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This work explores the use of a tree tensor network ansatz to simulate theground state of a local Hamiltonian on a two-dimensional lattice. By exploitingthe entropic area law, the tree tensor network ansatz seems to producequasi-exact results in systems with sizes well beyond the reach of exactdiagonalisation techniques. We describe an algorithm to approximate the groundstate of a local Hamiltonian on a L times L lattice with the topology of atorus. Accurate results are obtained for L={4,6,8}, whereas approximate resultsare obtained for larger lattices. As an application of the approach, we analysethe scaling of the ground state entanglement entropy at the quantum criticalpoint of the model. We confirm the presence of a positive additive constant tothe area law for half a torus. We also find a logarithmic additive correctionto the entropic area law for a square block. The single copy entanglement forhalf a torus reveals similar corrections to the area law with a further termproportional to 1/L.

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