2) Quantum oscillations in insulators with neutral Fermi surfaces

1) Unraveling the hidden link between composite fermions and exciton condensate

Fractionalization in quantum matter: past, present and future

IMPRS RETREAT

September 20, 2017

Inti Sodemann Max Planck Institute for the Physics of Complex Systems

Debye and the birth of quasiparticles

• Debye model (1912)

• Quantized a la Planck-Einstein black body photons.

• Sound ~ Light. Phonon ~ Photon.

Debye

Bohr model (1913) Bose paper (1924)

!

k

Landau and the quasiparticle paradigm

• Charged quasiparticles: Fermions.

t = �1

�Q = +1 �Q = +1

Vint = 0

Vint 6= 0 + -

+ -+-

+-• Neutral quasiparticles (quanta of collective oscillations):

Bosons.

PhononSpin waves

Sound

Magnon

Landau

~B

n = 1, 2, 3...

�xy

= ne2

h

N� =BA

�0

von Klitzing• Electrons under strong magnetic fields display the quantum Hall

effect:

Landau levels

Quantum Hall revolution

Landau level filling:

⌫ =Ne

N�= n

Fractional Quantum Hall effect

• Plateau at 1/3?

Tsui, Stormer, Gossard, PRL (1982).

Eisenstein, Stormer, Science (1990).

�xy

=e2

3h

???

Stormer Tsui Gossard

• Laughlin liquid at filling 1/3

Laughlint = �1�Q = +1 �Q = +1

Vint = 0

Vint 6= 0 + -

+-+-

z = x+ iy

=Y

i<j

(zi � zj)3e�

|zi|2

4l2

⌫ =Ne

N�=

1

3

+ + · · ·

Fractional Quantum Hall effect

+ -

Fractional Quantum Hall effect• Laughlin liquid at filling 1/3

Laughlin

+ + · · ·

⌫ =Ne

N�=

1

3

~B

~E

�Q = �xy

��

@ ~B

@t= �r⇥ ~E

jx

= �xy

Ey

�xy

=e2

3h

�0 =h

e �Q0 = ±e

3

~j

Fractional Statistics• Only fermions and bosons in 3D:

HalperinWilczek

(ei')2 = 1

ei' = ±1 Bosons or Fermions

• In 2D “any-ons” are allowed:

time

e/3e/3

' = ⇡/3Laughlin anyons

• Non-trivial degeneracy on closed manifolds:

Fractionalization and topology

• Non-abelian anyons: “irrational” size of Hilbert space.

Wen

�1 �2

D�1�2 = 2 D� =p2

Majorana fermions

(nontrivial quasiparticles + 1)

G

1 non-trivial qps+1G = 3

Read Moore Kitaev

G = 0 G = 1

Experimentally realized in:⌫ = 5/2-GaAs at

-1D chains superconducting p-wave.

D =

• BEC - BCS crossover a powerful unification in physics of quantum matter:

The hidden link between composite fermions and the exciton condensate

Bose Einstein condensate molecules

SuperconductorFermions

• Unification between two celebrated quantum Hall phases of matter: the exciton condensate and the composite fermion metal.

• No tunneling but strong interactions

Exciton condensate

top

bottom~n

|topi+ e

i�|bottomi

• Exciton condensate:

⌫ = ⌫top

+ ⌫bottom

= 1/2 + 1/2

hc†bottom

ctop

i / ei�

Long range XY order

• Superfluidity for charge imbalance:

• Half-charged vortices (merons):

Properties of exciton condensate

Q� = Qtop

�Qbottom

[Q�,�] = i

• Linearly dispersing Goldstone mode of (pseudo-spin wave).

�

Q+ = (vnz)e

2

nz = ±1

• Moon, Mori, Yang, Girvin, MacDonald, Zheng, Yoshioka, Zhang, PRB 51, 5138 (1995).• Wen, Zee, PRL 69, 1811 (1992).

v = 1

Q+ = e/2

2⇡ winding

E

q

Spin-wave

v 2 Z

• Emergent 2-dimensional “gauge field” (analogous to the electro-magnetic field in 2D).

Composite fermion metal• Fractionalized metal for half filled landau level:

Ne =1

2N�

• Composite fermion: electron bound to two vortices

~Beff = 0~B 6= 0

electrons composite fermions

px

py

composite fermion fermi surface

Halperin, Lee, Read, PRB (1993).Jain, PRL (1989).

Duality in 1+1D QFT’sHaldaneColeman Luttinger

1

2

(@�)2 + (m/�)2 cos(��) (i/@ �m) � g

2( �µ )

2

4⇡

�2= 1 +

g

⇡

��

x

2⇡

S. Coleman,

PRD 1975

† =�

2⇡@x

�

Sine�Gordon

1 + 1 Massive� Thirring1 + 1⌘

Fermion vortex duality Physical

Dirac fermion

Le = e(i/@ � /A) e + Lint

Dirac composite fermion vortex

Lcf = cf (i/@ � /a) cf +adA

4⇡+ Lint

aµ

�nelec(r) =r⇥ ~a

4⇡

†e $ M4⇡

z ⇥~jelec(r) =ra0 + @t~a

4⇡

Electron creation is flux insertion operator

Son PRX (2015). Wang, Senthil PRB (2016). Metlitski, Vishwanath, PRB (2015). Mross, Alicea, Motrunich PRL (2016). Seiberg, Wang, Senthil, Witten Ann. Phys. (2016).

Bilayer exciton condensate and Composite fermion metal• Are zero and infinite

distance connected?

d = 1d = 0

Exciton condensate Two C-Fermion metals?

Exciton condensate

Theory

Bonesteel et al. PRL(1996)

Numerics

Möller et al. PRL (2008)

• PrecedentsExperiment

Eisenstein, ARCMP (2014)

⌫ = ⌫top

+ ⌫bottom

= 1/2 + 1/2

Bilayer exciton condensate and Composite fermion metal• A special particle-hole invariant “cooper pairing” of composite fermions is

equivalent to exciton condensate:

Composite fermion Cooper pairs

Exciton condensate

d top

bottom

top

� = i †�y

⌧x

† ⇠ i †top

�y

†bottom

I. Sodemann, I. Kimchi, C. Wang, T. Senthil,Phys. Rev. B 95, 085135 (2017).

• Symmetric gauge field is gapped via Higgs.

• Anti-symmetric gauge field remains gapless. 2+1 Maxwell theory has a spontaneously broken symmetry:

†⌧x

(i�y

) †

aglobal U(1)

µ

arelative U(1)µ

!(q)

c�q

q

p+ip like !+0

hM�(r)M†�(0)i

|r|!1����! const

a+ =a1 + a2

2

a� =a1 � a2

2

netop

� nebottom

=r⇥ ~a�

2⇡

hc†bottom

ctop

i / ei� The state is an exciton condensate!

Exciton condensate from CF pairing

Relative u(1) photon = Goldstone mode

• Photon is exciton condensate “spin-wave”.

†⌧x

(i�y

) †

aglobal U(1)

µ

arelative U(1)µ

!(q)

c�q

q

p+ip like !+0

• Electric charges under field are vortices of condensate order parameter:

a�

4⇡q� $ vorticity

z ⇥ (~jtop

(r)�~jbottom

(r)) =ra

0

+ @t~a

4⇡

• Abrikosov vortices carry half charge:

⇡ � vortex

Q = 1/2

Q⇡ = ±1

2netop

+ nebottom

=r⇥ ~a

+

2⇡

• Abrikosov vortices have a complex fermion zero mode:

|1i ⌘ †0|0i

|0i |1i

|0i

Layer X-change

|1i

|0iq� (vorticity)1/2

�1/2 �2⇡

2⇡

Abrikosov vortices = merons

Abrikosov vortices = merons

⇡ � vortex

Q = 1/2

• Their fusion is a fermion:

The electron (with layer charge imbalance neutralized by condensate).

• Two Abrikosov vortices of opposite vorticity are mutual semions

⇡

|1i ⌘ †0|0i

|0i

Bogoliubov fermion

• Fusion is Bogoliubov fermion

• Consider fusing two Abrikosov vortices of opposite flux but same charge (order parameter vorticity): a�

4⇡

Q = �1/2Q = 1/2

Neutral Vortex

4⇡

Q = 1/2XY vortex

2⇡ winding

Abrikosov vortex

Dictionary

E

q

Spin-wave Photon

E

qa1 � a2

Exciton condensateComposite fermion

superconductor

⇡ flux

Q = 1/2

Q = �1/2Q = 1/2

neutral vortex4⇡

Charge neutral Dipole carrying

Composite fermion

Fractionalization w/out magnetic fields• Spin liquids in frustrated magnets.

• Fractionalization beyond the realm of frustrated magnets or quantum Hall?

• Bosonic Laughlin state can be viewed as chiral spin liquid after mapping bosons to spins.

• “Smoking gun” experimental signatures?

Anderson Kitaev Laughlin

Puzzles of SmB6• Simple cubic structure.

• All action happens in Samarium.

• Traditional picture of mixed valence insulator:

[Xe]4f65d06s2

5d1 + 4f5 ↵ 4f6

Atomic limit

✏F

d✏(k)

Mixed valence

f

✏F

✏(k)

SmB6 puzzling behavior• Insulating behavior from charge

transport:

B. S. Tan et al., Science (2015).

⇢ ⇡ ⇢0e�T � ⇡ 10meV

• De Haas-van Alphen effect visible at B ⇠ 5T

SmB6 puzzles• Could be magnetic breakdown?

✏F

✏(k) Knolle and Cooper, PRL (2015).

� ⇠ 10meV !c ⇡ 0.2meV B[T ]Gap: Cyclotron:

Experiment oscillations visible at B ⇠ 5T

• Other anomalies:

� =C

T

Specific heat to temperature ratio has finite intercept:

Like in a fermi seaC

phonon

/ T 3Cfermions

/ �T

Zhang, Song, Wang, PRL (2016).

Theory oscillations visible at B ⇠ 50T

“Composite exciton Fermi liquid”d

✏(k)electron

✏F

Fermi-bose mixture:

b† : spinless boson

�†� : neutral spinfull fermion

d†� : d-electron

Nd

electrons

= N b = N�

�

One option:

hbi 6= 0

=> Metal (“boring”)

bosons condense

More “interesting” option:

Bosons bind with d electrons

�Udf

X

i

nfi n

di

b and d attract: Composite fermionic exciton:

Bound state of “f-holon" and d electron.

Properties of “Composite exciton Fermi liquid”

D. Chowdhury, I. Sodemann, T. Senthil, arXiv:1706.00418 (2017).

f-hole spinon Fermionic exciton Fractionalized

fermi sea with two pockets (“semi-metal”)

Some properties:• Essentially linear specific heat:

• Sub-gap optical conductivity:

Upturn might indicate other

physics at lower

temperature

B. S. Tan et al., Science (2015).I. Sodemann, D. Chowdhury, T. Senthil, arXiv:1708.06354 (2017).

The end of the beginning!Conceptual frontier: • Topological matter beyond free fermions. • Fractionalization and topology in 3D. • Gapless fractionalized phases in 2D and 3D. • Novel non-pertubative approaches to interacting systems.

Real world frontier: • New probes for fractionalized matter. • Fractionalization beyond quantum Hall and frustrated magnets. • More cross talk between materials and models.

Non-equilibrium and transport frontier: • Transport in fractionalized and topological matter. • Collective behavior and broken symmetries in topological and

fractionalized matter. • Dynamics of nearly conserved quantities (hydrodynamics).