## Tom Jackson, Condensed matter theory

#### tsjackson@physics.ucla.edu

# Research interests

I’m a postdoc working with Prof. Rahul Roy on issues related to the role of interactions and/or disorder in topological insulators. Current and past research interests include

- Extending the current theory of the fractional quantum Hall effect to fractional Chern insulators by incorporating geometric degrees of freedom.
- Combinatorial models for the algebraic structure of conformal field theories and applications to conformal block wavefunctions for the fractional quantum Hall effect.
- The uses of quantum mechanical entanglement for characterizing many-body wavefunctions: entanglement spectrum and multipartite generalizations of the von Neumann entropy.
- New perspectives on the statistical mechanics of systems with quenched randomess: in particular, spin glasses (both classical and quantum) and their relationship to combinatorial optimization problems.

More broadly, I switched fields from astronomy to condensed matter physics because “more is different”: strongly interacting many-body systems represent a complexity frontier that, unlike other frontiers of physics, is a boundary set purely by our theoretical knowledge rather than technological capabilities. The sort of research I aspire to do addresses this problem through novel applications of mathematical techniques: my goal is to not merely perform computations, but to introduce new tools that provide a fresh perspective on established areas of study.

CV (PDF download)### Education and training

- Postdoctoral scholar, University of California, Los Angeles
- Postdoctoral research associate, University of Virginia
- Ph.D. in Physics, Yale University
- Master of Advanced Study (part III of the Mathematical Tripos), Churchill College, University of Cambridge
- A.B. in Physics, Princeton University

### Publications

- T. S. Jackson, G. Möller and R. Roy, “Geometric stability of topological lattice phases,” arxiv: 1408.0843.
The fractional quantum Hall effect (FQHE) is predicted to give rise to many phenomena unlike anything we encounter in everyday life — see the next entry for details. It's notoriously difficult to create in the lab, though, requiring intense external magnetic fields, which is why many people are interested in fractional Chern insulators (FCIs), which display all the physics of the FQHE without the need for large magnets. In this paper, we develop principles put forward in a previous paper by Rahul Roy concerning the most likely scenario under which an FCI becomes unstable. We find remarkable agreement in numerical tests against a range of the best-studied models of FCIs in the literature, which led us to propose what we call the “geometric stability hypothesis:” that the robustness or fragility of an FCI phase can be determined from single-particle quantities that are easy to calculate, despite the fact that the phase itself is strongly interacting.
- T. S. Jackson, N. Read and S. H. Simon, “Entanglement subspaces, trial wavefunctions, and special Hamiltonians in the fractional Quantum Hall effect,” Editor's Suggestion, Phys. Rev. B 88, 075313 (2013) [arxiv] [slides].
The fractional quantum Hall (FQH) effect is an example of the highly nontrivial effects that can arise in a system with both interactions and topological order, and has led to one Nobel prize so far. A point that’s frequently glossed over is that the term refers to not just a single phenomenon, but a wide variety of phases with different topological behavior. The ‘big picture’ description of an FQH phase — its universality class — is specified by a conformal field theory (CFT) which serves a dual role as the effective theory of low-energy edge excitations and as a generator of “trial wavefunctions” describing the electrons in the bulk. A major and long-standing gap in our theoretical understanding is how to relate this ‘big picture’ to the ‘details’: if we want to get a phase that has a particular kind of topological order (for example, because we want to build a quantum computer that’s protected from errors by topological principles), what sort of microscopic effective interaction between electrons do we need?

In this paper, we present a fundamentally new approach to this problem; although the specific cases we consider are rather technical, the picture that emerges should hold generally. We look at a class of “clustered states” with continuous parameters, which are an extension of the existing cases for which the interactions→order connection is known. Trying to obtain the order using the repulsive interaction one would naïvely expect doesn’t work: a certain class of excitations proliferates without bound, meaning that the phase is unstable, and it turns out that additional, subleading repulsion terms are needed to obtain a stable phase matching the CFT description. This situation should occur for any phase with a complicated clustering pattern: the required interaction will require a high degree of fine-tuning, meaning that these phases are unlikely to be produced in laboratory systems where we don't have complete control over the interaction. - L. Fidkowski, T. S. Jackson, and I. Klich, “Model characterization of gapless edge modes of topological insulators using intermediate Brillouin-zone functions,” Phys. Rev. Lett. 107, 036601 (2011) [arxiv].
Topologically ordered materials typically admit description by different theories either living in the bulk of a sample, or defined only on its boundary. Understanding how these descriptions complement each other is necessary for relating the mathematical definition of topological order to quantities that can be measured experimentally. Here we show that an abstract quantity defined in the bulk (essentially the first Čech cohomology class of the vector bundle of electron wavefunctions) has the same spectrum as a very physical quantity defined on the boundary, the energy spectrum of edge excitations. These are also related to the “entanglement spectrum,” which can be analogized as the edge excitations arising from a fictitious cut that splits the sample in half. We give illustrative examples of this correspondence for a few entries in the “periodic table” (aka the “ten-fold way”) of non-interacting topological orders.
- T. S. Jackson and N. Read, “Theory of minimum spanning trees. I. Mean-field theory and strongly disordered spin-glass model,” Phys. Rev. E 81, 021130 (2010) [arxiv] [slides].
The minimum spanning tree (MST) problem is one of the oldest optimization problems studied in computer science. Here we considered a variation where the underlying graph is a simple grid, but the edge costs are independent random variables: can anything be said about the “average” MST arising from many different instances of this input? This scenario is relevant for physics, since it describes the strong disorder limit of spin glasses — random magnetic materials — when the couplings between the individual atomic magnetic moments become infinitely broad. In this paper we build on an analogy with percolation theory to obtain a mean-field description of the fractal structure of the average-case MST, which is a good approximation for high-dimensional graphs.
- T. S. Jackson and N. Read, “Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold,” Phys. Rev. E 81, 021131 (2010) [arxiv] [slides].
Continuation of the preceding paper: we extend our mean-field results by deriving a Feynman diagram expansion describing corrections to the various conndectedness functions in low dimensions. The Feynman rules arise from combinatorial principles, rather than the expansion of an action functional. Despite the fact that the minimum spanning tree problem is inherently non-local (i.e. in order to construct the MST in a small patch, any algorithm might need to access information on arbitrarily distant parts of the graph), our diagram series is sufficiently well-behaved to be renormalizable by local operators: in other words, this random system can’t be described by a local field theory, but it still acts that way in important aspects — the general principle (if any) behind this is still unclear. Our results were recently verified in simulations by Sweeney and Middleton [paper] [arxiv].
- T. S. Jackson, M. Werner, and T. N. Gautier III, “A catalog of bright filamentary structures in the interstellar medium,” Astrophys. J. Suppl. Ser. 149, 365 (2003).
This publication describes work done during a SURF fellowship at JPL with the Spitzer space telescope (SIRTF at the time) group. The polarization of the cosmic microwave background (CMB) yields important information about the state of the early universe, but the polarization of a CMB photon can be rotated as it passes through the Milky Way galaxy’s magnetic field. Experiments need a way to estimate this source of bias, and in this paper we explored the idea of using the distributions of interstellar gas clouds as a proxy for actual magnetic field measurements. This works because of “flux freezing”: if we see a diffuse elongated cloud, we can (with some caveats) infer that it’s been stretched out parallel to the local magnetic field. I implemented a simple feature detection algorithm to extract elongated cloud structures from existing infrared survey data (similar to this) in an unbiased way.
- T. S. Jackson, Ž. Ivezić, and G. R. Knapp, “The galactic distribution of asymptotic giant branch stars,” Monthly notices R.A.S. 337, 749 (2002) [arxiv].
The asymptotic giant branch (AGB) is a stage encountered by stars slightly more massive than the sun at the end of their life span, after they’ve burned through almost all of their hydrogen and helium fuel. We were interested in this stellar population because the intrinsic luminosity of stars in the AGB is roughly independent of their age and mass, which makes them useful as “standard candles”: a central problem in observational astronomy is figuring how far away things are, but if you know all objects of a given type have the same intrinsic luminosity then you know that the ones that appear less bright are actually further away. We used this principle to map the structure of the Milky Way galaxy and comment on the validity of a few models for its disk.
- B. Willman et. al., “An SDSS survey for resolved Milky Way satellite galaxies. I. Detection limits,” Astron. J. 123, 848 (2002) [arxiv].
This was a preliminary attempt to detect undiscovered satellite galaxies of the Milky Way using early data from the Sloan Digital Sky Survey. Despite being nearby, many of the Milky Way’s satellites were only discovered recently due to their low surface brightness, making them difficult to pick out visually, especially if nearer stars belonging to the Milky Way are present in the foreground. To address this, we attempted simple Voronoi clustering on star positions, brightnesses and colors, using the latter two quantities to differentiate foreground and background stars.

### Unpublished

- T. S. Jackson, “Properties of minimum spanning trees and fractional quantum Hall states,” PhD thesis, Yale University, 2010 [pdf].
- T. S. Jackson, “Causality in discrete models of quantum spacetime,” Part III essay, University of Cambridge, 2004 [pdf].
Models of spacetime have been proposed which are mathematically similar to the spin models studied in condensed matter physics: the ultimate goal is to obtain general relativity as an emergent phenomenon, analogous to the way the spin wave dispersion in an antiferromagnet arises from the individual magnetic interactions.
- T. S. Jackson, “Quantum entanglement in random antiferromagnetic spin chains,” Undergraduate thesis, Princeton University, 2003 [pdf].
- T. S. Jackson, “Statistical structure in natural language,” Independent work submitted to the Program in Applied and Computational Mathematics, Princeton University, 2003 [pdf] [slides].
Summer research with William Bialek. Using novel unbaised estimators, we found that the mutual information between two words in several text sources decays as a power law. We were motivated by an information-theoretic approach to compression and prediction.