Unified topological phase diagram of quantum Hall and superconducting vortex-lattice states

TL;DR

The paper constructs a comprehensive global topological phase diagram for a two-dimensional electron gas in a quantizing magnetic field proximized by a superconducting vortex lattice, incorporating Landau-level mixing and arbitrary ratios of the pairing amplitude $|\Delta|$, cyclotron energy $\hbar\omega_c$, and chemical potential $\mu$. It shows that LL mixing splits the conventional IQHE transition lines $\mu \approx E_N$ into multiple, symmetry-protected trajectories, producing dome-shaped regions with rich topological content and large Chern-number jumps up to $\mathcal{C}=12$; these jumps can have either sign and require careful counting of gap-closure points. The authors develop two complementary methods to compute the Chern number—TKNN Berry-curvature integration and gap-closure analysis across transition lines— while elaborating the role of space-group symmetries (particle-hole, half-cell translation, rotations, and space-time reflection) in shaping the degeneracies and pattern of gap closures. They further show that lattice distortions and potential disorder can split high-magnitude transitions, discuss extensions to Zeeman and spin-orbit coupling (which could realize Majorana modes in class D), and outline experimental implications for proximitized two-dimensional systems and related materials.

Abstract

We present the global topological phase diagram of a two-dimensional electron gas placed in a quantizing magnetic field and proximitized by a superconducting vortex lattice. Our theory allows for arbitrary ratios of the pairing amplitude, magnetic field, and chemical potential. By analyzing the Bogoliubov--de Gennes Hamiltonian, we show that the resulting phase diagram is highly nontrivial, featuring a plethora of topological superconducting phases with chiral edge modes of quasiparticles. Landau-level mixing plays an essential role in our theory: even in the weak-pairing limit, it generically splits the integer quantum Hall transition lines into a sequence of transitions with larger Chern number jumps of both signs protected by the symmetries of the superconducting vortex lattice. Interestingly, we find that weak pairing induces trivial or topological superconductivity when chemical potential is tuned to a Landau level energy, depending on the Landau level index.

Figures (7)

• Figure 1: (Triangular vortex lattice) Topological phase diagram of the two-dimensional electron gas in the presence of magnetic field and superconducting order parameter with Abrikosov vortex lattice in two different axes choice. Here $\omega_c$ is the cyclotron frequency, $\mu$ is the chemical potential and $\Delta$ is the superconducting order parameter amplitude. The dashed line ($\omega_c \mu \sim \Delta^2$) is an estimate for the boundary of the region containing topological domes, see Eq. \ref{['top-trivial-boundary']}. The dotted line shows a pair of closely lying transition lines, illustrated in the zoom-in. The black squares mark the tricritical points. Inset: configuration of the degenerate gap closure points in the magnetic Brillouin zone; filled circles mark Dirac points while crosses stand for the cubic band touchings. The colors correspond to the colors of the lines in the main plot. As shown in Sec. \ref{['sec:symmetry']}, blue points and red crosses coincide with the $C_6$ rotation axes; the other groups of same-color points are symmetric under this operation. When moving along the transition lines in the main plot, the green, orange, and purple points move in the radial direction retaining the $C_6$ symmetry.
• Figure 2: (a, b) Superconducting vortex lattices with the (a) square and (b) triangular configurations. Magnetic unit cell have the area $2\pi l_B^2$ in both cases and is highlighted in grey. (c, d) Magnetic Brillouin zones for the (c) square vortex lattice and (d) triangular vortex lattice; reciprocal lattice vectors ${\bf G}_1$ and ${\bf G}_2$ are shown in purple. A half of each BZ is highlighted in turquoise; the dispersion in the other half is identical due to the symmetry \ref{['half-magnetic-transform']}. Stars represent the centers of the inversion and rotation symmetries ($C_4$ and $C_6$ correspondingly); dashed line depicts one of the mirror symmetries of the dispersion.
• Figure 3: Gap closure points in the limit of a single Landau level studied in Section \ref{['sec:single-LL']} for the (a) square vortex lattice; (b) triangular vortex lattice. This regime requires $\omega_c \gg \Delta$ and chemical potential close to the energy of the $N$-th Landau level, $N = 0$--$4$. Colors of the dots match the colors of the topological phase transition lines of Figs. \ref{['fig:tr-phase-diagram']} and \ref{['fig:sq-phase-diagram']}.
• Figure 4: (Square vortex lattice) Topological phase diagram of the two-dimensional electron gas in the presence of magnetic field and superconducting order parameter with Abrikosov vortex lattice in two different axes choice. Here $\omega_c$ is the cyclotron frequency, $\mu$ is the chemical potential and $\Delta$ is the superconducting order parameter amplitude. The dashed line ($\omega_c \mu \sim \Delta^2$) is an estimate for the boundary of the region containing topological domes, see Eq. \ref{['top-trivial-boundary']}. The black squares mark the tricritical points. Inset: configuration of the degenerate gap closure points in the magnetic Brillouin zone; filled circles mark Dirac points while empty circles stand for the quadratic band touchings. The colors correspond to the colors of the lines in the main plot. As shown in Sec.\ref{['sec:symmetry']}, blue points coincide with the $C_4$ rotation axes; the other groups of same-color points are symmetric under this operation. When moving along the transition lines in the main plot, the green, orange, and purple points move retaining the $C_4$ symmetry.
• Figure 5: (Anisotropic triangular vortex lattice) Topological phase diagram for the anisotropically stretched triangular vortex lattice, compare with the Fig. \ref{['fig:tr-phase-diagram']} for the undistorted case. Reduction of the rotation symmetry from $C_6$ to $C_2$ leads to the splitting of the transition lines with a high Chern number jump into several lines with a smaller one.