Disordered holographic systems: Marginal relevance of
imperfection
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Adams, Allan, and Sho Yaida. “Disordered Holographic Systems:
Marginal Relevance of Imperfection.” Phys. Rev. D 90, no. 4
(August 2014). © 2014 American Physical Society
As Published
http://dx.doi.org/10.1103/PhysRevD.90.046007
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PHYSICAL REVIEW D 90, 046007 (2014)
Disordered holographic systems: Marginal relevance of imperfection
Allan Adams and Sho Yaida
Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
(Received 25 July 2014; published 25 August 2014)
We continue our study of quenched disorder in holographic systems, focusing on the effects of mild
electric disorder. By studying the renormalization group evolution of the disorder distribution at subleading
order in perturbations away from the clean fixed point, we show that electric disorder is marginally relevant
in (2 þ 1)-dimensional holographic conformal field theories.
DOI: 10.1103/PhysRevD.90.046007
PACS numbers: 11.25.Tq, 71.23.-k
The standard lore on disorder holds that, in the absence
of an external magnetic field, all states are localized in d ≤
2 þ 1 spacetime dimensions, regardless of the strength of
the disorder. This was codified in a beautiful scaling
argument by the gang of four [1]. For Fermi liquids, the
scaling argument can be substantiated by explicit calculations of the beta-function for the conductance [2].
However, it was shown that the spin-orbit coupling, for
example, can lead to an important sign change of the betafunction [3], invalidating the naive scaling argument for
localization. When confronted with strongly correlated
systems which cannot be described in terms of weakly
interacting quasiparticles, such calculations are generally
lacking and it is thus completely unclear whether systems
flow toward localization or not.
An interesting test case for localization in strongly
correlated systems is disorder in holographic quantum field
theories [4]. Despite having no quasiparticle descriptions,
these models are computationally tractable thanks to a dual
description via classical gravity in one higher dimension.
The extra dimension encodes the renormalization group
scale of the field theory. Quenched disorder can be
implemented by imposing disordered boundary conditions
for the classical fields in the emergent spacetime. Tracing
the renormalization group flow of the disorder distribution
then reduces to solving the classical equations of motion of
the dual gravitational system subject to disordered boundary conditions and studying how the effective distribution
varies along the extra dimension. This strategy was used in
[5] to show that the classical Harris criterion holds. In
particular, Gaussian quenched electric disorder is marginal
in two spatial dimensions at leading order in perturbations
around the clean fixed point.
In this paper we show that quenched electric disorder in
our holographic conformal field theory (CFT) is in fact
marginally relevant in two spatial dimensions. This is
demonstrated by computing subleading corrections to the
bulk energy-momentum tensor and the moments of the
disorder distribution; resumming the resulting logarithms
yields marginal relevance. Our discussion will rely heavily
1550-7998=2014=90(4)=046007(5)
on the formalism developed in [5], to which we refer the
reader for details and further references.
As in [5], we focus on a d-dimensional strongly
correlated CFT which is holographically dual to (d þ 1)dimensional classical Einstein-Maxwell theory, whose
equations of motion are
pffiffiffiffiffi
1
pffiffiffiffiffi ∂ Q ½ jgjgQP gMN FPN ¼ 0;
jgj
ð1Þ
1
dðd − 1Þ
RMN − RgMN −
gMN
2
2L2
8πG
1
¼ 2 N FMP FN P − FPQ FPQ gMN ≡ 8πGN T MN :
4
gdþ1
ð2Þ
The dimensionless constants
Ld−1
GN
≡ N 2c and
Ld−3
g2dþ1
≡ N 2f are
determined by the parameters of the boundary CFT. We
take a large-N c limit to ensure classicality of the bulk
theory, but keep N f ∼ N c to bring the role of gravitational
backreaction to the fore. Recalling that the chemical
potential of the CFT is encoded by the boundary value
of the electric potential in the bulk, we see that quenched
electric disorder can be implemented by fixing the boundary value of A0 ðz; x0 ; xÞ to take a time-independent
random value, VðxÞ, dictated by a suitable distribution
PV ½WðxÞ over functions WðxÞ [5].
In the limit of mild disorder, we can expand our bulk
fields in perturbations around the clean background,
ð0Þ
ð2Þ
ð4Þ
gMN ¼ gMN þ gMN þ gMN þ ;
ð1Þ
ð3Þ
ð3Þ
AM ¼ AM þ AM þ :
ð0Þ
ð4Þ
ð1Þ
Here, gMN is the unperturbed background metric, AM is the
gauge field sourced by electric disorder potential, VðxÞ, in
ð2nÞ
ð2nþ1Þ
the ultraviolet, and gMN and AM
for n ≥ 1 are generated
by higher-order backreaction. We fix gauge by putting the
metric in the form,
046007-1
© 2014 American Physical Society
ALLAN ADAMS AND SHO YAIDA
gMN dxM dxN ¼
PHYSICAL REVIEW D 90, 046007 (2014)
d−1
L2 2 X
dz
þ
gμν dxμ dxν
z2
μ;ν¼0
−jkjz 2
ð5Þ
ð2nÞ
and setting Az ¼ 0. In particular, gzM ¼ 0 for n ≥ 1.
We must also specify boundary conditions in the infrared
and ultraviolet. We require all field to be regular at the
infrared horizon. In the ultraviolet, we may consider a
simple model of quenched electric disorder [5],
limAμ ¼ δ0μ V ∞ ðxÞ;
ð6Þ
z→0
with ðe
Þ jz¼Λ1 correctly reproducing the soft cutoff
stipulated above.
At second order, Að1Þ generates an inhomogeneous
energy-momentum tensor
ð2Þ
8πGN T MN ðk; zÞ ¼ −
gμν jz¼Λ1 ¼ L2 Λ2 δμν
ð8Þ
and
Aμ jz¼Λ1 ¼ δ0μ V Λ ðxÞ;
ð9Þ
PV Λ ½WðxÞ ¼ N 0♯ e
1
2f Λ
dis
−
R dd−1 k
2
X
L
ð0Þ
M
N
2
2
i 2
gMN dx dx ¼ 2 dz þ dτ þ
ðdx Þ :
z
i¼1
2
ð15Þ
tττ ðk1 ; k2 Þ ¼jk1 jjk2 j − k1 · k2 ;
ð16Þ
and
−ðk1 Þi ðk2 Þj − ðk2 Þi ðk1 Þj :
ð2Þ
2
ð17Þ
ð2Þ
This in turn activates gμν ðk; zÞ ≡ Lz2 ϕμν ðk; zÞ through the
Einstein equations. There are two cases to be dealt with
separately: k ¼ 0 and k ≠ 0.
For k ¼ 0, the only relevant ingredient for our
later calculations turns out to be the disorder-averaged
metric. Solving the homogeneous Einstein equation
ð2Þ
½8πGN T zz d:a: ¼0,
ð2Þ
3fΛ G
½8πGN T ττ d:a: ¼− 2g2disL2 zN2 ,
4
and
yields
ð2Þ
½ϕττ d:a:
Λ
f dis GN
log ðΛzÞ
¼−
g24 L2
ð2Þ
½ϕij d:a:
δij f Λdis GN
log ðΛzÞ:
¼þ
2
g24 L2
and
ð18Þ
ð19Þ
The overall sign in front is crucial for marginal relevance of
the quenched electric disorder.
For k ≠ 0, solving the inhomogeneous linearized
Einstein equation [7], we obtain ϕτi ¼ 0,
ð2Þ
ϕττ ðk; zÞ ¼ −Bðk; zÞ − Cðk; zÞ;
ð11Þ
ð2Þ
ϕij ðk; zÞ
ð1Þ
At first order, the Maxwell equations give Ai ¼ 0 and
ð1Þ
tzi ðk1 ; k2 Þ ¼ −ijk1 jðk2 Þi − ijk2 jðk1 Þi ;
tij ðk1 ; k2 Þ ¼ δij ð−jk1 jjk2 j þ k1 · k2 Þ
2jkj
Aτ ðk; zÞ ¼ ð−iÞV Λ∞ ðkÞe−jkjz ;
ð14Þ
ð10Þ
Note that the e Λ softly cuts off high-momentum modes. As
will be clear shortly [cf. Eqs. (11) and (12)], for d ¼ 2 þ 1
at zero temperature, this is tantamount to choosing
ð1Þ
the condition (6) with f dis ¼ f Λdis for Aμ and setting
ð2nÞ
ð2nþ1Þ
jz¼Λ1 ¼ 0 for n ≥ 1.
gμν jz¼Λ1 ¼ 0 and Aμ
We now solve the equations of motion order by order,
focusing on the marginal case, d ¼ 2 þ 1. Working at zero
temperature and in Euclidean time, τ ¼ þix0 , the leadingorder background is pure anti–de Sitter space,
ð13Þ
tzz ðk1 ; k2 Þ ¼ jk1 jjk2 j þ k1 · k2 ;
4
:
0
with
3f Λ G
ð2Þ
½8πGN T ij d:a: ¼ 4g2disL2 zN2 δij
2jkj
Λ
WðkÞWð−kÞe
ð2πÞd−1
d2 k0 Λ
V ∞ ðk − k0 ÞV Λ∞ ðk0 Þ
4π 2
0
with
with the disorder distribution functional
Z
× tMN ðk − k0 ; k0 Þe−ðjk−k jþjk jÞz
where V ∞ ðxÞ is governed by the Gaussian disorder distribution functional
R d−1
− 1
d xWðxÞ2
PV ∞ ½WðxÞ ¼ N ♯ e 2fdis
R dd−1 k
− 1
WðkÞWð−kÞ
¼ N ♯ e 2fdis ð2πÞd−1
:
ð7Þ
However, this simple choice leads to a divergence in the
disorder-averaged metric for d ≥ 2 þ 1 [5]. To regulate this
divergence we instead impose the following Dirichlet
boundary condition on the hypersurface at z ¼ Λ1 ,
4πGN z2
g24 L2
ð12Þ
where V Λ∞ is governed by the functional (7) with
f dis ¼ f Λdis . We see that it is regular at the Poincaré horizon
(z → ∞) and is governed by the functional (10) at z ¼ Λ1 ,
046007-2
and
ð20Þ
ðkÞi ðkÞj
Bðk; zÞ
¼ −δij þ 3
k2
ðkÞi ðkÞj
þ δij −
Cðk; zÞ
k2
ðkÞi ðkÞj
Dðk; zÞ
þ
k2
− iðkÞi Ej ðk; zÞ − iðkÞj Ei ðk; zÞ
ð21Þ
DISORDERED HOLOGRAPHIC SYSTEMS: MARGINAL …
with
z
iðkÞi
ð2Þ
0
dz0
8πG
T
ðk;
z
Þ
;
Bðk; zÞ ¼ −
N zi
1
k2
Λ
Z z
Z ∞
z02
G ðjkjz00 Þ
dz0 2
dz00 2 002
Cðk; zÞ ¼ −G2 ðjkjzÞ
0
1
z
G2 ðjkjz Þ z0
Λ
ðkÞi ðkÞj ð2Þ
ð2Þ
ð2Þ
ðk; z00 Þ;
× 8πGN T ττ − T ii þ
T ij
k2
2
z
1
1
ð2Þ
Dðk; zÞ ¼ − þ 2 8πGN T zz k;
Λ
2 2Λ
Z z0
Z z
ð2Þ
8πGN T zz ðk; z00 Þ
dz0 z0
dz00
; and
−2
1
1
z″
Λ
Λ
Z z ðkÞi ðkÞj
ð2Þ
0 2
0
8πGN T zj ðk; z Þ ;
dz 2 δij −
Ei ðk; zÞ ¼
1
k
k2
Λ
Z
where we defined G2 ðyÞ ≡ ð1 þ yÞe−y .
ð3Þ
Finally, at third order, Ai ¼ 0 and, solving
ð3Þ
½∂ 2z − k2 Aτ ðk; zÞ ¼ Sð3Þ ðk; zÞ
ð22Þ
with
1
ð2Þ
ð2Þ
ð1Þ
Sð3Þ ≡ f∂ z ðϕττ − ϕjj Þgð∂ z Aτ Þ
2
1
ð2Þ
ð2Þ
ð1Þ
þ f∂ i ðϕττ − ϕjj Þgð∂ i Aτ Þ
2
ð2Þ
ð1Þ
þ ∂ i fϕij ð∂ j Aτ Þg;
−jkjz
Z
¼ −e
z
1
Λ
0 2jkjz0
dz e
Z
z0
∞
ð23Þ
00
dz00 e−jkjz Sð3Þ ðk; z00 Þ
ð24Þ
which is regular at z ¼ ∞ and vanishes at z ¼ Λ1 .
To diagnose the relevance of the quenched electric
disorder, we define
I dis ðzÞ ≡ ½Aμ ðx; zÞAμ ðx; zÞd:a: ;
Z
¼
d2 k
4π 2
Z
d2 k0
½Aμ ðk; zÞAμ ðk0 ; zÞd:a: ;
4π 2
ð25Þ
ð2Þ
Z
d2 k Λ z2
fΛ
f
ð−iÞ2 e−2jkjz ¼ − dis2 ;
2 dis 2
4π
L
8πL
ð4Þ
z2 ð1Þ
ð3Þ
½Aτ ðx; zÞAτ ðx; zÞd:a:
L2
z2 ð1Þ
ð2Þ
ð1Þ
− 2 ½Aτ ðx; zÞϕττ ðx; zÞAτ ðx; zÞd:a:
L
f Λdis f Λdis GN
× I~ ð4Þ ðΛzÞ:
¼−
8πL2 g24 L2
I dis ðzÞ ¼ 2
∂ ~ ð4Þ
3
log ðΛzÞ
I j½g½AA ¼ þ þ O
:
∂Λ
2
Λz
ð29Þ
ð2Þ
The other two contractions involve one factor of V Λ∞ in gμν
ð1Þ
ð2Þ
contracting with one of the Aμ and the other V Λ∞ in gμν
ð1Þ
contracting with the other Aμ . It turns out that neither
contributes a logarithmic term, as we show explicitly in the
Appendix. All in all we find
ð4Þ
I dis ðzÞ ¼ −
f Λdis f Λdis GN
3
×
þ
log ðΛzÞ þ c0 þ ;
2
8πL2 g24 L2
ð30Þ
where c0 is a constant of order 1 and dots indicate the terms
ð2Þ
which asymptote to zero for z ≫ Λ1. We see that I dis ðzÞ þ
ð4Þ
I dis ðzÞ grows toward infrared. In other words, the disorder
is marginally relevant, with an effective disorder strength
Λ
Λ 2
f eff
dis ðzÞ ¼ f dis þ ðf dis Þ
ð27Þ
ð28Þ
A mathematical trick to extract the logarithmic term, the
sign of which tells relevancy apart from irrelevancy, is to
∂ ð4Þ
I dis and send Λz → ∞, while pretending that V Λ∞
take Λ ∂Λ
are Λ independent.
∂ ð4Þ
In evaluating Λ ∂Λ
I dis , there are three different ways
Λ
of contracting four V ∞ ’s involved at this order. First, we
ð2Þ
may disorder-average gμν and then disorder-average the
ð1Þ
remaining two factors of Aμ [8]. Both averages are
straightforward, leading to a contribution of the form
ð26Þ
where we used the fact that disorder-averaged quantities are
translationally invariant. This characterizes the intensity of
the disorder as a function of energy scale 1z.
At the leading order
I dis ðzÞ ¼
the constancy of which reflects the marginality of the
disorder at this order. The same calculation for d ≠ 2 þ 1
reproduces the Harris criterion, which has also been
confirmed for the holographic CFT by computing disorder
corrections to thermodynamic quantities [5].
At the next order we have
Λ
we get
ð3Þ
Aτ
PHYSICAL REVIEW D 90, 046007 (2014)
3 Nf
log ðΛzÞ:
2 Nc
ð31Þ
We can check this result by studying the disorderaveraged bulk energy-momentum tensor, whose growth
would indicate large corrections to the geometry in the
infrared. A similar, if more involved, analysis shows that
we again receive a logarithmic contribution only from
ð2Þ
contractions involving ½gμν d:a: . The results are [9]
046007-3
ALLAN ADAMS AND SHO YAIDA
½8πGN T ð4Þz z d:a:
½8πGN T ð4Þτ τ d:a:
½8πGN T ð4Þi j d:a:
−9 f Λ GN 2
¼ 2 dis
½c1 þ ;
4L
g24 L2
−9 f Λdis GN 2
¼ 2
½log ðΛzÞ þ c2 þ ;
4L
g24 L2
þ9δi j f Λdis GN 2
¼
½log ðΛzÞ þ c3 þ :
8L2
g24 L2
The ci ’s are constants of order 1 satisfying c1 þ c2 −
c3 ¼ 0. The absence of the logarithmic term in
½8πGN T ð4Þz z d:a: is consistent with ½8πGN T ð4Þz z d:a: ¼ 0,
while comparison with ½8πGN T ð2Þμ ν d:a: once again reveals
the “golden number,” þ 32 NNcf , which adds confidence to the
claim that we are looking at right quantities to diagnose
relevance of the disorder.
We conclude that quenched electric disorder is marginally relevant around the clean fixed point of our (2 þ 1)dimensional holographic CFT. Together with the Harris
criterion found at leading order in [5], we see that the
standard lore is consistent with our results for mild electric
disorder in the holographic setting: quenched electric
disorder is irrelevant for d > 2 þ 1, relevant for
d < 2 þ 1, and marginally relevant for d ¼ 2 þ 1.
Given this close parallel with simple weakly-coupled
disordered electronic systems around their clean fixed
points [2,10], for which the standard lore holds, it is
natural to wonder whether localization is universal in
holographic CFTs with strong electric disorder in arbitrary
spacetime dimension. If so, there must be an unstable
disordered critical point governing a metal-insulator
transition in holographic CFTs in d > 2 þ 1 spacetime
dimensions, with the insulating phase dual to a strongly
inhomogeneous black hole horizon [11]. We leave this
question for future work.
We thank Steven A. Kivelson and Stephen H. Shenker
for helpful discussions and Omid Saremi for collaboration
on related questions. The research of A. A. is supported by
the DOE under Contract No. DE-FC02-94ER40818. S. Y.
is supported by a JSPS Postdoctoral Fellowship for
Research Abroad.
APPENDIX
Let us explicitly work out the contribution to the
∂ ~ ð4Þ
cross-contracted piece Λ ∂Λ
I jcross coming from Bðk; zÞ.
2
ð1Þ
ð2Þ
ð1Þ
∂
ϕττ ÞAτ d:a: , we receive
From − Lz 2 ½Aτ ðΛ ∂Λ
Z 2 Z 2
ðf Λdis Þ2 8πGN z2
d k1 d k2
−
2
2 2
3
L
g4 L Λ
4π 2
4π 2
−ðjk1 jþjk2 jÞðzþΛ1 Þ ðjk1 jjk2 j þ k1 · k2 Þ
e
ðjk1 j þ jk2 jÞ :
ðk1 þ k2 Þ2
PHYSICAL REVIEW D 90, 046007 (2014)
After rescaling to k~ i ≡ zki , we see that this term is of order
1
OðΛ31z3 Þ. Note that there is no singularity from ðk þk
2,
1
2Þ
thanks to the numerator ðjk1 jjk2 j þ k1 · k2 Þ.
ð3Þ
As for terms involving Aτ , note that
Z ∞
∂ ð3Þ
0
−jkjzþjkjΛ2 1
Aτ ¼ −e
dz0 e−jkjz Sð3Þ ðk; z0 Þ
Λ
1
∂Λ
Λ Λ
Z z
0
− e−jkjz
dz0 e2jkjz
Z
×
z0
1
Λ
∞
00
dz00 e−jkjz Λ
∂Sð3Þ
ðk; z00 Þ:
∂Λ
∂ ~ ð4Þ
The first term makes a contribution to Λ ∂Λ
I of the
form
jk1 jjk2 j þ k1 · k2 z2 −2jk1 jz−2jk2 j 1
Λ
e
Λ
ðk1 þ k2 Þ2
jk2 j sB ðk1 ; k2 Þ
2
2
1
;
þ
×
þ
þ
4
4ka
Λ2 ka Λ k2a
−8
π2
Z
d2 k1 d2 k2
where we defined ka ≡ jk1 j þ jk2 j and sB ðk1 ; k2 Þ≡
Þ·k1 gfðk1 þk2 Þ·k2 g
−jk2 j2 − 2k1 · k2 þ 3fðk1 þk2ðk
. This time,
þk Þ2
̬
̬
1
2
rescaling to k1 ≡ zk1 and k2 ≡ kΛ2 , one can argue that it
ð3Þ
1
Þ. The other term coming from Aτ
is of order OðΛz
contributes
8
π2
2 jk1 jjk2 j þ k1 · k2
z
sB ðk1 ; k2 Þ
2
ðk1 þ k2 Þ
Λ3
−ðjk jþjk jÞðzþ 1 Þ
1 Λ − e−2jk1 jz−2jk2 jΛ Þ
ðe 1 2
×
:
jk2 j − jk1 j
Z
d2 k1 d2 k2
Note that there is no singularity from jk1 j → jk2 j as
1
1
e−ðjk1 jþjk2 jÞðzþΛÞ and e−2jk1 jz−2jk2 jΛ approach each other.
1
Then, the term involving e−ðjk1 jþjk2 jÞðzþΛÞ , upon rescaling
to k~ i , gives the contribution of order OðΛ31z3 Þ, while the term
1
̬
involving e−2jk1 jz−2jk2 jΛ , upon rescaling to ki , gives the
1
contribution of order OðΛz
Þ.
The cross-contracted terms involving Bðk; zÞ thus do not
contribute any logarithms. Similar logic applies to the
Dðk; zÞ and Ei ðk; zÞ contributions. For Cðk; zÞ it turns out
to be easier to simply show that setting Λ1 ¼ 0 yields a finite
contribution to I~ ð4Þ , which in particular implies the absence
of any divergent logarithms.
046007-4
DISORDERED HOLOGRAPHIC SYSTEMS: MARGINAL …
PHYSICAL REVIEW D 90, 046007 (2014)
[1] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V.
Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[2] For a review, see P. A. Lee and T. V. Ramakrishnan, Rev.
Mod. Phys. 57, 287 (1985).
[3] S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor. Phys.
63, 707 (1980).
[4] For reviews with a view toward condensed matter
physics, see S. A. Hartnoll, Classical Quantum Gravity
26, 224002 (2009); C. P. Herzog, J. Phys. A 42, 343001
(2009); J. McGreevy, Adv. High Energy Phys. 2010,
723105 (2010).
[5] A. Adams and S. Yaida, arXiv:1102.2892.
[6] S. A. Hartnoll and J. E. Santos, Phys. Rev. Lett. 112, 231601
(2014).
ð2Þ
[7] One way to arrive at gMN ðkÞ for k ≠ 0 is to first work in a
ð2Þ
ð2Þ
coordinate system where gzz ¼ 0, gμμ ¼ 0, and
[9] A quick path to get ½T ð4Þz z d:a: is to use T M M ¼ 0 for d ¼
2 þ 1 and the energy-momentum conservation ∇M T M z ¼ 0,
together with the regularity at the horizon. Incidentally, the
same logic implies that ½T ð2Þz z d:a: ¼ 0.
[10] One may be concerned with the fact that the Uð1Þ symmetry
of the holographic CFT, whose current is coupled to the
random potential, is global on the field theory side.
Although we do not fully justify it, this may be a red
herring: at least, for electronic systems, one can often
proceed without explicitly taking Uð1Þ gauge field into
account. For holographic systems, the literature on the issue
of gauging Uð1Þ symmetry is surprisingly scarce and it calls
for careful development.
[11] The approach developed herein has recently been pushed
further in a paper [6], where authors attempted to resum
perturbative series in order to access randomly disordered
fixed points. While it is not clear that the resummation is
reliable, or indeed whether the Fefferman-Graham expansion is valid, the result is suggestive.
ð2Þ
ðkÞ ðkÞ
ðkÞi gil ðδlj − kl 2 j Þ ¼ 0 and then transform back to the
radial coordinate.
ð2Þ
ð1Þ
[8] Note that Að3Þ is essentially a product of gμν and Aμ .
046007-5