Manipulation and amplification of the Casimir force through surface fields using helicity

We present both exact and numerical results for the behavior of the Casimir force in $O\left(n\right)$ systems with a finite extension $L$ in one direction when the system is subjected to surface fields that induce helicity in the order parameter. We show that for such systems, the Casimir force in certain temperature ranges is of the order of $L^{-2}$, both above and below the critical temperature, $T_c$, of the bulk system. An example of such a system would be one with chemically modulated bounding surfaces, in which the modulation couples directly to the system's order parameter. We demonstrate that, depending on the parameters of the system, the Casimir force can be either attractive or repulsive. The exact calculations presented are for the one-dimensional $XY$ and Heisenberg models under twisted boundary conditions resulting from finite surface fields that differ in direction by a specified angle, and the three-dimensional Gaussian model with surface fields in the form of plane waves that are shifted in phase with respect to each other. Additionally, we present exact and numerical results for the mean-field version of the three-dimensional $O\left(2\right)$ model with finite surface fields on the bounding surfaces. We find that all significant results are consistent with the expectations of finite-size scaling.

Figure 1

The scaling function $X_{\mathrm{Cas}}$ of the $XY$ model as a function of the scaling variable $x$ [see Eq. (2.12)] for different values of the phase change $\psi$.

Figure 2

The surface of the scaling function $X_{\mathrm{Cas}}(\psi ,x)$ of the $XY$ model as a function of the scaling variables $x$ and $\psi$. The horizontal plane marks the $X_{\mathrm{Cas}}=0$ value.

Figure 3

The scaling function $X_{\mathrm{Cas}}$ of the Heisenberg model as a function of the scaling variable $x$ [see Eq. (2.21)] for different values of the phase change $\psi$.

Figure 4

The surface of the scaling function $X_{\mathrm{Cas}}(\psi ,x)$ of the Heisenberg model as a function of the scaling variables $x$ and $\psi$. The horizontal plane marks the $X_{\mathrm{Cas}}=0$ value.

Figure 5

The scaling function $X_{\mathrm{Cas}}^{(0,\perp )}\left(x_t\right)$ as a function of the temperature-dependent scaling variable $x_t$. The horizontal line marks the Casimir amplitude $X_{\mathrm{Cas}}^{(0,\perp )}\left(0\right)=-\zeta \left(3\right)/\left(8\pi \right)$.

Figure 6

The scaling function $X_{\mathrm{Cas}}^{(h,\perp )}(w,x_1,x_L)$ [see Eq. (3.34)] as a function of $w\in (0,10]$ and $\left(\mathbf{k}·\mathrm{\Delta }\right)\in [0,2\pi ]$ for $x_1=x_L=1$. As we see, $X_{\mathrm{Cas}}^{(h,\perp )}(w,x_1,x_L)$ can be both positive and negative, depending on the values of its arguments.

Figure 7

The scaling function $X_{\mathrm{Cas}}^{(h,\perp )}(w,x_1,x_L)$ [see Eq. (3.35)] as a function of $w\in (0,10]$ and ${\mathrm{\Delta }}_x+{\mathrm{\Delta }}_y\in [0,1]$ for $x_1=x_L=1$. As we see, also in this case $X_{\mathrm{Cas}}^{(h,\perp )}(w,x_1,x_L)$ can be both positive and negative, depending on the values of its arguments. Let us recall that in this subcase $w=x_t/2$.

Figure 8

The scaling function $X_{\mathrm{Cas}}^{(\perp )}(x_t,x_k,x_1,x_L)$ as a function of $x_t\in (0,10]$ and $\mathbf{k}·\mathrm{\Delta }\in [0,2\pi ]$ for $x_k=0.1,x_1=x_L=1$. As we see, $X_{\mathrm{Cas}}^{(\perp )}$ can be both positive and negative, depending on the values of its arguments.

Figure 9

The scaling function $X_{\mathrm{Cas}}^{(\perp )}(x_t,x_k,x_1,x_L)$ as a function of $x_t\in (0,10]$ and $\mathbf{k}·\mathrm{\Delta }\in [0,2\pi ]$ for $x_k=0.1,x_1=10x_L=1$. As we see, the scaling function in that case is predominantly positive.

Figure 10

The scaling function $X_{\mathrm{Cas}}^{(\perp )}(x_t,x_k,x_1,x_L)$ as a function of $x_t\in (0,10]$ and $\mathbf{k}·\mathrm{\Delta }\in [0,2\pi ]$ for $x_k=0.1,x_1=-x_L=1$. As we see, the scaling function in that case can be both positive and negative, depending on the values of its arguments.

Figure 11

The scaling function $X_{\mathrm{Cas}}^{(\perp )}(x_t,x_k=0,x_1,x_L)$ as a function of $x_t\in (0,10]$ and ${\mathrm{\Delta }}_x+{\mathrm{\Delta }}_y\in [0,1]$ for $x_1=x_L=1$. As we see, $X_{\mathrm{Cas}}^{(\perp )}$ can be both positive and negative, depending on the values of its arguments.

Figure 12

The scaling function $X_{\mathrm{Cas}}^{(\perp )}(x_t,x_k=0,x_1,x_L)$ as a function of $x_t\in (0,10]$ and ${\mathrm{\Delta }}_x+{\mathrm{\Delta }}_y\in [0,1]$ for $x_1=10x_L=1$. As we see, the scaling function in that case is predominantly positive.

Figure 13

The scaling function $X_{\mathrm{Cas}}^{(\perp )}(x_t,x_k=0,x_1,x_L)$ as a function of $x_t\in (0,10]$ and ${\mathrm{\Delta }}_x+{\mathrm{\Delta }}_y\in [0,1]$ for $10x_1=x_L=1$. As we see, the scaling function in that case can be both positive and negative.

Figure 14

The scaling function $X_{\mathrm{Cas}}^{(\perp )}(x_t,x_k=0,x_1,x_L)$ as a function of $x_t\in (0,10]$ and ${\mathrm{\Delta }}_x+{\mathrm{\Delta }}_y\in [0,1]$ for $x_1=-x_L=1$ or $x_1=-x_L=-1$. As we see, the scaling function in that case can be both positive and negative.

Figure 15

The scaling function $X_{\mathrm{Cas}}^{(h,\alpha )}(w,x_1,x_L)$ [see Eq. (3.44)] as a function of $w\in (0,3]$ and $\left(\mathbf{k}·\mathrm{\Delta }\right)\in [0,2\pi ]$ for $x_1=x_L=1$.

Figure 16

The scaling function $X_{\mathrm{Cas}}^{(h,\alpha )}(x_t,x_1,x_L)$ [see Eq. (3.45)] as a function of $w\in (0,3]$ and ${\mathrm{\Delta }}_x+{\mathrm{\Delta }}_y\in [0,1]$ for $x_1=x_L=1$. Let us recall that in this subcase, $w=x_t/2$.

Figure 17

The scaling function $X_{\mathrm{Cas}}^{\left(\alpha \right)}\left(x_t\right)$ of the $XY$ model under twisted boundary conditions as a function of $x_t$ and $\alpha$ for $h=0$. The plane surface marks the $X_{\mathrm{Cas}}^{\left(\alpha \right)}\left(x_t\right)=0$ value of the force: the force is repulsive above it and attractive below it.

Figure 18

Scaled Casimir force, $L^4{F}_{\mathrm{Cas}}$, as a function of the scaled reduced temperature, $t{L}^2$, and the scaled surface field amplitude, $h_{s}L$. The number of layers in the two films is $L=50$ and 100, and $\alpha$, the angle between the surface fields, is $\pi /3$. The difference between the two plots is barely discernible, indicating that the difference between the scaling function for $L=50$ and the infinite $L$ limit is quite small.

Figure 19

Scaled Casimir force, $L^4{F}_{\mathrm{Cas}}$, as a function of the scaled surface field, $h_{s}L$, for various scaled reduced temperatures, $t{L}^2$. Here, $L=50$ and $\alpha =\pi /3$. When $t{L}^2>-{\pi }^2$, the small $h_s$ dependence of the Casimir force is quadratic, consistent with (4.18). Below that value of the scaled reduced temperature, the small $h_s$ dependence is linear in the absolute value of that quantity, as exemplified by the curve for $t{L}^2=-15$.

Figure 20

Dependence of the scaled Casimir force on the scaled surface field for two values of scaled reduced temperature above the point, $t{L}^2=-{\pi }^2$, at which spontaneous ordering occurs in the film. Here, $L=50$ and $\alpha =\pi /3$. The plots illustrate the saturation of the influence of the surface fields, at odds with the amplification effect seen in Sec. 3. The figure also illustrates the fact that the Casimir force can change sign as the temperature is varied. This is due to the fact that there is a range of temperatures below the bulk critical temperature in which the bulk system orders while the film remains disordered. For $T>T_c$, both the bulk and the finite system are disordered. For $|h_s|\gg 1$, the Casimir force approaches its value for fixed boundary conditions, the case considered in Sec. 4a.

Figure 21

Scaled Casimir force, $L^2{F}_{\mathrm{Cas}}$, as a function of $L$, for fixed values of temperature $t=0.001$, helicity $\alpha =\pi /3$, and the value of the surface field amplitude, $h_s=0.1$.

Figure 22

Illustrating the $L$ dependence of the Casimir force for a negative value of reduced temperature, $t=-0.05$, with surface field amplitude $h_s=0.05$ and $\alpha =\pi /3$. The plot is generated by varying the film thickness $L$ for fixed values of $t, h_s$, and $\alpha$. The large graph shows how $L^2{F}_{\mathrm{Cas}}$ varies over an extended range of film thicknesses $L$, and the inset shows the $L$ dependence over a much smaller range.

Figure 23

The scaled Casimir force, $L^4{F}_{\mathrm{Cas}}$, as a function of the scaled variable $t{L}^2$. The thickness of the film is $L=50$, the surface field amplitudes have been set to 0.01, and the angle between them, $\alpha$, is $\pi /3$.

Figure 24

Scaled Casimir force, $L^4{F}_{\mathrm{Cas}}$, as a function of the scaled reduced temperature, $t{L}^2$, and scaled surface field amplitude, $h_{s}L$. The number of layers in the film is $L=50$. The values of $\alpha$, the angle between the surface fields, are, reading left to right and then top to bottom, (a) 0, (b) $\pi /2$, (c) $2\pi /3$, and (d) $\pi$.