Few-body systems capture many-body physics: Tensor network approach

Due to the presence of strong correlations, theoretical or experimental investigations of quantum many-body systems belong to the most challenging tasks in modern physics. Stimulated by tensor networks, we propose a scheme of constructing the few-body models that can be easily accessed by theoretical or experimental means, to accurately capture the ground-state properties of infinite many-body systems in higher dimensions. The general idea is to embed a small bulk of the infinite model in an “entanglement bath” so that the many-body effects can be faithfully mimicked. The approach we propose is efficient, simple, flexible, sign-problem free, and it directly accesses the thermodynamic limit. The numerical results of the spin models on honeycomb and simple cubic lattices show that the ground-state properties including quantum phase transitions and the critical behaviors are accurately captured by only $O\left(10\right)$ physical and bath sites. Moreover, since the few-body Hamiltonian only contains local interactions among a handful of sites, our work provides different ways of studying the many-body phenomena in the infinite strongly correlated systems by mimicking them in the few-body experiments using cold atoms/ions, or developing quantum devices by utilizing the many-body features.

Figure 1

The system on an infinite lattice is transformed into one defined on finite clusters embedded in the entanglement bath. We take the $(8+8)$-site and $(18+12)$-site clusters for the simulations on honeycomb lattice, where the first contains 8 physical (blue balls) and 8 bath sites (red balls), and the second contains 18 physical and 12 bath sites. The entanglement bath is calculated by choosing two sites as the supercell (small circle). The legs stand for the interactions between the connecting sites.

Figure 2

The ground-state energy $E$ (per site) of the Heisenberg model on honeycomb lattice. The cluster we choose is ($N_p+N_b$) site where $N_p$ and $N_b$ denote the number of physical and bath sites, respectively (see Fig. 1). The ED on the 18-site cluster with periodic boundary condition suffers severe finite-size effects, and the tree approximation (simply from the bath calculations) underestimates long-range correlations. Our results are consistent with second renormalization group (SRG) of TN, showing that both finite-size effects and the error from the tree approximation are largely reduced.

Figure 3

“Finite-size effects” of AOP from the ground-state magnetization (absolute value) and the bond energies $e=\langle {\widehat{S}}_i{\widehat{S}}_j\rangle$ of the Heisenberg model on honeycomb lattice. Here, we show the cluster with $N_p=18$ physical and $N_b=12$ bath sites. Each peak shows the absolute values of the local magnetization of the physical sites, which ranges from $M=0.329$ (center) to 0.347 (boundary). We take $D=8$ and $\chi =60$. For comparison, the results from the tree approximation in the first stage are $e=-0.360$ and $M=0.347$, and those from SRG are $e=-0.363$ and $M=0.310$.

Figure 4

The ground-state energy $E$ of the Heisenberg model on simple cubic lattice versus $\chi$ is shown in the left figure. The simulation by QMC on a ($10\times 10\times 10$) lattice with periodic boundary condition gives $E=-0.902$. The inset shows the cluster with $N_p=8$ physical (blue balls) and $N_b=24$ bath sites (red balls) used in the second stage in our AOP approach for the simulations on cubic lattice. The legs stand for the interactions between the connecting sites. In the right one, we show the staggered magnetization $M_s$ and correlation length $\xi$ of the ground state. We take $D=2$ and 3.

Figure 5

Staggered magnetization per site $M_s$ and dynamic correlation length $\xi$ of the ground state of the transverse Ising model on simple cubic lattice. The results obtained in stage one (Bethe approximation) and stage two are shown for comparison. We take $D=2,\chi =10$ and $D=3,\chi =20$. The quantum phase transition is found to occur at $h_c=2.66$. The critical behaviors are obtained by fitting the data from the results of stage two near $h_c$, where we have $M_s\propto {(h_c-h_x)}^{0.48}$ and $\xi \propto {(h_c-h_x)}^{-0.25}$.

Figure 6

The left figure shows Eq. (4). The right one shows the construction of the cell tensor given by Eq. (5).

Figure 7

The graphic representations of $\widehat{\mathcal{H}}(i,j)$ in Eq. (6) and $M^{\left[3\right]}$ in Eq. (9) are given in the left and middle figures, respectively. The QR decomposition [Eq. (11)] of the central tensor $\left|A\right(i,j)\rangle$ is shown in the right figure, where the arrows indicate the orthogonality of $|{\tilde{A}}^{\left[3\right]}(i,j)\rangle$ [Eq. (12)].

Figure 8

The left figure shows the bath Hamiltonian $\widehat{\mathcal{H}}{}^{\partial }$ [Eq. (14)] that gives the interaction between the corresponding physical and bath sites. The few-body Hamiltonian in Eq. (15) is formed by the shifted bulk Hamiltonian and $\widehat{\mathcal{H}}{}^{\partial }$ between every physical site on the boundary and a neighboring bath site. For simplicity, the middle figure only illustrates four of the $\widehat{\mathcal{H}}{}^{\partial }$'s. The right one shows the ground-state ansatz of AOP approach, which is the bulk state of the few- body Hamiltonian entangled with several branches of infinite tree PEPS. In fact, the number of tree branches should equal to the number of the physical sites on the boundary (i.e., the number of $\widehat{\mathcal{H}}{}^{\partial }$). For conciseness, we only illustrate four of the tree branches.

Figure 9

The illustration of the self-consistent eigenvalue equations of 1D AOP approach.

Figure 10

The illustration of the encoding scheme for a 2D TN.

Figure 11

The illustrations of three kinds of possible few-body Hamiltonians that contain several interacting physical and bath sites. All the physical interactions (black lines) inside the chosen cluster should be fully considered. The left figure illustrates the one by using the tree DMRG for the physical-bath interactions $\widehat{\mathcal{H}}{}^{\partial }$ (blue dashes), where there are no bath-bath interactions. By choosing other algorithms (e.g., SRG or CTMRG) to calculate $\widehat{\mathcal{H}}{}^{\partial }$, it is possible to also have nearest-neighbor (middle figure) or even long-range (right figure) bath-bath interactions (red dots).

Figure 12

The illustrations of the computation of the correlation functions in 1D AOP.

Figure 13

Relations between AOP and several existing algorithms (PEPS, DMRG, and ED) for the ground-state simulations of 2D and 3D Hamiltonians. The corresponding computational setups in the first (bath calculation) and second (solving the few-body Hamiltonian) stages of AOP algorithm are given above and under the arrows, respectively.