Correction: Klimchitskaya et al. Nonequilibrium Casimir–Polder Interaction between Nanoparticles and Substrates Coated with Gapped Graphene. Symmetry 2023, 15, 1580

The authors wish to make the following corrections in their paper [1].

1. Equation (28) is replaced with

$$\begin{array}{ccc}& & {\tilde{\mathsf{\Pi }}}_{00}^{\left(1\right)}(t,\tilde{\Delta },{\tau }_g)=\frac{8\alpha }{{\tilde{v}}_F^2{t}^2}\underset{\tilde{\Delta }}{\overset{\infty }{\int }}dz\mathrm{\Psi }(z,{\tau }_g)\left[1+\frac{1}{2}\sum _{\lambda =\pm 1}\lambda \frac{{\tilde{v}}_F^2{t}^2-{(z+\lambda )}^2}{\sqrt{[{\tilde{v}}_F^2{t}^2-{(z+\lambda )}^2]({\tilde{v}}_F^2{t}^2-1)+{\tilde{v}}_F^2{t}^2{\tilde{\Delta }}^2}}\right],\hfill \\ & & \hfill \\ & & {\tilde{\mathsf{\Pi }}}^{\left(1\right)}(t,\tilde{\Delta },{\tau }_g)=\frac{8\alpha }{{\tilde{v}}_F^2{t}^2}\underset{\tilde{\Delta }}{\overset{\infty }{\int }}dz\mathrm{\Psi }(z,{\tau }_g)\left[1+\frac{1}{2}\sum _{\lambda =\pm 1}\lambda \frac{{(z+\lambda )}^2({\tilde{v}}_F^2{t}^2-1)-{\tilde{v}}_F^2{t}^2{\tilde{\Delta }}^2}{\sqrt{[{\tilde{v}}_F^2{t}^2-{(z+\lambda )}^2]({\tilde{v}}_F^2{t}^2-1)+{\tilde{v}}_F^2{t}^2{\tilde{\Delta }}^2}}\right].\hfill \end{array}$$

(28)

This equation differs from Equation (28) in Ref. [1] by the additional multiple $-\lambda$ under the summation signs, which is responsible for the correct choice of branches of the square root in denominators when performing the analytic continuation of the polarization tensor to the real frequency axis.

This error does not influence the main results of the Ref. [1] but makes a quantitative impact on Figures 5 and 6.

2. Section 4, paragraph 8, starting from the words “As shown in Figure 5a,b”, should be formulated as:

As shown in Figure 5a,b, the quantity $F_r$, unlike $F_M$, substantially depends on the value of $\Delta$ at both $T_g=77$ and 500 K. The point is that, for $T_g<T_E$, the contributions to $F_r$ from the regions (14) and (15) are positive, i.e., decrease the force magnitude. The opposite situation occurs for $T_g>T_E$, i.e., the contributions of Equations (14) and (15) to $F_r$ are negative, leading to the increase in force magnitude.

Section 4, paragraph 10, the text starting with “In so doing, …” should be omitted, as well the full paragraphs 11 and 12.

The corrected Figure 5 is

Figure 5. The ratio of the second contribution to the nonequilibrium Casimir–Polder force acting on a nanoparticle from a SiO₂ plate coated by a graphene sheet (a) at $T_g=77$ K and (b) $T_g=500$ K to the equilibrium one from an ideal metal plane at $T_p=T_E=0$ is shown as the function of separation (a) by the two lines for $\Delta =0.1$ and 0.2 eV, where the latter coincides with that for an uncoated plate shown by the dashed line, and (b) by the two solid lines for $\Delta =0.1$ and 0.2 eV, where the latter coincides with that for an uncoated plate shown by the dashed one.

Figure 5. The ratio of the second contribution to the nonequilibrium Casimir–Polder force acting on a nanoparticle from a SiO₂ plate coated by a graphene sheet (a) at $T_g=77$ K and (b) $T_g=500$ K to the equilibrium one from an ideal metal plane at $T_p=T_E=0$ is shown as the function of separation (a) by the two lines for $\Delta =0.1$ and 0.2 eV, where the latter coincides with that for an uncoated plate shown by the dashed line, and (b) by the two solid lines for $\Delta =0.1$ and 0.2 eV, where the latter coincides with that for an uncoated plate shown by the dashed one.

3. Next, Figure 6 should be corrected, and in the third paragraph from the bottom in Section 4 the values of the ratio $F_{\mathrm{neq}}/F_{\mathrm{neq}}^{{\mathrm{SiO}}_2}$ equal to 1.03 and 1.35 at separations $a=0.2$ and 2 μm for a graphene sheet with Δ = 0.1 eV should be replaced with 1.05 and 0.94.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.

The corrected Figure 6 is

Figure 6. The ratio of the nonequilibrium Casimir–Polder force acting on a nanoparticle from a SiO₂ plate coated by a graphene sheet with Δ = 0.1 eV to the equilibrium one from an ideal metal plane (a) at $T_p=T_E=0$ and (b) at $T_p=T_E=300$ K (the classical limit) is shown as the function of separation. The bottom and top lines are for the graphene plate temperatures $T_g=77$ K and 500 K, respectively. The middle lines demonstrate similar ratio, where $T_g=T_E=300$ K.

Figure 6. The ratio of the nonequilibrium Casimir–Polder force acting on a nanoparticle from a SiO₂ plate coated by a graphene sheet with Δ = 0.1 eV to the equilibrium one from an ideal metal plane (a) at $T_p=T_E=0$ and (b) at $T_p=T_E=300$ K (the classical limit) is shown as the function of separation. The bottom and top lines are for the graphene plate temperatures $T_g=77$ K and 500 K, respectively. The middle lines demonstrate similar ratio, where $T_g=T_E=300$ K.