# Topological insulators and superconductors (11w5053)

Arriving in Banff, Alberta Sunday, February 6 and departing Friday February 11, 2011

## Organizers

Marcel Franz (University of British Columbia)

Zahid Hasan (Princeton University)

Joel Moore (University of California - Berkeley)

Shoucheng Zhang (Stanford University)

## Objectives

Prediction and the subsequent experimental discovery of topological insulators in the recent years illustrates the potential for startling new physics and mathematics in areas previously thought well understood. The proposed workshop will bring together top researchers in theoretical and experimental physics, mathematics and quantum information, interested in topological insulators and superconductors. The objectives of the proposed workshop are twofold. Firstly, the workshop will provide a platform for the exchange of ideas and studies of the physical nature of the topological phases of matter recently discovered. The participants will address the most important theoretical and experimental issues connected with the underlying physical models, real materials, as well as their potential practical applications. Secondly, we expect that the workshop will present an opportunity for the participants to reflect upon other condensed matter systems where a unique combination of physical insights and ideas from topology, homotopy theory, and other areas of mathematics could lead to discoveries of new forms of quantum matter. Specifically, we plan to focus on the following topics:

1) Theoretical models of topological insulators and superconductors. Standard models have been successfully constructed for simple topological insulators HgTe, Bi2Te3 and Bi2Se3. These theoretical models demonstrate the basic physical mechanism of topological insulator behavior based on band inversion, and make quantitative physical predictions. However, as investigations have now focused on topological insulators in transition metal oxides, more theoretical models are needed, for different lattices (honeycomb and kagome in 2D, cubic, diamond and pyrochlore in 3D). For various parameters these realize all possible topological classes and can be used to study their universal physical properties. Standard models have also been constructed for know topological superfluid state, the He3B phase, however, more realistic models are needed in the search for topological superconductors in Nature. The proposed workshop will help in formulating new models of topological superconductors and insulators, clarify some of their properties that are still in doubt, and will stimulate participants to think about new forms of topologically non-trivial quantum matter.

2) Mathematical structures of the underlying topological invariants. The first descriptions of the new topological invariants that arise with the condition of time-reversal invariance used either Chern classes of complex line bundles on the d-dimensional torus, representing wavefunctions on the Brillouin zone, or homotopy of finite- or infinite-dimensional Hermitian matrices with additional time-reversal conditions, representing the "Bloch Hamiltonians" on the Brillouin zone. Since that time, there have been several mathematical developments at a higher level of sophistication: for example, a powerful mathematical description using K-theory was developed by Kitaev, enabling a rigorous and complete classification of topological insulators of free particles. However, these descriptions are based on non-interacting band theory. The most general definition of the concept of topological insulator is based on the topological field theory of the electromagnetic response, which is generally valid for interacting systems and in the presence of disorder. This topological field theory is constructed from the mathematical concepts of the second Chern class, and directly gives all physically measurable topological responses of the real system. Another topic deals with the question whether the non-Abelian particles that exist in certain quantum Hall states, corresponding to representations of the braid group rather than the familiar bosons and fermions that represent the permutation group, can be realized using topological insulators as a starting point. Some more detail on one proposal to create such particles is given below.

3) Real materials, experimental signatures. As of today a handful of real materials have been confirmed as topological insulators but no topological superconductors have yet been identified. There is little doubt that many more insulating crystals will prove to be topological insulators and the workshop will help communicate ideas essential to the new discoveries. Many theoretical predictions have been made of striking phenomena occurring in topological insulators, such as the topological magneto-electric effect, image magnetic monopoles, axion electrodynamics, spin-charge separation and fractionalization, exciton condensation etc. These remain untested and the workshop will provide a forum for discussions between theorists and experimentalists about the ways to test such predictions in a laboratory.

4) Future practical applications. Among the grand challenges for the future science and technology is realization of a quantum computer, device that could achieve exponential speedup of certain computational tasks, such as factoring large numbers or sorting large databases. One possible realization of a `topological' quantum computer takes advantage of a curious occurrence in nature of exotic particles called non-Abelian anions. These exhibit an interesting property that upon exchange of two such particles their internal quantum state changes in a specific manner. This internal space can be used to store and manipulate quantum information in a way that is protected from the influence of the environment. The problem is that there are at present no real systems known to contain such non-Abelian anions although several candidate systems have been theoretically proposed. Of these arguably the most promising is the proposal made in 2008 by Fu and Kane to create such a non-Abelian anion by nucleating a vortex in a thin superconducting film deposited on a surface of topological insulator. Several related proposals have been made very recently that show ways to test for these elusive particles. The workshop will stimulate discussions and studies of this and other ways to use topological states of matter as building blocks of future quantum computers as well as other practical devices.

5) Beyond the single-electron physics. Rapid progress in this field occurred in part because much of the underlying physics is that of non-interacting electrons and well established powerful methods exist to study such systems. Initially topological insulator state has been defined within the non-interacting band theory, but now a general definition based on topological field theory can be applied to all interacting systems. A big question in the field concerns the role of interactions and strong correlations in topological states of matter. Can we have 'fractional' topological insulator (in analogy with the fractional quantum Hall liquid)? What would be the properties of such a system and can it be realized in a laboratory? Although some initial steps have recently been made towards answering these questions the field is wide open and there is a definite potential for major discoveries. We intend to devote a significant portion of the workshop to these and related questions.

1) Theoretical models of topological insulators and superconductors. Standard models have been successfully constructed for simple topological insulators HgTe, Bi2Te3 and Bi2Se3. These theoretical models demonstrate the basic physical mechanism of topological insulator behavior based on band inversion, and make quantitative physical predictions. However, as investigations have now focused on topological insulators in transition metal oxides, more theoretical models are needed, for different lattices (honeycomb and kagome in 2D, cubic, diamond and pyrochlore in 3D). For various parameters these realize all possible topological classes and can be used to study their universal physical properties. Standard models have also been constructed for know topological superfluid state, the He3B phase, however, more realistic models are needed in the search for topological superconductors in Nature. The proposed workshop will help in formulating new models of topological superconductors and insulators, clarify some of their properties that are still in doubt, and will stimulate participants to think about new forms of topologically non-trivial quantum matter.

2) Mathematical structures of the underlying topological invariants. The first descriptions of the new topological invariants that arise with the condition of time-reversal invariance used either Chern classes of complex line bundles on the d-dimensional torus, representing wavefunctions on the Brillouin zone, or homotopy of finite- or infinite-dimensional Hermitian matrices with additional time-reversal conditions, representing the "Bloch Hamiltonians" on the Brillouin zone. Since that time, there have been several mathematical developments at a higher level of sophistication: for example, a powerful mathematical description using K-theory was developed by Kitaev, enabling a rigorous and complete classification of topological insulators of free particles. However, these descriptions are based on non-interacting band theory. The most general definition of the concept of topological insulator is based on the topological field theory of the electromagnetic response, which is generally valid for interacting systems and in the presence of disorder. This topological field theory is constructed from the mathematical concepts of the second Chern class, and directly gives all physically measurable topological responses of the real system. Another topic deals with the question whether the non-Abelian particles that exist in certain quantum Hall states, corresponding to representations of the braid group rather than the familiar bosons and fermions that represent the permutation group, can be realized using topological insulators as a starting point. Some more detail on one proposal to create such particles is given below.

3) Real materials, experimental signatures. As of today a handful of real materials have been confirmed as topological insulators but no topological superconductors have yet been identified. There is little doubt that many more insulating crystals will prove to be topological insulators and the workshop will help communicate ideas essential to the new discoveries. Many theoretical predictions have been made of striking phenomena occurring in topological insulators, such as the topological magneto-electric effect, image magnetic monopoles, axion electrodynamics, spin-charge separation and fractionalization, exciton condensation etc. These remain untested and the workshop will provide a forum for discussions between theorists and experimentalists about the ways to test such predictions in a laboratory.

4) Future practical applications. Among the grand challenges for the future science and technology is realization of a quantum computer, device that could achieve exponential speedup of certain computational tasks, such as factoring large numbers or sorting large databases. One possible realization of a `topological' quantum computer takes advantage of a curious occurrence in nature of exotic particles called non-Abelian anions. These exhibit an interesting property that upon exchange of two such particles their internal quantum state changes in a specific manner. This internal space can be used to store and manipulate quantum information in a way that is protected from the influence of the environment. The problem is that there are at present no real systems known to contain such non-Abelian anions although several candidate systems have been theoretically proposed. Of these arguably the most promising is the proposal made in 2008 by Fu and Kane to create such a non-Abelian anion by nucleating a vortex in a thin superconducting film deposited on a surface of topological insulator. Several related proposals have been made very recently that show ways to test for these elusive particles. The workshop will stimulate discussions and studies of this and other ways to use topological states of matter as building blocks of future quantum computers as well as other practical devices.

5) Beyond the single-electron physics. Rapid progress in this field occurred in part because much of the underlying physics is that of non-interacting electrons and well established powerful methods exist to study such systems. Initially topological insulator state has been defined within the non-interacting band theory, but now a general definition based on topological field theory can be applied to all interacting systems. A big question in the field concerns the role of interactions and strong correlations in topological states of matter. Can we have 'fractional' topological insulator (in analogy with the fractional quantum Hall liquid)? What would be the properties of such a system and can it be realized in a laboratory? Although some initial steps have recently been made towards answering these questions the field is wide open and there is a definite potential for major discoveries. We intend to devote a significant portion of the workshop to these and related questions.