Maissam Barkeshli


Department of Physics

University of Maryland, College Park

Email: maissam at umd dot edu


I am a theoretical physicist studying condensed matter physics, quantum field theory, quantum topology, and quantum information theory.  More recently I also work on the science of deep learning, studying the fundamental properties underlying modern AI models. 

I received my PhD in Physics from MIT in 2010 under the supervision of Xiao-Gang Wen. From 2010 - 2013 I was a Simons postdoctoral fellow at Stanford University, where I worked with Steven Kivelson, Xiao-Liang Qi, and Sean Hartnoll. From 2013 - 2016 I was a postdoctoral researcher at Microsoft Station Q in Santa Barbara, CA, where I worked with mathematicians and physicists: Michael Freedman, Zhenghan Wang, Kevin Walker, Chetan Nayak, Matthew Fisher, and Parsa Bonderson. In 2018 I received an Alfred P. Sloan Research Fellowship and a CAREER award from the National Science Foundation. 

Before MIT, I studied Electrical Engineering and Computer Science (EECS) and Physics at UC Berkeley. As an undergraduate I developed models of quantum-limited SQUID amplifiers in the experimental group of John Clarke and I worked on the fundamental bounds on heat dissipation in computation

I am also a Fellow of the Joint Quantum Institute (JQI) at UMD and an Affiliate Fellow of the Joint Center for Quantum Information and Computer Science (QuICS).

Curriculum Vitae

Google Scholar

ArXiv papers

General theory of topological phases of matter with symmetry

Many major discoveries in the past 40 years in condensed matter physics centered on the experimental or theoretical discoveries of quantized topological invariants that can distinguish quantum phases of matter. For example, the Chern number, which sets the quantized Hall conductivity in integer and fractional quantum Hall systems, or various Z/2 invariants that can distinguish topological insulators and superconductors.  

This raises the question: How do we understand the complete set of possible topological invariants that can describe a gapped quantum phase of matter? What theory should we use to systematically characterize such systems? This question has been answered in the context of free fermion systems, where interactions are completely ignored, through the development of topological band theory. But how do we understand systems in the presence of interactions? Over the past decade, my collaborators and I have provided an answer to this question for topological phases of matter in two spatial dimensions. 

In 2014, my collaborators and I wrote down a precise set of algebraic data and consistency conditions, which define a G-crossed braided tensor category, and which provides the basic mathematical framework to characterize and classify (2+1)D topological phases of matter:

Just like group theory is used to characterize crystals based on their space group symmetries, G-crossed braided tensor categories can be used to characterize topological phases of matter with symmetry in two spatial dimensions.  

Our original theory was developed for bosonic topological phases of matter; that is, phases of matter where the constituent degrees of freedom are bosons. Recently, we have understood how to generalize the theory to describe fermionic topological phases as well:

In particular, invertible topological phases of matter, like integer quantum Hall states and topological insulators, are a special class of topological phases of matter that do not contain any topologically non-trivial excitations (that is, no anyons).  40 years since the experimental discovery of the integer quantum Hall effect, we finally understand how to systematically characterize and classify symmetric fermionic invertible topological phases of matter, by simply solving a set of consistency equations. 

Anomalies of topological phases of matter and 4-manifold invariants

One of the most important phenomena in quantum field theory is that of a quantum anomaly ('t Hooft anomaly to be specific). The reason is that anomalies, along with symmetry, are two properties of a quantum system that remain invariant no matter what energy or length scale we study the system at. The technical statement is that anomalies are invariant under renormalization group flow. For continuous symmetries, anomalies can often be understood using perturbation theory. However for discrete symmetries, anomalies necessarily require non-perturbative techniques. 

In contemporary physics, anomalies in d dimensions are understood to be equivalent invertible topological phases of matter in d+1 dimensions. The reason is that a system in d dimensions with an anomaly cannot exist on its own purely in d dimensions, but can exist at the boundary of a d+1 dimensional invertible topological phase of matter. Therefore, given a description of a d-dimensional quantum field theory, computing the anomaly means finding a way to extract a d+1 dimensional invertible topological phase. 

One important reason that we are interested in anomalies in condensed matter physics is that in the last few years we understood that the celebrated Lieb-Schultz-Mattis theorem (which goes back to the 1960s), is closely related to the physics of anomalies. In particular, a condensed matter system with a non-trivial LSM constraint has a mixed anomaly between translation symmetry and the on-site symmetry. This was first observed in a paper I wrote with my collaborators, where we used this insight to significantly strengthen the LSM theorem and its higher-dimensional generalizations:

In the past few years my collaborators and I have been pioneering techniques to systematically compute anomalies for general symmetry groups in (2+1)D topological quantum field theories:

Phys. Rev. B 100, 115147 (2019)

Phys. Rev. B 98, 115129 (2018)

SciPost Phys. 8, 028 (2020)

Phys. Rev. Research 2, 043033 (2020)

One of the interesting byproducts of our above results is that we are naturally led to some new constructions of topological invariants of 4-manifolds defined on principal G bundles, with or without spin structures. In particular, in Tata et. al. , we even found explicit topologically invariant combinatorial state sums that can distinguish 4-manifolds that are homeomorphic but not diffeomorphic (e.g. real vs. fake RP4). In Kobayashi et. al, we found new combinatorial state sums for invariants of topologically non-trivial, curved U(1) bundles over 4-manifolds, including spin^c structures. 

Classifying crystalline topological states and quantized crystalline response theories

My student Naren Manjunath and I have been using these theories to provide systematic classifications of topological phases, like fractional Chern insulators, in the presence of crystalline space group symmetries. Our work has led to predictions of several new kinds of topological invariants and fractional quantized responses:

We are currently studying the predictions of our theory through numerical calculations of concrete microscopic models. 

Extracting topological invariants from many-body wave functions

Suppose we are given a ground state wave function of an interacting many-body topological phase of matter. Is it possible to extract all possible topological invariants that fully characterize the phase of matter, entirely from this ground state wave function? More specifically, suppose we have access to the equal-time correlators of a ground state wave function defined on a simply connected patch of space. Can we, without any knowledge of the underlying Hamiltonian, extract all of the topological invariants of the phase? 

I have recently been tackling this research direction. We have so far written two papers:

Phys. Rev. B 103, 075102 (2021)

In the first paper, we showed how the many-body Chern number can be extracted from a single ground state wave function. In the second paper, we then designed a protocol to extract the Chern number through the statistical correlations of randomized measurements of a quantum state, without requiring any knowledge of any Hamiltonian. 

Twist defects and topological domain walls in topologically ordered states

Physical Review B 81, 045323 (2010)

Phys. Rev. X 2, 031013 (2012)

Phys. Rev. B 87, 045130 (2013) 

Phys. Rev. B 88, 235103 (2013)

       Phys. Rev. B 88, 241103(R) (2013)

Phys. Rev. X 4, 041035 (2014)

 Phys. Rev. Lett. 114, 026401 (2015)

Phys. Rev. Lett. 113, 236804 (2014)

Science, 346 6210 (2014)

Phys. Rev. Lett. 117, 096803 (2016) 

Quantum criticality

Phys. Rev. Lett. 115, 026802 (2015) 

Phys. Rev. B 86, 075136 (2012) 

Phys. Rev. B 89, 235116 (2014)

Phys. Rev. Lett. 105, 216804 (2010)

Phys. Rev. B 86, 085114 (2012)

Quantum error correction and fault-tolerant quantum computation 

Quantum 3, 180 (2019)

Phys. Rev. B 102, 075105 (2020)

Phys. Rev. Lett. 125, 050502 (2020)

JQI News article on our work above:

Phys. Rev. A 98, 052319 (2018)

Phys. Rev. Research 2, 013285 (2020)

Phys. Rev. B 94, 165108 (2016)

Phys. Rev. X 4, 041035 (2014)

Entanglement transitions in monitored random quantum circuits 

 circuits, arXiv:2011.06595,  Phys. Rev. Lett.  127, 235701 (2021)

Nature Physics 17, 342 - 347 (2021)

A "News and Views" article in Nature Physics on our work:

Brayden Ware and Romain Vasseur, Measurements make the phase, Nature Physics 17 298-299 (2021)

Quantum spin liquids 

Science, 346 6210 (2014)

Phys. Rev. B 87, 140402(R) (2013)


Phys. Rev. B 100, 155146 (2019)

arXiv:1802.10099, Phys. Rev. B 97, 235112 (2018)