Theory of interacting topological insulators and superconductors

Harvard 2014 Shoucheng Zhang, Stanford University

Search for a new electronic states of matter

Semiconductors Magnets Superconductor

The search for new elements led to a golden age of chemistry.

The search for new particles led to the golden age of particle physics.

Complex states of matter from the simplicity of the building blocks!

In the classical world we have solid, liquid and gas. The same H2O molecules can condense into ice, water or vapor.

In the quantum world we have metals, insulators, superconductors, magnets etc.

Quantum Hall effect and quantum spin Hall effect

Two independent proposals: Kane+Mele: graphene model Bernevig+Zhang: strained GaAs model

V Klitzing: QHE in GaAs

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.76.6

1.0

1.5

0.5

0.0

-0.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

6.0

5.5

Bandgap vs. lattice constant(at room temperature in zinc blende structure)

Ba

nd

ga

p e

ner

gy

(eV

)

lattice constant a [臸0

All

subsequent

TIs are

predicted

theoretically

based on the

principle of

band

inversion

driven by

spin-orbit

coupling

Experimental observation of the QSH edge state (Konig et al, Science 2007)

x

x

Edge current distribution in HgTe (Yacoby group 2013)

• In HgTe, the band inversion occurs

intrinsically in the material. However, in

InAs/GaSb quantum wells, a similar

inversion can occur, since the valance

band edge of GaSb lies above the

conduction band edge of InAs.

• A small hybridazation gap opens up

due to tunneling at the interface.

• Theoretical work show that the QSH

can occur in InAs/Gab quantum wells.

This material can be fabricated

commercially in many places around the

world.

• InAs can also be used for

superconducting proximity effect.

QSH state in InAs/GaSb type II quantum wells (theoretically predicted by Liu et al PRL 100, 236601 (2008), Zhang group)

QSH state in InAs/GaSb type II quantum wells (experimentally observed in Ruirui Du group)

New prediction of the QSH in stanene (2D tin)! (Zhang group, PRL 111, 136804 (2013))

Electronic structure (Zhang group, PRL 111, 136804 (2013))

3D topological insulators

(a) Sb2Se3 (b) Sb2Te3

(c) Bi2Se3 (d) Bi2Te3

Model for topological insulator Bi2Te3, (Zhang et al, 2009)

Pz+, up, Pz-, up, Pz+, down, Pz-, down

Single Dirac cone on the surface of Bi2Te3

Surface of Bi2Te3 = ¼ Graphene !

General theory of topological insulators

• Topological band theory based on Z2 topological band invariant of single particle states. (Fu, Kane and Mele, Moore and Balents,

Roy)

• Topological field theory of topological insulators. Generally valid for interacting and disordered systems. Directly measurable physically. Quantized magneto-electric effect (Qi, Hughes and Zhang)

• For a periodic system, the system is time reversal symmetric only when q=0 => trivial insulator q=p => non-trivial insulator

FM FM

chiral interconnect

3D topological insulator

Gapped Dirac fermions on the surface, chiral fermions on the domain wall

QAH can be realized in magnetic TI (Qi, Hughes, Zhang, PRB 2008)

History of the AHE

Intrinsic vs extrinsic mechanism for AHE in metals and semiconductors are still debated today, more than 100 years after the discovery of the AHE!

Theoretical developments of the QAH Haldane model of circulating currents

on honeycomb lattice (PRL61, 2015, 1988). Qi, Wu and Zhang introduced a model of the QAH (PRB74, 085308, 2006), which contains spin-orbit coupling and magnetization, both essential ingredients of the AHE.

QWZ model of the QAH. (PRB74, 085308, 2006).

The QAH conductance is determined by the topological winding number from a torus to a sphere.

Theoretical developments of the QAH

Liu et al showed that spin-split band inversion is the fundamental mechanism for QAH in magnetic TI. (PRL101, 146802, 2008).

Spin-split band inversion mechanism of the QAH. (PRL101, 146802, 2008).

QAH can be realized in magnetic TI like Cr doped Bi2Te3. (Science 329, 61, 2010).

Li et al proposed magnetic TIs in Bi2Te3 class of materials doped with 3d magnetic elements. (Nature Physics 6, 284 (2010). Yu et al proposed that 3d magnetic elements can order magnetically in Bi2Te3 class of TIs, leading to the QAH. (Science 329, 61, 2010).

Mechanism of magnetic TI

Ferromagnetic coupling through the RKKY interaction on the TI surface. (Zhang group, PRL2009).

Spin-orbit coupling and band inversion increases magnetic susceptibility, leading to magnetic order in the TI state. (IOP+Zhang groups, Science 2010).

Completion of the quantum Hall trio!!!

Discovery of the QAH (Science 340, 167 (2013))

Topological invariants for time-reversal-invariant topological

insulators

Fu-Kane

A notable feature shared by all topological invariants is:

Wavefunction plays a more important role than the energy spectrum.

With inversion

symmetry:

Fu-Kane-Mele

Qi-Hughes-Zhang

Beyond free fermions?

However, electrons do

have interactions, and in

many problems

interactions are

essential.

Free fermions: Free

and easy

• High Tc superconductivity

• Kondo and heavy fermions

• Quantum critical points

• Fractional quantum Hall

• ……

Topological invariants for interacting insulators?

• e.g. Hubbard model

• How to define topological invariants?

The difficulty is that there is no simple matrix like h(k) to work with.

(See Hubbard model).

• Ground-states wavefunction approach (Niu-Thouless-Wu)

Theoretically appealing, but requires a great amount of computational power.

Green’s function is a matrix

• Green’s function

propagate birth death

• Free fermion Green’s function

• For strongly interacting systems, Green’s function can be obtained

from quantum Monte Carlo, dynamical mean field theory, etc.

• Many physical quantities, e.g. density of states, can be directly obtained from Green’s functions. We expect that Green’s function

also contains topological information.

Green’s function approach to topological invariants: QH/QAH

p+q

p

q

Hall conductance C1 is determined by the Chern-Simons term (Zhang S

C et al)

The formula with correct numerical coefficient reads

Compare the coefficient of A(q)A(-q), we have roughly

Free-fermion as a special case

TKNN

invariant

If

If

Green’s function approach seems to be too complicated because

all frequencies (and even more) are needed

There were generalizations of this formula to time-reversal invariant topological

insulators, which are Wess-Zumino-Witten terms. Again frequency integral causes

considerable difficulty:

Wang,Qi&Zhang, PRL

The frequency integral is too complicated:

Simplified topological invariants

Define topological invariants from zero-frequency Green’s function?

is a Hermitian matrix, just like a k-space Hamiltonian for free fermion system.

For free fermion we know how to define the topological invariants (e.g.

TKNN invariant) , so we can define a `generalized TKNN invariant’ using

zero-frequency Green’s function:

This is much simpler than

Hall

conductanc

e

?

Zero-frequency Green’s function contains all topological information

A proof exists. We can show that the interpolation

smoothly connects the two Green’s functions

So these two Green’s functions are topologically equivalent, therefore, any

topological invariants defined from G can also be defined from G’ !

The above interpolation is very general. It is applicable to other topological

insulators/superconductors as well!

Technical details in Wang&Zhang PRX, 2, 031008 (2012)

A systematic approach of topological invariants in interacting systems

The topological invariants of an interacting insulator can be calculated from Green’s function

at zero-frequency. Nonzero frequencies can be safely ignored. (Wang&Zhang PRX, 2, 031008)

This is a general framework instead of a single formula. It is applicable to many

different topological insulators and superconductors.

Applications in real materials

Green’s function from DMFT

Applications in models

Topological invariants as tools to investigate (very) strongly

interacting topological superconductors

• Fidkowski-Kitaev

model

You,ZW, Oon, Xu (arXiv: 1403.4938)

……

u v u v 1 2

7 8

……

w term

• The Green’s function

• Phase diagram inspired

by topological

invariants

Topological insulators and superconductors

Full pairing gap in the bulk, gapless Majorana edge and surface states

Chiral Majorana fermions Chiral fermions

massless Majorana fermions massless Dirac fermions

Qi, Hughes, Raghu and Zhang, PRL, 2009

Topological superconductors

Chiral topological superconductivity from QAHE

1 chiral fermion=2 identical majorana fermion

X. L. Qi et al, PRB 82, 184516 (2010)

Phase diagram and realization • Vortex of TSC host

majorana zero mode.

• Majorana fermion at edge

To realize TSC:

1. Finite chemical

potential.

2. Top and bottom

surface better have

coupling, otherwise

fine tuning of chemical

potential into gap is

needed.

3. SC proximity only to

one surface. (top and

bottom have different

SC pairing order).

From traffic jam to info-superhighway

Traffic jam inside chips today Info highways for the chips in the future

0

100

200

300

400

500

0.5 0.35 0.25 0.18 0.13 0.1 0.07 0.05

Active Power

Passive Power (Device Leakage)

350 250 180 130 100 70 50

500

500

400

300

200

100

0

Technology node (nm)

Po

we

r d

ensity (

W/c

m)

2

Energy applications: thermal electrics

TI & TSC in relation to other branches of physics

• Semiconductor physics

• narrow gap semiconductors, dopant and defect physics, MBE • Magnetism: Spintronics, DMS, ferromagnetic layers on TI, half-metals • Superconductivity and superfluidity

• Novel proximity effects, topological superconductors, vortex states, He3B is a topological superfluid! Oxide interface superconductivity

• Quantum Hall effect

• Quantum spin Hall effect, Quantum anomalous Hall effect, graphene • Heavy fermions: Mixed valence and the d-f band inversion • Cold atoms: Artificially engineered spin-orbit interaction

• Exotic particles

• Magnetic monopoles, axions, Majorana fermions • String theory

• Modular invariance and anomaly determine topological stability

• Standard model • Is the vacuum a topological insulator?