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Kai Sun JQI-NSF-PFC and CMTC, University of Maryland, College Park

Topological Insulators, Topological Semimetals, and Quantum Hall Effects without Landau Levels. Kai Sun JQI-NSF-PFC and CMTC, University of Maryland, College Park. Outline. Introduction Phenomena and theories of topological insulators Realization in band insulators

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Kai Sun JQI-NSF-PFC and CMTC, University of Maryland, College Park

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  1. Topological Insulators, Topological Semimetals, andQuantum Hall Effects without Landau Levels Kai Sun • JQI-NSF-PFC and CMTC, University of Maryland, College Park

  2. Outline • Introduction • Phenomena and theories of topological insulators • Realization in band insulators • Magnetic field/spin-orbit effects • Interaction induced topological states • 2D integer topological states • 2D fractional topological states • 3D topological states

  3. Collaborators 2DCold Gases Hong Yao (Berkeley) W. Vincent Liu (Pittsburgh) Eduardo Fradkin (UIUC) Andreas Hemmerich (Hamburg) Steve Kivelson (Stanford) Sankar Das Sarma (Maryland) 2D Fractional Topological insulators Zhengcheng Gu (KITP, UCSB) Donna Sheng (California State) Hosho Katsura (Gakushuin University) Li Sheng (NanJing) Sankar Das Sarma (Maryland) 3D Elasticity in classical systems Maxim Dzero (Kent State) Anton Souslov (Gatech) Victor Galitski(Maryland) Xiaoming Mao (U Penn) Piers Coleman (Rutgers) Tom C. Lubensky (U Penn)

  4. What is a topological insulator? Dissipationlesscurrent: more efficient electronic devices (proposal) High precision measurement: The best way to determine (reality) • May have fractional quasi-particles Topological quantum computation (proposal)

  5. Integer Quantum Hall Effect ρxy=E y/jx~B σxy=j x/E y= n e2/h σxx =jx/Ex= 0 • Why is it an insulator? • Why quantized?

  6. A semi-classical picture and the chiral edge states Edge state stable against impurities Insulating Bulk σxy=j x/E y= n e2/h Conducting chiral edge (half of a 1D quantum wire)

  7. Topological insulators with T-reversal symmetry • Idea: • B>0 for • B<0 for Q: How? A: Spin-orbit Compare: • Spin is no longer a good quantum number • T-symmetry: Z2 topological index • even or odd (pairs of) edges • Kane and Mele, PRL, 2005 Konig et. al., 2007

  8. Topology in a nut shell • Total Gaussian curvature and Euler characteristic • Integer value for closed manifold • Measures the topology: “the number of holes” Topological Index

  9. Topology in insulators • Total Berry curvature in the momentum space • First Chern number: Measures the topology of GS wave-function • Quantized (integer) for insulators • Ordinary Insulators: 0 • Each filled Landau level: 1 Bloch waves n is summed over fully occupied bands Topological insulator: Insulators with non-zero topological index • Other topological indices give other types of TIs.

  10. Topology and edge states Quantum Hall Insulator topologically nontrivial Chern number =1 Vacuum (Insulator) topologically trivial Chern number =0 Quantum Hall Insulator topologically nontrivial Chern number =1 Metallic edge

  11. Rigorous Theory We need one “heavy” theoretical tool. Weight: 2.9 pounds!

  12. Rigorous Theory E&M fields in an isotropic insulating media • Question: • Can there be any other terms? • Answer: • No • Yes! • Reasons: • Gauge symmetry • Time-reversal symmetries • Gauge symmetry

  13. Chern-Simons gauge theory • A topological field theory: • k is quantized = the Chern number in IQHE • Open manifold: gauge invariant in the bulk but NOT on the edge • good or bad? • Cure from high-energy physics: gauge anomaly • 1D chiral fermion also violates gauge symmetry • Two effects cancel each other, if there are k edges states • Gauge anomaly in (2n-1) space dimensions and the Chern-Simons terms in 2n space dimensions

  14. Cyclotron motion vs. Topologywhich one is more fundamental? Vacuum (Insulator) topologically trivial Chern number =0 Quantum Hall Insulator topologically nontrivial Chern number =1 Q: Which picture is more fundamental? A: Topology is the key and B field is not essential. Q: How to demonstrate this? A: Find a model with no B fields (no cyclotron motion), but still shows the same physics. Metallic edge

  15. Haldane’s model on honeycomb lattice • Staggered B field in each hexagon • +B near center, -B near the bond • total B=0 • Insulator • Topology same as IQH • Properties same as IQH • No cyclotron motion • First example demonstrating that only thing • crucial here is topology • Spin up + spin down (opposite B pattern) • = the Kane-Mele model (T-invariant TI)

  16. “Family Tree” of topological insulators • Integer Quantum Hall Integer topological insulators • 2D T-invariant TI • (QSH) • Quantum Anomalous Hall effect • 3D T-invariant TI

  17. Interactions induce TI? • All known TIs need B field/SO couplings • Q: Can TIs be stabilized without B-field or SO couplings? • Focus: Interaction induced TIs without B or S-O couplings • Why interactions? • History: free interactions • Noninteracting systems: well-understood • In the presence interactions:full of puzzles • Experiments • may reveal to us new families of topological materials.

  18. Honeycomb lattice • Honeycomb lattice: graphene K K’ Condensed matter realization of the Dirac theory, with c reduced by a factor of 1000 Semi-metal at half-filling

  19. Topology behind 2D Dirac Points • If symmetry prevents the presence of all three Pauli matrices • Combination of time-reversal and space-inversion (T I) • Defines a vector field in the k-space: • Dirac Point: +1 -1

  20. How to make it into an insulator? • Annihilate two vortices: • Topologically trivial insulator • Allows all the three-Pauli matrix • Mapping from BZ to 3D vectors • Kronecker Index: the Chern number • Topological insulator • Band crossing points: vortex • Break T-symmetry: get the third Pauli matrix • Then we get a topological insulator!

  21. Interaction induce topological insulator without B?Graphene (Dirac point) is a bad choice • Dimension counting Kenneth G. Wilson The Nobel Prize in Physics 1982 renormalization group theory Nobel Prize winner This is part of the studies, which wins him the prize • g: irrelevant in d>1 • Weak coupling: Dirac points asymptotically free • Strong coupling: Competing orders/uncontrollable • Hard to get: NNN repulsion> NN repulsion • (Raghu, Qi, Honerkampand Zhang, PRL, 2008). • Interaction destroys TI in Haldane’s model • (Varney, KS, Rigol and Galitski, PRB, 2010). Wilson, PRD 7, 2911 (1973).

  22. Vortex with higher winding number QBCP • Half-filling: Topological Semi-metal • Protected with 4-fold/6-fold rotational symmetry! KS, Yao, Fradkin and Kivelson, PRL (2009) KS, Liu, Hemmerich and Das Sarma, Nature Physics, (2012)

  23. Example: CuO2 lattice in high Tccompounds Emery lattice/Lieb Lattice KS, Yao, Fradkin and Kivelson, PRL (2009)

  24. Example: Checkerboard lattice Checkerboard lattice Thing films of LiV2O4, MgTi2O4, Cd2Re2O7,… KS, Yao, Fradkin and Kivelson, PRL (2009)

  25. Example: Kagome lattice Kagome lattice KS, Yao, Fradkin and Kivelson, PRL (2009)

  26. Frustrated spin systems Topologically protected quadratic band crossing point in spinon spectrum C. Xu, F. Wang, Y. Qi, L. Balents and M. P. A. Fisher, 2011

  27. Topological Semi-metal vs. Dirac Points Total winding=0 Total winding=2 +1 , • Open gap = topological insulators -1

  28. Topological semi-metal • D-wave version of Dirac point: • Infinitesimal repulsion: induce topological insulator KS, Yao, Fradkin and Kivelson, PRL (2009)

  29. Interaction effect in topological semi-metals • Action: • Dimension counting • 2+1D: RG flows for • t0, t1 , g and Ψ • t0, t1, and Ψ don’t flow at the one-loop level • doesn’t depend on momentum KS, Yao, Fradkin and Kivelson, PRL (2009)

  30. Graphene, Dirac point and topological defect T + Zero Sound (bubble) Normal BCS t Tc QBCP unstable: open gap QBCP stable: remain gapless g Topological 0 0 KS, Yao, Fradkin and Kivelson, PRL (2009)

  31. Order parameters • Fermion bilinears (order parameters) Nematic: main-axis Nematic: diagonal Topological insulator Quantum Nematic Semi-metal Topological insulator Topological insulators: favored at weak coupling • Similar to the BCS theory • Infinitesimal instability • no other competing orders at weak-coupling

  32. Real Space Picture: spontaneous symmetry breaking • Spinless fermions: QH insulator with Chern number=1 • Spin-1/2 fermions: depends on spin-spin interactions • QH insulator with Chern number =2 • Z2 topological insulator KS, Yao, Fradkin and Kivelson, PRL, 2009 KS, Liu, Hemmerich and Das Sarma, Nature Physics, 2012

  33. Bilayer Graphene K +2 K’ -2 • 2 vortex: what is the Chern number? • C=(1-1)X2=0X2=0: trivial insulator • C=(1+1)X2=2X2=4: topological insulator • Insulating behavior: observed • (transport gap, compressiblity) • Yacobygroup, Science 2010, PRL 2010. • Schonenberger2011, Lau 2011, Fuhrer 2011 … • C=? Open question (0 or 2X2=4) • Vafek and Yang, Levitov group,McDonald group, • Das Sarma group… Yacoby group PRL 2010

  34. TI in ultracold gases • Topological insulators have never been achieved in cold gases! • Main Challenge: no B field and no spin-orbit coupling. • Possible solution #1: mimic B field (using rotations or laser beams) • Possible solution #2: TS + interactions =TI (without B or SO couplings) In cold gases, we can design the band structure and interactions! KS, Liu, Hemmerich and Das Sarma, Nature Physics, 2012

  35. “Family Tree” of topological insulators • Integer Quantum Hall • Fractional • Quantum Hall Integer topological insulators Fractional topological insulators • 2D T-invariant TI • Quantum Anomalous Hall effect • 3D T-invariant TI

  36. Fractional quantum Hall effect • Strong interactions: necessary! • Fractionalization (charge and statistics) • Fractional Charge • Fractional Statistics • Topological degeneracy • Chern-Simons term: intrinsic gauge Stormer, et. al., RMP, 1999 • Insulating bulk • Chiral metallic edge • Quantized Hall conductivity (fractional ) • Topological insulator Goldman et. al., 1995

  37. “Family Tree” of topological insulators • Integer Quantum Hall • Fractional • Quantum Hall Integer topological insulators Fractional topological insulators • 2D T-invariant TI • Quantum Anomalous Hall effect • 3D T-invariant TI Can we introduce any new member here? FQHE with B=0?

  38. Why do we need a new setup for FQHE? • Wanted for 20+ years • Theoretically • Which properties are universal in FTIs? • Different paths towards the same topology • B field? Strong Lattice? • T-invariant fractional topological insulator • Experimentally • new materials • FQHE is very fragile: due to B field! T<< e B/m~ 1K • If no B field, gap can be large ~ 10000 K • FQHE at room temperature! (Wen and coworkers) New FQH state

  39. Key ingredients of the FQHE

  40. Landau level without B field? • Start from a model with topological bands • Tune hopping strengths to reduce the bandwidth Topologically nontrivial nearly-flat bands • Checker-board lattice with NN, NNN and NNNN hoppings • Square lattice with NN hoppings • Kagome lattice with NN and NNN hoppings • Honeycomb lattice with NN and NNN hoppings • … • Hopping range ~ (gap/effective mass)1/2 Tang, et. al., PRL (2011). KS, et. al., PRL (2011). Neupert, et. al., PRL (2011).

  41. Numerical Results (Exact diagonalization) Sheng, Gu, KS, and Sheng, Nature Communications, (2011)

  42. Signatures of the fractional states Sheng, Gu, KS, and Sheng, Nature Communications, (2011)

  43. Knowledge and puzzles • Numerics: every topological property agrees with the FQHE • Topological degeneracy (KS, et. al, to be published) • Entanglement spectrum (Regnault, et.al.2011, KS, et. al, to be published) • Abelian FQHE: fermion with odd denominator or bosons with even denominator (Wang, et.al. 2011) • Nonabelian FQHE: SU(2)k, with k+1 body interaction (Bernevig, et.al., 2011) (TQC, universal TQC) • Theory: both progresses and puzzles • Girvin-MacDonald-Platzman algebra (Parameswaran et. al. 2011) • Wave-function on open cylinder/thin torus (Qi 2011) • no numerical confirmation yet • Composite fermions picture: not available • Mean-field: possible • Gauge fluctuations: no Chern-Simon’s gauge theoryavailable on frustrated lattices yet • Parton: mean-field studies • no numerical confirmation yet • Gauge fluctuations: no Chern-Simon’s gauge theory available on frustrated lattices yet

  44. Topological quantum computation

  45. “Family Tree” of topological insulators • Integer Quantum Hall • Fractional • Quantum Hall Integer topological insulators Fractional topological insulators • 2D T-invariant TI • 2D T-invariant FTI • Quantum Anomalous Hall effect • Anomalous • FQHE • 3D T-invariant TI Quadratic Band Crossing Topological flat band Kondo couplingin heavy fermion compounds Topological Kondo Insulator

  46. 3D Topological Insulator in a Nut Shell 3D TI can only exist with T-symmetry Brillouin zone of a 3D lattice Hasan and Kane, RMP (2010). Qi and Zhang, RMP (2011).

  47. Heavy Fermion in a Nut Shell • Experimental signature: • Fermi liquid with very large effect electron mass ~ 100 me • Theoretical description: • Electrons moving on a lattice of local spins Dzero, KS, Galitski and Coleman, PRL, (2010).

  48. Why topological?Topological Insulator and mass of a Dirac theory • Toy model of TI (X-L Qi, etc, PRB 2008) • TI: Dirac mass changes sign as k increases from 0 to ∞ • Conclusion remains if k is changed into odd function of k • Effective Hamiltonian of a Kondo insulator • Compare with model above and To get a Topological insulator: • is an odd function of k • Local spins and conducting electrons have opposite parities • “Dirac” mass changes sign Dzero, KS, Galitski and Coleman, PRL, (2010).

  49. Topological Kondo Insulators: TI from Kondo couplings Conducting electrons and spins must have opposite parity Strong TI Strong TI Trivial Insulator Weak TI Trivial Insulator SmB6: Strong TI CeNiSn: Weak TI • On-going experiments: • Sample: Johnpierre Paglione group (UMD) • Transport (indirect evidence): supportive. • ARPES (the smoking gun): ongoing. Dzero, KS, Galitski and Coleman, PRL, (2010).

  50. “Family Tree” of topological insulators • Integer Quantum Hall • Fractional • Quantum Hall Integer topological insulators Fractional topological insulators • 2D T-invariant TI • 2D T-invariant FTI • Quantum Anomalous Hall effect • Anomalous • FQHE • 3D T-invariant TI Quadratic Band Crossing Topological flat band Kondo couplingin heavy fermion compounds Topological Kondo Insulator

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