Entanglement under the renormalization-group transformations on quantum states and in quantum phase transitions

We consider quantum states under the renormalization-group (RG) transformations introduced by Verstraete et al. [Phys. Rev. Lett. 94, 140601 (2005)] and propose a quantification of entanglement under such RGs (via the geometric measure of entanglement). We examine the resulting entanglement under RG transformations for the ground states of “matrix-product-state” Hamiltonians constructed by Wolf et al. [Phys. Rev. Lett. 97, 110403 (2006)] that possess quantum phase transitions. We find that near critical points, the ground-state entanglement exhibits singular behavior. The singular behavior within finite steps of the RG obeys a scaling hypothesis and reveals the correlation length exponent. However, under the infinite steps of RG transformation, the singular behavior is rendered different and is universal only when there is an underlying conformal-field-theory description of the critical point.

Figure 1

Entanglement density ${\mathcal{E}}_L(g)$ of the ground state of Hamiltonian (69). Top: For $N=10$ spins and $L=1$ (top); labels $a$, $b$, and $c$ denote the results from the identical, alternating, and arbitrary ansatz states, respectively. Bottom: $L=2$ (bottom panel); here labels $a$ and $b$ denote the results from the identical and arbitrary ansatz states, respectively.

Figure 2

Entanglement behavior of the ground state of Hamiltonian (69). Top: Entanglement ${\mathcal{E}}_{\infty }(g)$. Bottom: Entanglement derivative $d{\mathcal{E}}_{\infty }(g)/\mathit{dg}$.

Figure 3

Top: Entanglement behavior ${\mathcal{E}}_L(g)$ of the ground state of Hamiltonian (69) for $L=1$, 2, 4, and 8, from bottom to top. Bottom: An enlargement of the region in $[-0.5,0.5]$. The dashed curve represents the results of ${\mathcal{E}}_{\infty }(g)$.

Figure 4

Nearest-neighbor concurrence behavior $C(g)$ of the ground state of Hamiltonian (69). The use of concurrence to infer critical points may incorrectly identify $g=1$ as a critical point.

Figure 5

Entanglement density ${\mathcal{E}}_{L=1}(g)$ of the ground state of Hamiltonian (83). (Top) For $N=4$; labels $a$, $b$, and $c$ denote the results from the identical, alternating, and arbitrary ansatz states, respectively. Bottom: For $N=4$, 6, and 8 using the alternating ansatz and the analytic expression from Eq. (91).

Figure 6

Entanglement ${\mathcal{E}}_L(g)$ for the ground state of Hamiltonian (83) for block sizes $L=1$, 2, 4, and 8, from bottom to top.

Figure 7

Fidelity measure $f(g_1,g_2)$ between ground states of the same model at two system parameters, $g_1$ and $g_2$. It turns out that both Hamiltonians, (69) and (83), have the same plot.