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.
Highlights
One of the most beautiful manifestations of emergent behavior in statistical physics can be found in the arena of frustrated spin systems [1]
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 that can be contracted using standard algorithms for infinite systems
Frustration in a classical spin system occurs whenever it is impossible to find a spin configuration which minimizes each and every term of the Hamiltonian simultaneously, leading to macroscopic ground-state degeneracies and giving rise to interesting zero-temperature physics such as effective realizations of gauge theories [2]. Exact results in this context were obtained for antiferromagnetic nearest-neighbor Ising models on triangular and kagome lattices [3,4] using Kauffman and Onsager’s method, for frustrated Ising models on all planar two-dimensional lattices with nearest-neighbor interactions using a mapping to free fermions [5,6], and for more general systems such as planar spin ice [7] using Bethe ansatz techniques [8]
Summary
One of the most beautiful manifestations of emergent behavior in statistical physics can be found in the arena of frustrated spin systems [1]. Exact results in this context were obtained for antiferromagnetic nearest-neighbor Ising models on triangular and kagome lattices [3,4] using Kauffman and Onsager’s method, for frustrated Ising models on all planar two-dimensional lattices with nearest-neighbor interactions using a mapping to free fermions [5,6], and for more general systems such as planar spin ice [7] using Bethe ansatz techniques [8]. To demonstrate the power of the method, we apply it to a frustrated Ising model with further-neighbour couplings on a kagome lattice, obtain the residual entropy with a very high accuracy, and use the local rules to make some exact statements regarding the physics of the ground-state manifold
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