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arXiv:1112.5647 · Nuclear Physics B

The O(n) loop model on a three-dimensional lattice

Qingquan Liu, Youjin Deng, Timothy M. Garoni, Henk W. J. Blöte

Published Feb 3, 2012• 26 citations
arXivHTMLPDF
DOI
10.1016/j.nuclphysb.2012.01.026

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The loop model is an important model of statistical mechanics and has been extensively studied in two-dimensional lattices. However, it is still difficult to simulate the loop model directly in three-dimensional lattices, especially in lattices with coordination numbers larger than 3. In this paper, a cluster weight Ising model is proposed by introducing an additional cluster weight n in the partition function of the traditional Ising model. This model is equivalent to the loop model on the two-dimensional lattice, but on the three-dimensional lattice, it is still not very clear whether or not these models have the same universality. By using a Monte Carlo method with cluster updates and color assignment, we obtain the global phase diagram containing the paramagnetic and ferromagnetic phases. The phase transition between the two phases is second order at 1≤n<n_{cri} and first order at n≥n_{cri}, where n_{cri}≈2. The thermal exponent y_{t} is equal to the system dimension d when the first-order transition occurs. For the second-order transitions, the numerical estimation of y_{t} and the magnetic exponent y_{m}, shows that the universalities of the two models on the three-dimensional lattice are different. Our results are helpful in the understanding of some traditional statistical mechanics models.
A statistical model of loops on the three-dimensional lattice is proposed and is investigated. It is O( n )-type but has loop fugacity that depends on global three-dimensional shapes of loops in a particular fashion. It is shown that, despite this nonlocality and the dimensionality, a layer-to-layer transfer matrix can be constructed as a product of local vertex weights for infinitely many points in the parameter space. Using this transfer matrix, the site entropy is estimated numerically in the fully packed limit.
A statistical model of loops on the three-dimensional lattice is proposed and is investigated. It is O(n)-type but has loop fugacity that depends on global three-dimensional shapes of loops in a particular fashion. It is shown that, despite this non-locality and the dimensionality, a layer-to-layer transfer matrix can be constructed as a product of local vertex weights for infinitely many points in the parameter space. Using this transfer matrix, the site entropy is estimated numerically in the fully packed limit.