Variational Numerical Renormalization Group: Bridging the gap between NRG and Density Matrix Renormalization Group

The numerical renormalization group (NRG) is rephrased as a variational method with the cost function given by the sum of all the energies of the effective low-energy Hamiltonian. This allows to systematically improve the spectrum obtained by NRG through sweeping. The ensuing algorithm has a lot of similarities to the density matrix renormalization group (DMRG) when targeting many states, and this synergy of NRG and DMRG combines the best of both worlds and extends their applicability. We illustrate this approach with simulations of a quantum spin chain and a single impurity Anderson model (SIAM) where the accuracy of the effective eigenstates is greatly enhanced as compared to the NRG, especially in the transition to the continuum limit.Comment: As accepted to PRL. Main text: 4 pages, 4 (PDF) figures; Supplementary material: 4 pages, 6 PDF figures; revtex4-

Fermionic Implementation of Projected Entangled Pair States Algorithm

We present and implement an efficient variational method to simulate two-dimensional finite size fermionic quantum systems by fermionic projected entangled pair states. The approach differs from the original one due to the fact that there is no need for an extra string-bond for contracting the tensor network. The method is tested on a bi-linear fermionic model on a square lattice for sizes up to ten by ten where good relative accuracy is achieved. Qualitatively good results are also obtained for an interacting fermionic system.Comment: As published in Phys. Rev.

Monte Carlo simulation with Tensor Network States

It is demonstrated that Monte Carlo sampling can be used to efficiently extract the expectation value of projected entangled pair states with large virtual bond dimension. We use the simple update rule introduced by Xiang et al. to obtain the tensors describing the ground state wavefunction of the antiferromagnetic Heisenberg model and evaluate the finite size energy and staggered magnetization for square lattices with periodic boundary conditions of sizes up to L=16 and virtual bond dimensions up to D=16. The finite size magnetization errors are 0.003(2) and 0.013(2) at D=16 for a system of size L=8,16 respectively. Finite D extrapolation provides exact finite size magnetization for L=8, and reduces the magnetization error to 0.005(3) for L=16, significantly improving the previous state of the art results.Comment: 6 pages, 7 figure

Tree tensor networks and entanglement spectra

A tree tensor network variational method is proposed to simulate quantum many-body systems with global symmetries where the optimization is reduced to individual charge configurations. A computational scheme is presented, how to extract the entanglement spectra in a bipartite splitting of a loopless tensor network across multiple links of the network, by constructing a matrix product operator for the reduced density operator and simulating its eigenstates. The entanglement spectra of 2 x L, 3 x L and 4 x L with either open or periodic boundary conditions on the rungs are studied using the presented methods, where it is found that the entanglement spectrum depends not only on the subsystem but also on the boundaries between the subsystems.Comment: 16 pages, 16 figures (20 PDF figures

Time evolution of projected entangled pair states in the single-layer picture

We propose an efficient algorithm for simulating quantum many-body systems in two spatial dimensions using projected entangled pair states. This is done by approximating the environment, arising in the context of updating tensors in the process of time evolution, using a single-layered tensor network structure. This significantly reduces the computational costs and allows simulations in a larger submanifold of the Hilbert space as bounded by the bond dimension of the tensor network. We present numerical evidence for stability of the method on an antiferromagnetic isotropic Heisenberg model where good agreement is found with the available accurate results

Real time evolution at finite temperatures with operator space matrix product states

We propose a method to simulate the real time evolution of one dimensional quantum many-body systems at finite temperature by expressing both the density matrices and the observables as matrix product states. This allows the calculation of expectation values and correlation functions as scalar products in operator space. The simulations of density matrices in inverse temperature and the local operators in the Heisenberg picture are independent and result in a grid of expectation values for all intermediate temperatures and times. Simulations can be performed using real arithmetics with only polynomial growth of computational resources in inverse temperature and time for integrable systems. The method is illustrated for the XXZ model and the single impurity Anderson model.Comment: 10 pages, 4 figures, published versio

Monte Carlo simulation with Tensor Network States

It is demonstrated that Monte Carlo sampling can be used to efficiently extract the expectation value of projected entangled pair states with large virtual bond dimension. We use the simple update rule introduced by Xiang et al. to obtain the tensors describing the ground state wavefunction of the antiferromagnetic Heisenberg model and evaluate the finite size energy and staggered magnetization for square lattices with periodic boundary conditions of sizes up to L=16 and virtual bond dimensions up to D=16. The finite size magnetization errors are 0.003(2) and 0.013(2) at D=16 for a system of size L=8,16 respectively. Finite D extrapolation provides exact finite size magnetization for L=8, and reduces the magnetization error to 0.005(3) for L=16, significantly improving the previous state of the art results