Hopf characterization of two-dimensional Floquet topological insulators

Ünal, F. Nur; Eckardt, André; Slager, Robert-Jan

doi:10.1103/PhysRevResearch.1.022003

Title: Hopf characterization of two-dimensional Floquet topological insulators

Abstract

We present a topological characterization of time-periodically driven two-band models in 2 + 1 dimensions as Hopf insulators. The intrinsic periodicity of the Floquet system with respect to both time and the underlying two-dimensional momentum space constitutes a map from a three-dimensional torus to the Bloch sphere. As a result, we find that the driven system can be understood by appealing to a Hopf map that is directly constructed from the micromotion of the drive. Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing the topology of knots in three dimensions. Moreover, after being cast as a Hopf insulator, not only the Chern numbers but also the winding numbers of the Floquet topological insulator become accessible in experiments as linking numbers. We exploit this description to propose a feasible scheme for measuring the complete set of their Floquet topological invariants in optical lattices.

Authors:

; Eckardt, André; Slager, Robert-Jan

Publication Date:: 2019-09-09

Research Org.:: Energy Frontier Research Centers (EFRC) (United States). Center for the Advancement of Topological Semimetals (CATS); Ames Lab., Ames, IA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Basic Energy Sciences (BES); German Research Foundation (DFG)

OSTI Identifier:: 1560766

Alternate Identifier(s):: OSTI ID: 1767792

Grant/Contract Number:: AC02-07CH11358; 277974659

Resource Type:: Published Article

Journal Name:: Physical Review Research

Additional Journal Information:: Journal Name: Physical Review Research Journal Volume: 1 Journal Issue: 2; Journal ID: ISSN 2643-1564

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; optics; charge transport; magnetism and spin physics; materials and chemistry by design; synthesis (novel materials); synthesis (predictive)

Citation Formats


                    Ünal, F. Nur, Eckardt, André, and Slager, Robert-Jan. Hopf characterization of two-dimensional Floquet topological insulators.  United States: N. p., 2019. 
Web.  doi:10.1103/PhysRevResearch.1.022003.

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                    Ünal, F. Nur, Eckardt, André, & Slager, Robert-Jan. Hopf characterization of two-dimensional Floquet topological insulators.  United States.  https://doi.org/10.1103/PhysRevResearch.1.022003

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                    Ünal, F. Nur, Eckardt, André, and Slager, Robert-Jan. Mon .  
"Hopf characterization of two-dimensional Floquet topological insulators".  United States.  https://doi.org/10.1103/PhysRevResearch.1.022003.

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@article{osti_1560766,

  title        = {Hopf characterization of two-dimensional Floquet topological insulators},

  author       = {Ünal, F. Nur and Eckardt, André and Slager, Robert-Jan},

  abstractNote = {We present a topological characterization of time-periodically driven two-band models in 2 + 1 dimensions as Hopf insulators. The intrinsic periodicity of the Floquet system with respect to both time and the underlying two-dimensional momentum space constitutes a map from a three-dimensional torus to the Bloch sphere. As a result, we find that the driven system can be understood by appealing to a Hopf map that is directly constructed from the micromotion of the drive. Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing the topology of knots in three dimensions. Moreover, after being cast as a Hopf insulator, not only the Chern numbers but also the winding numbers of the Floquet topological insulator become accessible in experiments as linking numbers. We exploit this description to propose a feasible scheme for measuring the complete set of their Floquet topological invariants in optical lattices.},

  doi          = {10.1103/PhysRevResearch.1.022003},

  journal      = {Physical Review Research},

  number       = 2,

  volume       = 1,

  place        = {United States},

  year         = {2019},

  month        = {9}

}

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Title: Hopf characterization of two-dimensional Floquet topological insulators

Abstract

Citation Formats

Unified bulk-boundary correspondence for band insulators journal, March 2018

Topological characterization of periodically driven quantum systems journal, December 2010

Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems journal, July 2013

Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms journal, December 2014

Colloquium: Atomic quantum gases in periodically driven optical lattices journal, March 2017

Three-dimensional two-band Floquet topological insulator with Z 2 index journal, February 2019

Topological insulators and superconductors journal, October 2011

Tomography of Band Insulators from Quench Dynamics journal, July 2014

Hopf insulators and their topologically protected surface states journal, November 2013

How to Directly Measure Floquet Topological Invariants in Optical Lattices journal, June 2019

Bloch state tomography using Wilson lines journal, May 2016

Colloquium: Topological insulators journal, November 2010

Measuring topology from dynamics by obtaining the Chern number from a linking number journal, April 2019

Impurity-bound states and Green's function zeros as local signatures of topology journal, August 2015

Experimental reconstruction of the Berry curvature in a Floquet Bloch band journal, May 2016

Phase vortices of the quenched Haldane model journal, August 2017

Observation of dynamical vortices after quenches in a system with topology journal, December 2017

Universal Probes of Two-Dimensional Topological Insulators: Dislocation and π Flux journal, March 2012

Topological Crystalline Insulators journal, March 2011

Periodic table for Floquet topological insulators journal, October 2017

Driving protocol for a Floquet topological phase without static counterpart journal, November 2017

Topological quantum chemistry journal, July 2017

Topological Surface States in Three-Dimensional Magnetic Insulators journal, October 2008

The translational side of topological band insulators journal, May 2019

Experimental realization of the topological Haldane model with ultracold fermions journal, November 2014

Bulk-boundary correspondence of topological insulators from their respective Green’s functions journal, September 2011

Topological Classification of Crystalline Insulators through Band Structure Combinatorics journal, December 2017

Symmetry-based indicators of band topology in the 230 space groups journal, June 2017

Topological Hopf-Chern insulators and the Hopf superconductor journal, July 2016

Inner topological structure of Hopf invariant journal, July 2007

Periodic table for topological insulators and superconductors conference, January 2009

State-recycling and time-resolved imaging in topological photonic lattices journal, October 2018

Symmetry-protected topological Hopf insulator and its generalizations journal, April 2017

Chern number and edge states in the integer quantum Hall effect journal, November 1993

The space group classification of topological band-insulators journal, December 2012

Uncover Topology by Quantum Quench Dynamics journal, December 2018

Observation of photonic anomalous Floquet topological insulators journal, January 2017

Measurement of a Topological Edge Invariant in a Microwave Network journal, February 2015

Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices journal, October 2013

Self-organized pseudo-graphene on grain boundaries in topological band insulators journal, June 2016

Observation of Topological Links Associated with Hopf Insulators in a Solid-State Quantum Simulator journal, June 2017

Probing topological protection using a designer surface plasmon structure journal, May 2016

Topological Bloch oscillations journal, July 2018

Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices journal, October 2013

Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice journal, January 2017

Dynamical classification of topological quantum phases journal, November 2018

Scheme to Measure the Topological Number of a Chern Insulator from Quench Dynamics journal, May 2017

Topological insulators and superconductors: tenfold way and dimensional hierarchy journal, June 2010

Unified bulk-boundary correspondence for band insulators
journal, March 2018

Topological characterization of periodically driven quantum systems
journal, December 2010

Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems
journal, July 2013

Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms
journal, December 2014

Colloquium: Atomic quantum gases in periodically driven optical lattices
journal, March 2017

Three-dimensional two-band Floquet topological insulator with $Z_{2}$ index
journal, February 2019

Topological insulators and superconductors
journal, October 2011

Tomography of Band Insulators from Quench Dynamics
journal, July 2014

Hopf insulators and their topologically protected surface states
journal, November 2013

How to Directly Measure Floquet Topological Invariants in Optical Lattices
journal, June 2019

Bloch state tomography using Wilson lines
journal, May 2016

Colloquium: Topological insulators
journal, November 2010

Measuring topology from dynamics by obtaining the Chern number from a linking number
journal, April 2019

Impurity-bound states and Green's function zeros as local signatures of topology
journal, August 2015

Experimental reconstruction of the Berry curvature in a Floquet Bloch band
journal, May 2016

Phase vortices of the quenched Haldane model
journal, August 2017

Observation of dynamical vortices after quenches in a system with topology
journal, December 2017

Universal Probes of Two-Dimensional Topological Insulators: Dislocation and $π$ Flux
journal, March 2012

Topological Crystalline Insulators
journal, March 2011

Periodic table for Floquet topological insulators
journal, October 2017

Driving protocol for a Floquet topological phase without static counterpart
journal, November 2017

Topological quantum chemistry
journal, July 2017

Topological Surface States in Three-Dimensional Magnetic Insulators
journal, October 2008

The translational side of topological band insulators
journal, May 2019

Experimental realization of the topological Haldane model with ultracold fermions
journal, November 2014

Bulk-boundary correspondence of topological insulators from their respective Green’s functions
journal, September 2011

Topological Classification of Crystalline Insulators through Band Structure Combinatorics
journal, December 2017

Symmetry-based indicators of band topology in the 230 space groups
journal, June 2017

Topological Hopf-Chern insulators and the Hopf superconductor
journal, July 2016

Inner topological structure of Hopf invariant
journal, July 2007

Periodic table for topological insulators and superconductors
conference, January 2009

State-recycling and time-resolved imaging in topological photonic lattices
journal, October 2018

Symmetry-protected topological Hopf insulator and its generalizations
journal, April 2017

Chern number and edge states in the integer quantum Hall effect
journal, November 1993

The space group classification of topological band-insulators
journal, December 2012

Uncover Topology by Quantum Quench Dynamics
journal, December 2018

Observation of photonic anomalous Floquet topological insulators
journal, January 2017

Measurement of a Topological Edge Invariant in a Microwave Network
journal, February 2015

Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices
journal, October 2013

Self-organized pseudo-graphene on grain boundaries in topological band insulators
journal, June 2016

Observation of Topological Links Associated with Hopf Insulators in a Solid-State Quantum Simulator
journal, June 2017

Probing topological protection using a designer surface plasmon structure
journal, May 2016

Topological Bloch oscillations
journal, July 2018

Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices
journal, October 2013

Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice
journal, January 2017

Dynamical classification of topological quantum phases
journal, November 2018

Scheme to Measure the Topological Number of a Chern Insulator from Quench Dynamics
journal, May 2017

Topological insulators and superconductors: tenfold way and dimensional hierarchy
journal, June 2010