Graphical representations and worm algorithms for the O(N) spin model

Abstract

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.