Parallel algorithm for spin and spin-lattice dynamics simulations

To control numerical errors accumulated over tens of millions of time steps during the integration of a set of highly coupled equations of motion is not a trivial task. In this paper, we propose a parallel algorithm for spin dynamics and the newly developed spin-lattice dynamics simulation [P. W. Ma et al., Phys. Rev. B 78, 024434 (2008)]. The algorithm is successfully tested in both types of dynamic calculations involving a million spins. It shows good stability and numerical accuracy over millions of time steps $\left(\sim 1\text{ns}\right)$. The scheme is based on the second-order Suzuki-Trotter decomposition (STD). The usage can avoid numerical energy dissipation despite the trajectory and machine errors. The mathematical base of the symplecticity, for properly decomposed evolution operators, is presented. Due to the noncommutative nature of the spin in the present STD scheme, a unique parallel algorithm is needed. The efficiency and stability are tested. It can attain six to seven times speed up when eight threads are used. The run time per time step is linearly proportional to the system size.

Figure 1

(Color online) Schematic picture of 1D spin chain containing ten spins, where the Suzuki-Trotter decomposition is (top) sequential and (bottom) parallelize.

Figure 2

(Color online) A two-dimensional system is cut into 25 link cells with periodic boundary condition. Cells are allocated into group A to group H.

Figure 3

(Color online) The actual run time of spin-only system (top) versus number of threads with $16000$, $54000$, $128000$, $250000$, and $1024000$ spins, (bottom left) versus number of spins with threads from 1 to 4, and (bottom right) versus number of spins with threads from 5 to 8.

Figure 4

(Color online) The efficiency of spin-only system (left) versus number of threads with $16000$, $54000$, $128000$, $250000$, and $1024000$ spins and (right) versus number of spins with threads from 1 to 8.

Figure 5

(Color online) The actual run time of spin-lattice system (top) versus number of threads with $16000$, $54000$, $128000$, $250000$, and $1024000$ spins, (bottom left) versus number of spins with threads from 1 to 4, and (bottom right) versus number of spins with threads from 5 to 8.

Figure 6

(Color online) The efficiency of spin and lattice coupled system (left) versus number of threads with $16000$, $54000$, $128000$, $250000$, and $1024000$ spins and (right) versus number of spins with threads from 1 to 8.

Figure 7

The total energy per atom versus time for a microcanonical ensemble simulation of the dynamical relaxation of a $180°$ domain wall. The energy is subtracted by their initial values. Two algorithms, the $\left(\mathcal{S},\mathcal{F},\mathcal{P},\mathcal{F},\mathcal{S}\right)$ and the $\left(\mathcal{S},\mathcal{P},\mathcal{F},\mathcal{P},\mathcal{S}\right)$, in both of which the parallel algorithm for STD is incorporated, are compared.