Continuous quantum phase transition between two topologically distinct valence bond solid states associated with the same spin value

We propose a one-dimensional quantum Heienberg spin-2 chain, which exhibits two topologically distinct valence bond solid states in two different solvable limits. We then construct the phase diagram and study the quantum phase transition between these two states using infinite time evolving block decimation algorithms. From the scaling relation between the entanglement entropy and the correlation length, we determine that the central charge for the underlying critical conformal field theory is around two.

Figure 1

Ground-state energy density varies as a function of $J_3$ for fixed $J_2$. The local matrix dimension $D$ is set to be $300$ in the calculation.

Figure 2

First derivative of the ground-state energy density varies as a function of $J_3$ for fixed $J_2$.

Figure 3

The second-order derivative of the ground-state energy density varies as a function of $J_3$ for fixed $J_2$.

Figure 4

Ground-state phase diagram of the model Hamiltonian, Eq. (1).

Figure 5

(a) The spin-spin correlation length varies as a function of $J_3$ for two typical values of $J_2$. (b) The entanglement entropy varies as a function of $J_3$ for two typical values of $J_2$. The local matrix dimension $D$ is fixed at $300$ in this calculation.

Figure 6

The 36 lowest values of the entanglement spectra for the VBS${}_{3/2}$ and VBS${}_1$ phases on two sides of the critical points. (a) $(J_2,J_3)=(0.1,0.029)$, (b) $(J_2,J_3)=(1.0,0.25)$, (c) $(J_2,J_3)=(3.0,0.53)$. A light gray vertical line is put on the critical point to separate the two phases. Each entanglement spectrum is represented by a small cross, and the degenerate spectra are spatially staggered a little bit in horizontal direction to distinguish and count them. In this calculations, the local matrix dimension $D$ is set to be $300$.

Figure 7

The scaling behavior of the entanglement entropy $S_e$ as the correlation length $\xi$ in the vicinity of different critical points. (a) $(0.1,0.029)$, (b) $(1.0,0.25)$, (c) $(2.0,0.42)$, (d) $(3.0,0.54)$. The black solid line is a reference with a central charge $c=2$ and the red squares are numerical results. In our calculation, the local matrix dimension increases from $12$ to $600$.