Efficient variational contraction of two-dimensional tensor networks with a non-trivial unit cell

A. Nietner1,2, B. Vanhecke3, F. Verstraete3, J. Eisert1,2, and L. Vanderstraeten3

1Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, D-14195 Berlin, Germany
2Helmholtz-Zentrum Berlin für Materialien und Energie, 14109 Berlin, Germany
3Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium

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Tensor network states provide an efficient class of states that faithfully capture strongly correlated quantum models and systems in classical statistical mechanics. While tensor networks can now be seen as becoming standard tools in the description of such complex many-body systems, close to optimal variational principles based on such states are less obvious to come by. In this work, we generalize a recently proposed variational uniform matrix product state algorithm for capturing one-dimensional quantum lattices in the thermodynamic limit, to the study of regular two-dimensional tensor networks with a non-trivial unit cell. A key property of the algorithm is a computational effort that scales linearly rather than exponentially in the size of the unit cell. We demonstrate the performance of our approach on the computation of the classical partition functions of the antiferromagnetic Ising model and interacting dimers on the square lattice, as well as of a quantum doped resonating valence bond state.

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[6] Joey Li, Amos Chan, and Thorsten B. Wahl, "Classification of symmetry-protected topological phases in two-dimensional many-body localized systems", Physical Review B 102 1, 014205 (2020).

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