Casimir forces in the flatland: Interplay between photoinduced phase transitions and quantum Hall physics

We investigate how photoinduced topological phase transitions and the magnetic-field-induced quantum Hall effect simultaneously influence the Casimir force between two parallel sheets of staggered two-dimensional (2D) materials of the graphene family. We show that the interplay between these two effects enables on-demand switching of the force between attractive and repulsive regimes while keeping its quantized characteristics. We also show that doping these 2D materials below their first Landau level allows one to probe the photoinduced topology in the Casimir force without the difficulties imposed by a circularly polarized laser. We demonstrate that the magnetic field has a huge impact on the thermal Casimir effect for dissipationless materials, where the quantized aspect of the energy levels leads to a strong repulsion that could be measured even at room temperature.

Figure 1

(a) Top and side views of the graphene family honeycomb structure. The red and blue colored atoms belong to inequivalent sublattices with a finite staggering $2\ell$ between them. (b) Schematics of the system under study. Two graphene family monolayers separated by a distance $d$ and subjected to externally applied electric ($E_z$) and magnetic ($B_z$) fields, and a circularly polarized laser ($\mathrm{\Lambda }$). (c) Low-energy band structure around a given $K$ or $K^{\prime }$ point. For a nonzero magnetic field, the quantized Landau levels (red circles) are built on top of the Dirac cone.

Figure 2

Casimir energy in the $(E_z,\mathrm{\Lambda })$ plane for (a) $\mu ={\lambda }_{\text{SO}}$ and $E_B=1.2{\lambda }_{\text{SO}}$ and (b) $E_B=0.8{\lambda }_{\text{SO}}$. (c) Normalized Casimir energy in the $(E_z,\mu )$ plane for $\mathrm{\Lambda }=0$ and $E_B={\lambda }_{\text{SO}}$. (d) Normalized Casimir energy as a function of the chemical potential for $\mathrm{\Lambda }=0$. In all plots the distance between the plates is given by $d{\lambda }_{\text{SO}}/\hslash c=10$ and the dissipation is set to zero.

Figure 3

Casimir energy as a function of the chemical potential of a single monolayer for $\{e\ell E_z,{\mu }_2\}/{\lambda }_{\text{SO}}=\{(1,0.5),(1,-0.5),(1.5,0.5)\}$ (solid blue, dashed red, and dash-dotted green, respectively) and $E_B={\lambda }_{\text{SO}}$. For all curves, the distance between the plates is long enough so that Eq. (10) is valid and $\mathrm{\Lambda },\mathrm{\Gamma }=0$.

Figure 4

(a) Long distance approximated expression given by Eq. (10) subtracted from the Casimir energy as a function of distance. The chosen parameters are $\mathrm{\Lambda }=0, \mathrm{\Gamma }=0, e\ell E_z={\lambda }_{\text{SO}}$, and $\mu =\{0.5,1.25,1.5\}{\lambda }_{\text{SO}}$ (solid blue, dashed red, and dash-dotted green lines, respectively). The inset shows the longitudinal and Hall conductivity at frequency $\hslash \xi =0.01{\lambda }_{\text{SO}}$ as a function of the chemical potential. (b) Casimir energy for $\mathrm{\Lambda }=0, e\ell E_z={\lambda }_{\text{SO}}, \mu =1.25{\lambda }_{\text{SO}}$, and $\hslash \mathrm{\Gamma }=\{0,0.01,0.02\}{\lambda }_{\text{SO}}$ (solid, dashed, and dash-dotted lines, respectively) as a function of distance. The gray dotted lines are the approximated solutions in the long distance regime, given by the sum of Eqs. (10) and (13).

Figure 5

(a) Dissipationless Casimir energy as a function of temperature for a separation distance of $d{\lambda }_{\text{SO}}/\hslash c=10$. The dotted lines correspond to the approximated value of the $n=0$ Matsubara frequency contribution presented in Eq. (14). The left inset is a zoom of the plot, showing that each curve goes to its correspondent zero-temperature limit (dashed gray lines). The right inset shows the Casimir force in the region where it becomes attractive. (b) Casimir energy as a function of distance for $k_{B}T={10}^{-3}{\lambda }_{\text{SO}}$ and $\hslash \mathrm{\Gamma }=0.01{\lambda }_{\text{SO}}$. The gray line corresponds to the $n=0$ Matsubara contribution [Eq. (15)]. In both plots, $\mathrm{\Lambda }=0, e\ell E_z={\lambda }_{\text{SO}}$, and $\mu =\{0.5,1.25,1.5\}{\lambda }_{\text{SO}}$ (solid blue, dashed red, and dash-dotted green lines, respectively).