Is the composite fermion a Dirac particle?

Transcription

1 Is the composite fermion a Dirac particle? Dam T. Son GGI conference Gauge/gravity duality 2015 Ref.:

2 Plan

3 Plan Fractional quantum Hall effect

4 Plan Fractional quantum Hall effect Composite fermion orthodoxy

5 Plan Fractional quantum Hall effect Composite fermion orthodoxy The old puzzle of particle-hole symmetry

6 Plan Fractional quantum Hall effect The solution to the puzzle Composite fermion orthodoxy The old puzzle of particle-hole symmetry

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

8 Fractional QH effect

9 Fractional quantum Hall effect Landau levels of 2D electron in B field n=3 n=2 n=1

10 Fractional quantum Hall effect Landau levels of 2D electron in B field n=3 n=2 n=1

11 Fractional quantum Hall effect Landau levels of 2D electron in B field n=3 n=2 n=1 Filling factor = n B/2

12 Fractional quantum Hall effect Landau levels of 2D electron in B field n=3 n=2 n=1 = 1 3 Filling factor = n B/2

13 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

14 Flux attachment (Wilczek 1982) Attaching flux changes statistics

15 Flux attachment (Wilczek 1982) Attaching flux changes statistics

16 Flux attachment (Wilczek 1982) Attaching flux changes statistics ( 1)

17 Flux attachment (Wilczek 1982) Attaching flux changes statistics ( 1)

18 Flux attachment (Wilczek 1982) Attaching flux changes statistics ( 1) exp(i ) = (+1)

19 Flux attachment (Wilczek 1982) Attaching flux changes statistics ( 1)

20 Flux attachment (Wilczek 1982) Attaching flux changes statistics ( 1)

21 Flux attachment (Wilczek 1982) Attaching flux changes statistics ( 1) exp(2i ) =( 1)

22 Flux attachment (Wilczek 1982) Attaching flux changes statistics ( 1) exp(2i ) =( 1) F = B F = F F = B

23 Composite fermion =1/3 FQH F F F per F

24 Composite fermion =1/3 FQH F F F per F

25 Composite fermion =1/3 FQH F F F per F average per F

26 Composite fermion =1/3 FQH F F F per F average per F FQHE for original fermions = IQHE for composite fermions

27 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)

28 Mathematically Lopez, Fradkin Halperin, Lee, Read L = i (@ 0 ia 0 + ia 0 ) 1 2m (@ i ia i + ia i ) p µ a a + # of attached flux quanta r a =2 p

29 Comments on flux attachment No small expansion parameter: p~1 Difficulty with energy scales, especially in the limit m 0 Nevertheless, explains a number of facts Jain sequences Gapless ν=1/2 state

30 Jain s sequences

31 ν=1/2 state After flux attachment: average magnetic field=0 Ground state: a gapless Fermi liquid described by Halperin-Lee-Read (HLR) field theory Substantial experimental evidence

32 Particle-hole symmetry Girvin 1984 PH symmetry! 1 Formalized as an anti-unitary transformation exact symmetry on the LLL, when mixing of higher LLs negligible

33 CF and particle-hole symmetry Comparing states on two Jain sequences ν=1/3 ν=2/3

34 CF and particle-hole symmetry Comparing states on two Jain sequences ν=2/5 ν=3/5

35 CF and particle-hole symmetry Comparing states on two Jain sequences ν=3/7 ν=4/7

36 CF and particle-hole symmetry Comparing states on two Jain sequences ν=3/7 ν=4/7 CF picture does not respect PH symmetry

37 PH symmetry of ν=1/2 Fermi liquid How can the inside of a circle be equivalent to the outside?

38 PH symmetric CFs? = n 2n +1 CF = n = n +1 2n +1 CF = n +1

39 PH symmetric CFs? = n 2n +1 = n +1 2n +1 CF = n + 1 2?

40 PH symmetric CFs? = n 2n +1 = n +1 2n +1 CF = n + 1 2? Can the filling factor of an IQH state be half-integer?

41 Relativistic model with FQHE S = Z d 3 xi µ (@ µ ia µ ) 1 4e 2 Z d 4 xf 2 µ (Graphene: 4 types of fermions, electrons and photons have different speeds)

42 Ground state in finite magnetic field Ground state is not determined without interaction When e 2 <<1: exactly the same FQH problem μ=0: half-filled Landau level

43 The offset of 1/2 =1/2 =1 = 1/2 = NR 1 2

44 Offset of 1/2 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 Discrete symmetries of (2+1)D Dirac fermion x! x y! y C P T PT CP B µ CP and PT are symmetries at μ=0 PH symmetry = CT = (CP)(PT) is a symmetry before projection to LLL

46 PH symmetry for Dirac fermion Jain sequences NR = n 2n +1! = 1 2(2n + 1) NR = n +1 2n +1! = 1 2(2n + 1) CF = n = 1 2 CF?

47 Particle-vortex duality? original fermion magnetic field density composite fermion density magnetic field

48 Particle-vortex duality? original fermion magnetic field density composite fermion density magnetic field This suggests the following effective action for CFs: S = Z d 3 x apple i µ (@ µ +2ia µ ) µ A a + 1 4e 2 Z d 4 xf 2 µ

49 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

50 Dirac composite fermions No Chern-Simons interaction ada ada would break CP and CT conflict with flux attachment idea? composite fermions have Berry phase π around Fermi surface

51 A few related works Composite Dirac liquid on surface of topological insulators Mross, Essin, Alicea 2014 Emergent Fermi surface from mirror symmetry Hook, Kachru, Torroba, Wang 2014

52 Consequences Exact particle hole symmetry in linear respose at = 1, xy = 1 exactly 2 2 New particle-hole symmetric gapped nonabelian state at ν=1/2: h i6=0

53 CS 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

54 Conclusion and open questions PH symmetry: a challenge for CF picture a CP-invariant Pfaffian-like state Proposal: Dirac CF with gauge, non-cs interaction Alternative: PH symmetry spontaneously broken Barkeshli Mulligan Fisher 2015 Open questions: derivation of the effective theory experimental measurement of the Berry phase