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This paper presents a novel hybrid Fourier-finite difference method for solving magnetic Bloch eigenvalue problems in nonuniform periodic fields, offering rigorous error estimates and numerical validation for quantum systems. The work advances the computational toolkit for simulating complex quantum phenomena by combining covariant discretization with efficient spectral solvers like LOBPCG.
Magnetic Bloch eigenvalue problems in nonuniform periodic magnetic fields: a Fourier-finite difference approach
arXiv:2611.05950v1 Announce Type: cross Abstract: We study magnetic Bloch eigenvalue problems for two-dimensional magnetic Schr\"odinger operators with nonuniform periodic magnetic fields and periodic scalar potentials, and develop a hybrid Fourier-finite difference discretization. A Landau-type gauge is constructed by solving a periodic Poisson equation. Under the integer magnetic flux condition, the magnetic Bloch boundary conditions are incorporated through shifts of Fourier-mode indices. The method combines link-variable-based covariant central differences in the $x$-direction with a Fourier pseudospectral discretization in the $y$-direction, yielding a Hermitian discrete Hamiltonian. The resulting eigenvalue problems are solved using a LOBPCG method. Under appropriate regularity assumptions, we establish a fully discrete consistency residual estimate of order $O(h_1^2+N_2^{-s})$. With suitable spectral separation and eigenspace dimension-preserving conditions, error estimates of the same order are derived for isolated eigenvalues and their associated eigenspaces. Numerical experiments show second-order convergence in the finite difference direction and rapid error reduction with increasing Fourier resolution, and illustrate band degeneracies and eigenstate distributions in nonuniform periodic magnetic fields.
Abstract:We study magnetic Bloch eigenvalue problems for two-dimensional magnetic Schrödinger operators with nonuniform periodic magnetic fields and periodic scalar potentials, and develop a hybrid Fourier-finite difference discretization. A Landau-type gauge is constructed by solving a periodic Poisson equation. Under the integer magnetic flux condition, the magnetic Bloch boundary conditions are incorporated through shifts of Fourier-mode indices. The method combines link-variable-based covariant central differences in the $x$-direction with a Fourier pseudospectral discretization in the $y$-direction, yielding a Hermitian discrete Hamiltonian. The resulting eigenvalue problems are solved using a LOBPCG method. Under appropriate regularity assumptions, we establish a fully discrete consistency residual estimate of order $O(h_1^2+N_2^{-s})$. With suitable spectral separation and eigenspace dimension-preserving conditions, error estimates of the same order are derived for isolated eigenvalues and their associated eigenspaces. Numerical experiments show second-order convergence in the finite difference direction and rapid error reduction with increasing Fourier resolution, and illustrate band degeneracies and eigenstate distributions in nonuniform periodic magnetic fields.
Numerical Analysis (math.NA); Quantum Physics (quant-ph)
arXiv:2611.05950 [math.NA]
(or arXiv:2611.05950v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2611.05950 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 254 words