Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law

TL;DR

This work develops a tree tensor network (TTN) variational approach to approximate ground states of local 2D lattice Hamiltonians, leveraging the entropic area law to enable quasi-exact results on modest lattices and to study entanglement scaling. It introduces an isometric TTN with coarse-grained layers tailored to torus topology, and provides methods to compute local observables, fidelities, and block entropies from the TTN. Benchmark results for the 2D Ising model show high accuracy for L up to 8 and qualitative reliability for larger L away from criticality, including explicit corrections to the area law: a positive constant for half-a-torus entanglement and a logarithmic term for quarter-torus blocks; single-copy entanglement exhibits related corrections with a 1/$L$ term. The paper situates TTN as a simple, complementary tool to PEPS/MERA for 2D quantum systems, particularly suited for entanglement studies and finite-size scaling, while clarifying its limitations and potential connections to real-space RG and DMRG concepts.

Abstract

This work explores the use of a tree tensor network ansatz to simulate the ground state of a local Hamiltonian on a two-dimensional lattice. By exploiting the entropic area law, the tree tensor network ansatz seems to produce quasi-exact results in systems with sizes well beyond the reach of exact diagonalisation techniques. We describe an algorithm to approximate the ground state of a local Hamiltonian on a L times L lattice with the topology of a torus. Accurate results are obtained for L={4,6,8}, whereas approximate results are obtained for larger lattices. As an application of the approach, we analyse the scaling of the ground state entanglement entropy at the quantum critical point of the model. We confirm the presence of a positive additive constant to the area law for half a torus. We also find a logarithmic additive correction to the entropic area law for a square block. The single copy entanglement for half a torus reveals similar corrections to the area law with a further term proportional to 1/L.

Figures (30)

• Figure 1: Example of TTN for a $2\times 2$ lattice and a $4\times 4$ lattice. Notice (right) that the TTN for a 2D lattice can always be represented as a planar graph, with the leaves or physical indices ordered on a line. The tensors labelled with $w_i$ are isometric tensors. They act locally by projecting the ground state onto its local support with dimension $\chi_i$ (see section \ref{['sect:ansatz']} for further explanation).
• Figure 2: (i) Diagrammatic representation of three types of isometric tensors in the TTN for a $4\times 4$ lattice in Fig. \ref{['fig:SmallTree']}. (ii) Graphical representation of the constraints in Eqs. \ref{['eq:const1']}-\ref{['eq:const3']} fulfilled by the isometric tensors.
• Figure 3: The isometric TTN of Fig. \ref{['fig:SmallTree']} for a $4\times 4$ lattice $\mathcal{L}_0$ is associated with a coarse-graining transformation that generates a sequence of increasingly coarse-grained lattices $\mathcal{L}_1$, $\mathcal{L}_2$ and $\mathcal{L}_3$. Notice that in this example we have added an extra index to the top isometry $w_{3}$, corresponding to the single site of an extra top lattice $\mathcal{L}_3$, which we can use to encode in the TTN a whole subspace of $\mathbb{V}^{\otimes N}$ instead of a single state $\hbox{$| \Psi \rangle$}$.
• Figure 4: Interacting boundaries $\Sigma_{1/2}$, $\Sigma_{1/4}$ and $\Sigma_{1/8}$ corresponding to one half, one quarter and one eighth of a lattice for three different choices of boundary conditions.
• Figure 5: Isometric TTN for lattices of $6\times 6$, $8\times 8$ and $10 \times 10$ sites as used in the manuscript for the purpose of benchmarking the performance of the algorithm of Sect. \ref{['sect:algorithm']}. Notice that all TTN have the same structure on the two top layers of isometries, whose manipulation dominates the computational cost of the algorithm, while they differ in the lower layers. In particular, in the $10\times 10$ lattice two lower layers of isometries are required, since a single layer of isometries mapping a block of $5\times 5$ sites directly into a single effective site would have been too expensive given the present capabilities of the desktop computers used for the simulations.