Publication list: See my ArXiv profile for the most updated list. My publications are organized below by topic. For a nontechnical summary, see the bottom of the page.

Characterization and classification of crystalline topological phases

2. Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry: link (2021)

3. Classification of fractional quantum Hall states with spatial symmetries: link (2020)

Crystalline topological phases exhibit a rich set of topological invariants that have no lattice analogs. A huge program of work over the last 15 years has been devoted to understanding how many such invariants exist given the symmetry of the system, and how they manifest in experiments or lattice models. One important class of crystalline topological phases comprises the Chern insulators, which have a quantized non-trivial Hall conductance, and are the lattice analogs of the quantum Hall states. A natural question of relevance to many current experiments is how to tell whether two Chern insulating states are in the same phase or not, assuming they have the same Chern number. This problem has so far not been fully addressed by theory, due to the absence of simple solvable models in the case where the underlying particles in the system are strongly interacting. 

Using tools from topological quantum field theory (TQFT) and G-crossed braided tensor category theory, we made progress on this question by writing down an effective response theory for general topological phases with charge conservation, translation and point group rotation symmetries; these are natural symmetries for a wide class of Chern insulators. The response theory is written in terms of gauge fields for the crystalline symmetries, which keep track of the properties of lattice defects, which are the dislocations and disclinations in the crystal. The terms in the response theory describe the quantized charge and other properties associated with these defects. The theory is non-perturbative, and can be applied to systems with and without topological order. 

We furthermore connect this response theory to other important classification schemes, which include ideal constructions of ground states in real space, as well as free fermion topological band theory. We explain how the more widely studied free fermion crystalline topological invariants are connected to the above TQFT invariants, thereby deriving a map between the free and interacting classifications. 

In summary, this work theoretically predicts many new crystalline topological invariants and suggests how they could potentially be measured in future experiments on crystalline topological states. The work below actually measures these invariants in numerical models, based on the intuition provided by TQFT.

Extracting crystalline topological invariants from microscopic models

The above methods based on topological quantum field theory predict a slew of crystalline topological invariants as well as the ways in which they affect various topological response properties. We first studied an invariant called the discrete shift, which manifests as a quantized fractional contribution to the electric charge at a lattice disclination (which is a type of curvature defect). We measured the discrete shift in the celebrated Hofstadter model of free fermions, which is known to contain many Chern insulating states in its phase diagram. We verified that there are many cases in which two states within the model have the same Chern number, but different values of the discrete shift (and therefore belong to distinct topological phases). This was the first new topological invariant measured in this classic model since 1982, when the Hall conductance was computed. 

The topological field theory predicts, in addition, that there is a fractional quantized contribution to the electric charge at another type of lattice defect, called a dislocation, which breaks the translation symmetry of the crystal. We were also able to measure this invariant numerically in the Hofstadter model. The surprising feature of this invariant is that it corresponds precisely to the charge polarization, which is a fundamental quantity in condensed matter physics. Since several previous attempts to define the polarization as a many-body invariant for Chern insulators have been problematic, this marks a conceptual breakthrough in the many-body theory of polarization.

The response theory predicts several more invariants that are associated only to the crystalline symmetries, and therefore do not manifest as an electric charge response. To measure these invariants, we used a different idea, which is to measure the ground state expectation value under suitable partial rotation operators. There turns out to be a deep connection between the complex arguments of these quantities (which are quantized) and the remaining topological invariants predicted by TQFT. We also measured these invariants in the square lattice Hofstadter model, thus achieving a complete characterization of its crystalline topological invariants.

Finally, it is well known that several bulk topological invariants also have a manifestation in terms of properties of the edge of the system. In this context, we studied the shift in continuum and lattice models (collaboration with Abhinav Prem and Yuan-Ming Lu) and showed that a non-trivial shift in an Abelian topological phase with zero Chern number and in the continuum results in counter-propagating edge states which cannot be gapped out by rotationally symmetric interactions. In the discrete case, the same invariant manifests through quantized fractional charges at corners and surfaces of polyhedra.


Classification of invertible fermionic topological phases

Invertible fermionic topological phases are characterized by having a unique, gapped ground state on any closed manifold; examples include topological insulators and superconductors, and the integer quantum Hall states. Our main achievement was to show how to incorporate the chiral central charge into the algebraic set of equations describing symmetry defects in invertible phases  (collaboration with Maissam Barkeshli, Yu-An Chen and Po-Shen Hsin). These equations can constrain the allowed topological invariants in interesting ways: for example, the theory gives non-perturbative constraints on when a given symmetry group permits invertible phases to have unpaired Majorana zero modes at symmetry defects (collaboration with Vladimir Calvera).


Non-technical summary

Solids, liquids, gases ... and so much more. Phases of matter are incredibly diverse in their properties. Some transport heat and electricity, others do not; some are magnetic, others are not; the list is endless. Just as for plants and animals in the biological world or for elements in the chemical world, it is useful to devise classification schemes that can distinguish different phases based on their observable properties. 

A basic property of a substance is its symmetry, i.e. the set of operations which leave the material invariant. (Think of an infinite square grid of atoms. It looks the same if we shift the entire grid by one atom in either direction, or rotate the whole grid by 90 degrees. These properties are referred to as translation invariance and rotational invariance respectively.) If two substances have different symmetries, they belong to different phases of matter, because the symmetry necessarily changes when we try to transform one substance into the other. This is the reason ice and water are in different phases: water is completely homogeneous and therefore has more translation symmetry than ice, which is a crystal and has a smaller translation symmetry.

But what if both substances have the same symmetry? They could still belong to different phases, but distinguishing them is harder. Such phases are called topological phases, and they generally only exist at extremely low temperatures. We care about them because they display some of the weirdest and most counterintuitive phenomena in physics. For example, in the 1980s, people discovered the 'quantum Hall effect' in the lab by placing a semiconductor between the poles of a very strong magnet. It turns out that applying a voltage along the X axis leads to a current flowing along the Y axis. Surprisingly, the value of the Hall conductance (Y current divided by X voltage) is extremely stable, up to one part in a hundred million. It is perhaps the most precisely quantized property in physics. But the real puzzle is that the Hall conductance is stable even if we add some dirt to the semiconductor, which we might expect would block the current somewhat and reduce the conductance. But no: it is still stable to one part in a hundred million! This is the mystery that inaugurated the field of topological phases of matter.

So, how to classify them? One fruitful idea (which is also the basis of my PhD work) is the following. Suppose we break the symmetry of the substance just a little. In a crystal, this could for example mean removing a line of atoms, so that the material no longer has perfect translation invariance. Such a modification is called a symmetry defect. Then, the big idea is simply:

If two substances have the same symmetry but respond differently upon inserting symmetry defects, then they are in different topological phases.

This is a bit like saying that if two plants which look visually the same respond differently to stimuli such as light and heat, then they should be treated as different species.

If a system has a fixed number of particles moving around on a square grid, and we introduce a defect of translation symmetry as above, then it could 'respond' by trapping a certain number of particles at the end points of the defect. This could be a fractional number, even though there is no way to break up particles into fractions in the original grid! This fraction turns out to encode a very important physical property called the electric polarization, which determines how electric current flows in the material when we put it under mechanical strain, or in a temperature gradient, or next to a magnet. Measuring the polarization through translation symmetry defects is difficult to do in real materials, but may be possible in engineered materials that are built up in the lab atom by atom.

The idea of symmetry defects underlies some powerful modern mathematical techniques, which go by the names of 'topological quantum field theory' and 'G-crossed braided tensor category theory'. In our work, we have used these mathematical methods to make predictions about the different response properties that could exist in crystals placed within a magnetic field. This includes in particular the electric polarization, as well as another interesting property called the 'shift'. We are now working out how to match these mathematical predictions to computer calculations. 

This work has led to some beautiful pictures, such as the one below! It shows the phase diagram of the shift for a well-known model of electrons in a magnet, called the Hofstadter model. Such pictures are called Hofstadter butterflies. Since 1982, only one such butterfly was known, and it was related to the Hall conductance. Our work has brought to light several more species of Hofstadter butterflies.

The application of tools involving symmetry defects in fact goes far beyond crystalline symmetries. Using them, we were able to classify topological phases of electrons that satisfy a technical property called 'invertibility', for any symmetry. This means that if you hand me a symmetry, I can write down a set of (complicated, but well-controlled and algebraic) equations that will describe the very low-temperature properties of invertible topological phases with that symmetry.