arXiv subjects

Yoshiyuki Matsuki

Publications and source records attributed to Yoshiyuki Matsuki.

11 recordsLinked to original sources

arXiv2024

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.

cond-mat.mes-hall↗

arXiv2024

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.

hep-lat↗

arXiv2023

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 boundary 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.

math-ph↗

arXiv2021

A physicist-friendly reformulation of the mod-two Atiyah-Patodi-Singer index

Gauge anomaly in 4-dimensions can be viewed as a current inflow into an extra-dimension, where the total phase of the fermion partition function is given in a gauge invariant way by the Atiyah- Patodi-Singer(APS) eta-invariant of a 5-dimensional Dirac operator. However, this formalism requires a non-local boundary condition, with which the physical roles of edge/bulk modes are unclear and how the causality of the theory is maintained is not obvious. In this work, we consider a special case where the Dirac operator is in a real representation and its eta invariant becomes the mod-two type APS index. We propose a physicist-friendly reformulation of the mod-two index using domain-wall fermion formalism, which naturally describes how the global anomaly is canceled between edge and bulk.

hep-th↗

arXiv2021

Hyperbolic Band Theory under Magnetic Field and Dirac Cones on a Higher Genus Surface

We explore the hyperbolic band theory under a magnetic field for the first time. Our theory is a general extension of the conventional band theory defined on a Euclidean lattice into the band theory on a general hyperbolic lattice/Riemann surface. Our methods and results can be confirmed experimentally by circuit quantum electrodynamics (cQED), which enables us to create novel materials in a hyperbolic space. To investigate the band structures, we construct directly the hyperbolic magnetic Bloch states and find that they form Dirac cones on a coordinate neighborhood, by which they can be regarded as a global quantum gravity solution detectable in a laboratory. Besides this is the first explicit example of a massless Dirac state on a higher genus surface. Moreover we show that the energy spectrum exhibits an unusual fractal structure refracting the negative curvature, when plotted as a function of a magnetic flux.

cond-mat.mes-hall↗

arXiv2021

Fractal defect states in the Hofstadter butterfly

We investigate the electronic properties in the Bloch electron on a square lattice with vacancies in the uniform magnetic field. We show that a single vacancy site introduced to the system creates a defect energy level in every single innumerable fractal energy gap in the Hofstadter butterfly. The wavefunctions of different defect levels have all different localization lengths depending on their fractal generations, and they can be described by a single universal function after an appropriate fractal scaling. We also show that each defect state has its own characteristic orbital magnetic moment, which is exactly correlated to the gradient of the energy level in the Hofstadter diagram. Probing the spatial nature of the defect-localized states provides a powerful way to elucidate the fractal nature of the Hofstadter butterfly.

cond-mat.mes-hall↗

arXiv2020

Mod-two APS index and domain-wall fermion

We reformulate the mod-two Atiyah-Patodi-Singer (APS) index in a physicist-friendly way using the domain-wall fermion. Our new formulation is given on a closed manifold, which is extended from the original manifold with boundary, where we instead give a fermion mass term changing its sign at the location of the original boundary. This new setup does not need the APS boundary condition, which is non-local. A mathematical proof of equivalence between the two different formulations is given by two different evaluations of the same index of a Dirac operator on a higher dimensional manifold. The domain-wall fermion allows us to separate the edge and bulk mode contributions in a natural but not in a gauge invariant way, which offers a straightforward description of the global anomaly inflow.

hep-th↗

arXiv2020

Atiyah-Patodi-Singer index on a lattice

We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS) index in lattice gauge theory, in which the index is given by the $η$ invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set of the free domain-wall fermion, we perturbatively show in the continuum limit that the curvature term in the APS theorem appears as the contribution from the massive bulk extended modes, while the boundary $η$ invariant comes entirely from the massless edge-localized modes.

hep-lat↗

arXiv2020

The Anomaly Inflow of the domain-wall fermion in odd dimension

In 1985, Callan and Harvey showed a view of gauge anomaly as a missing current into an extra-dimension, and the total contribution, including the Chern-Simons current in the bulk, is conserved. However in their computation, the edge and bulk contributions are separately evaluated and their cross correlations, which should be relevant at boundary, are simply ignored. This issue has been solved in many approaches. In this work, we revisit this issue with a complete set of eigenstates of free domain-wall Hamiltonian and give the systematic evaluation, easy to take in the higher mass correction and extend to the higher dimension.

hep-th↗

arXiv2020

A lattice formulation of the Atiyah-Patodi-Singer index

Atiyah-Singer index theorem on a lattice without boundary is well understood owing to the seminal work by Hasenfratz et al. But its extension to the system with boundary (the so-called Atiyah- Patodi-Singer index theorem), which plays a crucial role in T-anomaly cancellation between bulk- and edge-modes in 3+1 dimensional topological matters, is known only in the continuum theory and no lattice realization has been made so far. In this work, we try to non-perturbatively define an alternative index from the lattice domain-wall fermion in 3+1 dimensions. We will show that this new index in the continuum limit, converges to the Atiyah-Patodi-Singer index defined on a manifold with boundary, which coincides with the surface of the domain-wall.

hep-lat↗

arXiv2018

Comments on the Fractal Energy Spectrum of Honeycomb Lattice with Defects

We address the energy spectrum of honeycomb lattice with various defects or impurities under a perpendicular magnetic field. We use a tight-binding Hamiltonian including interactions with the nearest neighbors and investigate its energy structure for two different choices of point defects or impurities. In the first case, we fix a unit cell consisting of 8 lattice points and survey the energy eigenvalues in the presence of up to 2 point defects. Then it turns out that the existence of the fractal energy structure, called Hofstadter's butterfly, depends on the choice of defect pairs. In the second case, we extend the size of a unit cell which contains a single point defect and up to 32 lattice points. The fractal structures indeed appear for those cases and there exist a robust gapless point in the $E=0$ eV line without depending on both the size of unit cells and the shape of lattices. Therefore we keep an immortal butterfly since such a robust point corresponds to the center of a butterfly. Consequently we predict that the presence of a butterfly in a graph is equivalent to that of fractality.

cond-mat.mes-hall↗