Rok Žitko

Ljubljana, 22. 7. 2011

Topological insulators and superconductors

• States of matter– insulators– quantum Hall effect

• Topological insulators (TI)– 2D TI and helical edge states– 3D TI and helical surface states

• Proximity effect and topological superconductors– Majorana edge states– Detections schemes

States of matter

• Characterized by – broken symmetries (long range correlations)– topological order

• Quantified by– order parameter– topological quantum number

• Described by– Landau theory of phase transitions– topological field theories

Solid-liquid phase transition

Broken translation invariance

Order parameter: FT of <(r)(0)>, Bragg peaks

FLandau=a2+b4 a=a0(T-Tc)

Insulators

• Anderson insulatorsdisorder electrons become localized

• Mott insulatorsCoulomb interaction (repulsion) between electrons motion suppressed

• Band insulatorsabsence of conduction states at the Fermi level forbidden band

Band insulators

• vacuum (“Dirac sea” model): Egap=2mc2=106 eV

• atomic insulators (solid argon):Egap=10eV

• covalent-bond semiconductors and insulators: Egap=1eV

Bloch, 1928

2D electron gas in strong magnetic field

Landau levels

c=eB/mc

Egap=ħc

Quantum Hall Effect

h

enxy

2

Jy=xyEx

edge stateschiral

von Klitzing et al. (1980)

Gauss-Bonnet theorem

KdS

2

1

Gaussian curvature, K=1/R1R2

Sphere =2 Torus =0

Topological insulators

• “Topological”: topological properties of the band structure in the reciprocal space

• “Insulators”: well, not really. They have gap, but they are conducting (on edges)!

• Quantum Hall effect: in high magnetic field, broken time-reversal symmetry (von Klitzing, 1980)

• Time-reversal-invariant topological insulators (Kane, Mele, Fu, Zhang, Qi, Bernevig, Molenkamp, Hasan and others, from 2006 and still on-going)

Smooth transformations and topology

Band structure: mapping from the Brillouin zone (k) to the Hilbert space (: k |(k)

Bloch theorem: (k)=eikr uk(r)

Smooth transformations: changes of the Hamiltonian such that the gap remains open at all times

See Fig.

uk(r)=uk(r+R)

TKNN (Chern) invariant

Fkdn 2

2

1

)()( kAkF k

Thouless-Kohmoto-Nightingale-den Nijs, PRL 1982

mk

N

mm uuikA

1

)(

Integernumber!

Berry curvature

Same n as in xy=ne2/h. An integer within an accuracy of at least 10-9!New resistance standard: RK=h/e2=25812.807557(18)

Spin-orbit coupling

SLr

rU

rmecH BSO

)(1

2

Nucleus

e-

2c

EvB

Stronger effect for heavy elements (Pb, Bi, etc.) from the bottom of the periodic system

Reinterpretation:

Quantum Spin Hall effect (QSHE)(“2D topological insulators”)

• Two copies of QHE, one for each spin, each seeing the opposite effective magnetic field induced by spin-orbit coupling.

• Insulating in the bulk, conducting helical edge states.

• Theoretically predicted (Bernevig, Hughes and Zhang, Science 2006) and experimentally observed (Koenig et al, Science 2007) in HgTe/CdTe quantum wells.

Edge states in 2D TIs

Helical modes: on each edge one pair of 1D modes related by the TR symmetry. Propagate in opposite directions for opposite spin.

3D topological insulators

• Generalization of QSHE to 3D.• Insulating in the bulk, conducting helical

surface states.• Theoretically predicted in 2006,

experimentally discovered in BiSb alloys (Hsieh et al., Nature 2008) and in Bi2Se3 and similar layered materials (Xia et al., Nature Phys. 2009).

Surface states on 3D topological insulators

• Conducting surface states must exist on the interface between two topologically different insulators, because the gap must close somewhere near the interface!

• Single Dirac cone = ¼ of graphene.

In graphene, there is spin and valley degeneracy, i.e., fourfold degeneracy.

Experimental detection in Bi2Te3

Chen et al. Science (2009)

Spin-momentum locking

Hsieh, Science (2009)

Spin-resolved ARPES

Topological field theory

=0, topologically trivial, =, topological insulator

Qi, Hughes, Zhang, PRB (2008), Wang, Qi, Zhang, NJP (2010).

Z2 invariants

Fu, Kane, Mele (2007)

Equivalence shown by Wang, Qi, Zhang (2010)

Time-reversal symmetry, t -t

• Time-reversal operator: T=K exp(iy)• Half-integer spin: rotation by 2 reverses the

sign of the state.• Kramer’s theorem: T2=-1 degeneracy!• Spin-orbit coupling does not break TR.• Magnetic field breaks TR: Zeeman splitting!

k

s

-k

-s

TTime-derivatives (momenta) are reversed!

Suppression of backreflection

Kramers doublet: |k↑=T|-k↓

k↑|U|-k↓=0 for any time-reversal-invariant operator U

Semiclassical picture: destructive interference.

Quantum picture: spin-flip would break TRI.

Magnetic impurities can open gap

Chen et al., Science (2010)

Kondo effect in helical electron liquids

• Broken SU(2) symmetry for spin, but total angular momentum (orbital+spin) still conserved

• Previous work: incomplete Kondo screening, residual degrees of freedom leading to anomalies in low-temperature thermodynamics

• My little contribution: complete screening, no anomalous features

R. Žitko, Phys. Rev. B 81, 241414(R) (2010)

General approach: reduce the problem to a one-dimensional tight-binding Hamiltonian (Wilson chain Hamiltonian) with the impurity attached to one edge

K. G. Wilson, RMP (1975)H. R. Krisnamurthy et al., PRB (1980)

The problem has time-reversal symmetry, so the persistance of Kondo screening seems likely. The Kramers symmetry, not the spin SU(2) symmetry, is essential for the Kondo effect.

R. Žitko, Phys. Rev. B 81, 241414(R) (2010)

Quantum anomalous Hall (QAH) state

See also Qi, Wu, Zhang, PRB (2006), Qi, Hughes, Zhang, PRB (2010)

= QHE without external magnetic field.

Proposal: magnetically doped HgTe quantum wells, Liu et al. (2008)

Chiral topological superconductor

Qi, Hughes, Zhang, PRB (2010)

= QAH + proximity induced superconductivity

One has to tune both the magnetization, m, and theinduced superconducting gap, .

chiral Majorana mode

Review: Qi, Zhang (2010), Hasan, Kane, RMP (2010)

Majorana fermionsTwo-state system: 0, 1

Complex “Dirac” fermionic operators and † defined as:

† 0= 1, 1= 0, 0=0, † 1=0

Canonical anticommutation relations: {,}=0, {†,†}=0, {,†}=1.

We “decompose” complex operator into its “real parts”:=(1+i2)/2, †=(1-i2)/2

Real operators: i †

=i

Canonical anticommutation relations: {1,1}=1, {2,2}=1, {1,2}=0.

Inverse transformation: 1=(+† )/2, 2=(-†

)/(2i)

Thus i2=1/2.

Is this merely a change of basis?

• Not if a single Majorana mode is considered! (Or several spatially separated ones.)

• Two separated Majorana fermions correspond to a two-state system (i.e., a qubit, cf. Kitaev 2001) where information is encoded non-locally.

• Many-particle systems may have elementary excitations which behave as Majorana fermions.

• Single Majorana fermion has half the degrees of freedom of a complex fermion → (1/2)ln2 entropy

Majorana excitations in superconductors

• Solutions of the Bogoliubov-de Gennes equation come in pairs: †(E) at energy E (E) at energy –E.

• At E=0, a solution with †= is possible. Majorana fermion level at zero energy

inside the vortex in a p-wave superconductor.

Reed, Green, PRB (2000), Ivanov, PRL (2001), Volovik

Non-Abelian states of matter

• In 2D, excitations with unusual statistics, anyons (= particles which are neither fermions nor bosons):12=ei21 with 0,

• Zero-energy Majorana modes degenerate ground state

• Non-Abelian statistics: 12=21U

unitary transformation within the ground state multiplet

Wilczek, PRL 1982

Majorana fermions in condensed-matter systems

• p-wave superconductors (Sr2RuO4, cold atom systems)

• =5/2 fractional quantum Hall state• topological superconductors• superconductor-topological insulator-magnet

heterostructures

Building blocks for topological quantum computers?

For a review, see Nayak, Simon, Stern, Freedman, Das Sarma, RMP 80, 1083 (2008).

Detection of Majorana fermions

• Problem: Majorana excitations in a superconductor have zero charge.

• Proposals:– electrical transport measurements in

interferometric setups (Akhmerov et al, 2009; Fu, Kane, 2009; Law, Lee, Ng 2009)

– “teleportation” (Fu, 2010)– Josephson currents (Tanaka et al. 2009)– non-Fermi-liquid kind of the Kondo effect

Interferometric detection

Akhmerov, Nilsson, Beenakker, PRL (2009); Fu, Kane, PRL (2009)

Electron can either be transmitted as an electron or as a hole (Andreev process), depending on the number of flux quanta enclosed.

2-ch Kondo effect – experimental detection in a quantum-dot system

Potok, Rau, Shtrikman, Oreg, Goldhaber-Gordon (2007)

Two-channel Kondo model

TKT

Can be solved by the numerical renormalization group (NRG), etc.

Bosonisation and refermionisation

One Majorana mode decouples!

Emery, Kivelson (1992)

Source-drain linear conductance:

Majorana detection via induced non-Fermi-liquid effects

Chiral TSC:single

Majoranaedge mode

R. Žitko, Phys. Rev. B 83,195137 (2011)

Parametrization:

Standard Andersonimpurity (=45º)

Impurity decoupled from one of the Majorana modes (=0)

Conclusion

• Spin-orbit coupling leads to non-trivial topological properties of insulators containing heavy elements.

• More surprises at the bottom of the periodic system?

• Great news for surface physicists: the interesting things happen at the surface.