Angular-time evolution for the Affleck-Kennedy-Lieb-Tasaki chain and its edge-state dynamics

We study the angular-time evolution that is a parameter-time evolution defined by the entanglement Hamiltonian for the bipartitioned ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) chain with an open boundary. In particular, we analytically calculate angular-time spin correlation functions $\langle S_n^{\alpha }\left(\tau \right)S_n^{\alpha }\left(0\right)\rangle$ with $\alpha =x,y,z$ using the matrix-product-state (MPS) representation of the valence-bond-solid state with edges. We also investigate how the angular-time evolution operator can be represented in the physical spin space with the use of gauge transformation for the MPS. We then discuss the physical interpretation of the angular-time evolution in the AKLT chain.

Figure 1

Setup of the system and reservoir parts in a half-infinite chain. The left edge corresponds to the $i=1$ site, and the system part contains $l$ sites from the left edge.

Figure 2

Angular-time spin correlations for $l=4$ and $n=4$ at $\theta =0$. The real and imaginary parts of the angular-time spin correlations for $\alpha =x$ and $y$ exhibit oscillating behavior. The frequency of the oscillation is ${\mathrm{\Omega }}_4\simeq 0.0247$ (the period is $T\simeq 254$), while those for $\alpha =z$ have no $\tau$ dependence. We note that the amplitude of the oscillation $D_{l,n}$ decays rapidly as $n$ shifts away from the entangle point ($n=l$).

Figure 3

Angular-time correlations of $M_l^{\alpha }\left(\tau \right)$ for $l=4$ at $\theta =0$. The real and imaginary parts of the angular-time spin correlations for $\alpha =x$ and $y$ exhibit oscillating behavior. The frequency of the oscillation is identical to that in Fig. 2. Meanwhile, the amplitude of the oscillation rapidly approaches $1/4$ as $l$ increases.

Figure 4

Effective $S=1$ spin operator with the angular-time evolution. The arrows represent that the time evolution operator $U$ in the auxiliary space can be gauged out toward the edge sites attached with $\mathrm{v}$.