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

Abstract

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.

23 pages, 8 Figures; Corrected eq. (41); Revised argument in A.2

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• Fermionic tensor network methods
• The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study
• An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation
• Tensor network approach to the fully frustrated XY model on a kagome lattice with a fractional vortex-antivortex pairing transition
• Unified tensor network theory for frustrated classical spin models in two dimensions
• Critical line of the triangular Ising antiferromagnet in a field from a $C_3$-symmetric corner transfer matrix algorithm
• Theory of charge-6e condensed phase in Kagome lattice superconductors
• Differentiable programming tensor networks for Kitaev magnets
• Ashkin-Teller phase transition and multicritical behavior in a classical monomer-dimer model
• Dynamics of two-dimensional open quantum lattice models with tensor networks
• Accelerating two-dimensional tensor network optimization by preconditioning
• Variational boundary based tensor network renormalization group
• Emergent Intermediate Phase in the $J_1$-$J_2$ XY model from Tensor Network Approaches
• Superfluid dome in the spatially modulated two-dimensional XY model