Accurate Determination of Tensor Network State of Quantum Lattice Models in Two Dimensions

Abstract

We have proposed a novel numerical method to calculate accurately the physical quantities of the ground state with the tensor-network wave function in two dimensions. We determine the tensor network wavefunction by a projection approach which applies iteratively the Trotter-Suzuki decomposition of the projection operator and the singular value decomposition of matrix. The norm of the wavefunction and the expectation value of a physical observable are evaluated by a coarse grain renormalization group approach. Our method allows a tensor-network wavefunction with a high bond degree of freedom (such as D = 8) to be handled accurately and efficiently in the thermodynamic limit. For the Heisenberg model on a honeycomb lattice, our results for the ground state energy and the staggered magnetization agree well with those obtained by the quantum Monte Carlo and other approaches.The application of the density matrix renormalization group (DMRG) proposed by White[1] has achieved great success in one dimension in both zero and finite temperatures [2,3,4]. However, in two dimensions, the application of the DMRG in both the real and momentum space [5,6] has been limited only to small lattices. The error resulting from the DMRG truncation increases extremely fast with increasing size of lattice. To resolve this problem, the tensor-network state, which is an extension of the matrix product in one dimension [7], was proposed [8]. In a tensor network state, a local tensor is directly entangled with the other local tensors on all the directions of the lattice. This leads to two problems in the treatment of the tensor network state. First, it is difficult to determine accurately all the elements of local tensors by any variational approach since the total degree of freedom of a local tensor increases exponentially with the dimension of the tensor. Second, it is even more difficult to calculate the expectation value of any physical observable even if we know the expression of the tensor network wave function, since the number of summations over the basis configurations increases exponentially with lattice size.In this paper, we propose a novel method to handle the tensor-network wave function in two dimensions. We will show that the tensor network wavefunction |Ψ of the ground state can be accurately determined by applying an iterative projection approach. This approach is similar to the time-evolving block decimation method that was used to determine the matrix product wave function of the ground state in one dimension[9, 10] Then we will generalize the classic coarse grain renormalization group approach proposed by Levin and Nave[11] to the quantum system, and use it to calculate the norm of the wave function and the expectation value of any physical observable. This provides an accurate and efficient tool to determine the expectation values of physical quantitise from the tensor-network wavefunction of the ground state.Below, we will take the S=1/2 Heisenberg model on a honeycomb latticeas an example to show how the method works...

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

“…In Refs. [7,10], a simple update scheme is proposed to determine the ground state tensor network wavefunction in two dimensions. This scheme is efficient and robust.…”

Plaquette renormalization scheme for tensor network states

“…The scheme may be a good approximation for some judiciously chosen truncated (renormalized) tensor, but how to find the optimal way to construct it is an open question. Gu, Levin, and Wen implemented a singular value decomposition (SVD) scheme [13] for the double tensors, and a similar method was proposed by Jiang, Weng, and Xiang [14]. In our scheme, the renormalization is instead accomplished with the aid of auxiliary 3-index tensors S n abc in the wave function, which transform and truncate pairs of indices of the plaquette tensors, as shown in Fig.…”

“…On the other hand, it should also be possible to construct special S tensors that effectively perform something very similar to a SVD (although globally optimized, not locally as in Ref. 13,14) and then the optimized T tensors by themselves should also form a good TNS when assembled into a standard 2D tensor network. Here we do not impose any such conditions on the S tensors.…”

“…To overcome this challenge, approximate ways to compute the contraction have been proposed [1,13,14]. Another approach is to use tree-tensor networks [4], or more sophisticated extensions of these [9], which can be efficiently contracted.…”

“…[ 13,14] that apply singular-value decompositions on the "double tensor" product obtained when the physical indices have been traced out. Our approach is strictly variational, and can be used also in combination with Monte Carlo sampling of the spins [16].…”