Abstract
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.
Highlights
In the past few years, tensor network states (TNS) have been revealed as a very promising choice for the numerical simulation of strongly correlated quantum many-body systems
In order to analyze the performance of the SL procedure, we study the accuracy of the norm contraction, ≡ 〈ψ|ψ〉, as a function of the truncation parameters, D′′ and d′, for a set of different Projected entangled pair states (PEPS), and compare the results to those from the original algorithm
We have shown how the different approximations applied to the environment explain the limitations of each method in the achievable ground state accuracy, an issue that we have studied quantitatively in the context of finite PEPS
Summary
In the past few years, tensor network states (TNS) have been revealed as a very promising choice for the numerical simulation of strongly correlated quantum many-body systems. It is possible to perform an approximate TN contraction with controlled error, albeit involving a much higher computational cost than in the case of MPS This limits the feasible PEPS simulations to relatively small tensor dimensions. It is possible to analyze the various approximations from the unifying point of view of how they treat the environment contraction, which in turn has a physical meaning This allows us to understand how a given strategy may attain only a limited precision approximation to the ground state, even when its computational cost allows for large bond dimensions.
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