Simulating strongly correlated quantum systems with tree tensor networks

We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chemistry versions of the density-matrix renormalization group (DMRG) method have long been applied to such systems, the spatial topology of DMRG-based methods allows efficient optimizations to be carried out with respect to one spatial dimension only. Extending the matrix-product-state picture, we formulate a more general approach by allowing the local sites to be coupled to more than two neighboring auxiliary subspaces. Following [Y. Shi, L. Duan, and G. Vidal, Phys. Rev. A 74, 022320 (2006)], we treat a treelike network ansatz with arbitrary coordination number $z$, where the $z=2$ case corresponds to the one-dimensional (1D) scheme. For this ansatz, the long-range correlation deviates from the mean-field value polynomially with distance, in contrast to the matrix-product ansatz, which deviates exponentially. The computational cost of the tree-tensor-network method is significantly smaller than that of previous DMRG-based attempts, which renormalize several blocks into a single block. In addition, we investigate the effect of unitary transformations on the local basis states and present a method for optimizing such transformations. For the 1D interacting spinless fermion model, the optimized transformation interpolates smoothly between real space and momentum space. Calculations carried out on small quantum chemical systems support our approach.

Figure 1

(Color online) Tree tensor network with (a) $z=3$ and (b) $z=4$. The structure of the tensors is shown in (c) and (d). The bonds indicate the virtual indices ${\alpha }_1,\dots ,{\alpha }_z$ and the circle the physical index $k$.

Figure 2

(Color online) (a) Separation of the state into $z$ blocks plus the site under optimization, as described by Eq. (3). (b) Natural freedom in the tensor network: insertion of a matrix $w$ and $w^{-1}$ at one bond leaves the state invariant. The contraction of $A$ with $w$ forms the new tensor $A^{\prime }$ on the left-hand side; the contraction of $B$ with $w^{-1}$ forms the new tensor $B^{\prime }$ at the right-hand side.

Figure 3

(Color online) (a) Formation of the effective Hamiltonian ${\mathfrak{h}}_{m,r}={\mathfrak{h}}_{m,r}^1\otimes {\mathfrak{h}}_{m,r}^2\otimes {\mathfrak{h}}_{m,r}^3$ with respect to the interaction $h_r={\tau }^{\left(7\right)}\otimes {\tau }^{\left(15\right)}$. The sites on which the interaction has support are marked by a filled circle. Each open (filled) circle in the tensor network corresponds to the contraction of the layered structure of tensors shown in (b).

Figure 4

(Color online) Branches with no support (marked by dotted lines and circles), which yield the identity when contracted and therefore do not have to be taken into account in calculating the effective Hamiltonian.

Figure 5

(Color online) Formation of the effective Hamiltonian ${\mathfrak{h}}_{m,r}={\mathfrak{h}}_{m,r}^1\otimes {\mathfrak{h}}_{m,r}^2\otimes {\mathfrak{h}}_{m,r}^3$ with respect to the fermionic interaction $h_r=a_7^{†}{a}_{15}$. The sites on which the interaction has support are marked in red.

Figure 6

(Color online) (a) Possible choice of an ordered tree graph with each site having two incoming and one outgoing bond. The sink site is marked in black. The tensors $A_m$ associated with each site with virtual indices consisting of pairs $\left({\alpha }_i,n_i\right)$ have the structure shown in (b).

Figure 7

(Color online) Formation of the effective Hamiltonian ${\mathfrak{h}}_{m,r}={\mathfrak{h}}_{m,r}^1\otimes {\mathfrak{h}}_{m,r}^2\otimes {\mathfrak{h}}_{m,r}^3$ with respect to the fermionic interaction $h_r=a_7^{†}{a}_{15}$ with parity symmetry taken into account. The sites on which the interaction has support are marked in red. All branches marked by dotted lines and circles yield the identity when contracted. The parity operator $\tilde{Z}$ is contracted to the virtual bond.

Figure 8

(Color online) Relative error $\Delta E_{rel}=\left|\left(E-E_{exact}\right)/E_{exact}\right|$ in the energy for Heisenberg model on (a) a $4\times 4$ lattice and (b) a $6\times 6$ lattice as a function of the optimization steps. The reference energy $E_{exact}$ was obtained by exact diagonalization in the $4\times 4$ case and by Monte Carlo simulations using Algorithms and Libraries for Physics Simulations (ALPS) [60] in the $6\times 6$ case. The calculations were performed with a fixed virtual dimension of $D=4$ and tree tensor networks with coordination numbers $z=2$ (blue curve), $z=3$ (green curve), and $z=4$ (red curve).

Figure 9

(Color online) Tree structures used for the study of models on a rectangular lattice. Structures with coordination number $z=3$ mapped on a rectangular lattices of size $4\times 4$ and $6\times 6$ are shown in (a) and (b), the structures with $z=4$ is depicted in (c) and (d). The sites marked in blue are virtual sites with no corresponding particle or mode.

Figure 10

(Color online) Relative error $\Delta E_{rel}=\left|\left(E-E_{exact}\right)/E_{exact}\right|$ in the energy for the interacting spinless fermion model on (a) a $4\times 4$-lattice and (b) a $6\times 6$-lattice as a function of the optimization steps. The calculations were performed with a fixed virtual dimension of $D=4$ and tree tensor networks with coordination numbers $z=2$ (blue curve) and $z=3$ (green curve). The chosen parameters are $J=1$, $U=0.5$ and the number of fermions is fixed to $N=3$.

Figure 11

(Color online) Relative error $\Delta E_{rel}=\left|\left(E-E_{exact}\right)/E_{exact}\right|$ in the energy for interacting spinless fermions with $M=7$ sites as a function of the optimization steps for the TTNS ansatz with (a) $z=2$ and (b) $z=3$. Here $U=1$, the particle number is fixed to $N=3$, and the virtual dimension is assumed to be $D=2$. Results with orbital optimization are compared to the results obtained in the real-space basis and in the momentum-space basis. The steps at which orbital optimizations are performed are marked with arrows.

Figure 12

(Color online) The relative error $\Delta E_{rel}=\left|\left(E-E_{exact}\right)/E_{exact}\right|$ in the ground-state energy for the Beryllium atom as a function of DMRG iteration steps for various values of the number of DMRG block states. The dashed line corresponds to the Hartree-Fock energy.

Figure 13

(Color online) Relative error $\Delta E_{rel}=\left|\left(E-E_{exact}\right)/E_{exact}\right|$ in the ground-state energy for the Be atom as a function of the optimization step for $D=2$. The results obtained using a TTNS ansatz with $z=3$ plus orbital optimization are compared to the TTNS-results without orbital optimization For comparison, the results with a MPS ansatz (corresponding to $z=2$) are also included. The steps at which orbital optimizations are performed are marked with arrows.

Figure 14

(Color online) Relative error $\Delta E_{rel}=\left|\left(E-E_{exact}\right)/E_{exact}\right|$ in the ground-state energy for the Beryllium atom as a function of the full bond dimension $\chi$ obtained using (a) the DMRG and (b) as a function of the virtual dimension $D$ obtained from a TTNS with $z=2$ and $z=3$.

Figure 15

(Color online) Graphical representation of the magnitude of the components of the two-dimensional entanglement matrix obtained from the two-site mutual information for the beryllium atom. For clarity, the one-dimensional chain of orbitals used in the DMRG calculation is plotted on a circle. Numbers next to the symbols label the ordering of the molecular orbitals within the chain.