Real time evolution in quantum many-body systems with unitary perturbation theory

We develop an analytical method for solving real time evolution problems of quantum many-body systems. Our approach is a direct generalization of the well-known canonical perturbation theory for classical systems. Similar to canonical perturbation theory, secular terms are avoided in a systematic expansion and one obtains stable long-time behavior. These general ideas are illustrated by applying them to the spin-boson model and studying its nonequilibrium spin dynamics.

Figure 1

The forward-backward transformation scheme induces a nonperturbative solution of the Heisenberg equations of motion for an operator. $U$ denotes the full unitary transformation that relates the $B=0$ to the $B=\infty$ basis (Ref. 15).

Figure 2

Real time evolution of the spin expectation value $⟨{\sigma }_z\left(t\right)⟩$ starting from a polarized spin in the $z$ direction with a relaxed Ohmic bath (see text) for two different values of $\alpha$ and ${\omega }_c/\Delta =10$. The full lines are the flow equation results, the dashed lines are TD-NRG curves for $\Lambda =2.0$, and the dotted lines are for $\Lambda =1.41$. The TD-NRG results are courtesy of Anders (see Ref. 11).The various curves agree extremely well except for very long times shown in the insets.

Figure 3

Ground-state expectation value of ${\sigma }_x$. Comparison of flow equation results (curves) and NRG data (squares) from Ref. 19 for an Ohmic bath with damping $\alpha$. The results are for ${\omega }_c/\Delta =25,28.6,33.3,40,50,66.7,100$ from top to bottom.