In school, students learn about three (or maybe four) phases of matter: solid, liquid, gas (and sometimes plasma). These states are classified based on relationships between the individual constituents of matter–e.g. are molecules regularly arranged or scattered, near or far apart? Scientists have historically had the most success classifying states (more than just those four, in fact) by looking at locally measurable quantities (distance to neighbors, local magnetism, etc) and the extent to which a given region affects regions elsewhere in the material (a notion called "correlation").

Over the last 30 years, a new method of classifying phases has emerged which doesn’t rely on local measurements or correlations, but rather depends on properties of the whole system, and is impervious to local changes. This new order is called “topological" order (TO), and has been instrumental in understanding quantum systems such as spin liquids and quantum Hall states, as well as providing a route towards quantum computation.

As currently defined, TO is inherently a phenomenon of quantum mechanics. We are re-examining TO in a classical setting which is more pedagogical and simpler to simulate. We have proposed a family of classical models which exhibit TO (defined as a massively degenerate ensemble of globally distinct, locally indistinguishable configurations). We have simulated several such models, and extracted statistical information about the types of configurations and the interactions between defects (see figures) important in classifying these models into new and different states – much as we would in a more “real-world” quantum mechanical model. In the process we have come to better understand which properties of TO are the result of quantum mechanics, and which are the result of any system sharing these interesting statistical properties.

Classical Topological Order in Non-Abelian Height Models

Christopher L. Henley, Cornell University, DMR 1005466

Ascending & Descending M.C. Escher.Counting steps up around the loop as +1’s and steps down as -1’s, a “real world” staircase should give us 0. In Escher’s world, though, we can have “defects” – loops around which the sum is not 0. Labeling the edges of a lattice by the “steps up” is the basis of a height model.

Lattice w/ defects.Here we have a more general version of the staircase setup above, but instead of numbers of stairs and addition we have elements of a group a, b, and c and the rules a x b = c, b x c = a, c x a = b, and a2 =b2 =c2 = the identity (like 0 in the staircase model). We can include defects (red circles a) and allow them to “walk around” from square to square. They interact through the other edges (along the red line)

Classical Topological Order in Non-Abelian Height Models

Christopher L. Henley, Cornell University, DMR 1005466

• Educationo C.L. Henley advised Cornell undergraduate Matt Lapa

(upper right) on his 2012 senior thesis project (an analysis of the ground states of the pyrochlore antiferromagnet). This year Matt enrolled as a graduate student at the University of Illinois at Urbana-Champaign.

• Outreacho Primary project GRA R. Zach Lamberty (lower right) has

participated in a number of local outreach activities aimed at educating the public – especially at-risk children and students from non-traditional education backgrounds – about science. Outreach examples include a series of demonstrations booths at the yearly Ithaca and “Dragon Days” festivals, a presentation at a University of Alabama workshop educating middle school teachers on the use of educational physics demonstrations available through Cornell University, and educational sessions with the Ithaca Big Brothers Big Sisters Organization.

• International Collaborationso Henley has collaborated with and mentored two

international postdoctorate fellows: Dr. Dimitris Galanakis (a SUSY lattice model, paper on the arXiv) and Dr. Mathieu Taillefumier (simulations of the classical Kagome lattice).

o Lamberty is in the process of establishing a mentorship (discussing frustrated magnetism) with Ms. Rika Noor, a student at National University of Yogyakarta in Indonesia.

Henley and Lapa

Henley and Cornell undergraduate thesis advisee Matt Lapa pose after Lapa’s graduation ceremony (Spring 2012)

Lamberty during outreachLamberty and a student from a local elementary school discuss “shrinky-dinks,” plastic polymer sheets which experience rapid shrinking when heated (Spring 2012)