Nematic Order and Geometry in Fractional Quantum Hall Fluids

Transcription

1 Nematic Order and Geometry in Fractional Quantum Hall Fluids Eduardo Fradkin Department of Physics and Institute for Condensed Matter Theory University of Illinois, Urbana, Illinois, USA Joint Condensed Matter Theory Symposium Urbana-Chicago Fest Urbana, April 2, 2016 Y. You, G. Y. Cho, and EF, Phys. Rev. X 4, (2014) Y. You, G. Y. Cho, and EF, arxiv: G. Y. Cho, Y. You, and EF, Phys. Rev. B 90, (2014) A. Gromov, G. Y. Cho, Y. You, A. Abanov, and EF, Phys. Rev. Lett. 114, (2014) 1

2 The Fractional Quantum Hall Effect 2 DEG B Al As Ga As heterostructure Energy 5/2 hω c 3/2 hω c E F 1/2 hω c Angular Momentum bulk edge 2

3 Phases of the 2DEG in magnetic fields Fractional quantum Hall fluids are preeminent at high fields (or high densities) in Landau levels N=0,1 On higher, N 2, Landau levels there are integer quantum Hall states At low densities Wigner crystals have been predicted (maybe seen) Compressible liquid crystal-like phases: nematic and stripe (`bubble ) phases are seen for N 2 and in N=1 in tilted magnetic fields Compressible nematic phases do not exhibit the fractional (or integer) Hall effect Nematic phases: uniform fluids that break rotational invariance spontaneously Stripe phases: break translation and rotational invariance spontaneously Crystalline, nematic and stripe phases are fragile: couple strongly to disorder 3

4 Compressible Nematic Phases J. Eisenstein et al, 1998 Compressible nematic phases in the middle of Landau levels N=2-6 For N=1 they appear in tilted fields replacing the non-abelian FQH state Anisotropic longitudinal transport Nematic order parameter: 2x2 real traceless symmetric tensor Q (quadrupole) Q changes sign under a 90 o rotation Q = Qxx Q xy Q xy Q xx / xx yy xy xy yy xx 4

5 Close competition between nematic and paired FQH states at ν=5/2 Observation of a transition from a topologically ordered to a spontaneously broken symmetry phase N. Samkharadze, K.A. Schreiber, G.C. Gardner, M.J. Manfra, E. Fradkin, G.A. Csáthy Nature Physics 12, 191 (2016) 5

6 Nature Physics 7, 845 (2011) This is a phase transition inside a FQH state! 6

7 }~ Transition to a Nematic FQH State Rapid but continuous rise in the anisotropy in the transport at temperatures where the FQH state is already well formed. The transition does not close the gap of the FQH state It happens for a range of tilt angles Continuous phase transition to an incompressible nematic FQH state ω c E Kohn mode (magnetoplasmon) GMP mode (magnetophonon) The transition is weakly rounded by the inplane component of the magnetic field which acts as a rotation symmetry breaking field k 0 magnetoroton k Possible mechanism: Condensation at k=0 of the Girvin-MacDonald-Platzman (GMP) collective mode (quadrupolar) Spectrum of collective modes of the Laughlin ν=1/3 FQH state 7

8 Electronic Nematic Fluids Kivelson, Fradkin, Emery, 1998; Fradkin and Kivelson 1998; Oganesyan, Kivelson and Fradkin (2001) Uniform phase of a fluid that breaks spontaneously rotational invariance Order parameter: traceless symmetric tensor (in 2D this is equivalent to a director) In an electronic system a nematic state is a phase with a large, sharp temperature-dependent transport anisotropy (2DEG near ν=9/2; Sr 3 Ru 2 O 7 for H~7-8T; YBCO in the pseudogap regime; BaFe 2 As 2 ) Two pathways to an electronic nematic state: a) by melting a charge stripe phase (i.e. quantum or thermal melting a unidirectional CDW), and b) by a Pomeranchuk instability of a Fermi liquid: particlehole condensate in the quadrupolar, l=2, channel) Q(x) = 1 k 2 F (x)(@ x y) 2 (x) (x)2@ y (x) (x)2@ y (x) (x)(@ y x) 2 (x) Smectic or stripe phase Via melting stripes Fermi Liquid Via Pomeranchuk Instability Nematic phases 8

9 Theories of the Nematic FQH state Mulligan, Nayak, and Kachru (2011) proposed an effective field theory of the phase transition to the nematic FQH state based on an extension of the hydrodynamic field theory of the FQH state (Wen, 1990). The nematic fluctuations entered in the parity even corrections It required that the coefficient of the leading parity even term ( Maxwell ) changes sign at the transition. But, this coefficient cannot be changed since it is given by the k=0 the gap of the Kohn (magneto-plasmon) mode, whose value is fixed to be the cyclotron energy by Galilean invariance Maciejko, Hsu, Kivelson, Park and Sondhi (2013) proposed a phenomenological theory of the nematic FQH state They showed that the (quadrupolar) Girvin-MacDonald-Platzman mode condensation can drive the transition You, Cho and Fradkin (2014) provided microscopic derivation of the effective field theory and connected it with the theory of the Hall viscosity and the geometric response of FQH fluids 9

10 Fractional Quantum Hall Effect and Nematicity Z S = apple d 2 xdt 1 (x)d 0 (x) (D (x)) (D (x)) 2m e Z d 2 x 0 d 2 xdt V ( x x 0 )( (x) )( (x 0 ) ) Z Z dt d 2 xd 2 x 0 F 2 ( x x 0 )Tr[Q(x)Q(x 0 )] F 2 (q) = F 2 1+appleq 2 Quadrupolar Landau interaction of Fermi Liquids D µ µ + ia µ Q(x) = Covariant derivative D 2 (x) x Dy 2 D x D y + D y D x D x D y + D y D x Dy 2 Dx 2 (x) Nematic order parameter: Quadrupolar fermionic density 10

11 Flux Attachment and Chern-Simons Gauge Theory Each fermion is attached to k Z (even) fluxes maps fermions to fermions S(,,A µ ) 7! S(,,A µ + a µ )+ 1 4 k D µ µ + ia µ 7! D µ µ + i(a µ + a µ ) Z López and EF, 1991 d 3 x µ a a Landau level at filling fraction ν =N e /N φ The average flux is reduced to N φ eff=n φ -2πkNe: ν eff =N e /N φ eff =p Z Composite fermions filling up p effective Landau levels (Jain, 1989) The allowed filling fractions are the Jain sequences: ν=p/(pk+1) Laughlin states: p=1 and k=m-1 ν=1/m (m odd) At one-loop σ xy =ν e 2 /h, and maps the composite fermions into anyons with e * =e/m, statistics: θ=π/m; this is an incompressible topological fluid If p, ν 1/(2p); in this limit the gap vanishes and the system is a composite Fermi liquid 11

12 Theory of the Nematic State for ν=1/3 (trivial to extend to all Jain states) M1 and M2 : Hubbard-Stratonovich fields δb= a S = Z apple d 2 xdt 1 (x)d 0 (x) (D (x)) (D (x)) 2m e Z d 2 x 0 d 2 xdt V ( x x 0 ) b(x) b(x 0 ) + 1 Z d 2 xdt µ a a 8 Z h + d 2 1 xdt 4F 2 m 2 M 2 + apple X e 4F 2 m 2 rm i 2 e i=1,2 + M 1 (Dx 2 D m y) 2 + M i 2 (D x D y + D y D x ) e m e Q ij = M1 M 2 M 1 M 2 The 2x2 traceless symmetric tensor Q couples to the composite fermions as a fluctuation of a two-dimensional metric tensor Nematic fluctuations are geometric degrees of freedom 12

13 Effective Action for the Nematic Fields L M = ij 2 M i@ 0 M j rm 2 apple 2 (rm i) 2 u 4 (M 2 ) 2 r = 1 4F 2 m 2 e! c 2 l 2 b, apple = apple 2F 2 m 2 e 5 2, u = b! c 4 First term: Berry phase that determines the commutation relations of the nematic fields This term requires that the dynamics has z=2 Its coefficient is the Hall viscosity of the composite fermions (not of the FQH fluid!) Nematic transition: r=0 and for F2=F2 c <0, F2 c = l 2 b 2! c m 2 e In the nematic phase <M> 0 A computation of the collective mode spectra shows that the GMP mode condenses at k=0 for r=0 13

14 The nematic spin connection and disclinations We can define a spin connection ωμ (Q) from the nematic fields (associated with local rotations) which couples to the gauge fields as a Wen-Zee term! 0 = ij M 0 M j,! x = ij M x M j (@ x M y M 1 ), L wz = 1 4 µ! ( a + A )! y = ij M y M j +(@ x M 1 y M 2 ), The flux of the spin connection defines a curvature which is equal to a disclination of the nematic order! The disclination is the vorticity of the nematic order It carries fractional charge and fractional statistics 14

15 The compressible nematic state Although similar ideas can, in principle, be applied to describe a compressible nematic state, there are problems In the compressible limit the gap vanishes and the flux attachment yields an uncontrollable amount of Landau level mixing There is no effective projection onto a Landau level There is no small expansion parameter to control the theory The fluctuations of a gauge field coupled to a Fermi surface lead to strong forward scattering infrared divergencies and to a breakdown of Fermi liquid theory: the quasiparticles are ill defined A naive computation of the Landau parameters of the CFL predicts that they are all equal to the Pomeranchuk instability value (in all channels!) Particle-hole symmetry is not possible Nevertheless we proceed and try to do our best 15

16 Effective Action of the Nematic Fields (one loop) S e [M] = R p, M i(p, )L ij (p, )M j (p, ) L ij (p, ) = apple F 2 p 2 cos 2 (2 p ) 1 2 `30 sin(2 p ) cos(2 p ) 1 2 `30 i p i p sin(2 p ) cos(2 p ) 1 2 `30 apple F 2 p 2 sin 2 (2 p ) 1 2 `30 i p i p 1 A Nematic fluctuations are Landau damped (Oganesyan et al (2001) Dynamic critical exponent z=3 No Berry phase term at one loop order Breaking of time-reversal is in the Chern-Simons term of the gauge fields Effective (quadrupolar) couplings mix the nematic and the gauge field fluctuations 16

17 Beyond one-loop: Berry phase! S = R p, ij M i (p, )M j ( p, ), 2 3 `20 Upon integrating-out the gauge field fluctuations we find a Berry phase term in the nematic action The coefficient χ is not universal The Berry phase is subleading to the Landau damping terms At criticality and in the nematic phase it leads to a mixing of longitudinal (underdamped) and transverse (Landau damped) modes (non-universal) Wen-Zee term induced at two loop order 17

18 Nematic phase: transport anisotropy The conductivity can be computed form the current-current correlation function in the nematic state The longitudinal conductivity σxx is xx!(1+m 1 )/V (0) (1+M 1 cos(2 q ))i! qv (0)l 0 +!2 V (0) 2 q 2 q 2 /4 Deep in the nematic phase M1~ O(1) and the damping term in the denominator is very small for θq=π/2 Resonance peak in the spectrum at 100 MHz (~4mK) coupling to the weak lattice anisotropy gaps-out the overdamped Goldstone mode Sambandmurthy et al (PRB 2008) 18

19 The Geometry of FQH fluids How does the topological fluid respond to changes in the geometry of the 2D surface? How is the concept of the Hall viscosity (i.e. a non-dissipative viscosity) related to a response to a change in the geometry? How is the nematic order related to changes in the actual geometry? We will now see that these concepts are actually related to each other and to the concept of flux attachment 19

20 }~ Flux Attachment and Geometry time charge worldline flux Flux attachment of k fluxes (k even) requires a topological spin If e1 and e2 denote a local frame tangent to the worldline This frame couples to the actual spin connection of the 2D surface 2D surface frame As a result, in addition to coupling to the metric of the 2D surface, the covariant derivative of the particles must be changed to D µ µ + i A µ + k 2! µ 20

21 Full Effective Hydrodynamic Action for the ν=1/(2k+1) Laughlin States L =+ A 0 + " µ 2 b µ@ 2k +1 2 A 2k +1! 0 4 "µ b b 2k +1" µ 2 2 b µ@! " µ 48! µ@! Cho, You & EF, 2014 Gromov, Cho, You, Abanov & EF, 2015 The Laughlin FQH state at ν=1/(2k+1) is described by a U(1)2k+1 Chern-Simons theory Hall viscosity H = L! 0 = s 2 = 2k The fractional spin s is the coefficient of the Wen-Zee term The last term is the gravitational Chern-Simons term for the abelian spin connection and controls the thermal transport (Stone 2012) This result generalizes to all FQH states, abelian and non-abelian 21

22 Nematicity and Geometry The application of this framework to the nematic transition shows that The nematic fields mix with the frame fields of the background metric Upon integrating out the nematic fluctuations one finds the correct value of the Hall viscosity In the isotropic phase the Hall viscosity is universal In the nematic phase the Hall viscosity is modified if disclinations of the nematic order are present in the form of a quadrupolar tail 22

23 Topological Quantum Hall Fluids Fractional quantum Hall fluids: uniform topological fluids with a topologically protected Hall conductivity σ xy =ν e 2 /h, where ν=n e /N φ is the filling fraction of the Landau level incompressible fluids with a finite energy gap a ground state degeneracy m g ; m Z, g is the genus of the 2D surface Excitations: `quasiparticles with fractional charge, fractional statistics (Abelian and non-abelian), topological spin, and a quantum dimension there are m distinct quasiparticle types (m=degeneracy on a torus) have chiral edge states with universal properties Entanglement entropy: S vn =const. l- ln D+ (with D 2 = k d k2 ) 23