Convex Hull of Planar H-Polyhedra

Suppose are planar (convex) H-polyhedra, that is, $A_i \in \mathbb{R}^{n_i \times 2}$ and $\vec{c}_i \in \mathbb{R}^{n_i}$. Let $P_i = \{\vec{x} \in \mathbb{R}^2 \mid A_i\vec{x} \leq \vec{c}_i \}$ and $n = n_1 + n_2$. We present an $O(n \log n)$ algorithm for calculating an H-polyhedron with the smallest $P = \{\vec{x} \in \mathbb{R}^2 \mid A\vec{x} \leq \vec{c} \}$ such that $P_1 \cup P_2 \subseteq P$

Phase transition for cutting-plane approach to vertex-cover problem

We study the vertex-cover problem which is an NP-hard optimization problem and a prototypical model exhibiting phase transitions on random graphs, e.g., Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes of the solution space structure, e.g, for the ER ensemble at connectivity c=e=2.7183 from replica symmetric to replica-symmetry broken. For the vertex-cover problem, also the typical complexity of exact branch-and-bound algorithms, which proceed by exploring the landscape of feasible configurations, change close to this phase transition from "easy" to "hard". In this work, we consider an algorithm which has a completely different strategy: The problem is mapped onto a linear programming problem augmented by a cutting-plane approach, hence the algorithm operates in a space OUTSIDE the space of feasible configurations until the final step, where a solution is found. Here we show that this type of algorithm also exhibits an "easy-hard" transition around c=e, which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem.Comment: 4 pages, 3 figure

Two-Dimensional Quantum XY Model with Ring Exchange and External Field

We present the zero-temperature phase diagram of a square lattice quantum spin 1/2 XY model with four-site ring exchange in a uniform external magnetic field. Using quantum Monte Carlo techniques, we identify various quantum phase transitions between the XY-order, striped or valence bond solid, staggered Neel antiferromagnet and fully polarized ground states of the model. We find no evidence for a quantum spin liquid phase.Comment: 4 pages, 4 figure

Staircase polygons: moments of diagonal lengths and column heights

We consider staircase polygons, counted by perimeter and sums of k-th powers of their diagonal lengths, k being a positive integer. We derive limit distributions for these parameters in the limit of large perimeter and compare the results to Monte-Carlo simulations of self-avoiding polygons. We also analyse staircase polygons, counted by width and sums of powers of their column heights, and we apply our methods to related models of directed walks.Comment: 24 pages, 7 figures; to appear in proceedings of Counting Complexity: An International Workshop On Statistical Mechanics And Combinatorics, 10-15 July 2005, Queensland, Australi

Topological Entanglement Entropy of a Bose-Hubbard Spin Liquid

The Landau paradigm of classifying phases by broken symmetries was demonstrated to be incomplete when it was realized that different quantum Hall states could only be distinguished by more subtle, topological properties. Today, the role of topology as an underlying description of order has branched out to include topological band insulators, and certain featureless gapped Mott insulators with a topological degeneracy in the groundstate wavefunction. Despite intense focus, very few candidates for these topologically ordered "spin liquids" exist. The main difficulty in finding systems that harbour spin liquid states is the very fact that they violate the Landau paradigm, making conventional order parameters non-existent. Here, we uncover a spin liquid phase in a Bose-Hubbard model on the kagome lattice, and measure its topological order directly via the topological entanglement entropy. This is the first smoking-gun demonstration of a non-trivial spin liquid, identified through its entanglement entropy as a gapped groundstate with emergent Z2 gauge symmetry.Comment: 4+ pages, 3 figure

Microscopic models for fractionalized phases in strongly correlated systems

We construct explicit examples of microscopic models that stabilize a variety of fractionalized phases of strongly correlated systems in spatial dimension bigger than one, and in zero external magnetic field. These include models of charge fractionalization in boson-only systems, and various kinds of spin-charge separation in electronic systems. We determine the excitation spectrum and show the consistency with that expected from field theoretic descriptions of fractionalization. Our results are further substantiated by direct numerical calculation of the phase diagram of one of the models.Comment: 10 pages, 4 figure

Phase Structure of d=2+1 Compact Lattice Gauge Theories and the Transition from Mott Insulator to Fractionalized Insulator

Large-scale Monte Carlo simulations are employed to study phase transitions in the three-dimensional compact abelian Higgs model in adjoint representations of the matter field, labelled by an integer q, for q=2,3,4,5. We also study various limiting cases of the model, such as the $Z_q$ lattice gauge theory, dual to the $3DZ_q$ spin model, and the 3DXY spin model which is dual to the $Z_q$ lattice gauge theory in the limit $q \to \infty$. We have computed the first, second, and third moments of the action to locate the phase transition of the model in the parameter space $(\beta,\kappa)$, where $\beta$ is the coupling constant of the matter term, and $\kappa$ is the coupling constant of the gauge term. We have found that for q=3, the three-dimensional compact abelian Higgs model has a phase-transition line $\beta_{\rm{c}}(\kappa)$ which is first order for $\kappa$ below a finite {\it tricritical} value $\kappa_{\rm{tri}}$, and second order above. We have found that the $\beta=\infty$ first order phase transition persists for finite $\beta$ and joins the second order phase transition at a tricritical point $(\beta_{\rm{tri}}, \kappa_{\rm{tri}}) = (1.23 \pm 0.03, 1.73 \pm 0.03)$. For all other integer $q \geq 2$ we have considered, the entire phase transition line $\beta_c(\kappa)$ is critical.Comment: 17 pages, 12 figures (new Fig. 2), new Section IVB, updated references, submitted to Physical Review

A fast Monte Carlo algorithm for site or bond percolation

We describe in detail a new and highly efficient algorithm for studying site or bond percolation on any lattice. The algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time which scales linearly with the size of the system. We demonstrate our algorithm by using it to investigate a number of issues in percolation theory, including the position of the percolation transition for site percolation on the square lattice, the stretched exponential behavior of spanning probabilities away from the critical point, and the size of the giant component for site percolation on random graphs.Comment: 17 pages, 13 figures. Corrections and some additional material in this version. Accompanying material can be found on the web at http://www.santafe.edu/~mark/percolation

Improved Classical Cryptanalysis of SIKE in Practice

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