Polaritonic Hofstadter butterfly and cavity control of the quantized Hall conductance

In a previous work [Rokaj et al., Phys. Rev. Lett. 123, 047202 (2019)] a translationally invariant framework called quantum-electrodynamical Bloch (QED-Bloch) theory was introduced for the description of periodic materials in homogeneous magnetic fields and strongly coupled to the quantized photon field in the optical limit. For such systems, we show that QED-Bloch theory predicts the existence of fractal polaritonic spectra as a function of the cavity coupling strength. In addition, for the energy spectrum as a function of the relative magnetic flux we find that a terahertz cavity can modify the standard Hofstadter butterfly. In the limit of no quantized photon field, QED-Bloch theory captures the well-known fractal spectrum of the Hofstadter butterfly and can be used for the description of two-dimensional materials in strong magnetic fields, which are of great experimental interest. As a further application, we consider Landau levels under cavity confinement and show that the cavity alters the quantized Hall conductance and that the Hall plateaus are modified as ${\sigma }_{xy}=e^2\nu /h(1+{\eta }^2)$ by the light-matter coupling $\eta$. Most of the aforementioned phenomena should be experimentally accessible, and corresponding implications are discussed.

Figure 1

Cartoon depiction of a two-dimensional periodic material confined inside a cavity with mirrors of length $L$ and area $S=L^2$. The distance between the cavity mirrors is $L_z$. The whole system is placed perpendicular to a classical homogeneous magnetic field ${\mathbf{B}}_{\text{ext}}$. We note that typically in experimental setups the space between the 2D material and the cavity is filled with a dielectric medium.

Figure 2

Energy spectra as a function of the light-matter coupling $\eta$ for the square cosine potential and for different values of the relative magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$. The inset in Fig. 2 displays the spatial profile of the applied square cosine lattice potential. (a) Energy spectrum as a function of the light-matter coupling $\eta ={\omega }_p/{\omega }_c$ for magnetic flux ratio $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.1$. (b) Energy spectrum as a function of the light-matter coupling $\eta ={\omega }_p/{\omega }_c$ for magnetic flux ratio $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.2$. (c) Energy spectrum as a function of the light-matter coupling $\eta ={\omega }_p/{\omega }_c$ for magnetic flux ratio $\mathrm{\Phi }/{\mathrm{\Phi }}_0=1$. (d) Energy spectrum as a function of the light-matter coupling $\eta ={\omega }_p/{\omega }_c$ for magnetic flux ratio $\mathrm{\Phi }/{\mathrm{\Phi }}_0=1$.

Figure 3

Energy spectra for the square cosine potential as a function of the relative magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ for different values of the diamagnetic frequency ${\omega }_p$. (a) Energy spectrum as a function of the relative magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ for ${\omega }_p={10}^{-3}\text{THz}$ (weak coupling). (b) Energy spectrum as a function of the relative magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ for ${\omega }_p={10}^{-1}\text{THz}$ (intermediate coupling). (c) Energy spectrum as a function of the relative magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ for ${\omega }_p=1\text{THz}$ (strong coupling).

Figure 4

Energy spectra as a function of the light-matter coupling $\eta$ for the hexagonal-cosine potential for different values of the relative magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$. The inset displays the spatial profile of the applied hexagonal-cosine lattice potential. (a) Energy spectrum as a function of the light-matter coupling $\eta ={\omega }_p/{\omega }_c$ for magnetic flux ratio $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.2$ (b) Energy spectrum as a function of the light-matter coupling $\eta ={\omega }_p/{\omega }_c$ for magnetic flux ratio $\mathrm{\Phi }/{\mathrm{\Phi }}_0=1$.

Figure 5

Energy spectra for the square cosine potential as a function of the relative magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ with higher Landau polariton states, for two different values of the diamagnetic frequency ${\omega }_p$ which correspond to weak (left) and strong (right) coupling to the cavity photons. (a) Energy spectrum as a function of $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ for a small diamagnetic frequency ${\omega }_p={10}^{-3}\mathrm{THz}$ with four Landau polariton states; (b) Energy spectrum as a function of $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ for a large diamagnetic frequency ${\omega }_p=1\mathrm{THz}$ with two Landau polariton states.

Figure 6

Comparison between the scaled and the unscaled spectrum for the square cosine potential. The potential strength is $V_0=3\mathrm{eV}$ and the lattice constant $a=2\AA$. We use a small lattice constant to make the linear dispersion of the Landau level in Fig. 6 clearly visible. (a) Scaled dimensionless energy spectrum obtained from the Harper Eq. (55) as a function of the reciprocal magnetic flux ${\mathrm{\Phi }}_0/\mathrm{\Phi }$. (b) Energy spectrum as function of the relative magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ as given by Eq. (53).

Figure 7

Schematic illustration of the duality between the minimal-coupling Hamiltonian and the tight-binding model with Peierls substitution. In both models the Hofstadter butterfly emerges but in a dual fashion. Namely, in the one case as a function of the reciprocal flux while in the other as a function of flux.

Figure 8

Scaled dimensionless energy spectrum for the square cosine lattice potential as a function of the magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ for ${\omega }_p=1\text{THz}$. The unscaled physical spectrum is shown in Fig. 3.

Figure 9

Cartoon depiction of a 2D electron gas (material in black) confined inside a cavity. The whole system is placed perpendicular to a classical homogeneous magnetic field ${\mathbf{B}}_{\text{ext}}$, and an external constant electric field ${\mathbf{E}}_{\text{ext}}$ is applied to the 2D material.

Figure 10

Depiction of the first quantum Hall plateau as a function of the light-matter coupling constant $\eta$. For $\eta =0$ we obtain the standard value of the Hall conductance. As the light-matter coupling increases, the Hall conductance decreases. This effect can be understood as a screening or renormalization effect due to the cavity. The shaded area indicates the regime in which typically experiments are performed.

Figure 11

First-quantized Hall plateau as a function of the ratio between the strength of the external magnetic field and the strength of the cavity magnetic field $B/B_{\text{cav}}$. In the region where the cavity strength is larger, the Hall conductance deviates strongly from its standard value $e^2/h$. As the external magnetic field increases and becomes larger than the strength of the cavity we obtain the standard quantized Hall plateau. The shaded area indicates the regime in which typically experiments are performed.