Lieb-Schultz-Mattis-Type and Laughlin-Type Argument for the Quantum Hall Effect in Lattice Fermions with Spiral Boundary Conditions

Abstract

We derive the condition for the occurrence of the quantum Hall effect in two-dimensional lattice systems, expressed as $φν-ρ\in\mathbb{Z}$, where $φ$, $ν$, and $ρ$ denote the magnetic flux, the Chern number, and the electron density, respectively. By employing spiral boundary conditions, which treat the system as an extended one-dimensional chain, this condition is obtained directly through a Lieb-Schultz-Mattis-type and Laughlin-type argument. This approach improves upon the preceding work based on conventional periodic boundary conditions, where the condition was derived indirectly containing redundant system-size dependence. The key to this approach is that the spatial directions of the external force and the response can be manipulated by a factor of the system size.

Figures (4)

• Figure 1: Several boundary conditions for 2D square lattices with $L_1\times L_2$ sites: (a) Conventional periodic boundary conditions (PBCs), (b) Spiral boundary conditions (SBCs). For the systems with SBCs, the lattices are labeled as extended 1D chains. $\bm{A}_i$ and $\bm{A}_i'$ are primitive vectors.
• Figure 2: The gauge field $\theta^{\alpha}$ in a square lattice in a magnetic field with (a) periodic boundary conditions, and (b) spiral boundary conditions.
• Figure 3: Sublattice structure of the system $(L_1,L_2,q)=(6,4,3)$ with a magnetic field ($\phi=p/q$). Here, $L_1/q$ is chosen to be an integer.
• Figure 4: Fluxes $\Phi_i$ for 2D lattice applied as $\mathcal{H}_i(\Phi)= U_i^{\Phi/2\pi} \mathcal{H} U_i^{-\Phi/2\pi}$. In the system with SBCs, the insertion of flux $\mathcal{H}_1'(\Phi_1)$ has the same effect of the flux $\mathcal{H}_2'(L_2\Phi_1)$ due to the relation $U_1'=U_2^{\prime L_2}$.