Solving frustrated Ising models using tensor networks

Abstract

Motivated by the recent success of tensor networks to calculate the residual entropy of spin ice and kagome Ising models, we develop a general framework to study frustrated Ising models in terms of infinite tensor networks %, i.e. tensor networks that can be contracted using standard algorithms for infinite systems. This is achieved by reformulating the problem as local rules for configurations on overlapping clusters chosen in such a way that they relieve the frustration, i.e. that the energy can be minimized independently on each cluster. We show that optimizing the choice of clusters, including the weight on shared bonds, is crucial for the contractibility of the tensor networks, and we derive some basic rules and a linear program to implement them. We illustrate the power of the method by computing the residual entropy of a frustrated Ising spin system on the kagome lattice with next-next-nearest neighbour interactions, vastly outperforming Monte Carlo methods in speed and accuracy. The extension to finite-temperature is briefly discussed.

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• 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
• Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices
• 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
• Matrix product state fixed points of non-Hermitian transfer matrices
• Differentiable programming tensor networks for Kitaev magnets
• Contrasting pseudo-criticality in the classical two-dimensional Heisenberg and ${\mathrm{RP}}^2$ models: zero-temperature phase transition versus finite-temperature crossover
• Phase coherence of charge-$6e$ superconductors via a frustrated Kagome XY antiferromagnet
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• Tensor network renormalization approach to antiferromagnetic 6-state clock model on the Union Jack lattice
• Efficient optimization of variational tensor-network approach to three-dimensional statistical systems
• Variational boundary based tensor network renormalization group
• Emergent Intermediate Phase in the $J_1$-$J_2$ XY model from Tensor Network Approaches
• Equivalence of residual entropy of hexagonal and cubic ices from tensor network methods