Transport in anisotropic model systems analyzed by a correlated projection superoperator technique

By using a correlated projection operator, the time-convolutionless (TCL) method to derive a quantum master equation can be utilized to investigate the transport behavior of quantum systems as well. Here, we analyze a three-dimensional anisotropic quantum model system according to this technique. The system consists of Heisenberg coupled two-level systems in one direction and weak random interactions in all other ones. Depending on the partition chosen, we obtain ballistic behavior along the chains and normal transport in the perpendicular direction. These results are perfectly confirmed by the numerical solution of the full time-dependent Schrödinger equation.

Figure 1

Partition schemes for investigating the transport perpendicular (a) or parallel (b) to the spin chains. Each spin is represented by a dot, solid lines indicate Heisenberg interactions along the chains, dashed lines represent random interactions. The diagonal couplings within each plane have been left out for clarity [except the lower left corner of (a)].

Figure 2

$N$ subunits with ground state and first excitation band of width $\delta \epsilon$ containing $n$ energy levels each. Black dots specify the initial states used.

Figure 3

Perpendicular transport: probability to find the excitation in subunit $\mu =1,2,3$. Comparison of the numerical solution of the Schrödinger equation (crosses) and second-order TCL (lines) ($N=3$, $n=600$, ${\lambda }_R=5\times {10}^{-4}\Delta E$, ${\lambda }_H=6.25\times {10}^{-2}\Delta E$).

Figure 4

Comparison of the eigenvalues of ${\widehat{H}}_L$ and a random matrix drawn from a Gaussian unitary ensemble ($n=600$, ${\lambda }_R=5\times {10}^{-4}\Delta E$).

Figure 5

Parallel transport: probability to find the excitation in the first subunit $(\mu =1)$. Comparison of the numerical solution of the Schrödinger equation (crosses) and second-order TCL (solid line). (Same parameters as for Fig. 3.)

Figure 6

Variance of an excitation initially at subunit ${\mu }_0=150$. Second-order TCL prediction (crosses) and quadratic fit (solid line) ($N=300$, $n=600$, ${\lambda }_R=5\times {10}^{-4}$, ${\lambda }_H=6.25\times {10}^{-2}$).