Detecting topological order in a ground state wave function

A large class of topological orders can be understood and classified using the string-net condensation picture. These topological orders can be characterized by a set of data (N, d_i, F^{ijk}_{lmn}, \delta_{ijk}). We describe a way to detect this kind of topological order using only the ground state wave function. The method involves computing a quantity called the ``topological entropy'' which directly measures the quantum dimension D = \sum_i d^2_i.Comment: 4 pages, 5 figures, homepage http://dao.mit.edu/~we

Education as a Complex System: Conceptual and Methodological Implications

Education is a complex system, which has conceptual and methodological implications for education research and policy. In this article, an overview is first provided of the Complex Systems Conceptual Framework for Learning (CSCFL), which consists of a set of conceptual perspectives that are generally shared by educational complex systems, organized into two focus areas: collective behaviors of a system, and behaviors of individual agents in a system. Complexity and research methodologies for education are then considered, and it is observed that commonly used quantitative and qualitative techniques are generally appropriate for studying linear dynamics of educational systems. However, it is proposed that computational modeling approaches, being extensively used for studying nonlinear characteristics of complex systems in other fields, can provide a methodological complement to quantitative and qualitative education research approaches. Two research case studies of this approach are discussed. We conclude with a consideration of how viewing education as a complex system using complex systems’ conceptual and methodological tools can help advance education research and also inform policy

In pursuit of Pomeron loops: the JIMWLK equation and the Wess-Zumino term

We derive corrections to the JIMWLK equation in the regime where the charge density in the hadronic wave function is small. We show that the framework of the JIMWLK equation has to be significantly modified at small densities in order to properly account for the noncommutativity of the charge density operators. In particular the weight function for the calculation of averages can not be real, but is shown to contain the Wess-Zumino term. The corrections to the kernel of the JIMWLK evolution which are leading at small density are resummed into a path ordered exponential of the functional derivative with respect to the charge density operator, thus hinting at intriguing duality between the high and the low density regimes.Comment: 8 pages, no figures. References added. Version to appear in Phys. Rev.

A mean field approach for string condensed states

We describe a mean field technique for quantum string (or dimer) models. Unlike traditional mean field approaches, the method is general enough to include string condensed phases in addition to the usual symmetry breaking phases. Thus, it can be used to study phases and phases transitions beyond Landau's symmetry breaking paradigm. We demonstrate the technique with a simple example: the spin-1 XXZ model on the Kagome lattice. The mean field calculation predicts a number of phases and phase transitions, including a z=2 deconfined quantum critical point.Comment: 10 pages + appendix, 15 figure