Locality optimization for parent Hamiltonians of tensor networks

Tensor network states form a powerful framework for both the analytical and numerical study of strongly correlated phases. Vital to their analytical utility is that they appear as the exact ground states of associated parent Hamiltonians, where canonical proof techniques guarantee a controlled ground space structure. Yet, while those Hamiltonians are local by construction, the known techniques often yield complex Hamiltonians which act on a rather large number of spins. In this paper, we present an algorithm to systematically simplify parent Hamiltonians, breaking them down into any given basis of elementary interaction terms. The underlying optimization problem is a semidefinite program, and thus the optimal solution can be found efficiently. Our method exploits a degree of freedom in the construction of parent Hamiltonians—the excitation spectrum of the local terms—over which it optimizes such as to obtain the best possible approximation. We benchmark our method on the AKLT model and the toric code model, where we show that the canonical parent Hamiltonians (acting on 3 or 4 and 12 sites, respectively) can be broken down to the known optimal two-body and four-body terms. We then apply our method to the paradigmatic resonating valence bond (RVB) model on the kagome lattice. Here, the simplest previously known parent Hamiltonian acts on all the 12 spins on one kagome star. With our optimization algorithm, we obtain a vastly simpler Hamiltonian: we find that the RVB model is the ground state of a parent Hamiltonian whose terms are all products of at most four Heisenberg interactions, and whose range can be further constrained, providing a major improvement over the previously known 12-body Hamiltonian.

Figure 1

Tensor networks (shown for a 2D square lattice) are constucted from a local tensors $A_{\alpha \beta \gamma \delta }^i$ (with physical index $i$ and virtual indices $\alpha ,\beta ,...$); they are arranged on a lattice and the virtual indices are contracted to yield the expansion coefficient $|\mathrm{\Psi }\rangle =\sum c_{i_1,\cdots ,i_N}|i_1,\cdots ,i_N\rangle$.

Figure 2

Schematics of the parent Hamiltonian construction. (a) The elementary tensors are blocked on a region $R$. The resulting tensor is interpreted as a linear map $\mathcal{P}$ from the virtual to the physical system. (b) If the blocked region is sufficiently large, the image of the map $\mathcal{P}$, and thus the support of the overall tensor network wave function, will be only a subspace of the full physical Hilbert space $\mathcal{H}$, leaving an orthogonal complement ${\left(\mathrm{Im}\mathcal{P}\right)}^{\perp }$ orthogonal to the state. (c) A parent Hamiltonian for the state is obtained by taking an arbitrary positive semi-definite operator $h$ which is zero of $\mathrm{Im}\mathcal{P}$ and strictly positive on ${\left(\mathrm{Im}\mathcal{P}\right)}^{\perp }$.

Figure 3

(a) Tensor Network representation of the AKLT state. (b) Blocking three sites of the MPS tensors yields the map $\mathcal{P}$, starting point of the parent Hamiltonian construction. In this case the number of blocked sites $N_s$ is larger than the minimum $N_s$ required to build the most local parent Hamiltonian Eq. (11).

Figure 4

[(a) and (b)] Cost function $F$, Eq. (3), during the minimization. The adaptive step-gradient descent procedure converges much faster ($\sim 200$ and 1500 steps to reach $F\sim {10}^{-12}$ for $N_s=3$ and 4, respectively) than the constant step one ($\eta =0.5$). [(c) and (d)] Coefficients of the operators $\left\{O_a\right\}$ in the optimal decomposition at convergence. The basis ordering is $\{\mathrm{Id},{\vec{S}}_i·{\vec{S}}_{i+1},\cdots ,{({\vec{S}}_i·{\vec{S}}_{i+1})}^2,\cdots \}$ with $i=1,\cdots ,N_s-1$. (e),(f) Eigenvalues of the matrix $P_{\alpha \beta }$ at convergence ($F<{10}^{-12}$) for $N_s=3\text{and}4$. The result shows that the locality-optimized Hamiltonian, acting on $N_s$ sites, is not the projector onto ${\left(\mathrm{Im}\mathcal{P}\right)}^{\perp }$.

Figure 5

(a) Vertex tensors providing the PEPS representation of the toric code ground state. The virtual index 1 signals the presence of a loop entering the vertex. $R_{\pi /2}$ stands for all the possible $\pi /2$ rotations of the vertex tensors. (b) Local projectors that map diagonally the virtual space of the vertex tensors to the physical space of a site on the tilted square lattice. (c) $N_s=12$ sites patch used for the parent Hamiltonian construction. Physical legs are omitted in the drawing for clarity; that is, each red dot additionally carries a physical index, as in (b). The linear map $\mathcal{P}$ obtained from this patch maps the virtual space ${\left({\mathbb{C}}^2\right)}^{\otimes 8}$ to the physical space ${\left({\mathbb{C}}^2\right)}^{\otimes 12}$. Plaquette terms are product of $X\mathrm{s}$ on the green line, cross terms are product of $Z\mathrm{s}$ on the blue line.

Figure 6

(a) Decomposition of the projector-valued parent Hamiltonian $h_{\mathrm{Id}}$ of the toric code in terms of $N$-body Pauli products; the plot shows the total weight of all $N$-body terms. Terms with all possible weights are needed in the decomposition. (b) Cost function during the minimization, using the adaptive step. Symmetric and nonsymmetric versions of the algorithm achieve convergence in the same number of steps, but the symmetric version reduces significantly the computation cost. (c) Coefficients of the operators $\left\{O_a\right\}$ in the optimal decomposition at convergence. When only cross and plaquette terms (made of $X,Y$ or $Z$) are included in the basis, the algorithm produces the expected result within machine precision. (d) Eigenvalues of the matrix $P_{\alpha \beta }$ at convergence, divided in the two physical ${\mathbb{Z}}_2$ symmetry sectors. The most local Hamiltonian is different from the projector on ${\left(\mathrm{Im}\mathcal{P}\right)}^{\perp }$.

Figure 7

TN representation of the spin-$1/2$ RVB state. The on-site tensors (a) and triangle plaquette tensors (b)—only nonzero entries are shown—are placed and contracted on the kagome lattice as shown in (c). The states 0 and 1 carry the spin-$1/2$ degree of freedom, while the state 2 signals the absence of a spin $1/2$. The on-site tensor selects either the left or right singlet and maps it to the physical spin $1/2$; the triangle plaquette tensor can either hold no singlet (222 configuration), or exactly one singlet (${\varepsilon }_{abc}$). Up- and downward-pointing blue tensors are related by rotation. Physical legs are omitted in the drawing for clarity; that is, each red dot additionally carries a physical index, just as in (a).

Figure 8

Operator basis dimension for the two cases considered here. (a) The basis contains all products of up to $r$ Heisenberg interactions. (b) The basis contains all products of up to $r=4$ Heisenberg interactions, restricted to the spins on the central hexagon and up to $N_{▵}$ continguous outer spins. In both cases, the operators are symmetrized w.r.t. the sixfold rotation and twofold reflection symmetry of the star, see text.

Figure 9

[(a) and (c)] Cost function $F$, Eq. (3), at the optimum as a function of $r$ and $N_{▵}$, respectively (see Fig. 8). A machine-precision approximation is obtained for $r=4$ and $N_{▵}=5$, respectively, but $N_{▵}=4$ already provides a fairly accurate approximation. [(b) and (d)] Eigenvalues of $P_{\alpha \beta }$ in the $S^z=0$ sector vs the value $s(s+1)$ of the total spin $S^2$, for $r=4$ and $N_{▵}$, respectively, showing that the optimal solution is not a projector.

Figure 10

(a) Periodic clusters employed for the computation of the spectrum of the parent Hamiltonian. The latter is obtained as a sum of the parent Hamiltonian $h$ on all stars of the cluster. [(b)–(e)] Comparison between the low-energy spectra of the RVB parent Hamiltonian obtained as a projector on ${\left(\mathrm{Im}\mathcal{P}\right)}^{\perp }$ (blue empty diamonds) and the optimized Hamiltonian (orange filled diamonds) obtained on a basis of operators with up to four Heisenberg interactions [(b) and (c)] and up to four triangles around the hexagon [(d) and (e)]. Periodic clusters with $N=12$ [(b) and (d)] and $N=18$ [(c) and (e)] are considered.