# P18: Tensor network algorithms

## Project Description

For many fundamental models of strongly correlated quantum many-body systems, a thorough understanding of their physical properties has still not been achieved. This holds in particular true for strongly correlated fermion systems, described e.g. by the Hubbard model, and for many basic models of quantum magnetism, in particular for systems with (geometric) frustration. In recent years, a series of new algorithmic approaches to such systems have been put forward, inspired to a large degree by insights gained from quantum information science on the entanglement properties of quantum many-body states. These methods provide a new approach to study basic models of strongly correlated quantum many-body systems. The basic ingredient to these methods are complete sets of quantum states that are defined in terms of networks of locally contracted tensors, which generalize the previously introduced matrix product states (MPS) ^{(1),(2)} that also underlie the density matrix renormalization group method used in P5 and P12. The MPS have thus been employed already very successfully to study lattice-based quantum many-body systems in one spatial dimension ^{(3)}. The more general class of states, also called projected entangled pair states (PEPS) ^{(4),(5)}, allow for an efficient representation of the entanglement properties of correlated quantum

systems also beyond one spatial dimension. Currently, they are intensively employed to study two-dimensional quantum systems. In the first part of this project, we will assess the reliability of PEPS-based methods in treating specific two-dimensional frustrated quantum spin systems ^{(6),(7)} and quantum phase transitions^{(8)}. For this purpose, a native version of the PEPS algorithms will be implemented, based on the itensor library.

In particular, we will perform a detailed comparison of the PEPS-based results with previous unbiased quantum Monte Carlo results that exist for these specific systems^{(9),(10)}. Of special interest here will be the non-variational aspects, that enter the iterative tensor-contraction. We also plan to examine these aspects from an analytical perspective. These studies will allow us to judge the robustness of these new approaches. It will also be important to explore in details the scaling behavior of the numerical data with the various control parameters of the numerical algorithms in order to provide robust means of extrapolations. In a second step, we plan to apply also optimization methods to the tensor network states employing Monte Carlo sampling, and extend these methods to the treatment of

fermion systems ^{(11)-(15)}.

The central goal of this project thus is to establish tensor network based algorithms as a reliable approach to the study of complex quantum many-body systems, in particular for studying ground state properties such as e.g. in project P2.

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