Two-dimensional heavy fermions in Kondo topological insulators Predrag Nikolić George Mason University Institute for Quantum Matter @ Johns Hopkins University Rice University, August 28, 2014
Acknowledgments Collin Broholm IQM @ Johns Hopkins Wesley T. Fuhrman Johns Hopkins Michael Levin University of Maryland Zlatko Tešanović IQM @ Johns Hopkins US Department of Energy National Science Foundation 2/35
Overview Introduction to strongly correlated TIs Proximity effect heterostructures Kondo TIs Correlations on Kondo TI boundaries Hybridized regime: SDW, SC, spin liquids... Local moment regime: AF metal, QED3... TI quantum wells Vortex states Fractional incompressible quantum liquids 3/35
Topological vs. Conventional Conventional states of matter Local properties (order parameter) + symmetry determine all global properties Topological states of matter Have global properties invisible to local probes (e.g. long-range entanglement) Examples of topological quantum states Quantum Hall states & topological insulators Spin liquids, string-net condensates... 4/35
The Appeal of Topological Macroscopic quantum entanglement What makes quantum mechanics fascinating... Still uncharted class of quantum states Non-local properties are resilient Standard for measurements of resistivity Topological quantum computation The math of topology is fancy 5/35
Quantum Hall Effect Integer QHE 2D 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 6/35
Quantum Hall Liquids 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: many-body quantum entanglement (fract. stat.) 7/35
TR-invariant topological Insulators Quantum (but not quantized) spin-hall effect Graphene HgTe, Bi2Se3, Bi2Te3, etc. quantum wells M. K nig, et.al, Science 318, 766 (2007) 8/35
3D Topological Insulators Materials Bi2Se3, Bi2Te3, etc. SmB6 and other Kondo insulators (?) TR symmetry Protected helical Dirac metal on the crystal surface Odd number of Dirac points M.Z.Hasan, C.L.Kane, Rev.Mod.Phys. 82, 3045 (2010) 9/35
Topology + Symmetry Symmetry-protected boundary states Charge conservation edge states in QHE TR symmetry edge states in QSHE, Dirac metal in TIs Crystal symmetry additional Dirac points on TI boundaries Quantum anomalies Local field theory on the boundary cannot be regularized global entanglement on the boundary at cut-off scales (through the bulk) Cancun, Mexico in June 2014 (proves that boundaries are most interesting) Is our universe a topological boundary? 10/35
Kondo Insulators Materials: SmB6, YbB12, Ce3Bi4Pt3, Ce3Pt3Sb3, CeNiSn, CeRhSb... Electron spectrum Hybridized broad d and narrow f orbitals of the rare earth atom (e.g. Sm) Heavy fermion insulators: Ef is inside the hybridization bandgap Coulomb interactions f-bandwidth Correlations T-dependent gap Collective modes 11/35
Kondo TIs (SmB6) Is SmB6 a topological insulator? Theoretical proposal by band-inversions: Dzero, Galtski, Coleman Residual T 0 resistivity & surface conduction Quantum Hall effect Latest neutron scattering G.Li, et al. D.J.Kim, J.Xia, Z.Fisk etc. arxiv:1306.5221 arxiv:1307.0448 A.Kebede, et al. Physica B 223, 256 (1996) 12/35
Collective modes in SmB6 Sharp peak in inelastic neutron scattering Nearly flat dispersion at 14 mev Protected from decay (inside the 19 mev bandgap) Theory: perturbative slave boson model Neutron scattering experiment: Broholm group at IQM/JHU Theory: PN 13/35
Collective Mode Theory Perturbative slave boson theory Applied to the Anderson model adapted by Dzero, et al. Input: renormalized band-structure + 3 phenomenological param. Output: mode dispersion & spectral weight in the 1st B.Z. Slave bosons no double-occupancy of lattice sites by f electrons 14/35
SmB6 Theory + Experiment Match the theoretical & experimental mode spectra Fit the band-structure and phenomenological param. deduce the renormalized fermion spectrum implied band inversion at X a TI experiment theory Implied qualitative FS of d electrons 15/35
Strongly Correlated TIs SmB6 is a strongly correlated TI? Coherent exciton (bound state) in 3D evidence of strong interactions Strong interactions on the TI's boundary? A 2D Dirac metal is only more sensitive to interactions & quantum fluctuations than a 3D insulator in the bulk A minimal lattice model of a TI boundary is desired Coulomb interactions & the S.O.C. can explore cut-off scales 2D Anderson lattice model is a good starting point for Kondo TIs 16/35
Rashba S.O.C. on a TI's Boundary Particles + static SU(2) gauge field SU(2) generators (spin projection matrices) Yang-Mills flux matrix ( magnetic for μ=0) D. Hsieh, et.al, PRL 103, 146401 (2009) Y. Zhang, et.al, Nature Phys. 6, 584 (2010) Rashba S.O.C. Dirac spectrum... Cyclotron: SU(2) flux: 17/35
Lattice Model of a TI boundary Tight-binding model with a Rashba SU(2) gauge field? Single boundary: fix the SU(2) charge τz=+1 or τz=-1 Problem: the number of Dirac points in the B.Z. is necessarily even Remedy: the model must describe two surfaces Inter-surface (bulk-mediated) coupling gaps out some Dirac points 18/35
Kondo TI Boundary Two-surface Anderson model with an SU(2) gauge field Bulk bands Bands bend near the boundary Details are not universal Hybridized regime Local moment regime 19/35
Hybridized Surface Regime The charge of f electrons is mobile The f orbital is less than half-filled & hybridized with the d orbital Coulomb: no double occupancy of any site by f electrons Slave bosons are condensed neutral: heavy fermion Dirac metal charged: exotic fractionalized superconductor band-structure is renormalized by the condensate 20/35
Instabilities of the Hybrid Metal 2D perturbative slave boson theory exciton Cooper Slave bosons mediate forces among electrons attractive in exciton, repulsive in Cooper ch. Basic process: f-d Hybridization assisted by a slave boson evolves into Kondo singlet upon localization slave boson renormalization Instabilities by nesting SDW (Hertz-Moriya-Millis) Dirac points perish (π,π) (π,0) Recall: we have a collective mode right at these wavevectors Superconductivity (s± or d-wave) Dirac points live 21/35
Localization of Hybridized Electrons Mott insulators in doped lattice models Localization at p/q particles per lattice site Broken translation symmetry Coulomb: may localize charge A flat band! (partially populated) Can't localize spin so easily on a TI surface! S.O.C. No spin back-scattering Compromise: spin liquid Algebraic: Dirac points of spinons Non-Abelian: fully gapped (Senthil, Metlitski, Viswanath...) 22/35
Surface Local Moment Regime The charge of f electrons is localized The f orbital is half-filled Low-energy dynamics: Kondo lattice model Kondo v.s. RKKY RKKY wins AF metal or insulator (gapped Dirac points) VBS, spin liquid, etc. Dirac metal Kondo wins (unrealistic over-screening) featureless conventional Dirac metal Kondo singlets frustrate the RKKY orders as doped holes spin liquid of f electrons + 2D QED of conduction electrons 23/35
Correlated TI quantum wells Perturb a QCP by the spin-orbit effect New phases emerge due to relevant scales PN, T.Duric, Z.Tesanovic PRL 110, 176804 (2013) PN, Z.Tesanovic PRB 87, 104514 (2013) PN, Z.Tesanovic PRB 87, 134511 (2013) 24/35
TI quantum wells 2D pairing + Rashba S.O.C. Cooper: TI+SC heterostructure Exciton: Kondo TI quantum wells? τ z = +1 τ z = -1 Controlled by gate Phonon (U<0) or Coulomb (U>0) interaction Rashba SU(2) gauge field 25/35
Competing orders Competition for 1% of free energy (F) PDW map (by ordering wavevectors): Vortex lattice wins? (tight S=1 Cooper pairs) commensurate pair density waves vortex lattices incommensurate plane waves 26/35
Helical spin currents on a lattice Main competitors for the ground state: Similar to: pair density wave (PDW) states W.S.Cole, S.Zhang, A.Paramekanti, N. Trivedi, usually win, but found only when looked for! PRL 109, 085302 (2012) SU(2) vortex lattices (type-i structures) always found in unconstrained minimization, sometimes win? VL PDW 27/35
Kondo TIs: heavy fermion boundary Condensation of triplet collective modes Rashba S.O.C.: momentum-dependent Zeeman A large-momentum mode has low energy Its condensate carries a helical spin current PN, arxiv:1406.1198 Helical spin currents in the continuum T1 phase can be TR-invariant breaks rotation and translation symmetries has metastable vortex clusters & lattices 28/35
Vortex lattice melting SU(2) vortex lattice Array of chiral vortices and antivortices Ideally one vortex per Yang-Mills flux quantum Frustrated by the crystal lattice Quantum fluctuations Positional fluctuations of vortices Grow when the condensate is weakened by tuning the gate voltage Eventually, 1st order phase transition (preempts the 2nd order one) PN, T.Duric, Z.Tesanovic, PRL 110, 176804 (2013) Vortex liquid Particles per flux quantum ~ 1 Fractional TI? PN, PRB 79, 144507 (2009) Cyclotron: SU(2) flux: 29/35
Effective theory of fractional TIs Landau-Ginzburg theory (dual Lagrangian of vortices) dual gauge field matrix spinor vortex field Maxwell term (density fluctuations) Topological term equation of motion drift currents in U(1)xSU(2) E&M field Levi-Civita tensor anti-commutators flux matrix PN, PRB 87, 245120 (2013) P.N, J.Phys: Cond.Mat. 25, 025602 (2013). 30/35
Duality Boson-vortex duality in (2+1)D Exact if: particles are bosons S z is conserved Conjecture: generalize by symmetry fermionic particles? Particle Lagrangian 31/35
Abelian quantum Hall liquids Low-energy dynamics: Chern-Simons Maxwell Fractionalization Measured charge, spin quantized Dual vorticity quantized Combine in quantum spin-hall liquids Laughlin sequence, quantized da dc 32/35
Hierarchical states Emergent symmetries in the low-energy dynamics Multiple vortex flavors Laughlin states: SU(2) hierarchy for spin-s particles 33/35
Rashba spin-orbit coupling Low-energy dynamics: Helical modes: S p Incompressible states: Local density constraint: one free local parameter ϕ (r) per mode quantized densities effective Chern-Simons theory w. constrained non-abelian gauge field 34/35
Conclusions Neutron studies of SmB6 Clear evidence of strong correlations Points to a topological band-structure Kondo TI boundaries Guarantied correlated states at the crystal boundary Hybridized regime: Dirac metal, SDW, SC, spin liquids... Local moment regime: AF metal/insulator, helical 2D QED Kondo (and other) TI quantum wells SU(2) vortex lattice, non-abelian fractional TIs 35/35