Operator space entanglement entropy in a transverse Ising chain

The efficiency of time-dependent density matrix renormalization group methods is intrinsically connected to the rate of entanglement growth. We introduce a measure of entanglement in the space of operators and show, for a transverse Ising spin-$1∕2$ chain, that the simulation of observables, contrary to the simulation of typical pure quantum states, is efficient for initial local operators. For initial operators with a finite index in Majorana representation, the operator space entanglement entropy saturates with time to a level which is calculated analytically, while for initial operators with infinite index the growth of operator space entanglement entropy is shown to be logarithmic.

Figure 1

Entanglement entropy for finite-index operators ${\mid X_1⟩}_{♯}$ (black) and ${\mid X_1{Y}_1⟩}_{♯}=i{\mid {\sigma }_1^z⟩}_{♯}$ (gray) compared to theoretic saturation value for $t\to \infty$ and $h=1$ (thick lines). Three different values of magnetic field are considered: $h=1$ (solid curve), $h=0.5$ (dotted curve), $h=2$ (dashed curve), and $h=3$ (dash-dotted curve). We set $2L=200$, such that no finite-size effects were noticable.

Figure 2

Entanglement entropy for infinite-index operators ${\mid F⟩}_{♯}$ (black) and ${\mid F{Y}_1⟩}_{♯}={\mid {\sigma }_1^y⟩}_{♯}$ (gray) and different magnetic fields (same styling as in Fig. 1) as they appear in the legend from top to bottom. Thick dashed lines correspond to $(1∕3)\ln t$ and $(1∕6)\ln t$.