### Theory of shift current in Anderson insulator

The shift current induced under light irradiation in a crystal without inversion symmetry is a current due to quantum geometry that is essentially different from the normal transport current. Therefore, it is expected that the tolerance to the Anderson localization effect due to the disorder will be significantly different. If only the transport current by the optical carrier is suppressed, it will lead to the improvement of the efficiency of the solar cell and the suppression of the noise in photodetector, so it has been recognized as an important problem in application. For a one-dimensional Anderson insulator model, the shift current is calculated by the Keldysh non-equilibrium Green’s function method. We have found that, when the disorder strength is moderate and the dissipation due to electron-phonon interaction is included, the magnitude of the shift current hardly depends on the localization length. This is a behavior that is completely different from the transport current. It was found that when the disorder is further strengthened to exceed the energy gap, the conduction band and the valence band are mixed, so that the shift current also decreases.

(Left figure) Conceptual diagram of shift current in Anderson insulator. (Right figure) The disorder strength (Vmd) dependence of the shift current. Up to Vmd = 0.5, the shift current hardly changes even though the wave function is already localized. If the disorder is further strengthened, the shift current will also decrease due to the mixing between the bands.

H. Ishizuka and N. Nagaosa, “Theory of bulk photovoltaic effect in Anderson insulator ”, Proc. Nat. Acad. Sci., 118(10) e2023642118 (2021).

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### Theory of emergent inductor

The emergent electric and magnetic fields originate from the Berry phase associated with the spin direction *n*. While the emergent magnetic field is detected as the topological Hall effect, the emergent electric field is given by the dynamics of the spin structures and can be detected as the voltage drop.

For the current-driven motion of the spiral spin structure, we find that the emergent electric field and the consequent voltage drop are proportional to the time-derivative of the current density, corresponding to the inductance given by

where *A* and ℓ are the cross-section and length of the sample, λ the period of the spiral, *j*_{int} is related to the magnetic anisotropy energy, and γ is a constant of the oder of unity. This emergent inductor does not require any complex structures such as coil and core, and the inductance is inversely proportoinal to the cross section of the sample. Therefore, it is advantageous for the miniturization of the devices.

(Left) Principles of emergent inductor. The spin structure of spiral magnet (upper), and the modification of the spin structure under the current-driven motion (lower). When the translational motion X occurs, the spins are tilted by the angle f. When ac current is applied, the voltage drop along the direction of the wavevector is induced proportional to the time-derivative of the current density. (Right) Various spin structures showing the inductance.

N. Nagaosa, “Emergent inductor by spiral magnets”, Japanese Journal of Applied Physics, 58(12) 12909 (2019)

### Theory of shift current in noncentrosymmetric materials

It has been known that the dc current is induced by the photo-excitation without the external electric field in noncentrosymmetric materials. This phenomenon, called ”shift current”, is driven by the geometrical Berry phase of the Bloch electrons in solids, and is one of the topological currents. We have studied theoretically the physical properties of this shift current such as the effects of the relaxation due to impurity scattering and electron-phonon interaction, the acceleration by external electric field, and quantum shot noise of shift current. We have revealed the followings: (i) the relaxation affects the dependence of the shift current on the intensity of photo-excitation, while shift current does not depend on it when the photo-excitation is weak. (ii) I-V characteristics of the shift current under the external electric field depends strongly on the relaxation, and the lower mobility of the photo-carriers gives the higher energy-conversion ratio, which is in sharp contrast to the conventional case. (iii) The noise of the shift current is reduced when the band-width is small. In addition, we have estimated the shift current in SbSI, and obtained the good agreement with the experimental result quantitatively including the energy dependence of the incident light.

Shift current in ferroelectric semiconductor SbSI. Left: Crystal structure of SbSI. Middle: Time dependence of the emitted THz light. Right: The incident-light energy dependence of the shift current. Theory and experiment shows good agreement quantitatively.

M. Sotome, et al., “Spectral dynamics of shift current in ferroelectric semiconductor SbSI” Proceedings of National Academy of Sciences of the United States of America 116, 1929-1933 (2019)

### Skyrmion dynamics in disordered magnet

Distinct from the magnetic textures such as domain wall, helical structure, and vortex, the topological particle in magnets, skyrmion, has unique features. A typical one is the extremely small critical current density *j*_{c} for the current-driven motion of the skyrmion. The finite *j*_{c} indicates the pinning effect due to the disorder such as impurities and defects. We study the current-driven skyrmion dynamics with disorders by a numerical simulation of Landau-Lifshitz-Gilbert equation, and four different skyrmion phases are found: (A) pinned state, (B) depinned state, (C) skyrmion multiplication/annihilation, and (D) segregation of skyrmions, as the current density increases, while only two phases (A) and (B) appear in the weak disorder case. The new phases (C) and (D) are due to the deformation of skyrmions and spin wave emission, respectively. In addition, we have developed the replica field theory of skyrmion glass state, and revealed the dramatic differences between helical and skyrmion phases by calculating the physical properties such as domain size, pinning frequency, nonreciprocal collective modes, optical conductivity, and magnetic resonance.

Skyrmion multiplication

Due to the strong impurity effect, the current driven skyrmion splits into two or more.

W. Koshibae and N. Nagaosa, Scientific Reports 8, 6328 (2018)