Worm Monte Carlo Method And Its Application In O(n) Loop Model

Among the many model systems studied in the field of statistical mechanics, two fundamental examples that continue to play a central role are the O(n) model and q-state Potts model. In this thesis, we summarize the current Markov-chain worm Monte Carlo study of the O(n) loop model on the honeycomb-lattice and three-dimensional hydrogen-peroxide lattice, and by using mathematical method and dynamical analysis, we prove the validity and efficiency of our worm algo-rithm. On the honeycome lattice, using known exact mappings when n=2, this algorithm can be used to simulate a number of zero-temperature Potts antiferro-magnets, including the3-state model on the kagome lattice and the4-state model on the triangular lattice. We then use this worm algorithm to perform a system-atic study of the honeycomb-lattice loop model on all the three critical branches. By comparing our numerical results with Coulomb gas theory, we identify a set of exact expressions for scaling exponents governing some fundamental geometric and dynamic observables. In particular, we show that for all n≤2, the scaling of a certain return time in the worm dynamics is governed by the magnetic dimension of the loop model. The case n>2is also considered, and we confirm the exis-tence of a phase transition in the3-state Potts universality class that was recently observed via numerical transfer matrix calculations. On the three-dimensional hydrogen-peroxide lattice, We perform Monte Carlo studies of the loop model for n=0,0.5,1,1.5,2,3,4,5and10and obtain the critical points and a number of critical exponents, including the thermal exponent yt, magnetic exponent yh, and loop exponent yl. The efficiency of the worm algorithms is reflected by the small value of the dynamic exponent z, determined from our analysis of the integrated autocorrelation times.In Chapter One, we summarized some background knowledge of phase tran- sition and critical phenomena. In addition, we introduced some famous Monte Carlo algorithm in statistical physics. In the last part of this chapter, we briefly introduced O(n) model and numerical study of such model among years.In Chapter Two, we introduced some Monte Carlo algorithms in statistical model, including Metropolis algorithm, Swendsen-Wang algorithm and standard worm algorithm. Next, we briefly introduce the development of the adapted worm algorithm for O(n) loop model. We mainly developed three kinds of worm algo-rithms:connectivity-checking worm algorithm, induced subgraph worm algorithm and worm algorithm for fully-packed loop model. As can be seen in the formu-lation of these worm algorithms, the detailed balance is automatically satisfied. Ergodicity can be strictly proved by mathematical methods. While for the effi-ciency of these algorithms, we leave it for Chapter Four.In Chapter Three, we studied the honeycomb-lattice O(n) loop model by using the adapted worm algorithm. For any real n>0, and any edge weight, including the fully-packed limit of infinite edge weight. We emphasize that by using known exact mappings when n=2, this algorithm can be used to simulate a number of zero-temperature Potts antiferromagnets for which the Wang-Swendsen-Kotecky cluster algorithm is non-ergodic, including the3-state model on the kagome lattice and the4-state model on the triangular lattice. We then use this worm algorithm to perform a systematic study of the honeycomb-lattice loop model on all the three critical branches. By comparing our numerical results with Coulomb gas theory, we identify a set of exact expressions for scaling exponents governing some fundamental geometric and dynamic observables. In particular, we show that for all n≤2, the scaling of a certain return time in the worm dynamics is governed by the magnetic dimension of the loop model, thus providing a concrete dynamical interpretation of this exponent. The case n>2is also considered, and we confirm the existence of a phase transition in the3-state Potts universality class that was recently observed via numerical transfer matrix calculations.In Chapter Four, we conducted worm Monte Carlo study of three-dimensional hydrogen-peroxide lattice O(n) loop model. The coordination number of this lattice is3. For integer n>0, these loop models provide graphical representations for n-vector models on the same lattice, while for n=0they reduce to the self-avoiding walk problem. We use worm algorithms to perform Monte Carlo studies of the loop model for n=0,0.5,1,1.5,2,3,4,5and10and obtain the critical points and a number of critical exponents, including the thermal exponent yt, magnetic exponent yh, and loop exponent yl. For integer n, the estimated values of yt and yh are found to agree with existing estimates for the three-dimensional O(n) universality class. The efficiency of the worm algorithms is reflected by the small value of the dynamic exponent z, determined from our analysis of the integrated autocorrelation times.