Density Matrix Renormalization Group in the Heisenberg Picture

In some cases the state of a quantum system with a large number of subsystems can be approximated efficiently by the density-matrix renormalization group, which makes use of redundancies in the description of the state. Here we show that the achievable efficiency can be much better when performing density-matrix renormalization group calculations in the Heisenberg picture, as only the observable of interest but not the entire state is considered. In some nontrivial cases, this approach can even be exact for finite bond dimensions.

Figure 1

The time evolution, $⟨{\sigma }_5^z⟩(t)$ for a model described by Eqs. (4) respectively (5) with parameters $N=10$, $B_z=0.8$, $J_x=0.5$, $J_y=0.4$, $J_z=0.01$, ${\Gamma }_u=0.1$, and ${\Gamma }_d=0.1$. (a) The exact solution, (b) ${\delta }_{\mathrm{DMRG}}$ for DMRG simulations in the Schrödinger picture for $d=20$ (blue), $d=40$ (green), $d=60$ (red), $d=80$ (cyan), and $d=100$ (magenta), (c) ${\delta }_{\mathrm{H}\mathrm{\text{−}}\mathrm{DMRG}}$ for DMRG simulations in the Heisenberg picture.

Figure 2

Dynamics of a chain with $N=100$ spins and $B_z=0.8$, $J_x=0.5$, $J_y=0.4$, $J_z=0.01$, ${\Gamma }_u=0.01$, and ${\Gamma }_d=0.1$. (a) $⟨{\sigma }_{50}^z⟩(t)$ as given by Eq. (4) (DMRG) for $d=20$ (blue), $d=40$ (green), $d=60$ (red), $d=80$ (cyan), and $d=100$ (magenta). (b) $⟨{\sigma }_{50}^z⟩(t)$ as given by Eq. (5) (H-DMRG) for $d=20$, 40, 60, 80, and 100. (c) $|[⟨{\sigma }_{50}^z⟩(t){]}_{d1}-[⟨{\sigma }_{50}^z⟩(t){]}_{d2}|$ as given by Eq. (4) (DMRG) for $(d1,d2)=(40,20)$ (blue), $(d1,d2)=(60,40)$ (green), $(d1,d2)=(80,60)$ (red), and $(d1,d2)=(100,80)$ (cyan). (d) $|[⟨{\sigma }_{50}^z⟩(t){]}_{d1}-[⟨{\sigma }_{50}^z⟩(t){]}_{d2}|$ as given by Eq. (5) (H-DMRG) for $(d1,d2)=(40,20)$, (60, 40), (80, 60), and (100, 80). With increasing bond dimension, results converge much faster in the Heisenberg than in the Schrödinger picture.

Figure 3

Truncation errors $ϵ$ at $t=10$ for the Schrödinger picture DMRG (blue) and H-DMRG (green) for $N=20$, 40, 60, and 80 and fixed bond dimension $d=60$. $B_z=0.8$, $J_x=-0.5$, $J_y=0.4$, $J_z=0.1$, ${\Gamma }_u=0.1$, and ${\Gamma }_d=0.2$. H-DMRG truncations saturate at $ϵ\le 0.96$.