Non-Local Actions and Anomalous Dimensions: Application to the Strange Metal

thanks to

NSF

Gabriele La Nave

arxiv:1605.07525/

Kridsangaphong Limtragool

strange metal

probe by Aharonov-Bohm effect on underdoped cuprates

fractional not d-1

[J ] = �

[Aµ] = dA 6= 1

non-local action

dimensions of current

�(x) ! �(x) + ��(x)

[J0(x),�(y)] = ��(x)�d(x� y)

[J0] = d

r · Jµ = @tJ0

⇠⌧ ⇡ ⇠

[Jµ] = d

can the dimension of the current change?

r · Jµ = @tJ0

⇠⌧ / ⇠z

d ! d� ✓

hyperscaling violation

Sachdev, Kiritsis

[Jµ] = d� ✓ + z � 1

Jµ = @⌫Z(�)Fµ⌫

dimension changes

consistent actions

S =

Zd

dx

p�g

⇢R� 1

2(@µ�)

2 + V (�)� 1

4Z(�)Fµ⌫

Fµ⌫ � W (�)

2A

2

�

At = Qr⇣�z

✓ 6= 0, z > 1, ⇣ 6= 0

W 6= 0

boundary terms?

are the boundary terms finite?

ra(Z(�)F ab)�W (�)Ab = 0

S

bound

= �1

2

Zd

3

x

p��Z(�)nµA⌫F

µ⌫

p��nr =

p�g = l�✓Lr2✓�z�3,

Z(�) = Z0

⇣rl

⌘4�✓,

AtFrtgrrgtt =

2(z � 1)

Z0r�3l2z+5L�2

S

bound

⇡ 2(z � 1)l2zL�1

Zd

3

xr

2✓�z�2

typically divergent J. Tarrio, W=0

are there Lifshitz-type solutions that modify the scaling of A that

have finite boundary actions? ⇣ 6= 0

units of gauge field

Aµ ! Aµ + @µG

has no units

[Aµ] =1

L= 1

dimensions of current can change without any change in the dimension of A

Jµ / � lnZ

�Aµ

dimensions of J and A are related

physics beyond HSV and Lifshitz geometries

how can the scaling of J and A be linked directly?

gauge symmetry

r↵ · Jµ = @tJ0

Aµ ! Aµ + @↵µG

both J and A have anomalous dimensions!

why would A have an anomalous dimension?

Hartnoll/Karch

[j] = d� ✓ + �+ z � 1

[jQ] = d� ✓ + 2z � 1

[A] = 1� �

[B] = 2� �

strange metal

� = �2/3✓ = 0

T-linear resistivity

Lxy

= xy

/T�xy

6= # / T

strange metal: experimental facts

cot ✓H ⌘ �xx

�xy⇡ T 2

all explained if

� = �2/3

Hall Angle

Hall Lorenz ratio

what else can be explained with an anomalous dimension for A and J?

n⌧e2

m

1

1� i!⌧

Drude conductivity

tan�2/�1 =p3

✓ = 60�

�(!) = C!��2ei⇡(1��/2)

� = 1.35

take experiments seriously

�i(!) =nie2i ⌧imi

1

1� i!⌧i

�(!) =

MZ

0

⇢(m)e2(m)⌧(m)

m

1

1� i!⌧(m)dm

continuous mass(scale invariance)

⇢(m) = ⇢0ma�1

Ma

e(m) = e0mb

M b

⌧(m) = ⌧0mc

M c

variable masses for everything

perform integral

�(!) =⇢0e20⌧0

Ma+2b+c

MZ

0

dmma+2b+c�2

1� i!⌧0mc

Mc

=⇢0e

20

cM

1

!(!⌧0)a+2b�1

c

!⌧0Z

0

dx

x

a+2b�1c

1� ix

Karch, 2015

a+ 2b� 1

c= �1

3

�(!) =⇢0e

20⌧

130

M

1

!

23

!⌧0Z

0

dx

x

� 13

1� ix

!⌧0 ! 1

�(!) =1

3(p3 + 3i)⇡

⇢0e20⌧130

M!23

tan� =p3

60�

momentum loss

anomalous dimension

hyperscaling violation⇢(m) = ⇢0

ma�1

Ma

e(m) = e0mb

M b

⌧(m) = ⌧0mc

M c

are anomalous dimensions necessary

a+ 2b� 1

c= �1

3

c = 1

a+ 2b = 2/3

b = 0

a = 2/3

No

but the Lorenz ratio is not a constant

LH =xy

T�xy⇠ T ⌘ T�2�/z

Hartnoll/Karch

� = bz = �2/3

⇢ / T�2�/z

experiments require and anomalous

dimension for the gauge field

no HSV (non-dilaton physics)

change underlying gauge transformation

Aµ ! Aµ + @↵µG

Aµ ! Aµ + @↵µG

(��)�f(x) = Cd,s

Z

Rd

f(x)� f(⇠)

|x� ⇠|d+2�d⇠

Reisz fractional Laplacian

non-local: f must be known everywhere

what theories have such non-local interactions?

hyperbolic spacetime

local CFT (operator locality)

1-1 state correspondence

any theory with gravity has less observables

than a theory without it!

how can a local CFT

emerge at the boundary?

quantum gravity boundary local QFT=?is

UV

QFTIR

(@µ�)2 +m2�2

standard holography

S = S(gµ⌫ , Aµ,�, · · · )

operatorsO

Zd

4x�0O

AdS=CFT claim: heRSd �0OiCFT = ZS(�0)

can be determined exactly in some cases?

O

O = CO limz!0

z

���(x, z) Polchinski: 1010.6134

� ⇡ z�

[O] = �

�0

smearing function

construct

exactly

O

consistent with Polchinski prescription

redo Witten’s massive scalar field calculation explicitly

S� =1

2

Zdd+1u

pg�|r�|2 +m2�2

�

to establish correspondence

}

dVg

heRSd �0OiCFT = ZS(�0)

Reisz fractional Laplacian(�r)��0

S� =1

2

Zdd+1u

pg�|r�|2 +m2�2

�

integrate by parts

S� =1

2

ZdVg

���@2

µ�+m2�2 + �@µ��

{���� s(d� s)� = 0 ��� = riri�equations

of motionm2 = �s(d� s) s =

d

2+

1

2

pd2 + 4m2

bound m2 � d2/4 BF boundm2 � �d2/4

solutions � = Fzd�s +Gzs, F,G 2 C1(H),

F = �0 +O(z2), G = g0 +O(z2)

restriction�0 = lim

z!0� boundary of AdS_{d+1}

Z

z>✏dVg�@µ�

S� =1

2

ZdVg

���@2

µ�+m2�2 + �@µ��

restriction

finite part from integration by parts

use Caffarelli-Silvestre extension theorem

(2006)

g(x, 0) = f(x)

�x

g +a

z

@

z

g + @

2z

g = 0

limz!0+

za@g

@z= Cd,� (�r)�f

� =1� a

2non-local

pf

Z

z>✏

�|@�|2 � s(d� s)�2

�dVg = �d

Z

z=0�0 g0

x

limz!0+

za@g

@z

z

Cd,� (�r)�f

g(z = 0, x) = f(x)

� =1� a

2

solves CS extension problem

� :=

pd2 + 4m2

2

g = z��d/2�

� solves massive scalar problem

the for massive scalar field

O = (��)��0O

O = CO limz!0

z

���(x, z)

consistency with Polchinski

limz!0+

za@g

@z= Cd,� (�r)�f

� =1� a

2

use Caffarelli/Silvestre

= CO limz!0

z

��+1@z�(x, z)

O = (��)��0

AdS-CFT correspondence but operators are

non-local !!

heRSd �0OiCFT = ZS(�0)

|x� x

0|�d�2�

2-point correlator

(�r)�

is not conformal if spacetime is

curved

Why should the boundary be conformal?

AdS metric: Euclidean signature

ds

2 =dz

2 +P

i dx2i

z

2

x

z

}

what is the length of this segment?

�s =

Z z0

0

dz

z= ln(z0/0) = 1

metric at boundary is not well defined

z

2ds

2 = dz

2 +X

i

dx

2i solves problem

works for any real wds2 ! e2wds2

boundary can only be specified conformally

bulk conformality

S = Sgr[g] + Smatter(�)

Smatter =

Z

Md

d+1x

pgLm

Lm := |@�|2 +✓m2 +

d� 1

4dR(g)

◆�2

scalar curvature

conformal sector

}`conformal mass’

on Riemannian (M,g) manifold of dimension

N=d+1

Lg = ��g +N � 2

4(N � 1)Rg = ��g +

d� 1

4dRg

conformal Laplacian

conformal changeAw(') = e�bwA(eaw')

g = y2 g

Lg(') = yd+32 Lg

⇣y�

d�12 '

⌘

RgH = �d(d+ 1)

LgH = ��gH � d2�14

Lg = ��g +N � 2

4(N � 1)Rg = ��g +

d� 1

4dRg

hyperbolic metric

m2 � d2�14 = �s(d� s)

s =d

2+

p4m2 + 1

2

stability independent of dimensionality

m2 > �1/4

Chang/Gonzalez 1003.0398 P� 2 (��g)

� + ��1

pseudo-differential operator

in general Pk = (��)

k+ lower order terms

P1 = ��+d� 1

4(d� 1)Rg

Panietz operator

P�f = d�S�d2 + �

�= d� h

scattering problem

fractional conformal Laplacian

pf

Z

y>✏[|@�|2 �

✓s(d� s) +

d� 1

4dR(g)

◆�2]dVg = �d

Z

@XdVhf P� [g

+, g]f

d+1 gravity

d-dimensional non-local

`QFT’

What about Maldacena conjecture?

S =

Zd

10x

p�g

✓e

�2�(R+ 4|r�|2)� 2e2↵�

(D � 2)F

2

◆

Type IIB String `action’

extremal solution

ds2L = H�1/2(r) ⌘µ⌫dxµdx⌫ +H1/2(r) �mndx

mdxn

D = 7

H = 1 +L4

r4, L4 = 4⇡gN↵02, r2 = �mnx

mxn

D3-braneshorizon at r=0

AdS

ds

2 = f

�1/2⌘µ⌫dx

µdx

⌫ + f

1/2�mndx

mdx

n

more generally

R3,1 ⇥K6

�f = (2⇡)4 ↵02g ⇢

density of D3-branes

N�(r)

y = y0

y = ✏

f(y0) = f(✏) = 0

D3-branesf is a harmonic

function

requires absolute-value singularity

|y| singular metrics (GI)

Randall-Sundrum

y 2 [�⇡R,⇡R]

ds

2 = �e

�2|y|/Lgµ⌫dx

µdx

⌫ + dy

2

Zd

4x

p�g

⇣g

µ⌫@µ�@⌫ �+m

2e

�2⇡R/L�

2⌘,

massive-particle action at Brane at ⇡R

� = e�⇡R/L�

limR/L!1

m2e�2⇡R/L ! 0

non-locality vanishes

� =1

2

B

y = y0

y = ✏

m2 = � 1

↵0 + (ln ✏)2/(2⇡↵0)2

non-locality vanishes

| ln ✏| > 2⇡p↵0

positive mass

Branes in Type IIB string theory

eliminate non-local boundary interactions

geodesic incompleteness

ds

2H =

1

y

2(dx2 + dy

2) H2

�x

xy

= �x

yx

= �1

y

�y

xx

= ��y

yy

=1

y{non-zero

Christoffel symbols

geodesics

x

ycover all spacetime

ds

2 = �e

�2|y|/Lgµ⌫dx

µdx

⌫ + dy

2

singularity

what if?

boundary locality

�⇢µ⌫ ill-defined (GI)

+ non-compactness

physical consequences of anomalous dimension for Aµ

Aµ ! Aµ + @↵µG

~r↵ ⇥ ~A = ~B

no Stokes’ theorem

I~A · d` 6=

Z

SB · d~S

↵Fµ⌫ = @↵µµ ↵⌫A⌫ � @↵⌫

⌫ ↵µAµ

flux through ring

�� =e

~

I~A · d`

�� =eB⇡r2

~

Aharonov-Bohm Effect must change

Stokes’ theoremI

~A · d` =Z

SB · d~S

ai ⌘ [@i, I↵i Ai] = @iI

↵i Ai

aµ ! aµ + @µ⇤

� ~22m

(@i � ie

~ai)2 = i~@t .

physical gauge connection

Aµ ! Aµ + @↵µ⇤

compute AB phase

compute AB phase

�� =e

~

I~a(r0) · ~d`0

use fractional calculus

��R =eB`2

~

✓b↵�1d↵�1

�2(↵)

◆c � l, d � l

��D =e

~⇡r2BR2↵�2

p⇡21�↵�(2� ↵)�(1� ↵

2 )

�(↵)�( 32 � ↵2 )

sin2⇡↵

22F1(1� ↵, 2� ↵; 2;

r2

R2)

!

is the correction large?

��R =eB`2

~ L�5/3/(0.43)2

yes!

↵ = 1 + 2/3 = 5/3

sum rules

sum rules

optical gap

Ne↵(⌦) =2mVcell

⇡e2

Z ⌦

0�(!)d!

Ne↵ / x

}x

Uchida, et al. Cooper, et al.

x

low-energy model for Ne↵ > x??

Ne↵(⌦) =2mVcell

⇡e2

Z ⌦

0�(!)d!

f-sum rule

K.E. = p2/2m

Ne↵ = x

what if?

K.E. / (@2µ)

↵

f-sum rule

↵ < 1W>n if

W (n, T )

⇡ce2= An1+ 2(↵�1)

d + · · ·

Type IIB String theory

AdS5 ⇥ S5

N D3-branes

local CFTOj =

X

jklm

Jjklm�k�l�m

SYK model

geodesic complete

non-local theories

combine AC+DC transport

fixes all exponents a,b,c

[J ] = dU

probe with fractional Aharonov-Bohm effect

boundary non-local action