Multipartite entanglement measures and quantum criticality from matrix and tensor product states

We compute the multipartite entanglement measures such as the global entanglement of various one- and two-dimensional quantum systems to probe the quantum criticality based on the matrix and tensor product states (MPSs and TPSs). We use the infinite time-evolving block decimation (iTEBD) method to find the ground states numerically in the form of MPSs and TPSs, and then evaluate their entanglement measures by the method of tensor renormalization group (TRG). We find that these entanglement measures can characterize the quantum phase transitions by their derivative discontinuity right at the critical points in all models considered here. We also comment on the scaling behaviors of the entanglement measures by the ideas of quantum-state renormalization group transformations.

Figure 1

Magnetizations vs transverse magnetic field of $\mathit{XY}$ spin chain using MPS with bond dimension $\chi =16$. It indicates a second-order phase transition from $F$ to $P$ phase at $h_c\simeq 1.005$(Ising, $\gamma =0$) and $h_c\simeq 1.005$($\mathit{XY}$, $\gamma =1/2$). However, it does not capture the phase transition from $O$ to $F$.

Figure 2

Entanglement measures vs transverse magnetic field of $\mathit{XY}$ spin chain using MPS with bond dimension $\chi =16$. (a) GE vs $h$. (b) $S_{\mathrm{BP}}$ vs $h$. Note that near the critical point $h_c\approx 1.005$, the first derivative of GE shows $\log {}_2|h-h_c|$ scaling behavior, see the small graph in (a). This agrees with the observation in [6].

Figure 3

Staggered magnetization of spin-1 $\mathit{XXZ}$ chain. (a) $M_s$ vs $D$ with fixed $J_z=2.59$, which indicates a phase transition from Néel to large-$D$ phase at $D\simeq 2.3$. (b) $M_s$ vs $J_z$ with fixed $D=0.3$, which indicates a phase transition from Haldane to Néel phase at $J_z\simeq 1.17$.

Figure 4

Entanglement measures of spin-1 $\mathit{XXZ}$ chain. Their derivative discontinuity characterizes the quantum phase transition as well as the staggered magnetization does. (a) Néel to large-$D$ phase; (b) Haldane to Néel phase.

Figure 5

2D transverse Ising model: $M_z$, GE, $S_{\mathrm{BP}}$, and $S_1$ vs the transverse magnetic field $h$ for a system of size $2^5\times 2^5$. We adopt TPS and TRG with $\chi =4$ and $D_{\mathrm{cut}}=16$. The derivatives of these quantities all have a discontinuity at $h\approx 3.25$. The GE and $S_{\mathrm{BP}}$ are scaled up by a factor of $25$ and $10$, respectively.

Figure 6

2D $\mathit{XYX}$ model with ${\Delta }_y=0.25$: $M_x$, $M_z$, GE, $S_{\mathrm{BP}},$ and $S_1$ vs the transverse magnetic field $h$ for a system of size $2^5\times 2^5$. We adopt TPS and TRG with $\chi =4$ and $D_{\mathrm{cut}}=16$. The derivative of these quantities shows a discontinuity at $h\approx 3.5$, and the entanglement measures vanish at the factorizing field value $h\simeq 3.16$. Here, the GE and $S_{\mathrm{BP}}$ are scaled up by a factor of $25$ and $10$, respectively.

Figure 7

2D $\mathit{XXZ}$ model: (a) Magnetizations vs the transverse magnetic field $h$. The derivative of all three magnetizations has a discontinuity at $h_c\simeq 1.8$ for the spin-flop phase transition, and only the $M_x$ and $M_z$ show discontinuity at $h_s\simeq 5$. (b) Entanglement measures vs $h$. Their derivative discontinuities characterize both quantum critical points. Here we adopt TPS and TRG with $\chi =4$ and $D_{\mathrm{cut}}=16$.

Figure 8

Block entanglement entropy $S_L$ for the spin-1 $\mathit{XXZ}$ chain with the fixed $J_z=2.59$ and the quantum critical point at $D\simeq 2.3$. We adopt the MPSs of $\chi =2,4,8$ for the ground state. (a) Off the critical point, the block entanglement entropy saturates around $L=4$. (b) Very near the critical point, the block entanglement entropy saturates around $L\simeq {\chi }^2$, which can be seen more clearly from the numerical data.

Figure 9

GE of block size $L$ for the spin-1 $\mathit{XXZ}$ chain with the fixed $J_z=2.59$ and the quantum critical point at $D\simeq 2.3$. We adopt the MPSs of $\chi =2,4$ for the ground state. (a) Off the critical point, the block GE saturates around $L=4$. (b) Very near the critical point, the block GE saturates around $L\simeq {\chi }^2$, which can be seen more clearly from the numerical data.

Figure 10

Scaling behavior of bipartite entanglement for the spin-1 $\mathit{XXZ}$ chain. The critical point is at $D\simeq 2.3$.

Figure 11

Spectra of singular values for initial bond dimension $\chi =4$. The index $\alpha$ is to number the singular values ${\lambda }_{\alpha }$’s. (a) Spectrum of singular values for bond dimension after performing the TRG to merge the four neighboring sites. (b) Spectrum of singular values for physical dimension after performing the SVD to find the nonlocal unitary transformation for quantum-state RG.