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arXiv:2511.13594 · Journal of the Physical Society of Japan

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

Masaaki Nakamura, Masanori Yamanaka

Published Jan 22, 2026• 0 citations
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Abstract

We derive the condition for the occurrence of the integer quantum Hall effect in two-dimensional lattice systems with interactions, 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 with redundant system-size dependence. The key to this approach is that the spatial directions of the external force and the response can be systematically controlled by a factor of the system size.

DOI
10.7566/jpsj.95.023702

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We consider many-body quantum systems on a finite lattice, where the Hilbert space is the tensor product of finite-dimensional Hilbert spaces associated with each site, and where the Hamiltonian of the system is a sum of local terms. We are interested in proving uniform bounds on various properties as the size of the lattice tends to infinity. An important case is when there is a spectral gap between the lowest state(s) and the rest of the spectrum which persists in this limit, corresponding to what physicists call a “phase of matter.” Here, the combination of elementary Fourier analysis with the technique of Lieb–Robinson bounds (bounds on the velocity of propagation) is surprisingly powerful. We use this to prove exponential decay of connected correlation functions, a higher-dimensional Lieb–Schultz–Mattis theorem, and a Hall conductance quantization theorem for interacting electrons with disorder.
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Symmetry-enforced quantum spin Hall insulators in πππ-flux models
We prove a Lieb-Schultz-Mattis theorem for the quantum spin Hall effect (QSHE) in two-dimensional πππ-flux models. In the presence of time reversal, U(1)U(1)U(1) charge conservation and magnetic translation (with πππ-flux per unit cell) symmetries, if a generic interacting Hamiltonian has a unique gapped symmetric ground state at half filling (i.e. an odd number of electrons per unit cell), it can only be a QSH insulator. In other words, a trivial Mott insulator is forbidden by symmetries at half filling. We further show that such a symmetry-enforced QSHE can be realized in cold atoms, by shaking an optical lattice and applying a time-dependent Zeeman field.
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