Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: General considerations and exact spherical-model results

We consider systems confined to a $d$-dimensional slab of macroscopic lateral extension and finite thickness $L$ that undergo a continuous bulk phase transition in the limit $L\to \infty$ and are describable by an $O\left(n\right)$ symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as $b{x}^{-(d+\sigma )}$ as $x\to \infty$, with $2<\sigma <4$ and $2<d+\sigma ⩽6$, on the Casimir effect at and near the bulk critical temperature $T_{c,\infty }$, for $2<d<4$. These interactions decay sufficiently fast to leave bulk critical exponents and other universal bulk quantities unchanged—i.e., they are irrelevant in the renormalization group (RG) sense. Yet they entail important modifications of the standard scaling behavior of the excess free energy and the Casimir force ${\mathcal{F}}_C$. We generalize the phenomenological scaling Ansätze for these quantities by incorporating these long-range interactions. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form $L^d{\mathcal{F}}_C∕k_{B}T\approx {\Xi }_0(L∕{\xi }_{\infty })+g_{\omega }{L}^{-\omega }{\Xi }_{\omega }(L∕{\xi }_{\infty })+g_{\sigma }{L}^{-{\omega }_{\sigma }}{\Xi }_{\sigma }(L∕{\xi }_{\infty })$. Here ${\Xi }_0$, ${\Xi }_{\omega }$, and ${\Xi }_{\sigma }$ are universal scaling functions; $g_{\omega }$ and $g_{\sigma }$ are scaling fields associated with the leading corrections to scaling and those of the long-range interaction, respectively; $\omega$ and ${\omega }_{\sigma }=\sigma +\eta -2$ are the associated correction-to-scaling exponents, where $\eta$ denotes the standard bulk correlation exponent of the system without long-range interactions; ${\xi }_{\infty }$ is the (second-moment) bulk correlation length (which itself involves corrections to scaling). The contribution $\propto g_{\sigma }$ decays for $T\ne T_{c,\infty }$ algebraically in $L$ rather than exponentially, and hence becomes dominant in an appropriate regime of temperatures and $L$. We derive exact results for spherical and Gaussian models which confirm these findings. In the case $d+\sigma =6$, which includes that of nonretarded van der Waals interactions in $d=3$ dimensions, the power laws of the corrections to scaling proportional to $b$ of the spherical model are found to get modified by logarithms. Using general RG ideas, we show that these logarithmic singularities originate from the degeneracy $\omega ={\omega }_{\sigma }=4-d$ that occurs for the spherical model when $d+\sigma =6$, in conjunction with the $b$ dependence of $g_{\omega }$.

Figure 1

Scaling function ${\mathsf{R}}_0({\check{r}}_{\infty },0)$ for $d=3$, as given by Eq. (4.72). The dotted line represents the asymptote ${\mathsf{R}}_{0,\mathrm{as}}({\check{r}}_{\infty },0)={\check{r}}_{\infty }$ that this function approaches for large values of ${\check{r}}_{\infty }$ in an exponential manner.

Figure 2

Scaling function ${\mathsf{R}}_{\omega }({\check{r}}_{\infty },0)$ for $d=3$ that one obtains by inserting Eq. (4.72) into (4.67).

Figure 3

Function ${\mathsf{R}}_{\sigma }({\check{r}}_{\infty },0;L)$ for $d=3$ and the indicated values of $L$, as obtained by insertion of Eq. (4.72) into (4.71).

Figure 4

Scaling function $Y_0({\check{r}}_{\infty },0)$ for $d=3$ (full line). The dashed line represents the asymptote (5.22). The value $Y_0(0,0)={\Delta }_C^{\mathrm{SM}}(d=3)$ which $Y_0({\check{r}}_{\infty },0)$ approaches as ${\check{r}}_{\infty }\to 0$ is known exactly: According to Ref. 82, it is given by ${\Delta }_C^{\mathrm{SM}}\left(3\right)=-2\zeta \left(3\right)∕\left(5\pi \right)=-0.15305\dots$.

Figure 5

Scaling function $Y_{\omega }({\check{r}}_{\infty },0)$ for $d=3$ (full line). The dashed line represents the asymptote (5.23).

Figure 6

Function $Y_{\sigma }({\check{r}}_{\infty },0;L)$ for $d=\sigma =3$ and the indicated values of $L$ (dashed-dotted and full lines). The dashed line represents the corresponding asymptote (5.24) with $d=\sigma =3$.

Figure 7

Scaled excess free-energy density (5.29) of the three-dimensional spherical model in an $L\times {\infty }^2$ slab with periodic boundary conditions for $L=50$, plotted versus the finite-size scaling variable $L∕{\xi }_{\infty }\equiv \sqrt{{\check{r}}_{\infty }}$ (a). The solid line corresponds to the case $\sigma =3$ of nonretarded van der Waals–type interactions with $g_{\sigma }\left(b\right)\equiv b=2∕3$; the dashed line shows results for the short-range case $b=0$ for comparison. In (b) the graphs displayed in (a) are plotted in a double-logarithmic manner. In this representation the asymptote (5.24) (dotted line) becomes a straight line with the slope $-2$. The nonuniversal constants $w_d$ and ${\tilde{w}}_d$ both have been set to unity.

Figure 8

Scaled excess free-energy densities (5.29) of the three-dimensional spherical model in an $L\times {\infty }^2$ slab with periodic boundary conditions and nonretarded van der Waals–type interactions $(\sigma =3)$. The results for various choices of $L$ including $L=\infty$ are shown as linear (a) and double-logarithmic plots (b). The asymptotes (dotted lines) correspond to the power-law behavior (5.24). The nonuniversal constants $w_d$ and ${\tilde{w}}_d$ both have been set to unity.

Figure 9

Scaled Casimir force (5.33) of the three-dimensional spherical model in an $L\times {\infty }^2$ slab with periodic boundary conditions for $L=50$, plotted versus the finite-size scaling variable $L∕{\xi }_{\infty }\equiv \sqrt{{\check{r}}_{\infty }}$ (a). The solid line corresponds to the case $\sigma =3$ of nonretarded van der Waals–type interactions with $g_{\sigma }\left(b\right)\equiv b=2∕3$; the dashed line represents results for the short-range case $b=0$ for comparison. In (b) the graphs displayed in (a) are plotted in a double-logarithmic manner. In this representation the asymptote (5.44) $\sim {\check{r}}_{\infty }^{-1}$ (dotted line) becomes a straight line with the slope $-2$. The nonuniversal constants $w_d$ and ${\tilde{w}}_d$ both have been set to unity.

Figure 10

Scaled Casimir force (5.33) of the three-dimensional spherical model in a $L\times {\infty }^2$ slab with periodic boundary conditions and nonretarded van-der-Waals-type interactions $(\sigma =3)$. The results for various choices of $L$ including $L=\infty$ are shown as linear (a) and double-logarithmic plots (b). The asymptotes (dotted lines) correspond to the power-law behavior $\sim {\check{r}}_{\infty }^{-1}$ of Eqs. (5.24, 5.44). The nonuniversal constants $w_d$ and ${\tilde{w}}_d$ both have been set to unity.

Figure 11

The functions $Q_{5,2}\left(y\right)$ (full line), $Q_{3,2}\left(y\right)$ (dashed), and $Q_{1,2}\left(y\right)∕10$ (dash-dotted), respectively.

Figure 12

The function $Q_{3,3}\left(y\right)$. The inset is a logarithmic-linear plot of this function, which illustrates the approach to the limiting value $-{\Delta }_{\sigma ,C}^{\mathrm{GM}}(3,3)=-{\pi }^2∕90$ implied by Eqs. (5.9, 5.3).