Berry Phase Effects in Dipole Density and the Mott Relation

Abstract

We provide a unified semiclassical approach for thermoelectric responses of any observable represented by an operatorθ that is well-defined in periodic crystals. The Mott relation is established, in the presence of Berry-phase effects, for various physical realizations ofθ in electronic systems, including the familiar case of the electric current as well as the currently controversial cases of the spin polarization and spin current. In our theory the dipole density of a physical quantity emerges and plays a vital role, which contains not only the statistical sum of the dipole moment ofθ but also a Berry-phase correction.

Energy magnetization and transport in systems with a non-zero Berry curvature in a magnetic field

“…In the absence of an external magnetic field, the total electric current density is given by 6 [11,32],…”

“…Similarly, in response to a temperature gradient, without any external magnetic field, it generate a transverse Hall voltage, known as the anomalous Nernst effect [9,[12][13][14]. Coupled with the Boltzmann transport theory, the modified semiclassical equations have been employed to study transport in topological insulators [15], Chern Insulators [16], Weyl Semi-Metals [13,14,[17][18][19][20][21][22], Kondo Insulators [23], Rashba systems [24,25], optical lattices and quasicrystals [26,27], superconductors [28], non-Hermitian systems [29,30], as well as in various other systems [31][32][33][34][35][36][37]. Non-linear effects in transport have also been studied within this formalism [38][39][40][41][42][43].…”

“…d. It can be shown that (see Appendix C 1) M ij is completely anti-symmetric [32]. Then, in 3D universe (the sample can still be 2 dimensional), we can construct a vector, whose components are, mi = 1 2 ijk M jk [32]. It can be shown (see Appendix C 2) that this is identical to the i-th component of the orbital magnetic moment m k , defined in Eq.…”

“…. d. It can be shown that (see Appendix C 1) M ij is completely anti-symmetric [32]. Then, in 3D universe (the sample can still be 2 dimensional), we can construct a vector, whose components are, mi = 1 2 ijk M jk [32].…”

Intrinsic Nonlinear Spin Magnetoelectricity in Centrosymmetric Magnets

“…To evaluate the spin response, a conventional way is to introduce a fictitious (homogeneous) Zeeman-like field m that couples to spin in the form of −ŝ • m, with ŝ the spin operator. This auxiliary field is to be distinguished from the genuine magnetization of the system and is set to zero at the end of the calculation [17]. Under the m field and the electric field, the electronic enthalpy of the magnetic insulator system follows the relation dH = −s • dm − P • dE, where s and P denote the spin magnetization and the electric polarization of electrons, respectively.…”