$ to $\rho =0$ corresponding to film geometry. The dotted lines represent (5.1) in the regime $\rho <{\rho }_{\max }$, where our perturbation theory is not applicable. The MC estimate for the $d=3$ Ising model from [16] (triangles), $\varepsilon$ expansion results for $n=1$ from [19] (squares), and from [20] (diamonds). See also Fig. 5.

Figure 8

Critical amplitudes (a) $\Phi (0,\rho )$ [Eq. (2.39)] and (b) $\Xi (0,\rho )$ [Eq. (5.11)] at $T_c$ in three dimensions as a function of the inverse aspect ratio $1/\rho$ for $n=1$ [thick lines, from (5.1)] and in the large-$n$ limit [thin lines, from (3.6)]. The maximum $-0.3658$ of the $n=1$ line in (a) is at ${(1/\rho )}_{\max }=0.3223$. The dashed lines are the extrapolations of the $n=1$ lines from ${(1/\rho )}_{\max }$ to $1/\rho =0$ corresponding to cylinder geometry. The dotted lines represent (5.1) in the regime of small $1/\rho <{(1/\rho )}_{\max }$, where our perturbation theory is not applicable. See also Fig. 6.

Figure 9

Scaling function (a) $F^{ex}(\tilde{x},\rho )$ [Eqs. (2.27) and (4.55)] and (b) $X(\tilde{x},\rho )$ [Eqs. (2.29), (2.27), and (4.55)] as a function of $\tilde{x}=t{(L/{\xi }_{0+})}^{1/\nu }$ for $n=1$ in three dimensions for slab geometry with finite aspect ratio $\rho =1/4$ (solid lines), $\rho =1/3$ (dotted lines), $\rho =1/2$ (dashed lines), $\rho =2/3$ (dotted-dashed line), $\rho =1$ (double-dotted-dashed line). Thin lines in (a): $\varepsilon$ expansion results for $\rho =0$ from [19, 21].

Figure 10

(a) Scaling function (a) ${\Phi }^{ex}({\tilde{x}}_{\parallel },\rho )$ [Eqs. (2.40), (2.27), and (4.55)] and (b) $\Xi ({\tilde{x}}_{\parallel },\rho )$ [Eq. (2.41)] as a function of ${\tilde{x}}_{\parallel }=t{(L_{\parallel }/{\xi }_{0+})}^{1/\nu }$ for $n=1$ in three dimensions for rod geometry with finite aspect ratio $1/\rho =1$ (double-dotted-dashed lines), $1/\rho =2/3$ (dotted lines), $1/\rho =1/2$ (dashed lines), and $1/\rho =2/5$ (solid lines).

Figure 11

Scaling function $F^{ex}(\tilde{x},1)$ [Eqs. (2.27) and (4.55)] for $n=1$ as a function of $\tilde{x}=t{(L/{\xi }_{0+})}^{1/\nu }$ in three dimensions for cubic geometry (solid line) and MC data for the $d=3$ Ising model by Hasenbusch [36]. See also Fig. 13c.

Figure 12

Scaling function $X(\tilde{x},\rho )$ [Eqs (2.29), (2.27), and (4.55)] for $n=1$ as a function of $\tilde{x}=t{(L/{\xi }_{0+})}^{1/\nu }$ in three dimensions for slab geometry with $\rho =1/4$ (solid line) and MC data for the $d=3$ Ising model with $\rho =1/6$ by Vasilyev et al. [16]. Thin lines: $\varepsilon$ expansion results for $\rho =0$ from [19, 21]. See also Fig. 14a.

Figure 13

Scaling functions $F^{ex}(\tilde{x},\rho )$ (a)–(c) and ${\Phi }^{ex}({\tilde{x}}_{\parallel },\rho )$ (d) as a function of $\tilde{x}=t{(L/{\xi }_{0+})}^{1/\nu }$ and ${\tilde{x}}_{\parallel }=t{(L_{\parallel }/{\xi }_{0+})}^{1/\nu }$, respectively, for $n=1$ in three dimensions for several values of the aspect ratio $\rho$. Thick lines: improved perturbation theory according to (4.55) and (2.40). Thin lines: one-loop perturbation theory according to (6.2) and (6.10). The thin lines diverge for $t\to 0$. The asymptotic value of the thin lines is $-{\rho }^2\ln 2$ for $\tilde{x}\to -\infty$ in (a)–(c) and $-(1/2)\ln 2$ for ${\tilde{x}}_{\parallel }\to -\infty$ in (d). MC data in (c) for the $d=3$ Ising model by Hasenbusch [36].

Figure 14

Scaling functions $X(\tilde{x},\rho )$ (a)–(c) and $\Xi ({\tilde{x}}_{\parallel },\rho )$ (d) as a function of $\tilde{x}=t{(L/{\xi }_{0+})}^{1/\nu }$ and ${\tilde{x}}_{\parallel }=t{(L_{\parallel }/{\xi }_{0+})}^{1/\nu }$, respectively, for $n=1$ in three dimensions for several values of the aspect ratio $\rho$. Thick lines: improved perturbation theory according to (4.55), (2.29), and (2.41). Thin lines: one-loop perturbation theory according to (6.2) and (6.10). The thin lines diverge for $t\to 0$. MC data in (a) for the $d=3$ Ising model with $\rho =1/6$ by Vasilyev et al. [16].