Algebra of Hyperbolic Band Theory under Magnetic Field

We explore algebras associated with the hyperbolic band theory under a magnetic field for the first time. We define the magnetic Fuchsian group associated with a higher genus Riemann surface. By imposing the magnetic bounday conditions for the hyperbolic Bloch states, we construct the hyperbolic magnetic Bloch states and investigate their energy spectrum. We give a connection between such magnetic Bloch states and automorphic forms. Our theory is a general extension of the conventional algebra associated with the band theory defined on a Euclidean lattice/space into that of the band theory on a general hyperbolic lattice/Riemann surface.Comment: 21 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:2104.1331

A lattice regularization of Weyl fermions in a gravitational background

We report on a lattice fermion formulation with a curved domain-wall mass term to nonperturbatively describe fermions in a gravitational background. In our previous work in 2022, we showed under the time-reversal symmetry that the edge-localized massless Dirac fermion appears on one and two-dimensional spherical domain-walls and the spin connection is induced on the lattice in a consistent way with continuum theory. In this work, we extend our study to the Shamir type curved domain-wall fermions without the time-reversal symmetry. We find in the free fermion case that a single Weyl fermion appears on the edge, and feels gravity through the induced spin connection. With a topologically nontrivial $U(1)$ gauge potential, however, we find an oppositely chiral zero mode at the center where the gauge field is singular.Comment: 9 pages, 6 figures. Talk presented at the 40th International Symposium on Lattice Field Theory (Lattice2023), 31 July - 4 August 2023, Fermi National Accelerator Laboratory, minor correction

A Microscopic study of Magnetic monopoles in Topological Insulators

In this article, we analyze a magnetic monopole in topological insulators. The monopole obtain a fractional electric charge because of the Witten effect. We consider this system with a microscopic view by adding the Wilson term to the ordinary Dirac Hamiltonian. The Wilson term yields the positive mass shift to the effective mass of the electrons, then the curved domain-wall is dynamically generated around the monopole. The zero-modes of the electrons are localized on the domain-wall, which can be identified as the source of the electric charge.Comment: 9 pages, contribution to the 40th International Symposium on Lattice Field Theory (Lattice2023), July 31st - August 4th, 2023, Fermi National Accelerator Laborator

Why magnetic monopole becomes dyon in topological insulators

The Witten effect predicts that a magnetic monopole acquires a fractional electric charge inside topological insulators. In this work, we give a microscopic description of this phenomenon, as well as an analogous two-dimensional system with a vortex. We solve the Dirac equation of electron field both analytically in continuum and numerically on a lattice, by adding the Wilson term and smearing the gauge field within a finite range to regularize the short-distance behavior of the system. Our results reveal that the Wilson term induces a strong positive mass shift, creating a domain-wall around the monopole/vortex. This small, yet finite-sized domain-wall localizes the chiral zero modes and ensures their stability through the Atiyah-Singer index theorem, whose cobordism invariance is crucial in explaining why the electric charge is fractional.Comment: 40 pages, 10 figures, minor corrections, references adde