George Mason University

Dept. of Physics and Astronomy

MS 3F3

Planetary Hall, Room 209

Fairfax, VA 22030

USA

703-993-5068

# Topological Insulators (TIs)

#### Updated on 09/29/2015

Symmetry and topology are fundamental governors and organizers of phenomena in physics. Yet, our insight about topology in quantum field theory looks almost anecdotal next to our systematic understanding of symmetry. The latest addition to our collection of topological systems are topological insulators (TIs) invariant under time-reversal. In my view, the most interesting questions and the future of the TI field live in the realm of strongly correlated topological states of matter that push hard on the limits of our knowledge. Here is a summary or my interests and work in this field:

- Introduction
- Kondo topological insulators
- Two-dimensional strongly correlated TIs
- Non-Abelian incompressible quantum liquids and fractional TIs

## Introduction

Topology in mathematics refers to the geometric properties of space that cannot be affected by any smooth deformations. Perhaps the simplest example of topologically non-trivial geometry is Möbius strip. Locally, it is equivalent to a cylinder: a very small ant walking on it would not be able to tell a difference. However, a Möbius strip is globally different than a cylinder. It has a different topology because it cannot be gently bent and twisted into a cylinder. Turning a Möbius strip into a cylinder takes nothing less than cutting it all the way across, untwisting it and then reconnecting.

In quantum mechanics, topology imposes specific boundary conditions on the particles' wavefunction, and thus profoundly affects the quantization of physical quantities and other aspects of dynamics. But, instead of a curved connected space, the origin of topology is dynamics. The coupling of matter to a gauge field with external flux, or another field that contains topological defects (e.g. vortices), introduces consistent local phase-shifts in the wavefunction as if the particle moves through a curved space with non-trivial topology. General relativity indirectly embraces this principle: apart from having more degrees of freedom, the space-time metric tensor is very similar to a gauge field (coordinate system transformations are "gauge" transformations) and has a direct relationship to space-time curvature.

Cylinder | Möbius strip |

While the geometric point of view might help us understand some topological states in quantum mechanics, the challenge posed by the enormous freedom of quantum fields to form exotic states of matter is far from being met, and may require many new ways of thinking. On the purely theoretical side, we have a few big open problems that my research tries to solve. One is explaining the microscopic mechanisms that stabilize topological states driven by interactions among particles. Our current understanding is ultimately phenomenological, but if we knew better we could perhaps predict or design material systems and conditions that realize some of the very interesting topological states. Another problem is predicting various manifestations of non-trivial topology, especially those that can be measured or manipulated in real systems (even in the absence of symmetries that protect a metallic system boundary and the quantization of conserved quantities). The most ambitious project is classifying all possible topological states of matter. We have benefited tremendously from the analogous complete classification of spontaneous symmetry breaking, and the appreciation of the extremely important role that symmetries play in physics.

The motivation to explore topological states of matter goes far beyond the desire to expand fundamental knowledge. Probably many experimental discoveries of new materials that host new types of topological states await us. So far, such major discoveries occurred twice, and both times caused a revolution in condensed matter physics. The robustness of topological states against perturbations and quantum decoherence is also viewed as a very promising feature for applications in quantum engineering and computing.

## Kondo topological insulators

It has been recently realized that spin-orbit coupling can give rise to topological states of electrons in both two- and three-dimensional crystals. Topological materials based on spin-orbit coupling have been discovered shortly after their theoretical prediction, and nowadays are called simply "topological insulators" (TIs). All these materials are band-insulators, interactions among electrons are too weak to affect dynamics or topology. However, a new brand of TIs is increasingly coming into spotlight: "heavy fermion" or Kondo insulators that combine a strong spin-orbit coupling with potent Coulomb interactions.

Unlike the typical "heavy fermion" metals, Kondo insulators have a bandgap in the spectrum of hybridized *d* and *f* electron orbitals that are contributed by a rare earth element. An energy gap is essential for topological protection. At least one material in the Kondo insulator family, samarium hexaboride (SmB_{6}), has been theoretically identified as a promising TI system. It might take some time to learn the whole truth, but many experiments already provide convincing evidence that SmB_{6} is a TI.

One obvious question is how to properly describe TIs based on Kondo insulators when electron interactions are strong and our classification of TIs is based on a non-interacting electron picture. The band-structure of SmB_{6} is not even known with enough accuracy to reliably decide if it has a non-trivial topology. My collaborators in Collin Broholm's neutron scattering group at Johns Hopkins University and I work together to understand SmB_{6} and solve this problem.

Neutrons are found to scatter from a neutral collective mode in SmB_{6} that has a very long lifetime (possibly infinite in a perfectly clean crystal) and a quite flat energy dispersion throughout the first Brillouin zone (1BZ). Scattering intensity is strongly peaked when transferred momentum lies near the R and X points of the 1BZ. Seeing this mode as an exciton (a bound state between a particle and a hole excitation of a band-insulator) indirectly yields important information about the dynamics of particles and holes. It turns out that third-neighbor electron hopping on the cubic sublattice of samarium atoms must be significant in both *d* and *f* orbitals in order for excitons to have the largest spectral weight at the R and X points. This alone determines where in the 1BZ the *d* and *f* orbitals hybridize by avoided energy crossing. Furthermore, since the cubic symmetry of the lattice forbids hybridization at high-symmetry wavevectors, the hybridization gap can exist only if all avoided crossings involve band-inversion. This provides enough information to conclude by a topological-invariant analysis that SmB_{6} is a TI.

The unit-cell of the SmB |
The qualitative energy dispersion of electrons in hybridized |

The relationship between the dispersions of quasiparticles and the collective mode is calculated by perturbation theory (RPA) from the periodic Anderson model that includes spin-orbit coupling. The effects of Coulomb interactions on "flat" *f* orbitals are captured by the standard but uncontrolled slave-boson approximation. This theory has to work with a few unknown microscopic parameters, including the tight-binding parameters for electron hopping between pairs of nearby atoms in both orbitals. We fit all such parameters to match the calculated mode dispersion with the experimentally measured one in the entire 1BZ. It is possible to achieve a very good qualitative match that only slightly overestimates the actual mode bandwidth. While the fitting procedure disqualifies this theory as a microscopic one, it serves an extremely important purpose: it extracts from the experiment the physical picture and quantitative information about *renormalized* excitations that exist in SmB_{6}. The dynamics of hybridized quasiparticles renormalized by Coulomb interactions exhibits a non-trivial topology. But SmB_{6} is a more exotic TI than any non-interacting one: it qualifies as a Mott insulator because its bulk dynamics is dominated by a collective bosonic mode that has lower energy than quasiparticles. The true nature of this correlated insulating state is still not clear and will be the subject of future study.

Feynman diagrams for processes included in the perturbative slave-boson calculation. (1) vertex functions for scattering of hybridized quasiparticles on slave bosons. (2) Slave-boson-mediated attractive interaction between a hybridized particle and a hole creates an exciton bound state. (3) A process that renormalizes and flattens the collective mode dispersion. |
A comparison between the measured neutron scattering intensity and the calculated collective mode spectrum (dotted black line). |

My research also explores novel physical phenomena on the boundaries of correlated TIs. Assuming that materials like SmB_{6} are indeed TIs, one would naively expect a topologically-protected metallic state of chiral (spin-momentum-locked) Dirac quasiparticles on their entire surface - this is a hallmark property of TIs with time-reversal symmetry. However, strong interactions found in Kondo TIs could bring a rich new "heavy fermion" phenomenology to this two-dimensional world.

Simple arguments based on the slave-boson theory and SmB_{6} findings point to the possible existence of spin-wave and superconducting (s^{±} and d-wave) instabilities of the surface metal in the regime of hybridized *d* and *f* orbitals. The former would break time-reversal symmetry and remove the protection of Dirac points. More exotic states involve localization of surface Dirac quasiparticles, made possible by repulsive interactions among the "flat" *f* electrons in reduced dimensionality. It is harder to localize spin than charge, since the spin-orbit coupling tends to prohibit the back-scattering of spin currents. This could perhaps stabilize an algebraic spin liquid, a surface Dirac metal of spinons that cannot conduct charge, but conducts spin. Surface states can also be completely gapped by a non-Abelian topological order without time-reversal symmetry breaking. The alternative local moment regime enables other surface states, such as Neel-ordered Dirac metals and insulators, as well as an unconventional metal of *d* electrons coupled to a U(1) gauge field with two-dimensional dynamics (a feedback from a spin-liquid of localized *f* electrons via Kondo singlets).

#### Papers:

- Two-dimensional heavy fermions on the strongly correlated boundaries of Kondo topological insulators (19 pages)
- In-gap collective mode spectrum of the topological Kondo insulator SmB
_{6}(14 pages) - Interaction Driven Subgap Spin Exciton in the Kondo Insulator SmB
_{6}(4+ pages) - Vortex states in a non-Abelian magnetic field (14 pages)

## Two-dimensional strongly-correlated TIs

A quantum well made from a TI material can be extremely interesting if significant interactions between electrons are naturally present or engineered. Here, interactions and Rashba spin-orbit coupling conspire to create novel vortex lattices of spin currents (invariant under time-reversal). And, in right conditions, quantum melting of such a vortex lattice can yield an *incompressible quantum liquid* - an SU(2) counterpart of fractional quantum Hall states. My work has led to the first prediction and characterization of such exotic states of matter. Even though it might take some time to observe these states, it is good to know that most, if not all, physical conditions required for their stability can be achieved in present-day experiments.

What physical mechanism can realize the above scenario? Consider a typical 3D TI crystal with a protected Dirac metal on its entire boundary. Suppose we gradually reduce its thickness until only a few unit-cell layers remain. As the Dirac metals on the opposite surfaces of this film become coupled by tunneling, their topological protection is lost and a bandgap opens up. We want to stop thinning the sample at the point where we obtain a good two-dimensional band-insulator whose conduction and valence bands remain the lowest energy degrees of freedom. These bands are shaped by a strong Rashba spin-orbit coupling. One can mathematically represent this spin-orbit coupling by a static effective SU(2) gauge field, which has a non-zero Yang-Mills flux coupled to electrons' spin - an SU(2) analogue of the ordinary U(1) magnetic field. The strength of the Rashba SU(2) flux is equivalent to about 1000 T in bismuth-based compounds.

In two spatial dimensions, any short-range attraction between non-relativistic particles always creates a bound state pair. Our two-dimensional band-insulator provides both particle and hole excitations for such bound states, in either exciton or Cooper channels. Suppose we have some interactions that do not discriminate between different spin states of electrons and holes (e.g. Coulomb or phonon-mediated). Then, the emergence of a bound state, which becomes the lowest energy excitation, promotes this band-insulator into a correlated "bosonic" one. The pairing interaction is free to explore the spin-triplet channels since the bound state wavefunction can be antisymmetrized in terms of the original electrons' surface index (top vs. bottom surface of the TI quantum well). The Rashba spin-orbit coupling can now boost the spin-triplet channel: if the spin and momentum of the bound state pair are properly locked, the pair can lower its energy in proportion to the momentum it carries, limited only by the momentum cutoff of the crystal lattice. Of course, life is complicated by competing instabilities and spin non-conservation, but TI quantum wells still provide a natural system where spinful condensates of excitons or Cooper pairs could form at large wavevectors.

The first proposal for realizing the above scenario involved engineering artificial correlations among electrons in a band-insulating TI quantum well. We showed that a TI quantum well interfaced with a conventional superconductor can interact with superconductor's phonons and acquire sufficiently strong attractive interactions for the above scenario in some cases. There are, however, significant practical limitations to how much this system could be tuned in a gated device. A much better solution is to make a quantum well from a naturally correlated Kondo TI material, perhaps SmB_{6}. There, the Rashba spin-orbit coupling could promote the dynamics of spin-triplet excitons.

The heterostructure device with a TI quantum well placed in proximity to a superconducting material |
The speculated phase diagram in the quantum well (Δ is the TI's bandgap tuned by the quantum well thickness, μ is the chemical potential tuned by the gate voltage) |

What is the nature of the finite-momentum spin-triplet condensate in this two-dimensional system? The mean-field approximation predicts two characteristic types of condensates, either at a single wavevector **Q**, or the wavevectors **Q** and -**Q** determined by the strength of the Rashba spin-orbit coupling in the continuum limit. While being somewhat mundane, these condensates have intriguing vortex excitations that often exhibit confinement (like quarks in quantum chromodynamics), or form closely-packed metastable structures in which singularities can be created or annihilated only in quadruplets. On a crystal lattice, however, vortex arrays of tiled metastable quadruplets can seemingly become truly stable mean-field states in certain conditions. Such a vortex lattice is an intriguing spin-current analogue of the Abrikosov lattice, brought about by a non-Abelian Rashba SU(2) flux.

A metastable quadruplet of SU(2) vortices in a condensate shaped by the Rashba spin-orbit coupling |
The pattern of |

If a vortex lattice can be stabilized in a correlated TI quantum well, then it can be also melted by quantum fluctuations. The mentioned quadruplet creations and annihilations are generally expected to reduce the average density of singularities and make room for surviving vortices to move around the crystal. Then, if one can destroy the condensate by tuning the gate voltage toward a band-insulating state, one must (almost) necessarily enter a correlated insulator by the virtue of having two-particle bound states in two dimensions. Since the Rashba spin-orbit coupling maintains a fixed density of SU(2) vortices, we could naively expect a first-order phase transition that produces some kind of a quantum vortex liquid. What is the nature of this state? Material and heterostructure properties look very promising for reaching a low number of particles per SU(2) vortex (of the order of one and smaller), which in U(1) magnetic fields can produce a fractional quantum Hall liquid. Field theory arguments point out that similar incompressible quantum liquids (but without any Hall effect) could arise from an SU(2) vortex lattice melting. Such a fractional TI would naturally respect time-reversal symmetry and exhibit a non-Abelian exchange statistics.

#### Papers:

- Fractional topological insulators of Cooper pairs induced by proximity effect (4+ pages)
- Pairing instabilities in topological insulator quantum wells (13 pages)
- Interaction proximity effect at the interface between a superconductor and a topological insulator quantum well (25 pages)
- Vortices and vortex states in Rashba spin-orbit-coupled condensates (20 pages)
- Vortex states in a non-Abelian magnetic field (14 pages)

## Non-Abelian incompressible quantum liquids and fractional TIs

The only incompressible quantum liquids confirmed by experiments without any dispute are fractional quantum Hall states (spin liquids are probably next in the line for definite confirmation). They possess topological order: a *fractionally quantized* exchange statistics of quasiparticles (neither fermionic nor bosonic) accompanied by ground-state degeneracy in topologically non-trivial spaces. The ubiquitous U(1) symmetry additionally gives rise to the conservation and fractional quantization of quasiparticle charge, and enables a quantized Hall effect via topologically protected gapless excitations confined to the system boundary.

Is it possible to obtain the physics analogous to *fractional* quantum Hall effect from SU(2) spin-orbit fluxes in 2D TIs? The fundamental physical principles suggest that such "fractional" TIs can indeed exist. What would be the properties of such states, and how to even think about them when spin-conservation, and hence a pristine fractional quantization of spin and Hall effect are absent? What is a realistic way to capture the non-commutative dynamics of the quantized SU(2) flux "attached" to particles? Where can one find, or how can one engineer TI quantum wells that could host "fractional" TIs? What experimental probes could unambiguously detect a fractional TI?

My research tries to answer these and other related questions, with an ultimate goal to discover new topological states of matter and contribute to their complete classification. The answers could be very valuable. Fractional TIs obtained from the Rashba spin-orbit coupling are completely new states of matter with natural non-Abelian statistics. If found, they would certainly expand the horizons of our fundamental knowledge and likely provide a new platform for quantum computing based on spintronics.

All qualitative properties of fractional quantum Hall states are captured by an effective Chern-Simons (CS) gauge theory. This very general theory provides a lot of predictive insight, yet its construction relies on the conservation of charge that couples to the external U(1) gauge field. Electron's spin similarly couples to an external SU(2) gauge field that describes the Rashba spin-orbit interaction. However, this SU(2) gauge field is non-commutative and does not conserve spin. Therefore, a generalization of the CS theory is needed to describe all possible incompressible quantum liquids shaped by spin-orbit couplings.

I recently constructed an effective topological theory of particles with arbitrary internal degrees of freedom coupled to any SU(N) gauge field. Since this theory represents particles by spinors, its dynamics has a conventional Landau-Ginzburg form that captures arbitrary disordered and symmetry-broken states. There is, however, an added topological term fully constrained by fundamental symmetries. It functions like a Berry's phase with a fixed coupling constant, but has a dynamical effect only in unconventional quantum states that allow both particles and their topological defects to occupy on average the same positions in space. The principle of duality makes it very hard to obtain such states in two dimensions: particles and vortices exclude each other. However, frustration caused by external gauge fluxes can stabilize incompressible quantum liquids in which particles and vortices become "attached" to each other. A quantum Hall state is precisely of this kind, and the theory I proposed predicts the existence of other such states.

An incompressible quantum vortex liquid is a state with topological order, where the simultaneous mobility of vortices and particles frustrates their relative motion and attaches them to each other |
Topological order features a ground-state degeneracy on topologically non-trivial manifolds like torus. This degeneracy is related to the vanishing energy cost of threading vortices through torus openings, and reflects many-body quantum entanglement. |

The main advantage of representing particles by spinors is to have a clear description of low-energy dynamics in any system. For example, if a bosonic spinor is coupled only to U(1) gauge fields and all of its components have only small density fluctuations about a fixed value, then the dynamics becomes effectively captured by a gauged XY model and the topological term reduces to a CS theory automatically tailored to the correct filling factor. In this picture, the gauge flux represents topological defects of the matter field, but the standard CS theory can be obtained by a duality mapping. On the other hand, if particles are coupled to an SU(2) gauge field that embodies a Rashba spin-orbit coupling, then the reduced effective theory specialized for incompressible states is a dynamically-extended non-Abelian CS gauge theory coupled to an external field and subjected to constraints. This describes new highly-entangled quantum liquids without a quantized Hall effect, whose fractional statistics, classification and hierarchy can be readily extracted from the theory.

There are obviously many open problems that can be addressed with this theory. The most important ones deal with the non-conservation of charge that couples to external flux, and the types of topological order that can exist in nature. It is already clear that non-Abelian gauge fields enable entirely new classes of topological order. Other directions that this work pursues is further generalization of topological field theories to non-minimal topological terms, higher dimensions, regularized spaces (lattices), gauge symmetries with discrete subgroups, etc.