Fractionalization in quantum matter: past, present and future

1) Unraveling the hidden link between composite fermions and exciton condensate 2) Quantum oscillations in insulators with neutral Fermi surfaces

IMPRS RETREAT

September 20, 2017

Inti Sodemann Max Planck Institute for the Physics of Complex Systems Debye and the birth of quasiparticles Debye • Debye model (1912) !

k

• Quantized a la Planck-Einstein black body photons.

• Sound ~ Light. Phonon ~ Photon.

Bohr model (1913) Bose paper (1924) Landau and the quasiparticle paradigm

• Charged quasiparticles: Fermions. Landau

Q =+1 Q =+1

- t = Vint =0 + -

1 6 +

Vint =0 + - + -

• Neutral quasiparticles (quanta of collective oscillations): Bosons. Magnon Spin waves Phonon Sound Quantum Hall revolution

• Electrons under strong magnetic fields display the quantum Hall effect: von Klitzing Landau B~ levels

e2 = n xy h n =1, 2, 3... Landau level filling:

Ne BA ⌫ = = n N = N 0 Fractional Quantum Hall effect

• Plateau at 1/3? Stormer Tsui Gossard

Eisenstein, Stormer, Science (1990).

e2 = xy 3h ??? Tsui, Stormer, Gossard, PRL (1982). Fractional Quantum Hall effect

Q =+1 Q =+1

-

t = V =0 + - Laughlin

1 int 6 +

+ - + Vint =0 -

• Laughlin liquid at filling 1/3

z 2 3 | i| = (z z ) e 4l2 i j N 1 i

+ + ··· Fractional Quantum Hall effect N 1 • Laughlin liquid at filling 1/3 ⌫ = e = N 3 Laughlin + + ···

@B~ 2 = E~ e @t r ⇥ xy = E~ 3h B~ jx = xyEy Q = xy h ~j 0 = e e Q = 0 ±3 Fractional Statistics

• Only fermions and bosons in 3D: Wilczek Halperin

(ei')2 =1

ei' = 1 Bosons or Fermions ±

• In 2D “any-ons” are allowed: e/3 e/3

time ' = ⇡/3 Laughlin anyons Fractionalization and topology

Wen Read Moore Kitaev • Non-trivial degeneracy on closed manifolds: = (nontrivial quasiparticles + 1)G D G =3 1 non-trivial qps+1

G =0 G =1 • Non-abelian anyons: “irrational” size of Hilbert space. Experimentally realized in: 1 2 -GaAs at ⌫ =5/2 p D12 =2 D = 2 -1D chains superconducting p-wave. Majorana fermions The hidden link between composite fermions and the exciton condensate

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

Bose Einstein condensate Superconductor molecules Fermions

• Unification between two celebrated quantum Hall phases of matter: the exciton condensate and the composite fermion metal. Exciton condensate • No tunneling but strong interactions

⌫ = ⌫top + ⌫bottom =1/2+1/2

• Exciton condensate: top top + ei bottom | i | i Long range XY order i c† ctop e ~n h bottom i/ bottom Properties of exciton condensate

• Superfluidity for charge imbalance:

Q = Qtop Qbottom [Q , ]=i Spin-wave • Linearly dispersing Goldstone mode of E (pseudo-spin wave).

• Half-charged vortices (merons): q 2⇡ winding v =1 e Q+ =(vnz) Q+ = e/2 2 v Z n 2= 1 z ±

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

• Fractionalized metal for half filled landau level: 1 N = N e 2 • Composite fermion: electron bound to two vortices

~ B~ =0 Beff =0 6 py

composite electrons px fermions composite fermion fermi surface • Emergent 2-dimensional “gauge field” (analogous to the electro-magnetic field in 2D).

Jain, PRL (1989). Halperin, Lee, Read, PRB (1993). Duality in 1+1D QFT’s

Coleman Luttinger Haldane 1+1 Sine Gordon 1+1 Massive Thirring 1 g 2 (@)2 +(m/)2 cos() ¯(i@/ m) ( ¯µ ) 2 2 4⇡ g ⌘ =1+ † = @ 2 ⇡ 2⇡ x

S. Coleman, 2⇡ PRD 1975

x Fermion vortex duality Physical Dirac composite Dirac fermion fermion vortex aµ

¯ adA = ¯ (i@/ A/) + cf = cf (i@/ a/) cf + + int Le e e Lint L 4⇡ L ~a ~ a0 + @t~a n (r)=r⇥ zˆ jelec(r)=r elec 4⇡ ⇥ 4⇡ Electron creation is flux † M e $ 4⇡ 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 ⌫ = ⌫top + ⌫bottom =1/2+1/2 distance connected?

d =0 d = 1 Exciton condensate ? Two C-Fermion metals

• Precedents Theory Numerics Experiment

Exciton condensate

Bonesteel et al. PRL(1996) Möller et al. PRL (2008) Eisenstein, ARCMP (2014) Bilayer exciton condensate and Composite fermion metal

• A special particle-hole invariant “cooper pairing” of composite fermions is equivalent to exciton condensate:

d toptop

bottom

Exciton Composite fermion ˆ = i † ⌧ † i † † condensate Cooper pairs y x ⇠ top y bottom

I. Sodemann, I. Kimchi, C. Wang, T. Senthil, Phys. Rev. B 95, 085135 (2017). Exciton condensate from CF pairing

global U(1) !(q) a + a aµ 1 2 p+ip like + a+ = !0 2 relative U(1) c q aµ †⌧ (i ) † a a x y a = 1 2q 2

• Symmetric gauge field is gapped via Higgs.

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

r ~a | |!1 e e (r) † (0) const ntop nbottom = r⇥ hM M i ! 2⇡ i c† c e The state is an exciton condensate! h bottom topi/ Relative u(1) photon = Goldstone mode !(q) global U(1) aµ p+ip like + !0 relative U(1) c q aµ †⌧x(iy) † q

• Photon is exciton condensate “spin-wave”. • Electric charges under field a are vortices of condensate order parameter: 4⇡q vorticity $ a + @ ~a zˆ (~j (r) ~j (r)) = r 0 t ⇥ top bottom 4⇡ Abrikosov vortices = merons

• Abrikosov vortices carry half charge:

⇡ vortex ~a 1 ne + ne = r⇥ + Q = top bottom 2⇡ ⇡ ±2

Q =1/2

• Abrikosov vortices have a complex fermion zero mode: Layer X-change q (vorticity) 0 1 0 1/2 2⇡ | i | i | i

1 † 0 0 1 1/2 2⇡ | i⌘ 0| i | i | i Abrikosov vortices = merons • Two ⇡ Abrikosov vortices of opposite vorticity are mutual semions ⇡ vortex 0 | i

1 † 0 | i⌘ 0| i Q =1/2

• Their fusion is a fermion:

The electron (with layer charge imbalance neutralized by condensate). Bogoliubov fermion • Consider fusing two Abrikosov vortices of opposite flux but same a charge (order parameter vorticity): 4⇡ Neutral Vortex • Fusion is Bogoliubov fermion

4⇡

Q =1/2 Q = 1/2 Dictionary Composite fermion Exciton condensate superconductor Spin-wave Photon E E a a 1 2 q q XY vortex Abrikosov vortex Q =1/2 ⇡ flux 2⇡ winding Q =1/2

4⇡ neutral vortex Composite fermion Charge neutral Dipole carrying Q =1/2 Q = 1/2 Fractionalization w/out magnetic fields

• Spin liquids in frustrated magnets. Anderson Kitaev Laughlin

• Bosonic Laughlin state can be viewed as chiral spin liquid after mapping bosons to spins. • “Smoking gun” experimental signatures? • Fractionalization beyond the realm of frustrated magnets or quantum Hall? Puzzles of SmB6

• Simple cubic structure.

• All action happens in Samarium.

• Traditional picture of mixed valence insulator: Atomic limit Mixed valence ✏(k) ✏(k) d [Xe]4f 65d06s2

1 5 6 5d +4f ↵ 4f ✏ F ✏F f SmB6 puzzling behavior

• Insulating behavior from charge • De Haas-van Alphen effect visible at transport: B 5T ⇠ ⇢ ⇢0e T 10meV ⇡ ⇡

B. S. Tan et al., Science (2015). SmB6 puzzles • Could be magnetic breakdown? Zhang, Song, Wang, PRL (2016). ✏(k) Knolle and Cooper, PRL (2015). Gap: Cyclotron: 10meV !c 0.2meV B[T ] ⇠ ⇡ Theory oscillations visible at B 50T ✏F ⇠ Experiment oscillations visible at B 5T ⇠ • Other anomalies: Specific heat to temperature ratio has finite intercept: C = T Like in a fermi sea C T C T 3 fermions / phonon / “Composite exciton Fermi liquid”

✏(k) N d = N b = N d electron electrons

Fermi-bose mixture: b† : spinless boson ✏F † : neutral spinfull fermion One option: : d-electron bosons condense d† b =0 More “interesting” option: h i6 => Metal (“boring”) Bosons bind with d electrons

b and d attract: Composite fermionic exciton:

U nf nd df i i i Bound state of “f-holon" X and d electron. Properties of “Composite exciton Fermi liquid”

Fractionalized Fermionic exciton fermi sea with two pockets f-hole spinon (“semi-metal”)

Some properties: • Essentially linear specific heat: Upturn might indicate other physics at lower temperature • Sub-gap optical conductivity:

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