`Anomalous dimensions for conserved currents from holographic dilatonic models to superconductivity. Gabriele La Nave. Thanks to: NSF, EFRC (DOE)

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1 `Anomalous dimensions for conserved currents from holographic dilatonic models to superconductivity Thanks to: NSF, EFRC (DOE) Gabriele La Nave Kridsangaphong Limtragool

2 Pippard s problem J s 6= c 4 2 A London Eq. failure of local London relations

3 inside superconductor r A =0 stable equilibrium Z 1 L m = L m0 2 around minimum C µ (x, x 0 )(A µ µ (x)) (A (x 0 (x 0 ))d 3 x 0 d 3 x +

4 Pippard Current J µ (x) = L m A µ = Z C µ (x, x 0 )(A (x 0 (x 0 ))d 3 x 0 Pippard kernel J s = 3 4 c 0 Z (~r ~r 0 )((~r ~r 0 ) ~A(~r 0 ))e (~r ~r0 )/ (`) (~r ~r 0 ) 4 d 3 ~r 0 non-local

5 Units of Current J µ (x) = L m A µ = Z C µ (x, x 0 )(A (x 0 (x 0 ))d 3 x 0 [J] =d d C d A anomalous dimension Standard Result Z d 3 xj µ A µ no anomalous dimension [J] =d d A = d 1 A µ! A µ µ J µ =0

6 Are there other examples of currents with anomalous dimensions? underlying electricity and magnetism? is symmetry breaking necessary?

7 ! 2/3 T Mott insulator 0 x

8 why is the problem hard?

9 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

10 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

11 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

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

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

14 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

15 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

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

17 gauge-gravity duality (Maldacena, 1997) gravity IR?? UV QF T coupling constant g =1/ego dg(e) dlne = (g(e)) locality in energy

18 model with anomalous dimensions claim [A] 6= 1 y =1 S = Z dv d dy y a F 2 + y =0 how?? F = da

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

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

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

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

23 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

24 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)

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

26 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

27 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

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

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

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

31 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

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

33 Noether s Theorems A µ! A µ µ G +,

34 is there a hidden broken symmetry?

35 1 2 (@ µ ) 2 + m 2 2 O = C O lim z!0 z (x, z) solves massive scalar problem g = z d/2 solves CS extension problem O =( ) 0

36 application: gauge fields with anomalous dimensions F µ F µ + m 2 A 2 y A? µ ( ) A?µ = p d 2 + m 2 1/2 dynamical `Higgs mode additional length scale

37 m IR J UV non-local E&M broken symmetry in higher dimension

38 experiments?

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

40 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.

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

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

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

44 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 U(1)! Z 2