non-local actions and anomalous dimensions: application to ... · (x,z) polchinski: 1010.6134 ⇡ z...
TRANSCRIPT
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Non-Local Actions and Anomalous Dimensions: Application to the Strange Metal
thanks to
NSF
Gabriele La Nave
arxiv:1605.07525/
Kridsangaphong Limtragool
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strange metal
probe by Aharonov-Bohm effect on underdoped cuprates
fractional not d-1
[J ] = �
[Aµ] = dA 6= 1
non-local action
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dimensions of current
�(x) ! �(x) + ��(x)
[J0(x),�(y)] = ��(x)�d(x� y)
[J0] = d
r · Jµ = @tJ0
⇠⌧ ⇡ ⇠
[Jµ] = d
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can the dimension of the current change?
r · Jµ = @tJ0
⇠⌧ / ⇠z
d ! d� ✓
hyperscaling violation
Sachdev, Kiritsis
[Jµ] = d� ✓ + z � 1
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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?
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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
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are there Lifshitz-type solutions that modify the scaling of A that
have finite boundary actions? ⇣ 6= 0
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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
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Jµ / � lnZ
�Aµ
dimensions of J and A are related
physics beyond HSV and Lifshitz geometries
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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!
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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
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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
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what else can be explained with an anomalous dimension for A and J?
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n⌧e2
m
1
1� i!⌧
Drude conductivity
tan�2/�1 =p3
✓ = 60�
�(!) = C!��2ei⇡(1��/2)
� = 1.35
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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)
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⇢(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
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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�
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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
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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
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experiments require and anomalous
dimension for the gauge field
no HSV (non-dilaton physics)
change underlying gauge transformation
Aµ ! Aµ + @↵µG
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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?
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hyperbolic spacetime
local CFT (operator locality)
1-1 state correspondence
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any theory with gravity has less observables
than a theory without it!
how can a local CFT
emerge at the boundary?
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quantum gravity boundary local QFT=?is
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UV
QFTIR
(@µ�)2 +m2�2
standard holography
S = S(gµ⌫ , Aµ,�, · · · )
operatorsO
Zd
4x�0O
AdS=CFT claim: heRSd �0OiCFT = ZS(�0)
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can be determined exactly in some cases?
O
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O = CO limz!0
z
���(x, z) Polchinski: 1010.6134
� ⇡ z�
[O] = �
�0
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smearing function
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construct
exactly
O
consistent with Polchinski prescription
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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
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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
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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��
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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
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x
limz!0+
za@g
@z
z
Cd,� (�r)�f
g(z = 0, x) = f(x)
� =1� a
2
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solves CS extension problem
� :=
pd2 + 4m2
2
g = z��d/2�
� solves massive scalar problem
the for massive scalar field
O = (��)��0O
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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)
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O = (��)��0
AdS-CFT correspondence but operators are
non-local !!
heRSd �0OiCFT = ZS(�0)
|x� x
0|�d�2�
2-point correlator
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(�r)�
is not conformal if spacetime is
curved
Why should the boundary be conformal?
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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
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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
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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’
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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 '
⌘
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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
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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
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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
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d+1 gravity
d-dimensional non-local
`QFT’
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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
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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)
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y = y0
y = ✏
f(y0) = f(✏) = 0
D3-branesf is a harmonic
function
requires absolute-value singularity
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|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�
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limR/L!1
m2e�2⇡R/L ! 0
non-locality vanishes
� =1
2
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B
y = y0
y = ✏
m2 = � 1
↵0 + (ln ✏)2/(2⇡↵0)2
non-locality vanishes
| ln ✏| > 2⇡p↵0
positive mass
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Branes in Type IIB string theory
eliminate non-local boundary interactions
geodesic incompleteness
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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
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ds
2 = �e
�2|y|/Lgµ⌫dx
µdx
⌫ + dy
2
singularity
what if?
boundary locality
�⇢µ⌫ ill-defined (GI)
+ non-compactness
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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
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flux through ring
�� =e
~
I~A · d`
�� =eB⇡r2
~
Aharonov-Bohm Effect must change
Stokes’ theoremI
~A · d` =Z
SB · d~S
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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
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compute AB phase
�� =e
~
I~a(r0) · ~d`0
use fractional calculus
��R =eB`2
~
✓b↵�1d↵�1
�2(↵)
◆c � l, d � l
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��D =e
~⇡r2BR2↵�2
p⇡21�↵�(2� ↵)�(1� ↵
2 )
�(↵)�( 32 � ↵2 )
sin2⇡↵
22F1(1� ↵, 2� ↵; 2;
r2
R2)
!
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is the correction large?
��R =eB`2
~ L�5/3/(0.43)2
yes!
↵ = 1 + 2/3 = 5/3
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sum rules
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sum rules
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optical gap
Ne↵(⌦) =2mVcell
⇡e2
Z ⌦
0�(!)d!
Ne↵ / x
}x
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Uchida, et al. Cooper, et al.
x
low-energy model for Ne↵ > x??
Ne↵(⌦) =2mVcell
⇡e2
Z ⌦
0�(!)d!
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f-sum rule
K.E. = p2/2m
Ne↵ = x
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what if?
K.E. / (@2µ)
↵
f-sum rule
↵ < 1W>n if
W (n, T )
⇡ce2= An1+ 2(↵�1)
d + · · ·
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Type IIB String theory
AdS5 ⇥ S5
N D3-branes
local CFTOj =
X
jklm
Jjklm�k�l�m
SYK model
geodesic complete
non-local theories
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combine AC+DC transport
fixes all exponents a,b,c
[J ] = dU
probe with fractional Aharonov-Bohm effect
boundary non-local action