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Charge and Spin Fractionalization in Strongly Correlated Topological Insulators

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Charge and Spin Fractionalization in Strongly Correlated Topological Insulators 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
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