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

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 quasiexact results in systems with sizes well beyond the reach of exact diagonalization techniques. We describe an algorithm to approximate the ground state of a local Hamiltonian on a L ϫ L lattice with the topology of a torus. Accurate results are obtained for L = ͕4,6,8͖, whereas approximate results are obtained for larger lattices. As an application of the approach, we analyze 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.

Efficient simulation of infinite tree tensor network states on the Bethe lattice

“…The critical field is found to be h c ≃ 1.115, in agreement with previous calculations. 25,26 It is also close to the critical field h c = 1.06625 (2) for the transverse Ising model on the honeycomb lattice. 28 Figure 4 shows the bipartite entanglement entropy S E and the correlation length ξ for the ground state.…”

“…Eqs. (1)(2)(3), is very useful in the calculations. First, the diagonal bond matrix describes the entanglement spectrum between the system and environment subblocks.…”

Tensor Network States and Geometry

“…The projected entangled-pair state (PEPS) [21][22][23][24][25][26][27][28][29][30] generalizes the MPS, whereas D > 1 versions of TTN 31,32 and MERA 33-39 also exist. Among those generalizations, PEPS and MERA stand out for offering efficient representations of many-body wave functions, thus leading to scalable simulations in D > 1 dimensions; and, importantly, for also being able to address systems that are beyond the reach of quantum Monte Carlo approaches due to the so-called sign problem, including frustrated spins 30,39 and interacting fermions [40][41][42][43][44][45][46][47][48][49][50] .…”

Tensor network states and algorithms in the presence of a global U(1) symmetry

“…16,21 For twodimensional (2D) lattices there are generalizations of these three tensor network states, namely projected entangled pair states [22][23][24][25][26][27][28][29][30][31] (PEPS), 2D TTN, 32,33 and 2D MERA, [34][35][36][37][38][39][40] respectively. As variational Ansätze, PEPS and 2D MERA are particularly interesting since they can be used to address large two-dimensional lattices, including systems of frustrated spins 31,40 and interacting fermions, [41][42][43][44][45][46][47][48][49][50] where Monte Carlo techniques fail due to the sign problem.…”