The Geometry of the Quantum Hall Effect

Transcription

1 The Geometry of the Quantum Hall Effect Dam Thanh Son (University of Chicago) Refs: Carlos Hoyos, DTS arxiv: DTS, M.Wingate cond-mat/

2 Plan Review of quantum Hall physics Summary of results Physical consequences Method: Nonrelativistic diffeomorphism Chern-Simons and Wen-Zee terms

3 Quantum Hall effect

4 Integer quantum Hall state simplest example: noninteracting electrons filling n Landau levels n=3 n=2 n=1 degeneracy B 2 Area The state is completely gapped

5 Fractional quantum Hall state n=3 n=2 n=1 Without interactions, ground state has huge degeneracy Interactions somehow lift the degeneracy, make system gapped at particular values of the filling factor = n B/2 = 1 3, 1 5,

6 High B limit in FQHE Theoretically interesting limit: e 2 r B m r = n 1/2 B 1/2 All interesting physics happens in the lowest Landau level Many properties captured by Laughlin s trial wave function (z) = Y hiji (z i z j ) 3 Y i e z i 2 /4`2

7 Effective field theory Effective field theory: captures low-energy dynamics What are the low-energy degrees of freedom of a quantum Hall state? there are none (in the bulk): energy gap Thus the effective Lagrangian is polynomial over external fields (we will consider only clean systems)

8 Chern-Simons action The orthodox point of view is that all universal information about transport is encoded in the Chern-Simons action: S = 4 d 3 x µ A µ A encodes Hall conductivity J µ = S A µ = 2 B J y = xy E x xy = 2 e 2

9 Universality beyond CS Higher-derivatives corrections: of dynamical, not topological nature, hence non universal? We will show that there is universality beyond the CS action

10 σxy(q) y v E E v x E x = E e iqx j y = xy (q)e x xy(q) = 2 apple 1+ S 4 1 (q`) 2 + O(q 4`4) S is the shift : topological property of a state S =1/ for Laughlin s state exact result, insensitive of interactions

11 Symmetries of NR theory Microscopic theory DTS, M.Wingate 2006 S 0 = dt d 2 x g i 2 D t g ij 2m D i D j D µ ( µ ia µ ) Gauge invariance: e i A µ A µ + µ General coordinate invariance: = k k L A 0 = k ka 0 L A 0 A i = k ka i A k i k g ij = k kg ij g kj i k L A i g ik j k L g ij Here ξ is time independent: ξ=ξ(x)

12 NR diffeomorphism These transformations can be generalized to be time-dependent: ξ=ξ(t,x) = L A 0 = L A 0 A k k A i = L A i mg k ik g ij = L g ij Galilean transformations: special case ξ i =v i t

13 Where does it come from Start with complex scalar field S = dx g (g µ µ + ) Take nonrelativistic limit: g µ = 1+ 2A 0 mc 2 A i mc A i mc g ij = e imcx0 2mc S = dt dx g i 2 t + A 0 g ij 2m ( i + ia i )( j ia j ).

14 Relativistic diffeomorphism x µ x µ + µ =0: gauge transform = e imcx0 2mc =i: general coordinate transformations

15 Interactions Interactions can be introduced that preserve nonrelativistic diffeomorphism interactions mediated by fields For example, Yukawa interactions S = S 0 + Z d 3 x p g Z d 3 x p g (g j + M 2 2 ) = k

16 Is CS action invariant?

17 Is CS action invariant? CS action is gauge invariant

18 Is CS action invariant? CS action is gauge invariant CS action is Galilean invariant

19 Is CS action invariant? CS action is gauge invariant CS action is Galilean invariant CS action is not diffeomorphism invariant

20 Is CS action invariant? CS action is gauge invariant CS action is Galilean invariant CS action is not diffeomorphism invariant S CS = 2 m dt d 2 x ij E i g jk k

21 Is CS action invariant? CS action is gauge invariant CS action is Galilean invariant CS action is not diffeomorphism invariant S CS = 2 m dt d 2 x ij E i g jk k A µ does not transform like a one-form

22 More on geometry System does not live in a 3D Riemann space 2D Riemann manifold at any time slice can parallel transport along equal-time slices, but not between different times

23 Velocity vector v t + t v t v i = [, v] i + i Can use v to transform objects from one time slice to another Different v s differ by a 2D vector: (ṽ i v i )= [, ṽ v] i

24 Connection with v one can define a connection: ĝ µ = µ = 1 2ĝ (@ µ g g g µ ) g ij g µ = v 2 v i v j g ij j 0i = 1 2 gjk (@ k v i v k +ġ ik ) j 00 = gjk v k + 1 kv 2

25 Improved gauge potentials With v one can construct a gauge potential that transforms as a one-form à i = A i + mv i à 0 = A 0 mv 2 2 à µ = k à µ à µ k B 1 à 2 à 1 Ẽ i i à 0 à i ij Ẽ j B transforms like v i

26 v as fluid velocity We can thus require v i = ij Ẽ j B This is a nonlinear equation for v v +(v r)v = E + v B hydrodynamic equation of an forced ideal fluid without pressure Can be solved by iteration: v i = ij E j B +

27 CS term corrected S = 4 Z d 3 x µ Ã Ã = S CS + 4 me 2 B + fixed coefficient Kohn s theorem but it is not the only Chern-Simons term

28 Spin connection e 2 e 1 Choose orthonormal basis at each point e a i (x) a =1, 2

29 The spin connection x+dx d parallel transporting from x to x+dx d =! i (x)dx i x! i = 1 2 ab e aj r i e b j

30 Spin connection: time component! 0 = 1 2 ab e ai r 0 e b i = 1 2 ab e t e b i i v j nonzero in flat space

31 Spin connection as U(1) gauge field Under local rotation of the vielbein (x)! µ!! µ curvature is flux of! = 1 2 gr! µ transforms like a (2+1)D one-form Two more CS terms µ Ã shift (Wen-Zee) µ!

32 Wen-Zee shift 2 µ A µ = 4 ga 0 R + Total particle number on a manifold: Q = d 2 x gj 0 = d 2 x g 2 B + 4 R = N + On a sphere: Q = (N + S), S = 2 IQH states: ν=n, κ=n 2 /2 Laughlin s states: ν=1/n, κ=1/2 shift

33 Shift for IQH states N +5 N +3 N +1 Q = nn + n 2 = n(n + n)

34 Back to flat space me2 B apple 2 Ã@! apple 4 B E rb ( B) (B) m B 00 (B)E rb

35 Back to flat space me2 B apple 2 Ã@! apple 4 B E rb ( B) (B) m B 00 (B)E rb q 2 correction to Hall conductivity ErB ra 0 rb q 2 A 0 B

36 σxy(q) y v E E v x E x = E e iqx j y = xy (q)e x xy(q) xy(0) =1+C 2(q ) 2 + O(q 4 4 ) C 2 = S 4 2 `2 ~! c B 2 00 (B)

37 Hall viscosity from WZ term S WZ = B 16 ij h ik t h jk + ht xy (T xx T yy )i i a! a = B 4 = 1 4 Sn derived by N.Read previously

38 What is Hall viscosity? Standard fluid dynamics: t + i j i =0 tj i + j T ij =0 continuity eq. Navier-Stokes eq. j i = v i T ij = v i v j + P ij V ij V ij = 1 2 ( iv j + j v i ) In 2 spatial dimensions, it is possible to write T ij = H( ik V kj + jk V ki ) Hall viscosity (Avron Seiler Zograf) breaks parity dissipationless

39 Hall viscosity in picture Hall shear stress

40 Physical interpretation First term: Hall viscosity y v E E v x xv y + y v x =0 T xx = T xx (x) =0 additional force Fx~ x Txx Hall effect: additional contribution to vy

41 Physical interpretation (II) 2nd term: more complicated interpretation Fluid has nonzero angular velocity (x) = 1 2 x v y = ce x(x) 2B B =2mc /e Coriolis=Lorentz Hall fluid is diamagnetic: d = MdB M is spatially dependent M=M(x) Extra contribution to current j = c ẑ M

42 Current ~ gradient of magnetization j = c ẑ M

43 High B limit In the limit of high magnetic field: ϵ(b) known: free fermions n Landau levels for IQH states first Landau level for FQH states with ν<1 Wen-Zee shift is known xy(q) xy(0) =1 3n 4 (q )2 + O(q 4 4 ) = n matches explicit calculations xy(q) xy(0) =1+2n 3 (q ) 2 + O(q 4 4 ), = 1 4 2n+1 exact

44 Conclusion Quantum Hall states live in a special type of geometric structure: global time, Symmetry determines the q^2 correction to Hall conductivity, related to Hall viscosity Open questions: edge states? Lowest Landau level: additional symmetries? AdS/CFT realizations? (g ij,v i )

45 Power counting

46 B gives rise to Power counting

47 B gives rise to Power counting magnetic length l ~ B-1/2

48 B gives rise to Power counting magnetic length l ~ B-1/2 Lamor frequency ω=b/m

49 B gives rise to Power counting magnetic length l ~ B-1/2 Lamor frequency ω=b/m Assume i~εb 1/2, 0~ε 2 B/m (z=2)

50 B gives rise to Power counting magnetic length l ~ B-1/2 Lamor frequency ω=b/m Assume i~εb 1/2, 0~ε 2 B/m (z=2) A i B 1/2, A 0 B m, g ij 1

51 B gives rise to Power counting magnetic length l ~ B-1/2 Lamor frequency ω=b/m Assume i~εb 1/2, 0~ε 2 B/m (z=2) B 1/2 B A i, A 0 m, g ij 1 Slowly varying, nonlinear external fields B B, A 0 µ, g ij 1

52 B gives rise to Power counting magnetic length l ~ B-1/2 Lamor frequency ω=b/m Assume i~εb 1/2, 0~ε 2 B/m (z=2) B 1/2 B A i, A 0 m, g ij 1 Slowly varying, nonlinear external fields B B, A 0 µ, g ij 1 E B 3/2 m