Interactions and disorder
in topological quantum matter
January 2012
Simon Trebst
University of Cologne
© 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 Kelvin
atoms = knotted tubes of ether
Knots might explain stability, variety, vibrations, ...
•
Maxwell: “It satisfies more of the
conditions than any atom hitherto
imagined.”
•
This inspires the mathematician
Tait 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
symmetry protected
Gapped phases that cannot be transformed
– without closing the bulk gap –
to “simple phases”
via any “paths”
Gapped phases that cannot be transformed
– without closing the bulk gap –
to “simple phases”
via any symmetry preserving “paths”.
quantum Hall states
spin liquids
topological band insulators
© Simon Trebst
Quantum Hall effect
Quantization of conductivity
for a two-dimensional electron gas
at very low temperatures
in a high magnetic field.
B
I
electron gas
V
AlGaAs
electron gas
GaAs
AlGaAs
Semiconductor heterostructure confines
electron gas to two spatial dimensions.
© Simon Trebst
Quantum Hall states
Landau levels
B
edge states
Landau level degeneracy
integer quantum Hall
fractional quantum Hall
filled level
partially filled level
Coulomb repulsion
orbital states
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incompressible liquid
incompressible liquid
Fractional quantum Hall states
“Pfaffian” state
Moore & Read (1994)
Charge e/4 quasiparticles
Ising anyons
SU(2)2
Nayak & Wilzcek (1996)
“Parafermion” state
Read & Rezayi (1999)
Charge e/5 quasiparticles
Fibonacci anyons
SU(2)3
Slingerland & Bais (2001)
J.S. Xia et al., PRL (2004)
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px+ipy superconductors
px+ipy superconductor
vortex
possible realizations
Sr2RuO4
p-wave superfluid of cold atoms
A1 phase of 3He films
fermion
Topological properties of px+ipy superconductors
Read & Green (2000)
Vortices carry characteristic “zero mode”
2N vortices give degeneracy of 2N.
© Simon Trebst
SU(2)2
Topological Insulators
2D – HgTe quantum wells
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3D – Bi2Se3
Proximity effects / heterostructures
Proximity effect
s-wave
superconductor
topological
insulator
© Simon Trebst
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.
Vortices, quasiholes, anyons, ...
Interactions
© Simon Trebst
Abelian vs. non-Abelian vortices
Consider a set of ‘pinned’ vortices at fixed positions.
Abelian
non-Abelian
single state
(degenerate) manifold of states
Example:
Manifold of states grows exponentially
with the number of vortices.
Laughlin-wavefunction + quasiholes
Ising anyons
(Majorana fermions)
© Simon Trebst
Fibonacci
anyons
Abelian vs. non-Abelian vortices
Consider a set of ‘pinned’ vortices at fixed positions.
Abelian
non-Abelian
single state
(degenerate) manifold of states
matrix
fractional phase
In general M and N do not commute!
© Simon Trebst
Topological quantum computing
Topological quantum computing
non-Abelian
Degenerate manifold = qubit
Employ braiding of non-Abelian
vortices to perform computing
(unitary transformations).
time
(degenerate) manifold of states
matrix
In general M and N do not commute!
© Simon Trebst
Vortex-vortex interactions
exponential decay
vortex separation
RKKY-like oscillation
© Simon Trebst
Vortex-vortex interactions
exponential decay
vortex separation
RKKY-like oscillation
© Simon Trebst
middle of
plateau
edge of
plateau
Vortex-vortex interactions
Vortex quantum numbers
Vortex pair
SU(2)k = ‘deformation’ of SU(2)
with finite set of representations
© Simon Trebst
fusion rules
Energetics for many vortices
example k = 2
“Heisenberg Hamiltonian”
for vortices
Vortex-vortex interactions
Microscopics
Vortex pair
Which channel is favored is
not universal, but microscopic detail.
p-wave SC, Kitaev model
Energetics for many vortices
vortex separation
Moore-Read state
“Heisenberg Hamiltonian”
for vortices
© Simon Trebst
The many-vortex problem
quantum liquid
quantum liquid
new quantum liquid
a
vortex-vortex
interactions
macroscopic degeneracy
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unique ground state
The collective state
quantum liquid
quantum liquid
bulk gap
SU(2)k-1
SU(2)k
U(1)
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Edge states
Finite-size gap
conformal field theory
description
© Simon Trebst
Entanglement entropy
central charge
Mapping & exact solution
The operators
form a representation of the
Temperley-Lieb algebra
for
The transfer matrix
is an integrable representation
of the RSOS model.
© Simon Trebst
Gapless theories
level k
2
3
4
5
k
∞
© Simon Trebst
‘antiferromagnetic’
‘ferromagnetic’
Ising
Ising
c = 1/2
c = 1/2
tricritical Ising
3-state Potts
c = 7/10
c = 4/5
tetracritical Ising
c = 4/5
pentacritical Ising
c = 6/7
c=1
c = 8/7
k-critical Ising
Zk-parafermions
c = 1-6/(k+1)(k+2)
c = 2(k-1)/(k+2)
Heisenberg AFM
Heisenberg FM
c=1
c=2
Gapless modes & edge states
SU(2)k liquid
critical theory
(AFM couplings)
interactions
SU(2)k liquid
gapless modes = edge states
nucleated liquid
© 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 liquid
thermal metal
a
disorder +
vortex-vortex
interactions
macroscopic degeneracy
© Simon Trebst
degeneracy is split
Interactions and disorder
quantum liquid
a
sign disorder
+ strong amplitude modulation
Natural analytical tool:
strong-randomness RG
separation a
© Simon Trebst
Unfortunately, this does not work.
The system flows away from strong
disorder under the RG.
No infinite randomness fixed point.
From Ising anyons to Majorana fermions
Ising anyon
SU(2)2
quantum number
interacting Ising anyons
“anyonic Heisenberg model”
© Simon Trebst
Majorana fermion
zero mode
Majorana operator
free Majorana fermion
hopping model
From Ising anyons to Majorana fermions
Majorana operators
Majorana fermion
zero mode
Majorana operator
free Majorana fermion
hopping model
© Simon Trebst
particle-hole symmetry
time-reversal symmetry
✓
✘
symmetry
class D
A disorder-driven metal-insulator transition
Griffith physics
gapped top. phase
Chern insulator ν = -1
Griffith physics
thermal metal
sign disorder
gapped top. phase
Chern insulator ν = +1
Density of states indicates phase transition.
© 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 metal
Inverse 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 liquid
thermal metal
a
disorder +
vortex-vortex
interactions
macroscopic degeneracy
© Simon Trebst
degeneracy is split
Experimental consequences
© Simon Trebst
Heat transport
middle of plateau
Caltech thermopower experiment
electrical transport remains unchanged
Bulk heat transport
diverges logarithmically
as T → 0.
Heat transport
along the sample edges
changes quantitatively
© Simon Trebst
Collective states – a good thing?
Topological quantum computing
Degenerate manifold = qubit
The interaction induced splitting
of the degenerate manifold = qubit states
is yet another obstacle to overcome.
Employ braiding of non-Abelian
vortices to perform computing
(unitary transformations).
time
The formation of collective states renders
all ideas of manipulating individual anyons
inapplicable.
Probably, a topological quantum computer
works best at finite temperatures.
© 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.
© Simon Trebst