Electrified magnetic catalysis in three-dimensional topological insulators

The gap equations for the surface quasiparticle propagators in a slab of three-dimensional topological insulator in external electric and magnetic fields perpendicular to the slab surfaces are analyzed and solved. A different type of magnetic catalysis is revealed with the dynamical generation of both Haldane and Dirac gaps. Its characteristic feature manifests itself in the crucial role that the electric field plays in dynamical symmetry breaking and the generation of a Dirac gap in the slab. It is argued that, for a sufficiently large external electric field, the ground state of the system is a phase with a homogeneous surface charge density.

Figure 1

The lowest and first Landau level parameters as functions of the magnetic field for fixed values of the external electric field $\mathcal{E}=1\mathrm{mV}/\AA$ and temperature $T=5\times {10}^{-3}{\mathrm{\Delta }}_{\mathrm{bulk}}\approx 20\text{K}$. The results for the (effective) electrochemical potentials ${\mathrm{\Delta }}_{\mathrm{eff},\pm }$ and ${\mu }_{1,\pm }$ are shown in the left panel, the gaps $m_{1,\pm }$ are shown in the middle panel, and Dirac and Haldane gaps $m_{1,D}, m_{1,H}$, and $\mathrm{\Delta }{m}_{1,H}$ are shown in the right panel.

Figure 2

The lowest and first Landau level parameters as functions of the external electric field for fixed values of the magnetic field $B=5\text{T}$ and temperature $T=5\times {10}^{-3}{\mathrm{\Delta }}_{\mathrm{bulk}}\approx 20\text{K}$. The results for the (effective) electrochemical potentials ${\mathrm{\Delta }}_{\mathrm{eff},\pm }$ and ${\mu }_{1,\pm }$ are shown in the left panel, the gaps $m_{1,\pm }$ are shown in the middle panel, and Dirac and Haldane gaps $m_{1,D}, m_{1,H}$, and $\mathrm{\Delta }{m}_{1,H}$ are shown in the right panel.

Figure 3

The electrochemical potentials ${\mu }_{n,\pm }$ (left panel), the gaps $m_{n,\pm }$ (middle panel), and the Dirac and Haldane gaps (right panel) as functions of the Landau level index $n$. The results for $\mathcal{E}=2\mathrm{mV}/\AA$ are represented by red solid and green dotted lines. The results for $\mathcal{E}=1\mathrm{mV}/\AA$ are represented by blue dashed and brown dashed-dotted lines. The values of the magnetic field and temperature are $B=5\text{T}$ and $T=5\times {10}^{-3}{\mathrm{\Delta }}_{\mathrm{bulk}}\approx 20\text{K}$, respectively.

Figure 4

The quasiparticle energies for the first three Landau levels as functions of the magnetic field at fixed $\mathcal{E}=1\mathrm{mV}/\AA$ (left panel) and as functions of the electric field at fixed $B=5\text{T}$ (right panel). Red and blue lines denote the quasiparticle energies on the top and bottom surfaces, respectively. Solid lines represent the LLL, dashed and dotted lines correspond to the first and second Landau levels, respectively. The temperature is $T=5\times {10}^{-3}{\mathrm{\Delta }}_{\mathrm{bulk}}\approx 20\text{K}$.

Figure 5

The energy spectrum as a function of $k_y$ (left panel), the chiral condensate (middle panel), and the charge density (right panel) as functions of $x$ in the Dirac problem with the inhomogeneous gap $m\left(x\right)=\left|m\right|\text{sign}\left(x\right)$. For simplicity, only the first six Landau levels are presented in the left panel. The red solid lines in the middle and right panels represent the numerical data. The blue dashed lines denote the corresponding fits. The model parameters are $m=5\text{meV}, \mu =0$, and $B=5\text{T}$.

Figure 6

The ratio of the width correction $\mathrm{\Delta }{l}_x$ to the stripe half-width $l_x$ as a function of the external electric field (red line). The model parameters are $m=5\text{meV}, \mu =0, B=5\text{T}$, and $l_x=500\text{Å}$.