Topological constraints on the electronic band structure of hexagonal lattice in a magnetic field

TL;DR

The paper investigates how projective lattice symmetry in a hexagonal lattice under a magnetic field constrains electronic band structures. By analyzing a Hofstadter-like model with rational flux $\phi=2\pi p/q$, it uncovers symmetry-enforced Dirac touchings at finite energy $E\neq 0$ for $\phi=\pi$ and zero-energy Dirac points whose number is tied to zero-energy solutions of the Hamiltonian, potentially yielding $q$ or $2q$ points. It derives topological constraints on Chern numbers for both isolated bands and Dirac-connected band multiplets, showing distinct relations from the square lattice, and discusses the impact of symmetry breaking and distortions. The results illuminate how hexagonal materials in magnetic fields host unique topological features and offer guidance for experimental exploration of topological transport in such systems. These findings highlight the rich interplay between projective symmetry, Dirac physics, and band topology in hexagonal lattices beyond the square-lattice archetype.

Abstract

The impact of projective lattice symmetry on electronic band structures has attracted significant attention in recent years, particularly in light of growing experimental studies of two-dimensional hexagonal materials in magnetic fields. Yet, most theoretical work to date has focused on the square lattice due to its relative simplicity. In this work, we investigate the role of projective lattice symmetry in a hexagonal lattice with rational magnetic flux, emphasizing the resulting topological constraints on the electronic band structure. We show that, at pi flux, the symmetry in the hexagonal lattice enforces novel Dirac band touchings at E not equal to zero, and for general rational flux it constrains the number of Dirac points at E = 0. We further analyze the symmetry-imposed constraints on the Chern numbers of both isolated gapped bands and band multiplets connected by Dirac-point touchings. Our results demonstrate that these constraints in the hexagonal lattice differ substantially from those in the square lattice.

Figures (5)

• Figure 1: (a) A real space hexagonal lattice, where $\mathbf{e}_1, \mathbf{e}_2$ are unit vectors of the primitive lattice. The gray shaded area is a magnetic unit cell with flux $\phi=\frac{p}{q}\phi_0, \phi_0=h/e$ and magnetic unit vectors $\tilde{\boldsymbol{e}}_{1} = \boldsymbol{e} _1, \tilde{\boldsymbol{e}}_{2}=q\boldsymbol{e}_{2}$ ($q=2$ in the figure as an example). The yellow rhombus is the center point of ${\cal C}_2$ rotation. (b) The first magnetic Brillouin zone (MBZ) of the Magnetic unit cell, and high symmetry points $\Gamma, X, Y, M$. The reciprocal lattice vectors are $\boldsymbol{g}_1=2\pi(\frac{1}{3}\hat{x}-\frac{1}{\sqrt{3}}\hat{y}),\boldsymbol{g}_2=\frac{2\pi}{q}(\frac{1}{3}\hat{x}+\frac{1}{\sqrt{3}}\hat{y})$.
• Figure 2: Electronic bands of the hexagonal Hofstadter model with hopping parameters $t_1=t_2=t_3=1$. (a) $\phi=\pi$ (b) $\phi=2\pi/3$ (c) $\phi=2\pi/4$ (d) $\phi=2\pi/5$.
• Figure 3: Electronic bands of the hexagonal lattice with broken sublattice symmetry under A/B site potentials of +0.5/-0.5 and hopping parameters $t_1 = t_2 = t_3 = 1$. (a) $\phi=\pi$ (b) $\phi=2\pi/3$ (c) $\phi=2\pi/4$ (d) $\phi=2\pi/5$.
• Figure 4: Energy bands with broken translation symmetry $\mathbf{T}_2$ for hopping parameters $t_1=t_2=t_3=1$: (a) $p/q=1/2$, on-site potential in a MUC on the $A$ and $B$ sublattice $V_A=\{+0.5,+1.0\}$, $V_B=\{-0.5,-1.0\}$ (b) $p/q=1/3$, on-site potential in a MUC $V_A=\{+0.2,+0.4,+0.6\}$, $V_B=\{-0.2,-0.4,-0.6\}$ (c) $p/q=1/4$, on-site potential in a MUC $V_A=\{+0.2,+0.4,+0.6,+0.8\}$, $V_B=\{-0.2,-0.4,-0.6,-0.8\}$ (d) $p/q=1/5$, on-site potential in a MUC $V_A=\{+0.05,+0.10,+0.15,+0.20,+0.25\}$, $V_B=\{-0.05,-0.10,-0.15,-0.20,-0.25\}$
• Figure 5: Electronic bands of the hexagonal Hofstadter model with $q$ Dirac points at $E=0$ in the first MBZ. (a) $\phi=\pi$, $t_1=2^{1/2}$, $t_2=t_3=1$, two Dirac points at $E=0$ and $\mathbf{k}=(0,0)$ and $(\pi,0)$. (b) $\phi=2\pi/3$, $t_1=2^{1/3}$, $t_2=t_3=1$, three Dirac points at $E=0$ and $\mathbf{k}=(-\pi/3,0)$, $(\pi/3,0)$ and $(\pi,0)$.