Is the composite fermion a Dirac particle?

Transcription

1 Is the composite fermion a Dirac particle? Dam T. Son (University of Chicago) Cold atoms meet QFT, 2015 Ref.:

2 Plan

3 Plan Composite fermion: quasiparticle of Fractional Quantum Hall Effect (FQHE)

4 Plan Composite fermion: quasiparticle of Fractional Quantum Hall Effect (FQHE) Berry phase: new characteristic of Fermi liquid

5 Plan Composite fermion: quasiparticle of Fractional Quantum Hall Effect (FQHE) Berry phase: new characteristic of Fermi liquid The old puzzle of particle-hole symmetry

6 Plan Composite fermion: quasiparticle of Fractional Quantum Hall Effect (FQHE) Berry phase: new characteristic of Fermi liquid The old puzzle of particle-hole symmetry Berry phase of composite fermions

7 Hall conductivity/resistivity j i = ij E j E i = ij j j i, j = x, y

8 Fractional QH effect

9 Integer quantum Hall state electrons filling n Landau levels n=3 n 2D = n eb 2 ~ n=2 n=1 xy = en 2D B = n e2 2 ~

10 Fractional QHE Landau levels of 2D electron in B field n=3 n=2 n=1 Large ground-state degeneracy without interactions

11 Fractional QHE Landau levels of 2D electron in B field n=3 n=2 n=1 Large ground-state degeneracy without interactions

12 Fractional QHE Landau levels of 2D electron in B field n=3 n=2 n=1 Large ground-state degeneracy without interactions Filling fraction = n B/2

13 Fractional QHE Landau levels of 2D electron in B field n=3 n=2 n=1 = 1 3 Large ground-state degeneracy without interactions Filling fraction = n B/2

14 Energy scales! c = eb mc eb mc e2 r IQH FQH Interesting limit: eb/mc >> Δ (m 0) only lowest Landau level (LLL) states survives No small parameter

15 QHE in cold atoms Rapidly rotating atomic systems Wilkin Gunn 2000 Lattice magnetic field by quadrupole potential and time modulation of tunneling Sørensen Demler Lukin 2005 Artificial magnetic field Jaksch Zoller 2003 Fractional Chern insulators Cooper Dalibard 2013, Yao et al 2013

16 Composite fermions Theoretical understanding of FQHE relies on the notion of the composite fermion e = CF

17 Mathematically Lopez, Fradkin Halperin, Lee, Read L = i (@ 0 ia 0 + ia 0 ) 1 2m (@ i ia i + ia i ) p µ a a + r a =2 p # of attached flux quanta At mean field level: B e = B b = B 2 pn 1 e = 1 p

18 Composite fermion =1/3 FQH e e e per e

19 Composite fermion =1/3 FQH cf cf cf per e

20 Composite fermion =1/3 FQH cf cf cf per e average per cf

21 Composite fermion =1/3 FQH cf cf cf per e average per cf IQHE of CFs

22 nu=1/2 state e e e per e

23 nu=1/2 state cf cf cf per e

24 nu=1/2 state cf cf cf per e Zero B field for cf

25 nu=1/2 state cf cf cf per e Zero B field for cf CFs form a Fermi liquid

26 Fermi liquid of CFs The theory of the nu=1/2 state as a Fermi liquid of CFs was developed by Halperin, Lee, Read (HLR) No small expansion parameter: p~1 Difficulty with energy scales in the limit m 0 Nevertheless, abundant experimental evidence for a Fermi liquid behavior of the nu=1/2 state

27 nu=1/2 state

28 nu=1/2 state ν=1/2

29 Despite its success, the HLR theory suffers from a flaw: lack of particle-hole symmetry

30 Particle-hole symmetry Girvin 1984 PH symmetry! 1 Can be formalized mathematically exact symmetry the Hamiltonian on the LLL, when mixing of higher LLs negligible

31 PH symmetry of a Fermi liquid?

32 PH symmetry of a Fermi liquid? PH

33 PH symmetry of a Fermi liquid? PH

34 PH symmetry of a Fermi liquid? =?

35 The particle-hole asymmetry of the HLR theory has been noticed early on Kivelson et al 1997 No conclusive resolution has emerged Maybe ground state at nu=1/2 breaks PH symmetry spontaneously? Barkeshli Mulligan Fisher 2015 By now, numerical and experimental evidence: nu=1/2 state is particle-hole symmetric

36 The proposal CF has Berry phase pi around the Fermi surface

37 The proposal PH CF has Berry phase pi around the Fermi surface

38 The proposal PH CF has Berry phase pi around the Fermi surface

39 The proposal PH CF has Berry phase pi around the Fermi surface

40 Berry phase in Fermi liquids Original Fermi liquid theory (Landau, 1956) Recent understanding: Landau s Fermi liquid theory has to be supplemented by the Berry phase of quasiparticles Niu, Haldane,...

41 Example: Dirac cone ( p)u p = p u p u pr p u p = ia(p) I A dp =

42 Jain s sequences = n +1 2n +1 = n 2n +1

43 Standard flux attachment: 1 e = 1 p = n 2n +1 e = n = n +1 2n +1 e = (n + 1) In the new picture, these two fractions correspond to CF = ± n + 1 2

44 IQHE in graphene xy = n + 1 e ~ Figure 4 QHE for massless Dirac fermions. Hall conductivity j xy and longitudinal resistivity r xx of graphene as a function of their concentration at B ¼ 14 T and T ¼ 4 K. j xy ; (4e 2 /h)n is calculated from the measured dependences of (V ) and (V ) as ¼ /( 2 þ 2 ). The

45 Alternative to flux attachment Flux attachment breaks PH symmetry Alternative: fermionic particle-vortex duality L A = i µ (@ µ ia µ ) L B = i µ (@ µ +2ia µ ) µ A a

46 Particle-vortex duality DTS; Metlitski, Vishwanath; Senthil, Wang original fermion magnetic field density composite fermion density magnetic field S = Z d 3 x apple i µ (@ µ +2ia µ ) µ A a + j µ = S A µ = 1 2 a S a 0 =0! h 0 i = B 4

47 Jain s sequences again 2 B = 1 2 A A = 1 2 = n 2n +1! B = n = n +1 2n +1! B = n + 1 2

48 Comments on particle-vortex duality Bosonic counterpart: duality between XY model and abelian Higgs model strong numerical evidence specific for d=3, N=1 Fermionic particle-vortex duality: no numerical evidence (yet?) at zero B field small N: chiral symmetry breaking in dual theory magnetic field quenches kinetic energy, strong interactions needed for original fermions?

49 Relativistic model with FQHE µ =0 S = Z d 3 xi µ (@ µ ia µ ) 1 4e 2 Z d 4 xf 2 µ Low-energy description of ground state at zero chemical potential, finite B field

50 Consequences Exact particle hole symmetry in linear response at = 1, exactly (HLR: ρxy=2) xy = New particle-hole symmetric gapped nonabelian state at ν=1/2: h i6=0 Pfaffian and anti-pfaffian states: pairing of Dirac CFs with angular momentum 2 and -2

51 Dirac composite fermions Emergent gauge field No Chern-Simons interaction ada ada would break CP and CT Composite fermion without flux attachment composite fermions have Berry phase π around Fermi surface

52 HLR theory as the NR limit μ When CP is broken, CF has mass In the NR limit: NR action for CF Integrating out Dirac sea: Chern- Simons interaction between CF ada Standard HLR theory is reproduced Particle-hole symmetry broken by the CF Dirac mass

53 Conclusion and open questions PH symmetry: a challenge for CF picture experimentally verifiable consequences Open questions: derivation of the effective theory Proposal: Dirac CF with gauge, non-cs interaction particle-vortex duality instead of flux attachment experimental measurement of the Berry phase: cold atoms?