CDL Talk Minnesota (Jason Alicea).pdf

Composite Dirac liquids

Jason Alicea, Caltech

David Mross, Andrew Essin, & JA, Physical Review X 5, 011011 (2015)

non-interacting 3D TI surface Interactions

Composite Dirac liquid

BComposite Fermi liquid

~

O(100) plateaus observed to date. How might we most efficiently capture these topological phases?

O(100) plateaus observed to date. How might we most efficiently capture these topological phases?

ν = 1/2(no plateau!)

Composite Fermi liquid at ν = 1/2Landau Level Energy

~!c

One state/flux quantum

Half-filled Landau level


~!c


B

electron



~!c


B

electron

Bcomposite fermion

“statistical flux quanta”

=



~!c


B

electron=

Bcomposite fermion


kx

ky

composite fermion Fermi sea!


Composite Fermi liquid at ν = 1/2

Composite Fermi liquid

B

electron=

Bcomposite fermion


kx

ky


B


Quantum oscillations of composite Fermi sea!

B

electron=

Bcomposite fermion


kx

ky


B


B

electron=

Bcomposite fermion


kx

ky


B

Pairing instability of composite Fermi sea yields non-Abelian

“Moore-Read” fractional quantum Hall state!Composite

fermion Cooper pairs

e/4non-Abelian

anyon

co-propagating Majorana + charge edge modes


B

electron=

Bcomposite fermion


kx

ky


B Composite fermion

Cooper pairs

Pairing instability of composite Fermi sea yields non-Abelian

“Moore-Read” fractional quantum Hall state!

Moral: emergent “metals” are interesting and provide

efficient way of capturing exotic topologically ordered

gapped phases.

e/4non-Abelian

anyon

co-propagating Majorana + charge edge modes

Anomalous 3D TI surface physics3D topological insulator: protected by time-reversal, U(1) particle conservation symmetries

Insulating bulk; assumed

symmetric throughout


Gapless surface with single Dirac cone

Impossible band structure in strict 2D systems with time-reversal, where only even # of Dirac cones can appear.



Impossible in strict 2D, since would imply fractionalization, topological order.

Ferromagnet

Integer quantum Hall state with half-integer Hall conductance

�xy

=e2

2h



Impossible in strict 2D, since would imply fractionalization, topological order.


“Up” Ferromagnet

�xy

=e2

2h

“Down” Ferromagnet

�xy

= � e2

2h

Magnetic domain walls bind integer quantum

Hall edge states



Impossible in strict 2D, since superconductor would always break T.


“Up” Ferromagnet

�xy

=e2

2h

“Down” Ferromagnet

�xy

= � e2

2h

s-wave super-conductor

Time-reversal-invariant cousin of topological “spinless” p+ip superconductor

Cooper pair

Without interactions, this is the full story: -Symmetry implies massless electron Dirac cone -broken symmetry implies anomalous gapped phases.

Does symmetry imply surface metallicity more generally?

Without interactions, this is the full story: -Symmetry implies massless electron Dirac cone -broken symmetry implies anomalous gapped phases.

Does symmetry imply surface metallicity more generally? No!

T-Pfaffian Pfaffian-antisemion


Clearly break time reversal symmetry in

2D, but not on surface!


Clearly break time reversal symmetry in

2D, but not on surface!

“Composite Dirac liquids”= new correlated

gapless surface states that unify these works.

Composite Dirac liquids: phenomenology

non-interacting 3D TI surface

electronic Dirac cone




Interactions strip electric charge from surface Dirac cone


Dirac cone for emergent neutral fermions!

-Charge sector gapped, so surface is electrical insulator

-Neutral fermions yield “metallic” thermal transport similar to original Dirac cone

-Wild deviation from Weidemann-Franz law!








kx

ky

composite fermion Fermi sea

~

-Charge sector gapped, so surface is electrical insulator

-Neutral fermions yield “metallic” thermal transport similar to original Dirac cone

-Wild deviation from Weidemann-Franz law!








kx

ky

composite fermion Fermi sea

~

Cooper pair composite fermions

Cooper pair neutral fermions [preserves U(1)]

non-Abelian Moore-Read FQH state

time-reversal-invariant cousin of Moore-Read

(T-Pfaffian)!

Accessing composite Dirac liquids

3D TI surface

In an ideal world: HDirac

+Hinteractions

⇒ Composite Dirac liquid


3D TI surface

In an ideal world: HDirac

+Hinteractions

⇒ Composite Dirac liquid (too hard)

Instead, will “cheat” and follow route with virtues of physical transparency, analytical control.


3D TI surface

Key technical idea: relax time-reversal to weaker “antiferromagnetic” symmetry (states still impossible in 2D)


Insert ferromagnets w/alternating magnetization

�xy

= � e2

2h

�xy

=e2

2h

�xy

=e2

2h

�xy

= � e2

2h

�xy

=e2

2h




�xy

= � e2

2h

�xy

=e2

2h

�xy

=e2

2h

�xy

= � e2

2h

�xy

=e2

2h

electron tunneling

electronic Dirac cone ✔




�xy

= � e2

2h

�xy

=e2

2h

�xy

=e2

2h

�xy

= � e2

2h

�xy

=e2

2h


Quasi-1D deformation allows rigorous analytical progress a la Teo & Kane.



�xy

= � e2

2h

�xy

=e2

2h

�xy

=e2

2h

�xy

= � e2

2h

�xy

=e2

2h

Insert 2D FQH

states

�xy

= � e2

2h

�xy

=e2

2h

�xy

=e2

2h

�xy

= � e2

2h

�xy

= � e2

2h




�xy

= � e2

2h

�xy

=e2

2h

�xy

=e2

2h

�xy

= � e2

2h

�xy

=e2

2h

Insert 2D FQH

states

�xy

= � e2

2h

�xy

=e2

2h

�xy

=e2

2h

�xy

= � e2

2h

�xy

= � e2

2h


Charge flow is non-chiral, so

interactions can gap charge

sector…

…but heat transport is chiral, so must have neutral modes left over!



Neutral fermions



Composite Dirac liquid!


Neutral fermions

neutral fermion tunneling

neutral Dirac cone

Nested Composite Dirac Liquids








(T-Pfaffian)


Nested Composite Dirac liquid

“2nd generation” neutral Dirac cone







Same interactions, but for neutral fermions


symmetric topological order w/more anyons(Pfaffian-antisemion)


(T-Pfaffian)


Nested Composite Dirac liquid

“2nd generation” neutral Dirac cone







Same interactions, but for neutral fermions


Topological orders captured previously, now in one unified

framework!

symmetric topological order w/more anyons(Pfaffian-antisemion)


(T-Pfaffian)

OutlookInterplay between interactions, symmetry-protected topological phases rich topic.


neutral Dirac cone

Provide unifying view on lots of physics, including quantum Hall; accessible via controlled analytical methods that deal with physical electrons.

Naturally extends to “weak topological insulators”. (D. Mross, A. Essin, JA, A. Stern, in preparation)

𝓣𝓣

𝑻𝑻𝒚𝒚�𝓣𝓣�𝓣𝓣

y

x

Spin

Spin

Spin

Spin

Spin

Spin

Spin

Spin

y

z

x

Extension to 3D topological superconductors?

Isotropic implementations?

Quasi-realistic Hamiltonians for such states?

CDL Talk Minnesota (Jason Alicea).pdf

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