Fractional quantum Hall effect and duality. Dam T. Son (University of Chicago) Canterbury Tales of hot QFTs, Oxford July 11, 2017

Transcription

1 Fractional quantum Hall effect and duality Dam T. Son (University of Chicago) Canterbury Tales of hot QFTs, Oxford July 11, 2017

2 Plan

3 Plan General prologue: Fractional Quantum Hall Effect (FQHE)

4 Plan General prologue: Fractional Quantum Hall Effect (FQHE) Composite fermions

5 Plan General prologue: Fractional Quantum Hall Effect (FQHE) Composite fermions The puzzle of particle-hole symmetry

6 Plan General prologue: Fractional Quantum Hall Effect (FQHE) Composite fermions The puzzle of particle-hole symmetry Dirac composite fermions

7 General Prologue QCD is a prime example of a strongly coupled theory The particle excitations of the vacuum are very different from the microscopic degree of freedom A very similar situation in FQHE

8 The setup H = X a (p a + ea a ) 2 2m + X ha,bi e 2 x a x b

9 Integer quantum Hall effect Ignore Coulomb interactions When electrons moving in 2D in a magnetic field, energy is quantized: Landau level IQHE: electrons filling n Landau levels n=3 n=2 = B m n=1 degeneracy BA 2

10 Plateaux require energy gap

11 Fractional QHE Assume we have less particles than states on LLL n=3 n=2 n=1 In the approximation of noninteracting electrons: exponential degeneracy of states

12 Fractional QHE Assume we have less particles than states on LLL n=3 n=2 n=1 In the approximation of noninteracting electrons: exponential degeneracy of states

13 Why the FQH problem is hard degenerate perturbation theory Starting point: exponentially large number of degenerate states Any small perturbation lifts the degeneracy no small parameter

14 Lowest Landau level limit H = X a (p a + ea a ) 2 2m + X ha,bi e 2 x a x b B m n=1 n=0

15 Lowest Landau level limit H = X a (p a + ea a ) 2 2m + X ha,bi e 2 x a x b m! 0 B m!1 n=1 n=0

16 Lowest Landau level limit H = X a (p a + ea a ) 2 2m + X ha,bi e 2 x a x b m! 0 n=1 H = P LLL X a,b e 2 x a x b B m!1 Projection to n=0 lowest Landau level

17 Experimental hints

18 Jain s sequences of QH plateaux = n +1 2n +1 = n 2n +1

19 Systematics of Jain s sequences Gapped states Energy gap goes down ~ 1/n for n n= : gapless, likely Fermi liquid state

20 A powerful theory with a flaw

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

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

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(i ) = (+1)

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

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

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

29 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

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

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

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

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

34 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)

35 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 flux attachment mean field: B e = B b = B 4 n = 1 2 B e =0

36 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

37 Prediction for nu=1/2 state Halperin Lee Read 1993 e e e per e

38 Prediction for nu=1/2 state Halperin Lee Read 1993 cf cf cf per e

39 Prediction for nu=1/2 state Halperin Lee Read 1993 cf cf cf per e Zero B field for cf

40 Prediction for nu=1/2 state Halperin Lee Read 1993 cf cf cf per e Zero B field for cf CFs form a Fermi liquid; HLR theory

41

42 ν=1/2

43 Is the composite fermion real? Composite fermion can be detected as a quasiparticle near half-filling large semiclassical orbit when magnetic fields do not exactly cancel

44 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) (Kamburov et al, 2014)

45 For a long time it was thought that the HLR theory (zoomed in the near Fermi surface region) gives the correct low-energy effective theory There is one crucial problem

46

47 The problem of particle-hole symmetry

48 Particle-hole symmetry emptyi = fulli PH symmetry c k 1 = c k i 1 = i! 1 exact symmetry the Hamiltonian on the LLL, when mixing of higher LLs negligible

49 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

50 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

51 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

52 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

53 PH symmetry of a Fermi liquid?

54 PH symmetry of a Fermi liquid? PH

55 PH symmetry of a Fermi liquid? PH

56 PH symmetry of a Fermi liquid? =?

57 PH symmetry in HLR HLR Lagrangian does not have any symmetry that can be identified with PH symmetry ~1997 The problem was considered hard as it requires projection to lowest Landau level PH conjugation acts nonlocally

58 Sharpening the problem Consider a 2-component massless Dirac fermion Can realize fractional quantum Hall effect Natural particle-hole symmetry at zero density E 0

59 The puzzle of QHE for Dirac fermion Half filled Landau level arises naturally at zero chemical potential Turn on a magnetic field: ground state is a Fermi liquid Volume of Fermi sphere ~ magnetic field Which conserved charge in Luttinger s theorem???

60 Solution to the problem of particlehole symmetry

61 Prelude to solution: particle-vortex duality Peskin; Dasgupta, Halperin L 1 µ 2 m L 2 = (@ µ a µ ) 2 m Goldstone boson particle photon vortex

62 Coupling to external gauge field L 1 = (@ µ A µ ) 2 m L 2 = (@ µ a µ ) 2 m µ A a j µ = 1 2 a

63 Hypothetical duality DTS 2015 Metlitski, Vishwanath 2015 Wang, Senthil 2015 electron theory L = i e µ (@ µ ia µ ) e CF theory L = i µ (@ µ ia µ ) 1 4 µ A a

64 Particle-vortex duality S = Z d 3 x apple i µ (@ µ ia µ ) 1 4 µ A a = S A 0 = b 4 S a 0 =0! h 0 i = B 4 Turn on magnetic field lead to a finite density Landau s reasoning: Fermi surface original fermion ψ magnetic field density composite fermion ψe density magnetic field

65 Dirac composite fermion Low energy dynamics of a half-filled Landau level is described by a low-energy effective theory of a new fermion ( composite fermion ) coupled to a dynamical gauge field The composite fermion is electrically neutral Density of composite fermion = physical magnetic field

66 Particle-hole symmetry as CT symmetry Magnetic field breaks C, P, T preserves PT, CT, CP Particle-hole symmetry of the n=0 Landau level can be identified with CT Effective theory of the composite fermion has CT symmetry

67 Action of CT A 0 (t, x)! A 0 ( t, x) A i (t, x)! A i ( t, x) a 0 (t, x)! a 0 ( t, x) a i (t, x)! a i ( t, x) (t, x)! i 2 ( t, x)

68 CT on composite fermion

69 CT on composite fermion PH

70 CT on composite fermion PH

71 CT on composite fermion PH

72 CT on composite fermion PH Particle-hole symmetry maps particle to particle k! k! i 2

73 CT on composite fermion PH Particle-hole symmetry maps particle to particle k! k! i 2 DTS 2015

74 Away from half filling S = Z d 3 x apple i µ (@ µ ia µ ) 1 4 µ A a = S A 0 = b 4 S a 0 =0! h 0 i = B 4 =0 Now the CFs move in large circular orbits

75 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) (Kamburov et al, 2014)

76 Mapping Jain s sequences = n 2n +1 CF = n = n +1 2n +1 CF = n +1

77 Mapping Jain s sequences = n 2n +1 = n +1 2n +1 CF = n + 1 2?

78 Mapping Jain s sequences = n 2n +1 CF = n + 1 2? = n +1 2n +1 CFs form an IQH state at half-integer filling factor: must be a Dirac fermion

79 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

80 (Particle-hole) 2 Θ 2 = ±1 Θ

81 On a single Landau level M

82 On a single Landau level 2 =( 1) M(M 1)/ M

83 On a single Landau level 2 =( 1) M(M 1)/ M M =2N CF 2 =( 1) N CF

84 On a single Landau level 2 =( 1) M(M 1)/ M M =2N CF 2 =( 1) N CF Dirac CF:! ( i 2 ) 2 = Geraedts, Zaletel, Mong, Metlitski, Vishwanath, Motrunich; Levin, Son

85 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

86 A new gapped state The composite fermions can form Cooper pairs Simplest pairing does not break particle-hole symmetry h i6=0 A new gapped state: PH-Pfaffian state

87 Consequences of Dirac CF Suppression of Friedel oscillations in correlations of particle-hole symmetric observables Ô =( 0 )r 2 k k k k Geraedts, Zaletel, Mong, Metlitsky, Vishwanath, Montrunich, 2015 Direct proof of Berry phase π of the composite fermion

88 A window to duality Fermionic particle-vortex duality can be derived of a more elementary fermion-boson duality Karch, Tong; Seiberg, Senthil, Wang, Witten small N version of duality between CS theories, tested at large N New dualities can be obtained Example: Nf=2 QED3 is self-dual Cenke Xu

89 The elementary duality L = L[,A] AdA L = L[,a]+ 1 4 ada Ada

90 Comments on duality One side of duality is a theory of a single 2- component Dirac fermion coupled to U(1) gauge field It is usually thought QED3 with Nf fermions is unstable with respect to SSB for Nf<Nf*. Nf* ~ 6 in Schwinger-Dyson One lattice simulation (Karthik, Narayanan 2015) indicates Nf* < 2. The stability of theory with Nf=1 is not required for fractional quantum Hall effect (finite density)

91 Conclusion The low-energy quasiparticle of half-filled Landau level is completely different from the electron Symmetries allowed to guess the form of the lowenergy effective theory Hints on new field-theoretic dualities in 2+1 D

92 References

93 References DTS, PRX 5, (2015) Wang, Senthil Metlitski, Vishwanath Geraedts et al., Mross, Alicea, Motrunich,