"Electromagnetism in matter" is a research field to study various responses of the charge current, charge polarization, and magnetization to electromagnetic fields, in which we assume that matter and electromagnetic fields are classical. On the other hand, realistic materials in which we are interested are crystals, and their properties are mainly determined by quantum-mechanical electrons. However, some of the physical quantities introduced in "electromagnetism in matter" become ill-defined in realistic materials. One of my interests is to define such quantities and to find relations to various phenomena.

Another keyword, "gravity", is the weakest among four fundamental interactions and not thought to be important in condensed matter physics. In fact, it is known that nonelectromagnetic forces such as temperature gradient, rotation, and strain are described by gravity. I consider "gravity in matter" as a research field to study thermal transport and viscoelastic properties systematically and predict new phenomena with the help of analogy to "electromagnetism in matter". 

The spin Hall (SH) and spin Nernst (SN) effects are phenomena in which the spin current flows perpendicular to an electric field and a temperature gradient, respectively. However, in the presence of a spin-orbit coupling, spin is not conserved, and the spin current is not well defined. Another problem is that the Kubo formula of the SN conductivity has unphysical divergence towards zero temperature. It is natural to focus on the well-defined and directly observable spin, rather than the ambiguous spin current. First, I calculated the spin response to an electric-field gradient and derived a generic fomula using the Bloch wave functions [1]. Since such a gradient naturally appears at the boundaries, this response directly describes the spin accumulation. I also calculated the response using the Green's functions for the Rashba model with short-range nonmagnetic disorder and found the nonzero spin accumulation in spite of the vanishing SH conductivity.

Next, I considered the spin accumulation in the SN effect and calculated the spin response to the gradient of a temperature gradient. Since the spin accumulation comes from a dissipative Fermi-surface term, there is no difficulty regarding the ambiguity of the spin current or the unphysical divergence of the Kubo formula. The generalized Mott relation holds almost obviously in the absence of inelastic scattering. I found the vanishing thermal spin accumulation for the Luttinger model with short-range nonmagnetic disorder.

1. Atsuo Shitade and Gen Tatara, "Spin accumulation without spin current", Phys. Rev. B 105, L201202 (2022); arXiv:2112.11043.

2. Atsuo Shitade, "Spin accumulation in the spin Nernst effect", Phys. Rev. B 106, 045203 (2022); arXiv:2205.05872.

Spin polarization was reported in chiral molecules, such as DNA, when a charge current is applied. This phenomenon is called chirality-induced spin selectivity. Since the conventional spin-orbit coupling (SOC) is O(1/m^2) and negligibly small in light elements, the existence of a large unknown SOC is indicated. In order to unveil the origin of the SOC, we need to start from the relativistic Dirac Lagrangian. By choosing a curvilinear coordinate system and taking the nonrelativistic limit, I found what I call geometric SOC of O(1/m). It is estimated to be 160 meV, using typical parameters of DNA. Furthermore, I investigated the Edelstein effect in a coupled-helix model and found that the current-induced spin polarization changes its sign by the chirality of the helix and is large enough to experimentally measure.

A relativistic system of chiral fermions has the chiral anomaly and exhibits the chiral vortical effect (CVE) in which the charge current flows parallel to the vorticity. I took the spin-vorticity coupling into account and calculated the transport charge current by subtracting the magnetization one from the local one [2]. I found that the transport charge current vanishes in the relativistic system, and the CVE cannot be observed in transport experiments. Furthermore, the CVE cannot be observed  even when  the chiral imbalance is dynamically generated applying both electric and magnetic fields. On the other hand, in nonrelativistic systems that belong to some chiral point groups, such as tellurium, the anisotropic CVE can be observed.

The axial magnetic effect (AME) is a phenomenon in which the energy current is induced by an axial magnetic field and reciprocal to the CVE in the relativistic case. I calculated the energy current by imposing the open boundary conditions and found that its average vanishes owing to the surface contribution [3]. The axial gauge field is in fact the spatially modulated Zeeman field and induces the spatially modulated energy magnetization. I also calculated the energy magnetization by dealing with such spatial dependence as a parameter and imposing the periodic boundary conditions and found that the AME energy current is just the magnetization one.

1. Atsuo Shitade and Emi Minamitani, "Geometric spin-orbit coupling and chirality-induced spin selectivity", New J. Phys. 22, 113023 (2020); arXiv:2002.05371.

2. Atsuo Shitade, Kazuya Mameda, and Tomoya Hayata, "Chiral vortical effect in relativistic and nonrelativistic systems", Phys. Rev. B 102, 205201 (2020); arXiv:2008.13320.

3. Atsuo Shitade and Yasufumi Araki, "Magnetization energy current in the axial magnetic effect", Phys. Rev. B 103, 155202 (2021); arXiv:2102.09236.

Multipole moments characterize the anisotropy of the charge and charge current densities in "electromagnetism in matter". They have been studied traditionally in strongly correlated electron systems and recently in higher-order topological insulators and locally noncentrosymmetric systems. In particular, the magnetic quadrupole moment (MQM) has been believed to be an important ingredient for the magnetoelectric (ME) effect. Nonetheless, we had not established how to calculate multipole moments, except for the lowest-order charge polarization and orbital magnetization, in realistic materials.

I defined the orbital MQM and proved its direct relation to the orbital ME susceptibility relying only on thermodynamics and "electromagnetism in matter" [1]. This relation indicates that the MQM is a microsopic origin of the ME effect. I derived a quantum-mechanical formula of the orbital MQM in crystals by using the gauve-covariant gradient expansion of the Keldysh Green's function. I also calculated the orbital MQM and ME susceptibility in a locally noncentrosymmetric antiferromagnet BaMn2As2 and found that the orbital contribution to the ME susceptibility is comparable with or even dominant over the spin contribution.

It is possible to define the spin MQM similarly and prove a direct relation to the spin ME susceptibility. Furthermore, I showed that the spin MQM plays an essential role in what I call gravito-ME effect where the magnetization is induced by a temperature gradient [2]. In order to obtain the correct gravito-ME susceptibility, we need to subtract the spin MQM from the Kubo formula, because it diverges at zero temperature and is unphysical. The obtained gravito-ME susceptibility vanishes at zero temperature and is related to the ME susceptibility by the Mott relation in electronic systems. I calculated it in a ferromagnetic metal with the Rashba spin-orbit interaction and found it dramatically enhanced at band edges.

In magnetic insulators, the gravito-ME effect is expected to occur if the symmetry allows, since magnons carry spin. I demonstrated that this magnon gravito-ME effect occurs only when a unit cell contains multiple magnetic ions with different g factors [3]. I calculated the gravito-ME susceptibility in an antiferromagnetic insulator based on the first ME material Cr2O3, which turns out to be very small.

Dr. Daido defined the electric quadrupole moment [4], which is a candidate order parameter of the electronic nematic phase. I contributed to this work by deriving the same result with use of the Kubo formula.

1. Atsuo Shitade, Hikaru Watanabe, and Youichi Yanase, "Theory of orbital magnetic quadrupole moment and magnetoelectric susceptibility", Phys. Rev. B 98, 020407(R) (2018); arXiv:1803.00217.

2. Atsuo Shitade, Akito Daido, and Youichi Yanase, "Theory of spin magnetic quadrupole moment and temperature-gradient-induced magnetization", Phys. Rev. B 99, 024404 (2019); arXiv:1811.05596.

3. Atsuo Shitade and Youichi Yanase, "Magnon gravitomagnetoelectric effect in noncentrosymmetric antiferromagnetic insulators", Phys. Rev. B 100, 224416 (2019); arXiv:1906.12031.

4. Akito Daido, Atsuo Shitade, and Youichi Yanase, "Thermodynamic approach to electric quadrupole moments", Phys. Rev. B 102, 235149 (2020); arXiv:2010.13999.

Orbital angular momentum (OAM) is one of the most important physical quantities in both classical and quatum mechanics. In condensed matter physics, it has been studied for a long time whether a chiral superconductor (SC), whose Cooper pairs have nonzero OAM, has nonzero OAM in the bulk. In crystals and infinite systems, the ordinary definition with use of the position operator does not make sense, and hence we need to seek for an alternative definition.

I noticed that the OAM can be regarded as a momentum analog of the charge polarization (CP) [1]. I introduced a gravitational gauge field coupled to the momentum and momentum current densities by imposing the local space translation symmetry. An angular velocity, which is conjugate to the OAM, is described by the antisymmetric part of a gravitational electric field, and hence the OAM is the momentum polarization. Following the seminal work on the CP, I defined the OAM by the momentum current density induced by an adiabatic deformation of the Hamiltonian and calculated it in chiral [1] and topological SCs [2,3]. 

1. Atsuo Shitade and Taro Kimura, "Bulk angular momentum and Hall viscosity in chiral superconductors", Phys. Rev. B 90, 134510 (2014); arXiv:1407.1877.

2. Atsuo Shitade and Yuki Nagai, "Orbital angular momentum in a nonchiral topological superconductor", Phys. Rev. B 92, 024502 (2015); arXiv:1501.06677.

3. Atsuo Shitade and Yuki Nagai, "Orbital angular momentum in a topological superconductor with Chern number higher than 1", Phys. Rev. B 93, 174517 (2016); arXiv:1512.07997.

The thermal Hall (TH) effect is a phenomenon in which the heat current flows perpendicular to a temperature gradient. Theoretically, we need to introduce Luttinger's gravitational potential to justify the Kubo formula, since the temperature gradient is a statistical force. Furthermore, in order to obtain the correct TH conductivity, we need to add the so-called heat magnetization (HM) to the Kubo formula, because it diverges at zero temperature and is unphysical. we had not established how to calculate the HM, which is a heat analog of the orbital magnetization.

I introduced a gravitational gauge field coupled to the Hamiltonian and energy current densities by imposing the local time translation symmetry [1,2]. This introduces a gravitational electric field that describes the temperature gradient and a gravitational magnetic field, with which the HM is defined thermodynamically. I also constructed a general formalism based on the Keldysh Green's functions to calculate the Kubo formula and HM. Later, I applied it to a disordered Weyl ferromagnet and reproduced the generalized Wiedemann-Franz law [3].

As a byproduct, I clarified the reason why Luttinger's gravitational potential is called gravity. The above local time translation symmetry means the invariance of a theory under spacetime distortion, which is indeed gravity. In a theory of gravity, the gravitational gauge and electromagnetic fields are called vielbien and torsion, respectively. 

1. Atsuo Shitade, "Theory of Charge and Heat Polarizations with the Keldysh Formalism", J. Phys. Soc. Jpn. 83, 033708 (2014); arXiv:1312.3448.

2. Atsuo Shitade, "Heat transport as torsional responses and Keldysh formalism in a curved spacetime", Prog. Theor. Exp. Phys. 2014, 123I01 (2014); arXiv:1310.8043.

3. Atsuo Shitade, "Anomalous Thermal Hall Effect in a Disordered Weyl Ferromagnet", J. Phys. Soc. Jpn. 86, 054601 (2017); arXiv:1610.00390.