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Register a new userLoop-Cluster Coupling and Algorithm for Classical Statistical Models
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Potts spin systems play a fundamental role in statistical mechanics and quantum field theory, and can be studied within the spin, the Fortuin-Kasteleyn (FK) bond or the $q$-flow (loop) representation. We introduce a Loop-Cluster (LC) joint model of bond-occupation variables interacting with $q$-flow variables, and formulate a LC algorithm that is found to be in the same dynamical universality as the celebrated Swendsen-Wang algorithm. This leads to a theoretical unification for all the representations, and numerically, one can apply the most efficient algorithm in one representation and measure physical quantities in others. Moreover, by using the LC scheme, we construct a hierarchy of geometric objects that contain as special cases the $q$-flow clusters and the backbone of FK clusters, the exact values of whose fractal dimensions in two dimensions remain as an open question. Our work not only provides a unified framework and an efficient algorithm for the Potts model, but also brings new insights into rich geometric structures of the FK clusters.
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In computer simulations, quantum delocalization of atomic nuclei can be modeled making use of the Path Integral (PI) formulation of quantum statistical mechanics. This approach, however, comes with a large computational cost. By restricting the PI modeling to a small region of space, this cost can be significantly reduced. In the present work we derive a Hamiltonian formulation for a bottom-up, theoretically solid simulation protocol that allows molecules to change their resolution from quantum-mechanical to classical and vice versa on the fly, while freely diffusing across the system. This approach renders possible simulations of quantum systems at constant chemical potential. The validity of the proposed scheme is demonstrated by means of simulations of low temperature parahydrogen. Potential future applications include simulations of biomolecules, membranes, and interfaces.
By interpreting the fusion matrix as an adjacency matrix we associate a loop model to every primary operator of a generic conformal field theory. The weight of these loop models is given by the quantum dimension of the corresponding primary operator. Using the known results for the O(n) models we establish a relationship between these models and SLEs. The method is applied to WZW, $c<1$ minimal conformal field theories and other coset models.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature/energy range around the critical point. By combining the replica-exchange algorithm with cluster updates and an adaptive routine to find the range of interest, we introduce a new flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain two further orders of magnitude in the performance for 2D and 3D Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.
We study a model of dilute oriented loops on the square lattice, where each loop is compatible with a fixed, alternating orientation of the lattice edges. This implies that loop strands are not allowed to go straight at vertices, and results in an enhancement of the usual O(n) symmetry to U(n). The corresponding transfer matrix acts on a number of representations (standard modules) that grows exponentially with the system size. We derive their dimension and those of the centraliser by both combinatorial and algebraic techniques. A mapping onto a field theory permits us to identify the conformal field theory governing the critical range, $n le 1$. We establish the phase diagram and the critical exponents of low-energy excitations. For generic n, there is a critical line in the universality class of the dilute O(2n) model, terminating in an SU(n+1) point. The case n=1 maps onto the critical line of the six-vertex model, along which exponents vary continuously.
We propose a simple procedure by which the interaction parameters of the classical spin Hamiltonian can be determined from the knowledge of four-point correlation functions and specific heat. The proposal is demonstrated by using the correlation and specific heat data generated by Monte Carlo method on one- and two-dimensional Ising-like models, and on the two-dimensional Heisenberg model with Dzyaloshinkii-Moriya interaction. A recipe for applying our scheme to experimental images of magnetization such as those made by magnetic force microscopy is outlined.
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