© Simon Trebst Interactions and disorder in topological quantum matter Simon Trebst University of Cologne Simon Trebst University of Cologne January 2012

© 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


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


• 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.

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Non-topological (quantum) matter

© Simon Trebst


• 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

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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

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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

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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

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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.

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Topological Insulators

2D – HgTe quantum wells 3D – Bi2Se3

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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

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Vortices, quasiholes, anyons, ...

Interactions

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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

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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

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exponential decay

RKKY-like oscillation middle of

plateau

middle of

plateau

edge of

plateau

edge of

plateau

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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

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Microscopics

Energetics for many vortices

“Heisenberg Hamiltonian”for vortices

Vortex pair


Which channel is favored is not universal, but microscopic detail.

p-wave SC, Kitaev model

Moore-Read state

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The many-vortex problem

quantum liquidquantum liquid

a

macroscopic degeneracy

vortex-vortexinteractions

unique ground state


new quantum liquidnew quantum liquid

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The collective state


bulk gapbulk gap


SU(2)k

SU(2)k-1

U(1)

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conformal field theory description

Edge states

Finite-size gap Entanglement entropy

central charge

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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

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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’

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Gapless modes & edge states

SU(2)k liquid

critical theory(AFM couplings)

SU(2)k liquidgapless modes = edge states

nucleated liquid

interactions

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Interactions + disorder

Work done with Chris Laumann (Harvard)David Huse (Princeton)

Andreas Ludwig (UCSB)

arXiv:1106.6265

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Disorder induced phase transition


a


disorder +vortex-vortexinteractions

degeneracy is split

thermal metalthermal metal

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Interactions and disorder


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

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From Ising anyons to Majorana fermions

Majorana operators

free Majorana fermionhopping model

Majorana fermionzero mode

Majorana operator

particle-hole symmetrytime-reversal symmetry

✓✘

symmetryclass D

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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

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Density of states

Oscillations in the DOS fit the prediction from random matrix theory for symmetry class D

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The thermal metal

Density of states diverges logarithmically at zero energy.

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The thermal metalInverse participation ratios (moments of the GS wavefunction)

indicate multifractal structure characteristic of a metallic state.

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A disorder-driven metal-insulator transition

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Disorder induced phase transition


a


disorder +vortex-vortexinteractions

degeneracy is split

thermal metalthermal metal

© Simon Trebst

Experimental consequences

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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.

Documents

© Simon Trebst Interactions and disorder in topological quantum matter Simon Trebst University of Cologne Simon Trebst University of Cologne January 2012