Figure 1

$f_1\left(x\right)$ (red dashed curve) and $f_2\left(x\right)$ (black full curve) as a function of the dimensionless distance $x$. The blue dotted curve is their common asymptotic limit to $\frac{23}{x}$ at large $x$. The limits at small $x$ of $f_1(x\to 0)\to 3\pi$ (yellow full curve) and $f_2(x\to 0)\to 6\pi$ (green full curve) are also plotted.Reuse & Permissions

Figure 2

Attraction versus repulsion in the classical limit ($T\to \infty$) and in the quantum limit ($T\to 0$) as a function of $w$ and $\left|\theta \right|$. For the classical limit, the green (light gray) region shows repulsion for all distances and the rest of phase space shows attraction for all distances. For the quantum limit, the red (dark gray) region shows repulsion for all distances and the white region shows parameters for which an equilibrium distance appears. Several curves of constant equilibrium dimensionless distance $x_{\mathrm{eq}}$ are also plotted. The black region is a forbidden region for the parameters due to the positive energy condition $|\overline{\alpha }|<\sqrt{\epsilon \left(0\right)}$.[25]Reuse & Permissions

Figure 3

Comparison between exact Casimir energies in units of $E_0=A\hslash c/{\left(2\pi \right)}^2{\left({\omega }_R/c\right)}^3$ (full curves) and PSA (points) at zero temperature for TI infinity plates with ${\theta }_1=-{\theta }_2=\pi$ and different $w=[\epsilon \left(0\right)-{\epsilon }_0]$ as a function of the dimensionless distance $\overline{d}$ between the plates (Ref. 10), from $w=0.2$ (blue upper curve) to $w=0.6$ (green lower curve). PSA tends to overestimate the absolute value of the energy, but captures the equilibrium distance without appreciable error. The overestimation of the energy increases with $w$, as expected because of the nature of the approximation.Reuse & Permissions

Figure 4

Dimensionless PSA Casimir energies in the sphere-plate TI system per unit of volume of the sphere $\overline{e}=\frac{c^3}{\hslash {\omega }_R^4}\frac{E}{V}$ with ${\theta }_1=-{\theta }_2=\pi$ and $w=0.45$ as a function of the dimensionless distance between the plate and the contact point of the sphere $\overline{h}=\overline{d}-{\overline{R}}_s$, where $\overline{d}$ is the dimensionless distance from the surface of the plate to the center of the sphere, from ${\overline{R}}_s\to 0$ (black upper line) to ${\overline{R}}_s=1$ (orange lower curve). Positive energies are the whole curves, while negative energies are the dashed curves. The equilibrium distance for each sphere radius is represented by a dot. These dots are joined by a soft line to have an idea of the dependence of the equilibrium distance with the radius of the sphere.Reuse & Permissions