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Physica A Statistical Mechanics and its Applications

Graphical representations and cluster algorithms I. Discrete spin systems

L. Chayes, Jonathan L. Machta

Published May 1, 1997• 62 citations
Publisher
DOI
10.1016/s0378-4371(96)00438-4

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We show that the two-dimensional (2D) Ising model is complete, in the sense that the partition function of any lattice model on any graph is equal to the partition function of the 2D Ising model with complex coupling. The latter model has all its spin-spin coupling equal to $i\frac{\ensuremath{\pi}}{4}$ and all parameters of the original model are contained in the local magnetic fields of the Ising model. This result has already been derived by using techniques from quantum information theory and by exploiting the universality of cluster states. Here we do not use the quantum formalism and hence make the completeness result accessible to a wide audience. Furthermore, our method has the advantage of being algorithmic in nature so that, by following a set of simple graphical transformations, one is able to transform any discrete lattice model to an Ising model defined on a (polynomially) larger 2D lattice.

arXiv:1207.6891•Physical Review A•Nov 2, 2012•13 citations
Abstract We present a family of graphical representations for the O(N) spin model, where N ≥ 1 represents the spin dimension, and N = 1, 2, 3 corresponds to the Ising, XY and Heisenberg models, respectively. With an integer parameter 0 ≤ ℓ ≤ N/2, each configuration is the coupling of ℓ copies of subgraphs consisting of directed flows and N − 2ℓ copies of subgraphs constructed by undirected loops, which we call the XY and Ising subgraphs, respectively. On each lattice site, the XY subgraphs satisfy the Kirchhoff flow-conservation law and the Ising subgraphs obey the Eulerian bond condition. Then, we formulate worm-type algorithms and simulate the O(N) model on the simple-cubic lattice for N from 2 to 6 at all possible ℓ. It is observed that the worm algorithm has much higher efficiency than the Metropolis method, and, for a given N, the efficiency is an increasing function of ℓ. Besides Monte Carlo simulations, we expect that these graphical representations would provide a convenient basis for the study of the O(N) spin model by other state-of-the-art methods like the tensor network renormalization.
We show that the two-dimensional (2D) Ising model is complete, in the sense that the partition function of any lattice model on any graph is equal to the partition function of the 2D Ising model with complex coupling. The latter model has all its spin-spin coupling equal to $i\frac{\ensuremath{\pi}}{4}$ and all parameters of the original model are contained in the local magnetic fields of the Ising model. This result has already been derived by using techniques from quantum information theory and by exploiting the universality of cluster states. Here we do not use the quantum formalism and hence make the completeness result accessible to a wide audience. Furthermore, our method has the advantage of being algorithmic in nature so that, by following a set of simple graphical transformations, one is able to transform any discrete lattice model to an Ising model defined on a (polynomially) larger 2D lattice.