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© Simon Trebst
Interactions and disorderin topological quantum matter
Simon TrebstUniversity of Cologne
Simon TrebstUniversity of Cologne
January 2012January 2012
© Simon Trebst
Overview
Topological quantum matter
Interactions
Disorder
Experimental signatures
© Simon Trebst
Topological quantum matter
The interplay between physics and topology takes its first roots in the 1860s.
The mathematician Peter Tait sets out to perform experimental studies.
© Simon Trebst
Topological quantum matter
• 1867: Lord Kelvinatoms = knotted tubes of etherKnots might explain stability, variety, vibrations, ...
• Maxwell: “It satisfies more of theconditions than any atom hithertoimagined.”
• This inspires the mathematicianTait to classify knots.
© Simon Trebst
Non-topological (quantum) matter
© Simon Trebst
Topological quantum matter
• Xiao-Gang Wen: A ground state of a many-body system that cannot be fully characterized by a local order parameter.
• A ground state of a many-body system that cannot be transformed to “simple phases” via local perturbations without going through phase transitions.
• Often characterized by a variety of “topological properties”.
© Simon Trebst
Topological matter / classification (rough)
Topological order
inherent
Gapped phases that cannot be transformed– without closing the bulk gap –
to “simple phases”via any “paths”
quantum Hall statesspin liquids
symmetry protected
Gapped phases that cannot be transformed– without closing the bulk gap –
to “simple phases”via any symmetry preserving
“paths”.
topological band insulators
© Simon Trebst
Quantum Hall effect
Semiconductor heterostructure confines electron gas to two spatial dimensions.
Quantization of conductivity for a two-dimensional electron gas
at very low temperatures in a high magnetic field.
AlGaAs
AlGaAs
GaAselectron gas
Belectron gas
I
V
© Simon Trebst
Quantum Hall states
BLandau levels
Landau level degeneracy
integer quantum Hall
fractional quantum Hall
incompressible liquid
incompressible liquid
edge states
filled level partially filled levelCoulomb repulsion
orbital states
© Simon Trebst
Fractional quantum Hall states
J.S. Xia et al., PRL (2004)
Charge e/4 quasiparticlesIsing anyons
Moore & Read (1994)
Nayak & Wilzcek (1996)
SU(2)2
“Pfaffian” state
Charge e/5 quasiparticlesFibonacci anyons
Read & Rezayi (1999)
Slingerland & Bais (2001)
SU(2)3
“Parafermion” state
© Simon Trebst
px+ipy superconductors
ppxx+ip+ipyy superconductor superconductorppxx+ip+ipyy superconductor superconductor
possible realizations
Sr2RuO4
p-wave superfluid of cold atoms
A1 phase of 3He films
vortex
fermion
SU(2)2
Topological properties of px+ipy superconductors
Read & Green (2000)
Vortices carry characteristic “zero mode”
2N vortices give degeneracy of 2N.
© Simon Trebst
Topological Insulators
2D – HgTe quantum wells 3D – Bi2Se3
© Simon Trebst
Proximity effects / heterostructures
Proximity effect
Proximity effect between an s-wave
superconductor and the surface states
of a (strong) topological insulator
induces exotic vortex statistics
in the superconductor.
Spinless px+ipy superconductor
where vortices bind a zero mode.
topologicalinsulator
s-wavesuperconductor
© Simon Trebst
Vortices, quasiholes, anyons, ...
Interactions
© Simon Trebst
Abelian vs. non-Abelian vortices
Abelian non-Abelian
single state (degenerate) manifold of states
Consider a set of ‘pinned’ vortices at fixed positions.
Example:
Laughlin-wavefunction + quasiholes
Manifold of states grows exponentiallywith the number of vortices.
Ising anyons(Majorana fermions)
Fibonacci anyons
© Simon Trebst
Abelian vs. non-Abelian vortices
Abelian non-Abelian
fractional phase
In general M and N do not commute!
matrix
single state (degenerate) manifold of states
Consider a set of ‘pinned’ vortices at fixed positions.
© Simon Trebst
Topological quantum computing
Employ braiding of non-Abelianvortices to perform computing
(unitary transformations).
Degenerate manifold = qubit
Topological quantum computingti
me
non-Abelian
In general M and N do not commute!
matrix
(degenerate) manifold of states
© Simon Trebst
Vortex-vortex interactions
vortex separationvortex separation
exponential decay
RKKY-like oscillation
© Simon Trebst
Vortex-vortex interactions
vortex separationvortex separation
exponential decay
RKKY-like oscillation middle of
plateau
middle of
plateau
edge of
plateau
edge of
plateau
© Simon Trebst
Vortex-vortex interactions
Vortex quantum numbers
SU(2)k = ‘deformation’ of SU(2)
with finite set of representations
Energetics for many vortices
“Heisenberg Hamiltonian”for vortices
Vortex pair
fusion rules
example k = 2
© Simon Trebst
Vortex-vortex interactions
Microscopics
Energetics for many vortices
“Heisenberg Hamiltonian”for vortices
Vortex pair
vortex separationvortex separation
Which channel is favored is not universal, but microscopic detail.
p-wave SC, Kitaev model
Moore-Read state
© Simon Trebst
The many-vortex problem
quantum liquidquantum liquid
a
macroscopic degeneracy
vortex-vortexinteractions
unique ground state
quantum liquidquantum liquid
new quantum liquidnew quantum liquid
© Simon Trebst
The collective state
quantum liquidquantum liquid
bulk gapbulk gap
quantum liquidquantum liquid
SU(2)k
SU(2)k-1
U(1)
© Simon Trebst
conformal field theory description
Edge states
Finite-size gap Entanglement entropy
central charge
© Simon Trebst
The operators form a representation of the Temperley-Lieb algebra
for
The transfer matrix is an integrable representation
of the RSOS model.
Mapping & exact solution
© Simon Trebst
level k2345k
∞
Isingc = 1/2
tricritical Isingc = 7/10
tetracritical Isingc = 4/5
pentacritical Isingc = 6/7
k-critical Isingc = 1-6/(k+1)(k+2)
Heisenberg AFMc = 1
Isingc = 1/2
3-state Pottsc = 4/5
Heisenberg FMc = 2
c = 1
c = 8/7
Zk-parafermionsc = 2(k-1)/(k+2)
Gapless theories
‘antiferromagnetic’ ‘ferromagnetic’
© Simon Trebst
Gapless modes & edge states
SU(2)k liquid
critical theory(AFM couplings)
SU(2)k liquidgapless modes = edge states
nucleated liquid
interactions
© Simon Trebst
Interactions + disorder
Work done with Chris Laumann (Harvard)David Huse (Princeton)
Andreas Ludwig (UCSB)
arXiv:1106.6265
© Simon Trebst
Disorder induced phase transition
quantum liquidquantum liquid
a
macroscopic degeneracy
disorder +vortex-vortexinteractions
degeneracy is split
thermal metalthermal metal
© Simon Trebst
Interactions and disorder
quantum liquidquantum liquid
a
separation aseparation a
sign disorder+ strong amplitude modulation
Natural analytical tool:strong-randomness RG
Unfortunately, this does not work.The system flows away from strong
disorder under the RG. No infinite randomness fixed point.
© Simon Trebst
From Ising anyons to Majorana fermions
interacting Ising anyons“anyonic Heisenberg model”
Ising anyon
SU(2)2
quantum number
free Majorana fermionhopping model
Majorana fermionzero mode
Majorana operator
© Simon Trebst
From Ising anyons to Majorana fermions
Majorana operators
free Majorana fermionhopping model
Majorana fermionzero mode
Majorana operator
particle-hole symmetrytime-reversal symmetry
✓✘
symmetryclass D
© Simon Trebst
A disorder-driven metal-insulator transition
Density of states indicates phase transition.
sign disorder
gapped top. phaseChern insulator ν = -1
gapped top. phaseChern insulator ν = +1
Griffith physics Griffith physics
thermal metal
© Simon Trebst
Density of states
Oscillations in the DOS fit the prediction from random matrix theory for symmetry class D
© Simon Trebst
The thermal metal
Density of states diverges logarithmically at zero energy.
© Simon Trebst
The thermal metalInverse participation ratios (moments of the GS wavefunction)
indicate multifractal structure characteristic of a metallic state.
© Simon Trebst
A disorder-driven metal-insulator transition
© Simon Trebst
Disorder induced phase transition
quantum liquidquantum liquid
a
macroscopic degeneracy
disorder +vortex-vortexinteractions
degeneracy is split
thermal metalthermal metal
© Simon Trebst
Experimental consequences
© Simon Trebst
Heat transport
Caltech thermopower experiment
middle of plateau
electrical transport remains unchanged
Heat transportalong the sample edges
changes quantitatively
Bulk heat transportdiverges logarithmically
as T → 0.
© Simon Trebst
Collective states – a good thing?
Topological quantum computing
Employ braiding of non-Abelianvortices to perform computing
(unitary transformations).
Degenerate manifold = qubit
tim
e
The interaction induced splitting of the degenerate manifold = qubit states
is yet another obstacle to overcome.
Probably, a topological quantum computer works best at finite temperatures.
The formation of collective states rendersall ideas of manipulating individual anyons
inapplicable.
© Simon Trebst
Summary
• Lord Kelvin was way ahead of his time.
• Topology has re-entered physics in many ways.
• Topological excitations + interactions + disorder can give rise to a plethora of collective phenomena.
• Topological liquid nucleation
• Thermal metal
• Distinct experimental bulk observable (heat transport) in search for Majorana fermions.
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