Fractional quantum Hall effect and duality. Dam Thanh Son (University of Chicago) Strings 2017, Tel Aviv, Israel June 26, 2017

Transcription

1 Fractional quantum Hall effect and duality Dam Thanh Son (University of Chicago) Strings 2017, Tel Aviv, Israel June 26, 2017

2 Plan Fractional quantum Hall effect Halperin-Lee-Read (HLR) theory Problem of particle-hole symmetry Dirac composite fermion theory Consequences, relationship to field-theoretic duality

3 References DTS, arxiv: Wang, Senthil, Metlitski, Vishwanath, Geraedts et al Karch, Tong, Seiberg, Senthil, Wang, Witten,

4 Fractional QHE Landau levels of 2D electron in B field n=3 n=2 n=1

5 Fractional QHE Landau levels of 2D electron in B field n=3 n=2 n=1

6 Fractional QHE Landau levels of 2D electron in B field n=3 n=2 n=1 Filling fraction = n B/2

7 Fractional QHE Landau levels of 2D electron in B field n=3 n=2 n=1 = 1 3 Filling fraction = n B/2

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

9 Lowest Landau level 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

10 Lowest Landau level 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

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

12 Flux attachment Experimental hint from late 80s: gapless state at ν=1/2, nontrivial low-energy effective theory late 80s - early 90s: idea of composite fermion electron = CF with 2 attached flux quanta

13 Flux attachment Experimental hint from late 80s: gapless state at ν=1/2, nontrivial low-energy effective theory late 80s - early 90s: idea of composite fermion electron = CF with 2 attached flux quanta e = CF

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

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

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

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

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

19 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

20 Fermi liquid at nu=1/2 Halperin Lee Read 1993 e e e per e

21 Fermi liquid at nu=1/2 Halperin Lee Read 1993 cf cf cf per e

22 Fermi liquid at nu=1/2 Halperin Lee Read 1993 cf cf cf per e Zero B field for cf

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

24 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

25 Reality of composite fermion confirmed in experiment (Kang et al, 1990) 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)

26 2 features of HLR theory Number of CFs = number of electrons (by construction) Chern-Simons term ada

27 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 problems were known one problem turns out to be crucial

28 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

29 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

30 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

31 FQHE for Dirac fermion FQHE for Dirac fermion sharpens the problem of particle-hole symmetry: Half filled Landau level at zero charge density ground state should be a Fermi liquid, volume of Fermi sphere ~ magnetic field Luttinger s theorem: Fermi volume = charge density which charge density?

32 Dirac composite fermion DTS 2015 electron theory L = i e µ (@ µ ia µ ) e CF theory L = i µ (@ µ ia µ ) 1 4 µ A a Note: no ada number of CFs number of electrons consistent with a large number of exp. constraints

33 Particle-vortex duality original fermion ψ magnetic field density composite fermion ψe density magnetic field 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 Fermi sphere from B

34 (Particle-hole) 2 Θ 2 = ±1 Θ

35 Necessity of Dirac CF On a single Landau level M

36 Necessity of Dirac CF On a single Landau level 2 =( 1) M(M 1)/ M

37 Necessity of Dirac CF On a single Landau level 2 =( 1) M(M 1)/ M M =2N CF 2 =( 1) N CF

38 Necessity of Dirac CF On a single Landau level 2 =( 1) M(M 1)/ M M =2N CF 2 =( 1) N CF Natural for Dirac CF Geraedts, Zaletel, Mong, Metlitski, Vishwanath, Motrunich; Levin, Son

39 More careful version of duality L = i e µ (@ µ ia µ ) e L = i µ (@ µ ia µ ) ada adb 2 4 bdb Adb AdA Naively integrating over b: b = 1 2 (A + a) L = i µ (@ µ a µ ) Ada Seiberg, Senthil, Wang, Witten,

40 Consequences of DCF Satisfies symmetry constraints on transport coefficient (conductivity, thermoelectric) at half filling A gapped particle-hole symmetric state: PH-Pfaffian Absence of Friedel oscillations in correlation of PHsymmetric operators Geraedts et al.

41 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

42 PH Pfaffian state The composite fermions can form Cooper pairs Simplest pairing does not break particle-hole symmetry h i6=0

43 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

44 A window to duality Fermionic particle-vortex duality is a consequence 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

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

46 Conclusion and open questions Dirac CF solves the 20-year old problem of PH symmetry of half-filled Landau level Distinct predictions, numerically checked A experimentally accessible window to fieldtheoretical duality between (2+1) dimensional theories