An Importance Sampling Algorithm for the Ising Model with Strong Couplings

We consider the problem of estimating the partition function of the ferromagnetic Ising model in a consistent external magnetic field. The estimation is done via importance sampling in the dual of the Forney factor graph representing the model. Emphasis is on models at low temperature (corresponding to models with strong couplings) and on models with a mixture of strong and weak coupling parameters.Comment: Proc. 2016 Int. Zurich Seminar on Communications (IZS), Zurich, Switzerland, March 2-4, 2016, pp. 180-18

Kirchhoff's Circuit Law Applications to Graph Simplification in Search Problems

This paper proposes a new analysis of graph using the concept of electric potential, and also proposes a graph simplification method based on this analysis. Suppose that each node in the weighted-graph has its respective potential value. Furthermore, suppose that the start and terminal nodes in graphs have maximum and zero potentials, respectively. When we let the level of each node be defined as the minimum number of edges/hops from the start node to the node, the proper potential of each level can be estimated based on geometric proportionality relationship. Based on the estimated potential for each level, we can re-design the graph for path-finding problems to be the electrical circuits, thus Kirchhoff's Circuit Law can be directed applicable for simplifying the graph for path-finding problems

Partition Function of the Ising Model via Factor Graph Duality

Abstract—The partition function of a factor graph and the partition function of the dual factor graph are related to each other by the normal factor graph duality theorem. We apply this result to the classical problem of computing the partition function of the Ising model. In the one-dimensional case, we thus obtain an alternative derivation of the (well-known) analytical solution. In the two-dimensional case, we find that Monte Carlo methods are much more efficient on the dual graph than on the original graph, especially at low temperature. I

Implementations and applications of Renyi entanglement in Monte Carlo simulations of spin models

Although entanglement is a well studied property in the context of quantum systems, the ability to measure it in Monte Carlo methods is relatively new. Through measures of the Renyi entanglement entropy and mutual information one is able to examine and characterize criticality, pinpoint phase transitions, and probe universality. We describe the most basic algorithms for calculating these quantities in straightforward Monte Carlo methods and state of the art techniques used in high performance computing. This description emphasizes the core principal of these measurements and allows one to both build an intuition for these quantities and how they are useful in numerical studies. Using the Renyi entanglement entropy we demonstrate the ability to detect thermal phase transitions in the Ising model and XY model without use of an order parameter. The scaling near the critical point also shows signatures identifying the universality class of the model. Improved methods are explored using extended ensemble techniques that can increase calculation efficiency, and show good agreement with the standard approach. We explore the "ratio trick" at finite temperature and use it to explore the quantum critical fan of the one dimensional transverse field Ising model, showing agreement with finite temperature and finite size scaling from field theory. This same technique is used at zero temperature to explore the geometric dependence of the entanglement entropy and examine the universal scaling functions in the two dimensional transverse field Ising model. All of this shows the multitude of ways in which the study of the Renyi entanglement entropy can be efficiently and practically used in conventional and exotic condensed matter systems, and should serve as a reference for those wishing to use it as a tool