Lisa Glaser discusses phase transitions, non-commutative geometry and how sitting in on a lecture can bring about new ideas as she talks about her recent paper in Journal of Physics A: Mathematical and Theoretical.
Who are you?
What prompted you to pursue this field of research?
My research is focused on exploring the path integral in different approaches to quantum gravity using computer simulations. I think that space-time should be discrete on a fundamental level, and discrete systems are just predestined to be explored with the computer. The idea to do this with non-commutative geometry happened kind of by accident. I was sitting in on a lecture series that John Barrett gave for his Ph.D. students in Nottingham, and realized then that the system he described would fit this framework very well. He liked the idea and we wrote a paper together. The current work is the logical next step in that direction. Our first paper explored a large class of systems, but only on a surface level, while now I go into much more depth for some of these.
What is this latest paper all about?
In non-commutative geometry we can describe a manifold through an algebra A, a Hilbert space H and a Dirac operator D. Fuzzy spaces are a special class of non-commutative geometry in which all of these are finite matrix spaces. The metric information is then contained in the Dirac operator, and we can use Monte Carlo simulations to integrate over these Dirac operators to obtain a class of random fuzzy spaces. In Monte Carlo Simulations of Random Non-Commutative Geometries by John W. Barrett and Lisa Glaser (J.Phys. A49 (May 11, 2016): 245001) we found that some of these fuzzy spaces showed phase transitions. To understand these phase transitions better I have here determined the location of the phase transition more precisely and determined that they are most likely of 2nd or higher order.
I determined the location of the phase transition by examining the variance of the action and the trace of the Dirac operator squared, Tr(D2). These two observables peak at the phase transition, and I found the location of their peaks to agree for all three geometries.
The strongest indication that the phase transition is of 2nd or higher order is the behavior of the correlations. We can calculate the covariance matrix for the eigenvalues of the Dirac operator before, at and after the phase transition. We can see that the correlation becomes much stronger, and involves more different eigenvalues at the phase transition. Since higher order phase transitions usually involve strong correlations this is additional evidence.
To better understand how the system behaves if we increase the size, I have also explored the scaling of the average of the action and Tr(D2). The scaling away from the phase transition can be predicted to be proportional to the number of eigenvalues of the Dirac operator, N2. For the variance the analytic estimation would lead us to expect a scaling like N4, however this estimate does not take into account that the correlations between different eigenvalues are weak away from the phase transition. Once we take this correctly into account we find that away from the phase transition the variance should also scale like N2, which our plots confirm.
The scaling at the phase transition can be determined through a fit. I find that the Variance at the phase transition diverges as N(2-2x) with x=0, -0.5, -1 for the three geometries ( (1,1) , (2,0), (1,3) ) I have examined.
What do you plan to do next?
I am currently writing a next paper on the same systems together with John Barrett and his student Paul Druce. While I only explored the phase transition and the scaling of the geometries the next step is to find new ways to quantify the geometry. We found some interesting ways to measure the dimension and the volume and applied a distance measure that has been defined before, and the results look very good. More long term we also want to construct coordinates on our random geometries, and couple matter to the system.
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Author Image owned by Lisa Glaser, used with permission. All other images copyright IOP publishing, all rights reserved to images.