Quantum anomalous Hall conductivity in altermagnets under applied magnetic field

Abstract

We investigate the emergence of quantum anomalous Hall conductivity in a two-dimensional $d$-wave altermagnet on a Lieb lattice under an external magnetic field. Altermagnetic order induces momentum-dependent spin splitting without net magnetization in the relativistic limit, producing distinct spin-resolved bands at the $X$ and $Y$ valleys. The phase diagram features a normal insulator and a spin Chern insulator separated by an accidental Dirac semimetal. The magnetic field breaks rotational symmetry between valleys while maintaining vanishing total magnetization, enabling independent valley contributions to topology. One valley supports Chern numbers $C=-1$ or $0$, while the other hosts $C=0$ or $+1$, governed by field strength and bandwidth. This competition yields valley-dependent topology. Berry curvature analysis reveals fully gapped phases with total Chern numbers $C=\pm1$, separated by valley-selective gap closings. We uncover a mechanism for rapid magnetic control of the quantum anomalous Hall effect near the semimetal phase and highlight key distinctions from ferro-valleytronic and quantum spin Hall systems.

Figures (6)

• Figure 1: Schematic illustration of topological surface states connecting valence and conduction bands for different topological phases and their associated spin-resolved Chern numbers and valleys. Chern numbers not reported in the panels are zero. Blue and red paraboloids represent the valleys with spin-up and down, respectively. For $t_d > 0$, the possible phases are: (a) $C_{\uparrow}^{X} = +1$, (b) $C_{\downarrow}^{Y} = -1$, and (c) the combination of the previous two cases with $C_{\uparrow}^{X} = +1$ and $C_{\downarrow}^{Y} = -1$. For $t_d < 0$, the possible phases are: (d) $C_{\downarrow}^{X} = +1$, (e) $C_{\uparrow}^{Y} = -1$, and (f) the combination of the previous two cases with $C_{\downarrow}^{X} = +1$ and $C_{\uparrow}^{Y} = -1$.
• Figure 2: Schematic of edge states in the real space of a monolayer QAH insulator with Chern number $C = C_{\uparrow} + C_{\downarrow}$ for different cases: (a) $C_{\downarrow} = +1$, (b) $C_{\uparrow} = +1$, (c) $C_{\downarrow} = -1$, and (d) $C_{\uparrow} = -1$. Red and blue arrows on the gray sample indicate the magnetic atoms. The arrows on the edges indicate the movement of the spinful electrons. For the case with C$_{s}$=2, we need to combine two of them.
• Figure 3: Global band gap $E_{\mathrm{gap}}$ as a function of the masses $(M_{0},M_{1})$ for two representative values of the altermagnetic hopping anisotropy $t_{d}$. (a),(b) $t_{d}=0.2<\Delta/4$ without and with SOC, respectively. The gap closes only along straight lines corresponding to Dirac mass inversions at the $M$, $X$, and $Y$ points in the absence of SOC, while SOC removes the $M$-point transition and confines all gap closings to $X$ and $Y$. (c),(d) $t_{d}=0.5>\Delta/4$ without and with SOC, respectively. For $t_{d}>\Delta/4$ a gapless region appears around the origin due to spin-polarized Weyl points on the Brillouin-zone edges; SOC gaps these Weyl points and collapses the gapless region onto the mass-inversion lines $m(X)=0$ and $m(Y)=0$, in agreement with the analytical theory.
• Figure 4: (a)–(c) Berry curvature $\Omega(\mathbf{k})$ of the occupied bands for $t_{d}=0.5$, $\lambda=0.5$, $M_{0}=-t_{d}$, and $M_{1}=t_{d},\,2t_{d},\,3t_{d}$, respectively. These three parameter sets realize Chern numbers $C=-1,0,+1$, as shown in Fig. S2. For $M_{1}=t_{d}$, the curvature is dominated by negative hot spots near the $Y$ valley, yielding a net negative flux. At $M_{1}=2t_{d}$ the valley contributions nearly cancel and the integrated curvature vanishes, while for $M_{1}=3t_{d}$ the hot spots reverse sign and produce a positive net flux. (d) Chern number $C$ as a function of $M_{0}$ for $t_{d}=0.5$, $\lambda=0.5$, and $M_{1}=0$. Quantized plateaus at $C=\pm1$ are separated by a region with $C=0$.
• Figure 5: (a) Chern-number phase diagram $C(M_{0}/t_{d},M_{1}/t_{d})$ for $t_{d}=0.5$ and $\lambda_{R}=0.5$. (b) Ribbon spectrum for a strip open along $x$ and periodic along $y$ at $(M_{0},M_{1})=(-t_{d},3t_{d})$, corresponding to a point inside the $C=1$ region. The color scale denotes edge-localization weight and reveals a single chiral edge mode per boundary. Edge states for region with $C=0$, but (c) and (e) nonzero $C_s$ and (d) $C_s=0$. Color bar denotes $S_z$. (f) Hall conductivity $\sigma_{xy}(E_{F})$ in units of $e^{2}/h$ for $(M_{0},M_{1})=(-t_{d},3t_{d})$. A quantized plateau at $\sigma_{xy}=1$ appears when the Fermi energy lies inside the bulk gap, consistent with the Chern number and the edge-state structure.