The Dirac composite fermions in fractional quantum Hall effect. Dam Thanh Son (University of Chicago) Nambu Memorial Symposium March 12, 2016

Transcription

1 The Dirac composite fermions in fractional quantum Hall effect Dam Thanh Son (University of Chicago) Nambu Memorial Symposium March 12, 2016

2 A story of a symmetry lost and recovered Dam Thanh Son (University of Chicago) Nambu Memorial Symposium March 12, 2016

3 Plan

4 Plan Fractional Quantum Hall Effect (FQHE)

5 Plan Fractional Quantum Hall Effect (FQHE) Composite fermions

6 Plan Fractional Quantum Hall Effect (FQHE) Composite fermions The old puzzle of particle-hole symmetry

7 Plan Fractional Quantum Hall Effect (FQHE) Composite fermions Dirac composite fermions The old puzzle of particle-hole symmetry

8 The TOE of QHE B e e e e H = X a (p a + ea a ) 2 2m + X ha,bi e 2 x a x b

9 Integer quantum Hall state electrons filling n Landau levels B m n=3 When = n B 2 n=2 n=1 energy gap: B m

10 1982: Fractional QH effect (Tsui, Stormer) Willett et al 1987 xy = 1 xy = p q e 2 h j x = xx E x + xy E y odd integer (most of the time)

11 Fractional QHE n=3 Filling fraction = B/2 n=2 n=1

12 Fractional QHE n=3 Filling fraction = B/2 n=2 n=1

13 Fractional QHE n=3 Filling fraction n=2 = B/2 n=1 = 1 3

14 Fractional QHE n=3 Filling fraction n=2 = B/2 n=1 = 1 3 Large ground-state degeneracy without interactions Experiments: energy gap for certain rational filling fractions

15 Energy scales B m B m e2 r IQH FQH Lowest Landau level limit: B/m >> Δ Perturbation theory useless

16 The Standard Model of FQHE H = P LLL X a,b e 2 x a x b Projection to lowest Landau level

17 Flux attachment D. Arovas

18 Flux attachment (Wilczek 1982, Jain 1989) Flux attachment: statistics does not change by attaching an even number of flux quanta

19 Flux attachment (Wilczek 1982, Jain 1989) Flux attachment: statistics does not change by attaching an even number of flux quanta

20 Flux attachment (Wilczek 1982, Jain 1989) Flux attachment: statistics does not change by attaching an even number of flux quanta ( 1)

21 Flux attachment (Wilczek 1982, Jain 1989) Flux attachment: statistics does not change by attaching an even number of flux quanta ( 1)

22 Flux attachment (Wilczek 1982, Jain 1989) Flux attachment: statistics does not change by attaching an even number of flux quanta ( 1) exp(i ) = (+1)

23 Flux attachment (Wilczek 1982, Jain 1989) Flux attachment: statistics does not change by attaching an even number of flux quanta ( 1)

24 Flux attachment (Wilczek 1982, Jain 1989) Flux attachment: statistics does not change by attaching an even number of flux quanta ( 1)

25 Flux attachment (Wilczek 1982, Jain 1989) Flux attachment: statistics does not change by attaching an even number of flux quanta ( 1) exp(2i ) =( 1)

26 Flux attachment (Wilczek 1982, Jain 1989) Flux attachment: statistics does not change by attaching an even number of flux quanta ( 1) exp(2i ) =( 1) e = CF

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

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

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

30 Composite fermion =1/3 FQH cf cf cf per e average per cf IQHE of CFs with ν=1

31 Composite fermion =2/3 FQH F F F F per F F average per F F FQHE for original fermions = IQHE for composite fermions (n=2)

32 Jain s sequence of plateaux Using the composite fermion most observed fractions can be explained Electrons Composite fermions = n 2n +1 CF = n = n +1 2n +1 CF = n +1

33 Jain s sequences of plateaux = n +1 2n +1 = n 2n +1

34 ν=1/2 ν=1/2

35 ν=1/2 state Halperin Lee Read e e e per e

36 ν=1/2 state Halperin Lee Read cf cf cf per e

37 ν=1/2 state Halperin Lee Read cf cf cf per e Zero B field for cf

38 ν=1/2 state Halperin Lee Read cf cf cf per e Zero B field for cf CFs form a gapless Fermi liquid: HLR field theory

39 HLR field theory L = i (@ 0 ia 0 + ia 0 ) 1 2m (@ i ia i + ia i ) µ a a b = r a =2 2 Can be understood as a low-energy effective theory. Two main features: number of CFs = number of electrons Chern-Simons action for gauge field a Both comes with the flux attachment procedure Halperin, Lee, Read 1993

40 End of flux attachment

41 Is the composite fermion real? The most amazing fact about the composite fermion is that it exists! One way to detect the composite fermion: go slightly away from half filling: composite fermions live in small magnetic field and move in large circular orbits Kang et al, 1993

42 How real are the CFs? R (k ) 3 2 (a) W = 40 nm a = 200 nm n = 1.74 x cm -2 T = 0.3 K ν = ν = 1/2 i = B (T) (Shayegan s group, 2014)

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

44 Girvin 1984 Particle-hole symmetry emptyi = fulli PH symmetry c k 1 = c k i 1 = i! 1 = 1 2! = 1 2 In the LLL limit: exact symmetry the Hamiltonian

45 PH symmetry in the CF theory PH conjugate pairs of FQH states = n 2n +1 = n +1 2n +1 ν=1/3 ν=2/3

46 PH symmetry in the CF theory PH conjugate pairs of FQH states = n 2n +1 = n +1 2n +1 ν=2/5 ν=3/5

47 PH symmetry in the CF theory PH conjugate pairs of FQH states = n 2n +1 = n +1 2n +1 ν=3/7 ν=4/7

48 PH symmetry in the CF theory PH conjugate pairs of FQH states = n 2n +1 = n +1 2n +1 ν=3/7 ν=4/7 CF picture does not respect PH symmetry

49 PH symmetry of CF Fermi liquid?

50 PH symmetry of CF Fermi liquid? PH

51 PH symmetry of CF Fermi liquid? PH

52 PH symmetry of CF Fermi liquid? PH Related: no simple description of the anti-pfaffian

53 The particle-hole asymmetry of the HLR theory has been noticed early on Kivelson, D-H Lee, Krotov, Gan The issue periodically came up, but no conclusive resolution has emerged 2007: anti-pfaffian state Levin, Halperin, Rosenow; Lee, Ryu, Nayak, Fisher

54 Why PH symmetry is difficult Particle-hole symmetry is not a symmetry of nonrelativistic fermions in a magnetic field (TOE) Only appears after projection to the lowest Landau level (SM) Not realized as a local operation on the fields

55 Three possibilities

56 Three possibilities Hidden symmetry of HLR theory

57 Three possibilities Hidden symmetry of HLR theory careful analysis of Kivelson et al (1997) showed no sign of it

58 Three possibilities Hidden symmetry of HLR theory careful analysis of Kivelson et al (1997) showed no sign of it Spontaneous breaking of particle-hole symmetry Barkeshli, Fisher, Mulligan 2015.

59 Three possibilities Hidden symmetry of HLR theory careful analysis of Kivelson et al (1997) showed no sign of it Spontaneous breaking of particle-hole symmetry Barkeshli, Fisher, Mulligan By now: considerable numerical evidence against it

60 Three possibilities Hidden symmetry of HLR theory careful analysis of Kivelson et al (1997) showed no sign of it Spontaneous breaking of particle-hole symmetry Barkeshli, Fisher, Mulligan By now: considerable numerical evidence against it The correct theory is a different theory, which realizes particle-hole symmetry in an explicit way

61 Dirac composite fermion DTS, arxiv:

62 Dirac composite fermion PH DTS, arxiv:

63 Dirac composite fermion PH DTS, arxiv:

64 Dirac composite fermion PH DTS, arxiv:

65 Dirac composite fermion PH The composite fermion is a massless Dirac fermion Particle-hole symmetry acts as time reversal k! k! i 2 DTS, arxiv:

66 First hint of Dirac nature of CFs = n 2n +1 CF = n = n +1 2n +1 CF = n +1

67 First hint of Dirac nature of CFs = n 2n +1 CF = n + 1 2? = n +1 2n +1

68 First hint of Dirac nature of CFs = n 2n +1 CF = n + 1 2? = n +1 2n +1 CFs form an IQH state at half-integer filling factor: must be a massless Dirac fermion

69 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 Novoselov et al 2005

70 (Particle-hole) 2 Θ 2 = ±1 Θ

71 Geraedts, Zaletel, Mong, Metlitski, Vishwanath, Motrunich Levin, Son 2015 (unpublished) M emptyi = fulli = c 1 c 2 c M emptyi c k 1 = c k anti-unitary

72 Geraedts, Zaletel, Mong, Metlitski, Vishwanath, Motrunich Levin, Son 2015 (unpublished) M emptyi = fulli = c 1 c 2 c M emptyi c k 1 = c k 2 =( 1) M(M 1)/2 anti-unitary

73 Geraedts, Zaletel, Mong, Metlitski, Vishwanath, Motrunich Levin, Son 2015 (unpublished) M emptyi = fulli = c 1 c 2 c M emptyi c k 1 = c k 2 =( 1) M(M 1)/2 anti-unitary M =2N CF 2 =( 1) N CF

74 Geraedts, Zaletel, Mong, Metlitski, Vishwanath, Motrunich Levin, Son 2015 (unpublished) M emptyi = fulli = c 1 c 2 c M emptyi 2 =( 1) M(M 1)/2 c k 1 = c k anti-unitary M =2N CF 2 =( 1) N CF Θ acts as time reversal on the Dirac composite fermion! (i 2 )! (i 2 ) 2 =

75 Problem with flux attachment For an even number of orbitals on the LLL, 2 anyi =( 1) M/2 anyi independent of the number of electrons in any> Number of CFs = half the degeneracy of the LLL does not coincide with the number of electrons away from half filling This goes against the philosophy of flux attachment

76 Tentative picture L = i µ (@ µ +2ia µ ) µ A a µ A A Instead of flux attachment, the CF appears through particle-vortex duality A known duality in the case of bosons, but in the case of fermion would be new ψ is a massless Dirac fermion ada and fermion mass term would break particle-hole symmetry Jain sequences not affected

77 Particle-vortex duality original fermion magnetic field density composite fermion density magnetic field L = i µ (@ µ +2ia µ ) µ A a µ A A j µ = S A µ = 1 2 a A S a 0 =0! h 0 i = B 4

78 Consequences of Dirac CF If Ô is particle-hole symmetric i.e., Ô =( 0 )r 2 h k Ô ki =0 Suppression of Friedel oscillations Geraedts et al, 2015

79 Consequences of PH symmetry j = xx E + xy E ẑ + xx rt + xy rt ẑ conductivities thermoelectric coefficients At exact half filling, in the presence of particle-hole symmetric disorders HLR xy = e2 2h xy = 2h e 2 xx =0 Potter, Serbyn, Vishwanath 2015

80 Taketani s three stages Phenomenon: fractional quantum Hall plateaux Substance: composite fermion Essence: fermionic particle-vortex duality?

81 How to get to the Essence Lowest Landau Level algebra from Dirac fermions Shankar Murthy Duality in surfaces of topological insulators at zero magnetic field: free fermion=qed3? Metlitskii, Vishwanath; Wang, Senthil; Alicea, Essin, Mross, Motrunich... From supersymmetric duality Kachru et al General principles: anomaly matching, spin/charge relation Seiberg, Witten Experimental progress Wei Pan, W. Kang...