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arXiv:1606.09170 · Physical review. B./Physical review. B

Gradient methods for variational optimization of projected entangled-pair states

Laurens Vanderstraeten, Jutho Haegeman, Philippe Corboz, Frank Verstraete

Published Oct 14, 2016• 206 citations
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Abstract

The authors present a new method for variationally optimizing projected entangled-pair states (PEPS) in the thermodynamic limit based on the global energy functional and novel tensor contraction strategies. It is successfully benchmarked using Ising and Heisenberg Hamiltonians. The method is completely independent from previous PEPS implementations and improves on the variational energies and order parameters as compared to other state-of-the-art PEPS simulations. The algorithm presented here should prove valuable in simulating ground-state properties of strongly correlated quantum spin systems, can be used with other variational ansatzes, and can be applied to tensor network states in general.

DOI
10.1103/physrevb.94.155123

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The authors present a new method for variationally optimizing projected entangled-pair states (PEPS) in the thermodynamic limit based on the global energy functional and novel tensor contraction strategies. It is successfully benchmarked using Ising and Heisenberg Hamiltonians. The method is completely independent from previous PEPS implementations and improves on the variational energies and order parameters as compared to other state-of-the-art PEPS simulations. The algorithm presented here should prove valuable in simulating ground-state properties of strongly correlated quantum spin systems, can be used with other variational ansatzes, and can be applied to tensor network states in general.
We present a conjugate-gradient method for the ground-state optimisation of projected entangled-pair states (PEPS) in the thermodynamic limit, as a direct implementation of the variational principle within the PEPS manifold. Our optimisation is based on an efficient and accurate evaluation of the gradient of the global energy functional by using effective corner environments, and is robust with respect to the initial starting points. It has the additional advantage that physical and virtual symmetries can be straightforwardly implemented. We provide the tools to compute static structure factors directly in momentum space, as well as the variance of the Hamiltonian. We benchmark our method on Ising and Heisenberg models, and show a significant improvement on the energies and order parameters as compared to algorithms based on imaginary-time evolution.
Projected entangled-pair states (PEPS) provide excellent variational wave functions for ground states of strongly correlated quantum systems in two dimensions. Determining the expectation value of such a PEPS wave function requires the contraction of a two-dimensional tensor network. Here, the authors show that this contraction can itself be formulated as a variational problem, allowing one to compare the accuracies of two commonly used contraction schemes. The authors find good agreement between the different schemes, demonstrating the robustness of PEPS simulations.
An infinite projected entangled-pair state (iPEPS) is a powerful variational tensor network ansatz for two-dimensional ground states in the thermodynamic limit, and can be seen as a natural generalization of a matrix-product state to two dimensions. One of the main challenges in iPEPS simulations is the optimization of the tensors, i.e., finding the optimal variational parameters, in order to have the best representation of the ground state of a given Hamiltonian. The author presents a variational optimization scheme, in which the energy is minimized in an iterative way by sweeping over all the tensors in the ansatz, in a similar spirit as done in the density-matrix renormalization group method. Benchmark results for challenging problems are presented that show that the variational scheme yields considerably more accurate results than the previously best imaginary-time evolution algorithm, with a similar computational cost and with a faster convergence towards the ground state.
Projected entangled pair states (PEPS) are a promising ansatz for the study of strongly correlated quantum many-body systems in two dimensions. However, due to their high computational cost, developing and improving PEPS algorithms is necessary to make the ansatz widely usable in practice. Here we analyze several algorithmic aspects of the method. On the one hand, we quantify the connection between the correlation length of the PEPS and the accuracy of its approximate contraction and discuss how purifications can be used in the latter. On the other hand, we present algorithmic improvements for the update of the tensor that introduce drastic gains in the numerical conditioning and the efficiency of the algorithms. Finally, the state-of-the-art general PEPS code is benchmarked with the Heisenberg and quantum Ising models on lattices of up to $21\ifmmode\times\else\texttimes\fi{}21$ sites.
The projected entangled pair state (PEPS) ansatz can represent a thermal state in a strongly correlated system. We introduce a variational algorithm to optimize this tensor network whose essential ingredient is an auxiliary tree tensor network (TTN). Since the full tensor environment is taken into account, with increasing bond dimension the PEPS-TTN ansatz provides the exact Gibbs state. Our presentation opens with a 1D version for a matrix product state (MPS-TTN) and then generalizes to PEPS-TTN in 2D. Benchmark results in the quantum Ising model are presented.
The approximate contraction of a tensor network of projected entangled pair states (PEPS) is a fundamental ingredient of any PEPS algorithm, required for the optimization of the tensors in ground state search or time evolution, as well as for the evaluation of expectation values. An exact contraction is in general impossible, and the choice of the approximating procedure determines the efficiency and accuracy of the algorithm. We analyze different previous proposals for this approximation, and show that they can be understood via the form of their environment, i.e. the operator that results from contracting part of the network. This provides physical insight into the limitation of various approaches, and allows us to introduce a new strategy, based on the idea of clusters, that unifies previous methods. The resulting contraction algorithm interpolates naturally between the cheapest and most imprecise and the most costly and most precise method. We benchmark the different algorithms with finite PEPS, and show how the cluster strategy can be used for both the tensor optimization and the calculation of expectation values. Additionally, we discuss its applicability to the parallelization of PEPS and to infinite systems.