Entanglement production by independent quantum channels

For the one-dimensional Hubbard model subject to periodic boundary conditions we construct a unitary transformation between basis states so that open boundary conditions apply for the transformed Hamiltonian. Despite the fact that the one-particle and two-particle interaction matrices link nearest and next-nearest neighbors only, the performance of the density-matrix renormalization-group (DMRG) method for the transformed Hamiltonian does not improve. Some of the new interactions act as independent quantum channels, which generate the same level of entanglement as periodic boundary conditions in the original formulation of the Hubbard model. We provide a detailed analysis of these channels and show that, apart from locality of the interactions, the performance of DMRG is effected significantly by the number and the strength of the quantum channels that entangle the DMRG blocks.

Figure 1

Schematic plot of the Hubbard model with periodic boundary conditions for a chain with $L_s=10$ lattice sites. Solid lines denote the nearest-neighbor hopping, while the on-site Coulomb interaction is shown by the gray shading. Numbers indicate the lattice-site indices.

Figure 2

Schematic plot of the transformed Hubbard model using the unitary transformation (2). Single solid lines denote single-particle couplings; double solid lines correspond to two-particle interactions. Shaded circles denote the on-site Coulomb interaction of strength $U$, while empty circles correspond to strength $U∕2$. Numbers indicate the lattice-site indices.

Figure 3

One-dimensional representation of the single-particle electron transfers ${\widehat{H}}_T$ (a) and the two-particle interactions ${\widehat{H}}_{U,d}$ (b), ${\widehat{H}}_{U,p}$ (c), ${\widehat{H}}_{U,s}$ (d) of the transformed Hubbard model. The loops in (b) denote on-site interactions.

Figure 4

Schematic plot of the system and environment block of DMRG. $B_l$ and $B_r$ denote the left and right blocks of length $l$ and $r$, and of dimension $M_l$ and $M_r$, respectively, where ● stands for the intermediate sites ($s_l$ and $s_r$) with $q_l$ and $q_r$ degrees of freedom. The blocks $B_L=B_l●$ and $B_R=●B_r$ have dimensions $M_L$ and $M_R$, respectively.

Figure 5

Two extreme orderings for the Hubbard chain.

Figure 6

Site entropy and block entropy for the half-filled Hubbard model for the two ordering criteria shown in Fig. 5 for $U=1$ and $\chi =0$ (exact calculation). The Schmidt number is also shown for $\chi ={10}^{-4}$ and $M_{\min }=4$. The lines are guides to the eyes.

Figure 7

Same as Fig. 6 for the Hubbard model with periodic boundary conditions and for the transformed Hubbard model with open boundary conditions for $\chi ={10}^{-4}$, $M_{\min }=64$ (PBC) and $M_{\min }=256$ (Transf.), $L_s=64$, and $U=0$.

Figure 8

Same as Fig. 7 for the transformed Hubbard model with open boundary conditions for $U=0$, $\chi ={10}^{-4}$, and $M_{\min }=64$ in independent-chain geometry.

Figure 9

Same as Fig. 7 for $U=10$ and $\chi ={10}^{-5}$.

Figure 10

Total quantum information for the original and transformed Hubbard models as a function of $U$ for various system sizes. For $L_s=8$ results are exact while for larger system sizes data were obtained by setting $\chi ={10}^{-5}$.

Figure 11

Site entropy, block entropy, and Schmidt number for the half-filled Hubbard model for the configuration shown in Fig. 5a as a function of $t^{\prime }$ for $t=1$ and $U=0$ for $L_s=\mathrm{16,32}$ sites, and $\chi ={10}^{-4}$, $M_{\min }=64$.

Figure 12

Same as Fig. 11 for the half-filled Hubbard model with open boundary condition as a function of the transfer integral $t^{\prime }$ between next-nearest neighbors. We set $t=1$ and $U=0$ for $L_s=16,32$ and $\chi ={10}^{-4}$, $M_{\min }=64$.

Figure 13

Same as Fig. 11 but as a function of $U$ for the local-density term only, ${\widehat{H}}_d=-{\widehat{H}}_T+U{\widehat{H}}_{U,d}$, as shown in Fig. 3b.

Figure 14

Same as Fig. 11 but as a function of $U$ for the pair-hopping term only, ${\widehat{H}}_p=-{\widehat{H}}_T+U{\widehat{H}}_{U,p}$, as shown in Fig. 3c.

Figure 15

Same as Fig. 11 but as a function of $U$ for the pair-hopping term only, ${\widehat{H}}_s=-{\widehat{H}}_T+U{\widehat{H}}_{U,s}$, as shown in Fig. 3d.

Figure 16

Site entropy and block entropy for the half-filled Hubbard model for periodic boundary conditions shown in Fig. 5a and for the supersite representation in Fig. 2 for $U=1$ and $\chi =0$ (exact calculation).

Figure 17

Same as Fig. 16 for $U=0$ for $N=34$, $\chi ={10}^{-4}$.