`Anomalous dimensions for conserved currents and Noether s Second Theorem

Transcription

1 `Anomalous dimensions for conserved currents and Noether s Second Theorem Thanks to: NSF, EFRC (DOE) Gabriele La Nave Kridsangaphong Limtragool

2 0 no order Mott insulator x

3 strange metal: experimental facts T-linear resistivity L xy = apple xy /T xy 6=#/ T (!) =C! 2 3 n e 2 m 1 1 i!

4 standard metals resistivity / T 2 Weidemann-Franz law apple xx = 2 T xx 3 optical conductivity < / 1/! 2

5 why is the problem hard?

6 single-parameter scaling / z 2 ln Z! A µ A µ 1 / T (2 d)/z (!, T) /! (d 2)/z 2/3 C v / T d/z anomalous dimension 2! 2d A

7 dimension of vector potential gauge invariance A µ!a µ µ G has no units [A µ ]= 1 L =1 vector potential cannot have an anomalous dimension conserved current

8 gauge invariance= current conservation Z S = d d x F 2 + J µ A µ + A! A Z S! S + d d xj integrate by parts Noether s Thm. µ J µ =0 current conservation

9 gauge invariance [A µ ]=1 current conservation [d d xja] =0 fixes dimension of current [J] =d 1

10 photon propagator ha i (k)a j ( k)i / ij k 2

11 strange metal explained! Hall Angle cot H xx xy T 2 T-linear resistivity Hall Lorenz ratio L xy = apple xy /T xy 6=#/ T all explained if [J µ ]=d + + z 1 Hartnoll/Karch [A µ ]=1 = 2/3

12 strange metal: strange E&M [J µ ]=d + + z 1 [A µ ]=1 = 2/3 [E] =1+z [B] =2 note r 2 B 6= flux ha i (k)a j ( k)i / ij k 2(1 ) non-local propagator

13 ha i (k)a j ( k)i / ij k 2(1 )

14 How is this possible - - if at all?

15 current algebra with an anomalous dimension? is there a direct experimental probe of this anomaly? how can A have an `anomalous dimension? what else can be explained with an anomalous dimension?

16 what is the new gauge principle? if [A µ ] 6= 1 A µ! A µ µ

17 µ J µ =0 current conservation what if [@ µ, Ŷ ]=0 new µ ŶJ µ µ J µ =0 [ J] =d 1 D Y

18 possible gauge transformations Z S = 1 d d xf 2 4 S = 1 2 Z d d k 2 d A µ(k)[k 2 µ M µ { k µ k ]A (k) k =0 zero eigenvector ik A µ! A µ µ

19 family of zero eigenvalues M µ fk =0 { generator of gauge symmetry 1.) rotational invariance 2.) A is still a 1-form 3.) [f,k µ ]=0

20 only choice f f(k 2 ) ( ) A µ! A µ +( ) ( 1) µ [A µ ]= what kind of E&M has such gauge transformations?

21 if [A µ ] 6= 1 but the current is conserved

22 extra dimensions

23 model with anomalous dimensions S = Z d d+2 x p g appler (@ µ ) 2 2 Z( ) 4 F 2 + V ( ) 8 < : Z( )!!1 Z 0 e V ( )!!1 V 0 e. e = r ±apple r F 2 Karch: Gouteraux:

24 claim [A] 6= 1 y =1 S = Z dv d dy y a F 2 + y =0 how?? F = da

25 if holography is RG then how can it lead to an anomalous dimension?

26 standard case A J Z d d xa J A(y = 0) = A bc does not satisfy A(y = 0) 6= A + d

27 alternatively (A + d = a + boundary theory has non-trivial gauge structure AdS/ Lifshitz Z dy/y = 1 large gauge transformation

28 construct boundary theory explicitly

29 S = Z dv d dy y a F 2 + eom d(y a? da) =0 y 6= 0 A! A what about the boundary?

30 Caffarelli-Silvestre extension theorem (2006) y g(x, y = 0) = f(x) xg + a y g y + g yy =0 r (y a rg(x, y)) = 0 lim y!0 y g? C d, ( ) f x g(z =0,x)=f(x) fractional Laplacian = 1 a 2

31 boundary is non-local! Z ( x) a f(x) =C d,a d (f(x) R d f( )) x d+2a Chang/Gonzalez P 2 ( ĝ) + 1 pseudo-differential operator

32 closer look r (y a ru) =0 scalar field (use CS theorem) d(y a? da) =0 holography similar equations generalize CS theorem to p-forms GL,PP: (CIMP)

33 generalize CS to p-forms y!0 ya i 1 i = C d, ( )! i1 i p fractional Maxwell equations at boundary!

34 boundary action: fractional Maxwell equations A? =0 A? = J boundary action has `anomalous dimension (non-locality)

35 if holography is RG then how can it lead to an anomalous dimension? Z S = dv d dy y a F 2 + [A] =1 a/2 dimension of A is fixed by the bulk theory: not really anomalous dimension

36 define F ij i A j A i d A = d 1 2 A, S = Z 1 4 F ijf ij integrate by parts S = Z 1 2 A i( ) 2 A i, non-local boundary action

37 fractional differential } (a 1)/2 d a = 1 2 (d(d d) (a 1)/2! +(dd ) (a 1)/2 d!) d =( 1) n(p+1)+1? d? dd : p (M)! p (M) does not change the order of the form

38 new gauge transformation A! A + d, d ( ) 1 2 d [A] = boundary does lies at infinity (large gauge transformation)

39 causality A µ =0 [, ] =0 share the same eigenfunctions e i ( ~ k ~x!t) =(k 2! 2 ) e i ( ~ k ~x!t)! = ck

40 from the bulk use CS theorem lim y!0 (x, y),y (x 0,y)] = [ (x, y, (x 0,y)] = (x x 0 )=0 no problem with causality

41 S = Z 1 2 A i( ) 2 A i, ha i (k)a j ( k)i / ij k 2

42 @ µ J µ =0 current conservation what if [@ µ, Ŷ ]=0

43 answer? µ, Ŷ ]=0 [d, ]=0 Ŷ = J! J [J] =d 1

44 current-current correlator ij k i k j C ij (k) / (k 2 ) k 2. standard Ward identity k i C ij (k) i C ij (k) =0 but k 1 k µ C µ µ ( ) 1 2 C µ =0 inherent ambiguity in E&M

45 family of zero eigenvalues M µ fk =0 most fundamental conservation µ ( r 2 ) ( 1)/2 J µ =0

46 Noether s Theorems A µ! A µ µ G +,

47 arxiv:

48 is this just a game? is there a consistent algebra for fractional currents?

49 Yes

50 Virasoro algebra L n := [L n,l m ]=(n m)l n+m + c 12 m(m2 1) m+n,0 Witt algebra central extension conformal transformations on unit disk V! W! 1

51 Fractional Virasoro algebra generators a a L a n = La n := z [L n,l m ](z ak )= (a(k + n) + 1) (a(k 1+n) + 1) (a(k + m) + 1) (a(k 1+m) + 1) L n+m (z ak ) =(A a n,m(k) L n+m )(z ak ) [L a m,l a n]=a m,n L a m+n + m,n h(n)cz a algebra for conformal nonlocal actions Z 2?(W a, H)/B 2?(W a, H)

52 experiments?

53 skin effect = r 2!µ

54 new result 1 2 r ~ B E ~ v = µ ~ J =1/k 2 = v 2! 1 2( +1) 1 sin 2( +1) + 2 n +1

55 ~B magnetic flux r 2 B should be dimensionless [B] =2 =2+2/3 6= 2 what s the resolution?

56 correct dimensionless quantity D i i e ~ a i fictitious gauge field a i [@ i,i i A i ]=@ i I i A i a µ! a µ µ A! A + d = e ~ I ~a(~r) d ~ l.

57 New Aharonov-Bohm Effect D = e p 2 ~ r2 BR (2 ) (1 2 )! ( ) ( ) sin F 1 (1, 2 ; 2; r2 R 2 )

58 is the correction large? =1+2/3 =5/3 R = eb`2 ~ L 5/3 /(0.43) 2 yes!

59 experiment Planckian dissipation = ~ k B T ybco, s

60 if in the strange metal [A µ ]=d A 6=1 1 2 r ~ B God said ~! E v 2 = µ ~ 1 2 r E ~ = 1 2 r ~ E ~ =0 1 2 r ~ B =0. Pippard Kernel Z J µ (x) = d d x 0 C µ ( x [J] 6= d 1 [A] 6= 1 x 0 )A fractional E&M in SC!! = ck