Semiclassical dynamics of spin density waves

We present a theoretical framework for equilibrium and nonequilibrium dynamical simulation of quantum states with spin-density-wave (SDW) order. Within a semiclassical adiabatic approximation that retains electron degrees of freedom, we demonstrate that the SDW order parameter obeys a generalized Landau-Lifshitz equation. With the aid of an enhanced kernel polynomial method, our linear-scaling quantum Landau-Lifshitz dynamics (QLLD) method enables dynamical SDW simulations with $N\simeq {10}^5$ lattice sites. Our real-space formulation can be used to compute dynamical responses, such as the dynamical structure factor, of complex and even inhomogeneous SDW configurations at zero or finite temperatures. Applying the QLLD to study the relaxation of a noncoplanar topological SDW under the excitation of a short pulse, we further demonstrate the crucial role of spatial correlations and fluctuations in the SDW dynamics.

Figure 1

Phase diagram of the half-filled triangular-lattice Hubbard model with NN hopping $t$. With increasing Hubbard repulsion, the four different phases are as follows: a paramagnetic metal, an incommensurate spiral metal, a collinear commensurate zigzag SDW, and the ${120}^{\circ }$ order. (a) average spin length $\langle \left|\mathbf{m}\right|\rangle$ as a function of $U/t$. The two commensurate SDW phases, collinear zigzag and ${120}^{\circ }$ orders, are shown in (b) and (c), respectively.

Figure 2

Comparison of the dynamical structure factor of the Hubbard model on a $4\times 4$ square lattice for $t=1$ and $U=7.33$. (a) The exact diagonalization result at $T=0$, where a Lorentzian broadening factor $\eta =0.02$ is used. (b) The semiclassical SDW calculation in the adiabatic approximation described by Eq. (9). (c) The real-space TDHF calculation. For (b) and (c), the temperature was set $T={10}^{-4}$ for generating initial configurations, and the results were averaged over 27 independent runs starting from different initial configurations.

Figure 3

Dynamical structure factor $\mathcal{S}(\mathbf{k},\omega )$ computed using the semiclassical SDW dynamics equation Eq. (9) for the ${120}^{\circ }$ order in (a) Hubbard model and (b) Anderson-Hubbard model with an on-site disorder. The lattice size is $120\times 120$. The Hubbard parameter is $U=7.33$ and the standard deviation of the on-site disorder is ${\sigma }_V=0.45$, in units of NN hopping $t_{\mathrm{nn}}$.

Figure 4

(a) Configuration of the topological triple-$\mathbf{Q}$ SDW with a quadrupled unit cell; the four spins in the unit cell point to corners of a regular tetrahedron. (b) Time dependence of the effective electron temperature. Also shown is the pump pulse with a Gaussian profile: $Q\left(t\right)\propto e^{-{(t-t_p)}^2/w^2}$, where $t_p=15$, and $w=5$, in units of $1/t_{\mathrm{nn}}$. The green dashed line marks the equilibrium transition temperature of the tetrahedral SDW. (c) The SDW order parameter $\mathcal{M}$, average normalized scalar spin chirality $\overline{{\chi }_{▵}}$, standard deviation of the scalar chirality ${\sigma }_{{\chi }_{▵}}$, and the electron energy gap ${\varepsilon }_{\mathrm{gap}}$ as a function of time. Both $\mathcal{M}$ and ${\varepsilon }_{\mathrm{gap}}$ are normalized to its maximum value. (d) The transverse and longitudinal conductivity (normalized to the quantized $e^2/h$) as a function of time. The (green) shaded area indicates the temporal window when the electronic gap is closed.

Figure 5

(Top panels) Spatial distribution of the normalized plaquette scalar spin chirality ${\chi }_{▵}$ at time (a) $t=5$ and (b) $t=40$ from QLLD simulations on a triangular lattice with $120\times 120$ sites. (c) Histogram of the normalized plaquette chirality ${\chi }_{▵}$ at varying simulation time. The corresponding electronic DOS near the Fermi level are shown in (d).