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Critical Behaviour of the Six Vertex F-model

N.B. Pannevis

Published Aug 29, 2012• 1 citation
OpenAlex

Abstract

The F-model is one of the six vertex models, a range of two-dimensional models designed to describe the positions of hydrogen atoms in ice. It exhibits a Kosterlitz-Thouless phase transition, and can easily be mapped onto many other models including the roughening transition in the body centered solid-on-solid model. In this thesis the critical behaviour in the F-model is examined through a computational approach, combined with finite-size scaling and real space renormalization. The simulations were run using two Monte Carlo algorithms; the so-called short loop algorithm, and the full lattice cluster algorithm. The ergodicity of the latter is revealed to be problematic, but this is overcome by combining both algorithms. Also, finite-size scaling is applied, returning a critical inverse temperature of \\epsilon \\beta = 0.79 plus or minus 0.02 compared to the known analytical value of log[2]. Furthermore, although the ice rules prevent local renormalization such as by Kadanoff block spins, a functioning coarse graining prescription based on the height function is proposed. This turns out to renormalise the high temperature states into states with temperatures below the critical point, thus preventing the determination of critical exponents from the renormalization flow.

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The F-model is one of the six vertex models, a range of two-dimensional models designed to describe the positions of hydrogen atoms in ice. It exhibits a Kosterlitz-Thouless phase transition, and can easily be mapped onto many other models including the roughening transition in the body centered solid-on-solid model. In this thesis the critical behaviour in the F-model is examined through a computational approach, combined with finite-size scaling and real space renormalization. The simulations were run using two Monte Carlo algorithms; the so-called short loop algorithm, and the full lattice cluster algorithm. The ergodicity of the latter is revealed to be problematic, but this is overcome by combining both algorithms. Also, finite-size scaling is applied, returning a critical inverse temperature of \epsilon \beta = 0.79 plus or minus 0.02 compared to the known analytical value of log[2]. Furthermore, although the ice rules prevent local renormalization such as by Kadanoff block spins, a functioning coarse graining prescription based on the height function is proposed. This turns out to renormalise the high temperature states into states with temperatures below the critical point, thus preventing the determination of critical exponents from the renormalization flow.
The six-vertex F model on the square lattice constitutes the unique example of an exactly solved model exhibiting an infinite-order phase transition of the Kosterlitz-Thouless type. As one of the few non-trivial exactly solved models, it provides a welcome gauge for new numerical simulation methods and scaling techniques. In view of the notorious problems of clearly resolving the Kosterlitz-Thouless scenario in the two-dimensional XY model numerically, the F model in particular constitutes an instructive reference case for the simulational description of this type of phase transition. We present a loop-cluster update Monte Carlo study of the square-lattice F model, with a focus on the properties not exactly known such as the polarizability or the scaling dimensions in the critical phase. For the analysis of the simulation data, finite-size scaling is explicitly derived from the exact solution and plausible assumptions. Guided by the available exact results, the careful inclusion of correction terms in the scaling formulae allows for a reliable determination of the asymptotic behaviour.
Using the x-y model and a nonlocal updating scheme called cluster Monte Carlo, we calculate the superfluid density of a two-dimensional superfluid on large-size square lattices L\ifmmode\times\else\texttimes\fi{}L up to 400\ifmmode\times\else\texttimes\fi{}400. This technique allows us to approach temperatures close to the critical point, and by studying a wide range of L values and applying finite-size scaling theory we are able to extract the critical properties of the system. We calculate the superfluid density and from that we extract the renormalization-group beta function. We derive finite-size scaling expressions using the Kosterlitz-Thouless-Nelson renormalization group equations and show that they are in very good agreement with our numerical results. This allows us to extrapolate our results to the infinite-size limit. We also find that the universal discontinuity of the superfluid density at the critical temperature is in very good agreement with the Kosterlitz-Thouless-Nelson calculation and experiments.
We present detailed Monte Carlo results for a two-dimensional grain boundary model on a lattice. The effective Hamiltonian of the system results from the microscopic interaction of grains with orientations described by spins of unit length, and leads to a nearest-neighbour interaction proportional to the absolute value of the angle between the grains. Our analysis of the correlation length ξ and susceptibility χ in the high-temperature phase favour a Kosterlitz-Thouless-like (KT) singularity over a second-order phase transition. Unconstrained KT fits of χ and ξ confirm the predicted value for the critical exponent ν, while the values of η deviate from the theoretical prediction. Additionally we apply finite-size scaling theory and investigate the question of multiplicative logarithmic corrections to a KT transition. As for the critical exponents our results are similar to data obtained from the XY model, so that both models probably lie in the same universality class. 1
We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz-Thouless phase transition. We determine the fixed point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single cluster algorithm, we simulate lattices up to L = 4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz-Thouless transition temperature on relatively small lattices. We test the new method at the example of the 2-component φ 4 model on the lattice.
We present a new type of cluster algorithm that strongly reduces critical slowing down in simulations of vertex models. Since the clusters are closed paths of bonds, we call it the loop algorithm. The basic steps in constructing a cluster are the breakup and the freezing of vertices. We concentrate on the case of the F model, which is a subset of the six-vertex model exhibiting a Kosterlitz-Thouless transition. The loop algorithm is also applicable to simulations of other vertex models and of one- and two-dimensional quantum spin systems.
We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz–Thouless phase transition. We determine the fixed-point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single-cluster algorithm, we simulate lattices up to L = 4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz–Thouless transition temperature on relatively small lattices. We test the new method on the example of the two-component ϕ 4 model on the lattice.