$SU(3)$ Semiclassical Representation of Quantum Dynamics of Interacting Spins

We present a formalism for simulating quantum dynamics of lattice spin-1 systems by first introducing local hidden variables and then doing semiclassical (truncated Wigner) approximation in the extended phase space. In this way, we exactly take into account the local on-site Hamiltonian and approximately treat spin-spin interactions. In particular, we represent each spin with eight classical $SU(3)$ variables. Three of them represent the usual spin components and five others are hidden variables representing local spin-spin correlations. We compare our formalism with exact quantum dynamics of fully connected spin systems and find very good agreement. As an application, we discuss quench dynamics of a Bose-Hubbard model near the superfluid-insulator transition for a 3D lattice system consisting of 1000 sites.

Figure 1

Comparison of the dynamics of $⟨{\widehat{S}}_x⟩$ for a spin-1 particle initially pointed in the $x$ direction subjected to the Hamiltonian $\frac{1}{2}{\widehat{S}}_z^2-{\widehat{S}}_z$. The dynamics are calculated with exact diagonalization (solid, blue), $SU(3)$ TWA (dashed, red), and $SU(2)$ TWA (dotted, yellow).

Figure 2

Comparison of the dynamics of $⟨{\widehat{S}}_z^2⟩$ for a fully connected system with $M$ spins due to the Hamiltonian (11) calculated with exact diagonalization (solid, blue), $SU(3)$ TWA (dashed, red), and $SU(2)$ TWA (dotted, yellow). Initially all spins are pointed in the $x$ direction. On the left are the results for various coupling strengths $Jz$, where $z=M-1$ is the connectedness (the inset in the bottom plot shows the same plot for times 0 to 50). On the right are the results for different system sizes.

Figure 3

The dynamics of the order parameter ${\rho }_s$ for the effective Bose-Hubbard model (15), calculated with $SU(3)$ TWA. The top row shows the results starting in the Mott insulator phase. The bottom row begins in the superfluid phase: in the left-hand plot, the system is quenched to $J=0$, while in the right-hand plot the system is quenched to $J\overline{n}z/U=0.1$.