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
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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
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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?