Charge and spin fractionalization in strongly correlated topological insulators Predrag Nikolić George Mason University October 26, 2011 Acknowledgments Zlatko Tešanović Michael Levin Tanja Duric IQM @ Johns Hopkins University of Maryland Max Planck Institute, Dresden • Affiliations and sponsors The Center for Quantum Science Charge and spin fractionalization in strongly correlated topological insulators 2/33 Overview • TIs with time-reversal symmetry – Introduction to topological band-insulators – Introduction to interacting TIs • Experimental realization of strongly correlated TIs – Cooper pair TI by proximity effect – Exciton TI • Theory of strongly correlated TIs – Topological Landau-Ginzburg theory – Charge and spin fractionalization Charge and spin fractionalization in strongly correlated topological insulators 3/33 Topological vs. Conventional • Conventional states of matter – Characterized by local properties (LDOS, order parameter...) • Topological states of matter – Characterized by non-local properties (entanglement, edge modes, torus spectra...) • Examples of topological quantum states – Quantum Hall states & “topological insulators” – Spin liquids, string-net condensates... Charge and spin fractionalization in strongly correlated topological insulators 4/33 The Appeal of Topological • Macroscopic quantum entanglement – What makes quantum mechanics fascinating... – Still uncharted class of quantum states • Non-local properties are hard to perturb – Standard for measurements of resistivity – Topological quantum computation • The math of topology is fancy Charge and spin fractionalization in strongly correlated topological insulators 5/33 Quantum Hall States • Integer QHE – Topological band-insulator of electrons in magnetic field – Bulk spectrum: Landau levels (continuum) or Hofstadter (lattice) – Boundary spectrum: chiral gapless edge modes – Non-local property: Chern #, quantized transverse (Hall) conductivity Charge and spin fractionalization in strongly correlated topological insulators 6/33 Quantum Hall States • Fractional QHE – Strongly correlated topological insulator of electrons in magn. field – Quasiparticles with fractional charge and exchange statistics – Ground-state degeneracy on a torus (no symmetry breaking) – Non-local property: quantum entanglement, quant. Hall conductivity Charge and spin fractionalization in strongly correlated topological insulators 7/33 Topological Insulators • Quantum spin-Hall systems (known so far) – Spin-orbit-coupled topological band-insulators – Boundary spectrum: 2D: a Kramers-pair of gapless chiral edge modes 3D: a massless helical Dirac cone on the surface (“strong” TIs) – Non-local property: Z2 topological invariant = 1 • Similar to integer QHE, but... – Respect the time-reversal (TR) symmetry – Spin plays the role of charge: SU(2) instead of U(1) gauge structure – Not an SU(2) gauge theory, though ⇒ non-degenerate spectrum – Spin is not conserved ⇒ Z2 topological invariant instead of Chern # ⇒ Only approximate quantization of the spin-Hall conductivity Charge and spin fractionalization in strongly correlated topological insulators 8/33 2D Topological Insulators • Materials – Graphene (unfortunately not) – HgTe, Bi2Se3, Bi2Te3, etc. quantum wells M. Knig, et.al, Science 318, 766 (2007) Charge and spin fractionalization in strongly correlated topological insulators 9/33 3D Topological Insulators • Materials – Bi2Se3, Bi2Te3, etc. quantum wells M.Z.Hasan, C.L.Kane Rev.Mod.Phys. 82, 3045 (2010) Charge and spin fractionalization in strongly correlated topological insulators 10/33 Quantum spin-Hall effect • TI quantum well: Bi2Se3, Bi2Te3 • Equation of motion: “cyclotron dynamics” • Approximate gauge theory with the same eq. of mot. – Static spin-orbit SU(2) gauge field with “magnetic” flux Charge and spin fractionalization in strongly correlated topological insulators 11/33 Real Tis vs. spin-Hall effect • Deviations from the SU(2) gauge symmetry 1) Gauge-symmetry-breaking “perturbations” • Fixing a gauge does not remove the SU(2) flux 2) Different spectra • No degeneracies... (important?) • The best gauge theory ⇔ the “smallest” • Similar energy landscapes ⇔ similar phase diagrams 3) Spin non-conservation • Profound consequences on “topological orders” - later... Charge and spin fractionalization in strongly correlated topological insulators 12/33 Typical scales: Bi2Se3, Bi2Te3 – Cyclotron energy: – Flux-quantum density: 1. 2. Intrinsic mass scale in zero spin-orbit coupling 1. D. Hsieh, et.al. Phys.Rev.Lett. 103, 146401 (2009) 2. Y. Zhang, et.al. Nature Phys. 6, 584 (2010) Charge and spin fractionalization in strongly correlated topological insulators 13/33 Interesting effects of interactions • Weak interactions – Instability of a topological (helical) Fermi surface ⇒ ordered state with “topological” quasiparticles (edge modes) and Majorana states in vortex cores... • Motness + spin-orbit effect – Spin liquid = topological band-insulator of spinons – Fractionalization (spin-charge separation) by Coulomb interactions • Narrow band TI + strong interactions – Fractional quantum spin-Hall effect? – Fractionalization by spin-orbit coupling Charge and spin fractionalization in strongly correlated topological insulators 14/33 Helical FS instability • Helical surface metal • Weak attractive interactions – Topological band-insulator of Bogoliubov quasiparticles – Gapless surface of edge Andreev modes – Majorana modes bound to topological defects Charge and spin fractionalization in strongly correlated topological insulators 15/33 Motness + spin-orbit effect • Spin liquid in a frustrated magnet – Mott insulator of electrons localized by Coulomb interactions – Topological order with spin-charge separation – Spinons: (gapped) neutral spinful (fermionic) excitations • Add strong spin-orbit coupling D.Pesin, L.Balents Nature Phys. 6, 376 (2010) – Topological band-insulator of spinons Charge and spin fractionalization in strongly correlated topological insulators 16/33 Fractional TIs • Fractionalization by spin-orbit effect (and interactions) – TR-invariant topological order with anyon excitations – Analogous to fractional quantum Hall states (FQHS) • Narrow band TI + strong interactions – Mimic the circumstances of FQHS – Must await new material discoveries • Perturb a QCP by the spin-orbit effect – Possible in heterostructures – New phases emerge due to relevant scales (often seen in heavy-fermion materials) N.D.Mathur et.al, Nature 394, 39 (1998) Charge and spin fractionalization in strongly correlated topological insulators 17/33 Correlated TI Devices • Two ways to create a QCP – Cooper pair localization – Exciton localization P.N, T. Duric; arXiv:1109.0017 • Cooper pair device – TI quantum well (say Bi2Se3) – Pairing by proximity to a superconductor – QCP tuned by SC's Tc, gate voltage... • Exciton device – TI quantum well in a capacitor – Biased capacitor creates an exciton gas – QCP tuned by capacitor bias Charge and spin fractionalization in strongly correlated topological insulators 18/33 The Nature of QCP • Short-range attraction in a 2D band-insulator – A two-particle bound state always exists – If the bound state energy is below the two-particle continuum... ⇒ Coherent bosonic low-energy mode exists • Localization QCP: condensate ↔ bosonic Mott insulator – mean-field or XY transition (fermions are gapped) Charge and spin fractionalization in strongly correlated topological insulators 19/33 Interesting channels for a QCP Cooper pair of two conduction band or two valence band electrons Cooper pair of a conduction and a valence band electron Exciton: bound singlet of a particle and a hole Assisted resonance? Extended s-wave resonating singlet (as in iron-based superconductors) Other: inter-orbital, p-wave, CDW, SDW... Theory of correlated superconductors: s- vs. d-wave 20/33 Renormalization group • Universality class of the SF-I transition – bosonic mean-field or XY: d=2, or strong coupling in d>2 – pair-breaking: weak coupling in d>2: P.N, PRB 83, 064523 (2011) Theory of correlated superconductors: s- vs. d-wave 21/33 Cooper pair TI • Proximity effect – SC is fully gapped (Anderson-Higgs) appart from possible nodal quasiparticles – SC is not a state with broken symmetry the global U(1) phase is not a degree of freedom – Virtual tunneling of Cooper pairs into the TI ⇒ Induced interactions in the TI quantum well – Short-range and attractive – Not a priori sensitive to spin (can make spin triplets) The microscopic Coulomb interaction is spin-blind (even in the phonon mechanism) ⇒ Cooper pair Mott transition in the TI – tunable by gate, Tc, etc. Charge and spin fractionalization in strongly correlated topological insulators 22/33 Pairing Channels • The minimal TR-invariant topological band-insulator – 2 spin states (σ = ±1) X 2 orbital states (τ = ±1) ⇒ The relevant Cooper channels (in the RG sense) Spin-singlet: , Spin-triplet: Charge and spin fractionalization in strongly correlated topological insulators 23/33 Effective Theory • The most generic boson action – Integrate out all fermion fields near the Cooper Mott transition – The spin-triplet Cooper pairs have spin-orbit coupling … plus terms that violate the SU(2) gauge symmetry and spin conservation... There is SU(2) “magnetic” flux: Charge and spin fractionalization in strongly correlated topological insulators 24/33 Spin-orbit Correlated States • Superconducting states – 〈φτ〉≠0 … conventional superconductor – 〈η0〉≠0 … “topological superconductor” – 〈η±〉≠0 … SU(2) flux lattices? • Quantum spin-Hall states? – Quantum