fractionalization in quantum matter: past, present and future · composite fermion metal •...
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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).