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Fundamental Journal of Modern Physics
ISSN: 2249-9768
Volume 20, Issue 1, 2023, Pages 25-35
This paper is available online at http://www.frdint.com/
Published online November 20, 2023
:esphras and Keywords high energy physics - theory.
Received April 26, 2023; Accepted May 24, 2023
© 2023 Fundamental Research and Development International
IDENTIFICATION OF UNIVERSAL FEATURES IN
THE CONDUCTIVITY OF CLASSES OF TWO-
DIMENSIONAL QFTs USING THE AdS/CFT
CORRESPONDENCE
MATTHEW JAMES STEPHENSON
Stanford University
353 Jane Stanford Way
Stanford, CA 94305
USA
e-mail: matthewjstephenson@icloud.com
Abstract
We study the electrical conductivity of strongly disordered, strongly
coupled quantum field theories, holographically dual to non-
perturbatively disordered uncharged black holes. The computation
reduces to solving a diffusive hydrostatic equation for an emergent
horizon fluid. We demonstrate that a large class of theories in two
spatial dimensions have a universal conductivity independent of
disorder strength, and rigorously rule out disorder-driven conductor-
insulator transitions in many theories. We present a (fine-tuned) axion-
dilaton bulk theory which realizes the conductor-insulator transition,
interpreted as a classical percolation transition in the horizon fluid. We
address aspects of strongly disordered holography that can and cannot
MATTHEW JAMES STEPHENSON
26
be addressed via mean-field modeling, such as massive gravity.
1. Introduction
We examined electrical transport in strongly coupled holographic
quantum field theories at zero charge density, constructing perfect metals
amidst disorder. Our findings have implications for realistic models of
disordered strange metals.
2. Conductivity
Consider a static, asymptotically anti-de Sitter space with a black
hole horizon sourced entirely by uncharged bulk matter and a dynamical
metric. We can choose the bulk metric using diffeomorphism invariance.
[ ]
,
2222 ji
ij
dxdxGQdtPdrLds +−=
(1)
,i j indices represent the spatial boundary directions, while ,M
N
represent all dimensions, and
L
is AdS radius. All functions in the metric
are functions of
r
and
.
x
We further choose bulk coordinate
,
0
∞
<
<
r
with
0
=
r
black hole horizon, and
∞
=
r
AdS boundary. Uncharged
matter not required, energy conditions obeyed.
We add a
(
)
1U
gauge field to the bulk, so the action of our theory is
.
4
2
uncharged
2
−−=
∫
+
F
Z
gxdS
d
L
(2)
Function
Z
is a parameter of (uncharged) scalar matter, but for our
purposes it is an arbitrary function of
r
and
.
x
Gauge field’s two-point
functions correspond to current-current correlation functions in the
boundary theory, including electrical conductivity matrix
.
ij
σ
The
conductivity may be related, via membrane paradigm [1], to data on the
horizon of the black hole alone. The expected value of the boundary
IDENTIFICATION OF UNIVERSAL FEATURES IN …
27
current is given by
[
( )
]
,α∂+γγ=σ=
jj
ij
j
iji
EZEJ
E
(3)
where
j
E
is the applied electric field,
[
]
...
E
denotes a uniform spatial
average,
(
)
0==γ rG
ijij
is the induced metric on the horizon, and
α
is
the unique function which obeys equation
( ( ))
α∂+γγ∂=
jj
ij
i
EZ0
(4)
with appropriate boundary conditions (for example, periodicity in
compact boundary spatial directions). The membrane paradigm was used
in holographic systems in [2], and similar computations appear in [3, 4, 5]
for black holes with translational symmetry broken only in one direction.
These results are special cases of this general formula. This formula may
break down if black hole horizon fragments and becomes disconnected, as
was considered in [6, 7].
We can interpret (4) as a hydrostatic equation enforcing local charge
conservation in an emergent horizon fluid. This is subtle - the local
“electric current” in (4) is not the same as the expected value of the local
current in the dual theory; only their spatial averages are equal. A
powerful set of techniques have been developed to understand the
qualitative behavior of transport in such fluids [8]; for example, it
immediately follows from (4) that
.
ij
ij
σ=σ
In particular, if
[
]
ij
Zγσ ;
is the conductivity matrix with given
Z
and
:
ij
γ
(
[ ]
)
.
1
;
1
det;det
8
e
Z
Z
ijij
=
γσγσ
(5)
If we set ,1=
Z
(5) gives
MATTHEW JAMES STEPHENSON
28
( )
.
1
det
4
e
=σ
(6)
If we expect that on average for a disordered sample, the conductivity
matrix is isotropic
( )
,
ij
ij
σδ=σ
that fixes conductivity to be
,1
2
e=σ
exactly the clean result!
A simple way to understand this result: suppose that in local
coordinates, the metric is given by
.
2222
dyadxadxdx
yx
ji
ij
+≈γ
(7)
Then we expect “locally”
xyxx
aa~σ
and
yxyy
aa~σ
[9]. On
average
y
a
and
x
a
should have identical distributions, so we expect that
xx
σ
and
xx
σ1
have the same distributions. This implies
;1
2
e=σ
analogous statements are known for random resistor lattices in
2=d
with analogous (e.g., log-normal) resistance distributions. And more
generally, if
Z
log is symmetrically distributed about
,
0
then in an
isotropic theory,
2
1e=σ
follows from (5) in the thermodynamic limit.
The robustness of
σ
in these strongly disordered
2=d
models is
remarkable, and deserves further comments. In models where
momentum dissipation is introduced through massive gravity [10] or “Q-
lattice” axions [11], one finds the hydrodynamic result [12]
,
2
P
Q
+ε
τ
+σ=σ Q
(8)
where
Q
is charge density,
ε
energy density,
P
pressure,
Q
σ
dissipative “quantum critical” conductivity without disorder, and
τ
a
“momentum relaxation time”, inversely related to graviton mass. Before
now, it was unclear whether the fact that (8) holds beyond the
hydrodynamic limit was an unrealistic feature of massive gravity or
IDENTIFICATION OF UNIVERSAL FEATURES IN …
29
similar theories. Our work confirms this is a sensible prediction of
massive gravity for many systems at
.
0
=
Q
(8) further implies another
mechanism,
,
0
→
τ
by which the conductivity can reach its lower bound,
.
Q
σ
The conductivity saturating this lower bound, at least qualitatively,
is likely to occur at strong disorder [8]. Confirmation that strongly-
disordered charged holographic models (with
)
1=Z
have a conductivity
no smaller than
2
1e
in
2=d
would be a further non-trivial test of
predictions of simple mean-field physics.
In
2≠d
and/or if
Z
is distributed more generically, it is valuable to
employ insight gained from equivalence between Markov chains on
lattices and resistance of a resistor lattice [13]. For arbitrary ,
Z
this
analogy can be leveraged to find lower and upper bounds to
,
σ
for a self-
averaging disordered sample: [8]
[ ]
.
2
2
1
2
2
d
Z
e
L
dZ
e
L
ii
d
ii
d
γγ
≤σ≤
γ
γ
−
−
−
E
E
(9)
It is straightforward to test these results and bounds by numerically
solving (4) for various disorder realizations. Good agreement with our
exact analytic results and consistency with our bounds is obtained.
3. Conductor-Insulator Transition
(9) constrains
σ
to deviate from the clean result by the strength of
fluctuations in
Z
and
.
ij
γ
It is evident from (9) that if
ij
γ
and
Z
are
finite at all points on the horizon, then the black hole necessarily
conducts electrical current, no matter how strong the disorder. This is a
remarkable result. In contrast, in non-interacting quantum field theory, a
conductor-insulator transition occurs at a finite disorder strength [14] in
,2>
d
and at arbitrarily small disorder in
2≤d
[15]. This transition
relates to the destructive interference of matter waves scattering off of
MATTHEW JAMES STEPHENSON
30
the disorder. Apparently, bulk fluctuations of the gauge field in
holographic theories do not suffer from such interference. While it is
known [16, 17] that metal-insulator transitions occur at a finite disorder
strength in an interacting quantum system, even such systems ultimately
succumb to (many-body) localization at strong disorder. Perhaps
holographic models have taken the “coupling
∞
→
” limit first, rendering
such a transition impossible.
Realizing a holographic conductor-insulator transition takes more
care. A “helical lattice” approach has generated such a transition in [18,
19], but there is no satisfying physical interpretation. However, even in
these papers, the conductivity in the insulating phase only decays as
algebraically in
T
as ,0→
T
in contrast to canonical insulators.
Assuming
2=d
and a probe limit with AdS-Schwarzschild geometry,
we need a large
[
]
Z1
E
for
,
0
=
C
requiring percolating
0→Z
bubbles
across the horizon. When these finite-
Z
regions disconnect, charge
transport is halted, causing a disorder-driven holographic metal-insulator
transition, similar to random resistor lattices [20].
Numerically compute conductivity for
Z
ansatz with ”bubbles” where
0→Z
percolate across horizon to test proposal. Numerics support this;
see Figures 1 and 2.
3.1. Holographic realizations
We now ask whether the percolation mechanism proposed above for a
disorder-driven metal-insulator transition can occur in a “realistic”
holographic model: a bottom-up Einstein-Maxwell-dilaton
(
)
Φ
-axion
(
)
α
theory with action
( ) ( )
22
2
22
1
16
2α∂+
Φ∂−
π
Λ−
−=
Φ−
+
∫
e
G
R
gxdS
dd
M
(
)
(
)
(
)
.
4
2
22
α
−
Φ+α
−F
e
Z
L
UV
(10)
IDENTIFICATION OF UNIVERSAL FEATURES IN …
31
Here
M
is a mass scale, whose precise value is unimportant - we choose
it so that
Φ
is strictly dimensionless, for simplicity, and
(
)
.
2
1
2
L
dd
+
−=Λ
(11)
At ,0→
G
generalizing choices yields similar results, but (10) with axio-
dilaton scalar kinetic terms is essential.
(
)
αZ
’s cosine potentials may suit
our needs, and arise due to instanton effects in effective actions (as in
QCD). In our holographic model,
(
)
αZ
is not suppressed by
d
LG
(the
scale of bulk’s quantum corrections).
Figure 1.
(
)
σdet
from a black hole horizon for a theory in ;2=
d
we set
,
1
=
e
and use periodic boundary conditions with
,, π≤yx
with a
discretized spatial grid of
2
701
points. We take
ijij
δ=γ
and exp=
Z
[
(
)
]
,21 ZBZ +−
where
( ( ( )
++φ−=
∑
=
2sinexp
2
1
xZ
jx
N
j
( ))
2sin
2
y
jy
+φ
)
,2
2
ξ
with
jx
φ
and
jy
φ
independent random phases, and
0>B
is a
random constant. We took various values of
B
and fixed
.70120π=ξ
When
[ ]
,28.0
*
~
ZZ ≡
>
E
curves at different
B
approximately collapse,
implying that current avoids the non-conducting bubbles; when
[ ]
,
*
~
ZZ <
E
the value of conductivity is sensitive to
.B
In the limit
∞→B
and
,
0
→
ξ
a metal-insulator transition appears at
.
*
Z
MATTHEW JAMES STEPHENSON
32
Figure 2.
Surface plots of
(
)
yxZ ,
for various bubble densities.
Depending on whether regions of high or low
Z
percolate across the
horizon determines whether we are in the metallic or insulating phase, as
is clear upon comparing with Figure 1.
For conductor-insulator transitions,
(
)
αV
must have at least two
minima,
c
α
and
,
i
α
with
(
)
0>α
c
Z
and
(
)
.0=α
i
Z
α
drives the
transition and
Φ
stabilizes it, although theories with finite Lifshitz or
hyperscaling-violating exponents may also work [21]. Insulators form
when bubbles of
i
α=α
percolate across the horizon; we aim to
demonstrate how to create and maintain these bubbles at low
temperatures. [21] is a citation. For this purpose, a simple choice of
potentials, though certainly not the only, is
( )
,
2
43
2
7
2
2
2
2
λΦ−λΦ−
λ
+λ−Φλ−
λ
=Φ eeU
(12a)
( )
,
2
2
0
4
2
α
α
+α−=αV
(12b)
( )
.1
2
0
α
α
−=αZ
(12c)
Using
Z
in [22], we set
,0
0
→α
,
2
>
λ
and
(
)
αV
marginal to avoid
axion backreaction on the dilaton. The Harris criterion [21] implies
IDENTIFICATION OF UNIVERSAL FEATURES IN …
33
inability to source disordered modes of all wavelengths without UV
geometry backreaction.
Let us begin by sourcing the dilaton with (positive)
δ
-like sources on
the AdS boundary - analogous to point-like impurities in the dual theory.
Each impurity produces an expanding bubble which becomes insulating;
width of the “bubbles” of
α
is
.1~ T
If density of the impurities is
,
n
then the bubbles percolate across the horizon when
.
~
nT
<
Within each
bubble,
,
0
α→α
and thus at low temperatures we obtain an insulator.
A second mechanism for obtaining the transition is as follows:
suppose
.
α
As
0→T
in the insulating phase, we predict:
.
8
exp~
2
ζ
λ
−σ
λ
T
T
(13)
4. Outlook
Recent models [23, 24, 25, 26] propose momentum non-conservation
in (quasi-2d) strange metals. We constructed perfect conductors in strong
disorder and predict finite charge density will not decrease conductivity.
We encourage extending holographic approach to charged black holes and
finding non-holographic field theories with disorder-resistant
.
Q
σ
Acknowledgements
We thank Ed Witten for discussions. We especially thank Veronica
Toro Arana and Anna Maria Wojtyra for providing some code for solving
elliptic partial differential equations. This research was funded through
Nvidia.
MATTHEW JAMES STEPHENSON
34
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