Microscopic theory of spin toroidization in periodic crystals

Using the semiclassical theory of electron dynamics, we derive a gauge-invariant expression for the spin toroidization in a periodical crystal. We show that the spin toroidization is comprised of two contributions: One is due to the configuration of a classical spin array, while the other comes from the coordinate shift of the electron as spin carrier in response to the inhomogeneous magnetic field. We then establish a direct and elegant relation between our spin toroidization and the antisymmetric magnetoelectric polarizability in insulators. Finally, we demonstrate our spin toroidization in a tight-binding model and show that it is a genuine bulk quantity.

Figure 1

Sample with nonzero surface magnetization. Red arrows show the direction of the surface magnetization on the left and right surfaces.

Figure 2

Spin toroidization of a tight-binding model. Panel (a) is part of a periodic crystal. $t_1$ and $t_2$ is the nearest neighbor hopping strength. The red arrow on each lattice site indicates the direction of the local exchange field. The lattice constant is $a/2$. Panel (b) is the calculated spin toroidization (in units of $g{\mu }_B/4a$) as a function of the chemical potential. The shaded areas correspond to energy gaps. The parameters are chosen as follows: $t_1=0.3\mathrm{\Delta }$ and $t_2=0.15\mathrm{\Delta }$.

Figure 3

Toroidization in the bulk and in finite samples for $\mu =-0.9\mathrm{\Delta }$ (inside the band gap). The horizontal axis is the number of lattice sites $N$ on one edge; the vertical axis is the toroidization in units of $g{\mu }_B/4a$. The black line is the bulk value. The red and blue symbols are the finite-sample results for the toroidization, and their quadratic fittings based on Eq. (25) are displayed as the red and blue curves, respectively. In the two equations, $x$ has the meaning of $1/N$. The inset illustrates the local exchange order in the finite sample, from which the hoppings can be derived via Fig. 2.

Figure 4

Toroidization for finite samples with surface perturbation. The axes and symbols have the same meaning as in Fig. 3. The inset shows the configuration of the finite sample. The layer with blue arrows is the additional perturbing layer. Here the original sample with red arrows has even $N$ along each edge. The odd $N$ case can be constructed in the same way. The hopping strength between the perturbation layer and the original sample is derived based on the known hopping strengths in the original sample in such a way that $t_{ij}$ still alternates between $t_1$ and $t_2$ for the whole sample.