Entanglement Entropy of Randomly Disordered System
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Entanglement Entropy of Randomly Disordered System
arXiv:1803.01064v1 [hep-th] 2 Mar 2018
Rajesh Narayanana,b∗ , Chanyong Parka,c,d† , and Yun-Long Zhanga‡
a
Asia Pacific Center for Theoretical Physics, Pohang 790-784, Korea
b
Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
c
Department of Physics, Postech, Pohang 790-784, Korea
d
Department of Physics and Photon Science, Gwangju Institute of Science and Technology,
Gwangju 61005, Korea
ABSTRACT
We investigate the entanglement entropy of a two-dimensional disordered system holograph-
ically. The disorder deforms a two-dimensional conformal field theory defined at a UV fixed
point to a Lifshitz field theory in an IR region. After decomposing the entanglement entropy
into short and long range correlations, we investigate how the entanglement entropy of the
disordered system evolves from UV to IR along the renormalization group flow. The long
range correlation interestingly leads to a universal critical exponent in the IR regime, which
is only very weakly dependent on the strength of the disorder. We also investigate a possible
phase transition and long range correlation between two subsystems by studying the mutual
information.
∗
rnarayanan@iitm.ac.in
†
cyong21@gist.ac.kr
‡
yunlong.zhang@apctp.org1 Introduction
It is now well known that the Entanglement Entropy (EE) or more specifically the von Neu-
mann entropy can be used as a measure to understand the properties of quantum many particle
systems: For instance, the EE can be used as a very good indicator of Quantum Phase Transi-
tions (QPT) in quantum systems [1–6]. Furthermore, they have been also used to characterize
topological order in many body systems [7, 8]. However, apart from a few notable exceptions
that include free fermions [9, 10], higher dimensional conformal field theories and a class of
quantum critical points where the dynamical exponent z = 1, the EE entropy has been no-
toriously difficult to calculate. One of the biggest barriers to making further progress in this
field is the difficulty in calculating the entanglement entropy of an interacting quantum field
theory (QFT). Thus, progress was stymied as interactions play role in many experimentally
relevant situations. However, further progress was achieved by realizing that the AdS/CFT
correspondence can shed light on the entanglement entropy of an interacting quantum system.
In particular, Ryu and Takayanagi proposed that the entanglement entropy of an interacting
QFT can be easily evaluated from a minimal surface area defined in the dual geometry [11–14].
In this work, following their suggestion, we will investigate the entanglement entropy of a CFT
deformed by a disordered source and analyze its flows to an IR fixed point.
The holographic entanglement entropy is useful to understand quantum aspects of an in-
teracting QFT from its dual geometry. Recently, it has been shown that the positivity of the
relative entropy leads to the entanglement entropy bound, and the thermodynamics-like law
appears when the entanglement entropy bound is saturated [15, 16]. The thermodynamics-like
law of the entanglement entropy allows us to reconstruct the AdS dual geometry from data of
the boundary conformal field theory (CFT) [17–21]. The holographic study on the entangle-
ment entropy is also useful to understand the Zamolodchikov’s c-theorem of higher dimensional
theories [22, 23]. In the holographic entanglement entropy contexts, it has been proven that
the a-type anomaly of an even dimensional theory monotonically decreases along the RG flow
due to the unitarity and strong subadditivity of the entanglement entropy [24–26]. In addition,
it has been further generalized to an odd dimensional theory where there is no a-type anomaly.
The F-theorem has been conjectured as an odd dimensional c-theorem in which the free energy
of an odd dimensional QFT decreases monotonically along the RG flow [25, 27].
Recently, the dual geometry of a CFT including the effects of disorder has been constructed
[28–34]. Its asymptotic geometry is given by an AdS space, whereas the geometry corresponding
to the IR regime approaches a Lifshitz geometry. Due to the specific scaling symmetry of the
Lifshitz geometry, its dual QFT is believed to become a Lifshitz field theory (LFT) which has
also the same scale symmetry (see eg.[35–42]).
This development of the dual geometry for a disordered CFT is particularly pertinent in
1the study of quantum many body systems. This is due to the fact that quenched or “frozen-
in” disorder is an ever present hazard in realistic quantum materials. This quenched disorder
can severely impact the physics near quantum critical points [43]. For instance, it has long
been known that disorder is a relevant operator if the Harris criterion [44] is violated. In
other words, under the violation of the Harris criterion [44] disorder correlations will drive
the renormalization group flow away from the clean critical point. Furthermore, disorder in
quantum systems triggers a host of very exotic effects like quantum Griffiths effects [45, 46], and
Infinite disorder fixed points wherein the scaling is of the activated kind [47, 48]. In some cases
disorder effects can be so stark so as to result in the ultimate destruction or smearing of the
transition itself [43, 49–53]. One of the most successful ways of theoretically capturing these
interesting quantum disorder effects is via a real space based renormalization group scheme
called the Strong Disorder Renormalization Group (SDRG) scheme. The scheme was first
introduced by Ma, Dasgupta and Hu [54, 55] and later refined by Fisher [47, 48]. This scheme
was lately adapted by Refael and Moore to calculate the entanglement entropy in the case of
the disordered spin chains [56]. They found that even in the case of the disordered spin chains
the entanglement entropy in 1-d still scales logarithmically with system size albeit with an
“effective” central charge that controls this logarithmic divergence which is different from its
clean counterpart. The SDRG scheme elucidated above has a particularly glaring weakness that
it cannot be well-controlled in d > 1 and thus cannot be used to easily unravel entanglement
properties associated with QPT higher dimensional disordered systems. It is in this context the
development of the dual geometrical description [28–34] of the disordered field theory provides
an elegant solution.
This paper looks at the flow of the entanglement entropy as we traverse from the ultra-
violet (UV) CFT to the infra-red (IR) LFT. Its dual gravity and geometric solution were
found in [28] in which a disordered source was represented as a massive bulk scalar field. On
this known dual geometry, we will study the effect of the randomly disordered source on the
UV entanglement entropy and investigate how the random disordered source modifies the UV
entanglement entropy to the IR one described by the LFT. We will also look into the mutual
information between two subsystems when the theory evolves from CFT to LFT.
The rest of the paper is organized as follows: In Section 2, we study several interesting and
important properties of a scale invariant theory through the holographic entanglement entropy
formula. In Section 3, we take account of a two-dimensional CFT deformed by a disordered
source. Intriguingly, its dual geometry allows a Lifshitz IR fixed point where the Lifshitz scale
symmetry appears. On this background, after investigating the long range quantum correlation
described by the entanglement entropy, we look into its critical behavior represented by a critical
exponent. In Section 4, we further investigate how the disordered source affects the mutual
2information in UV and IR regions. In Section 5, we finish this work with some concluding
remarks.
2 Entanglement entropy of a scale invariant theory
Let us reiterate several important features of a scale invariant QFT and its dual geometry: The
most well-known scale invariant theory is a conformal field theory (CFT). Another scale invari-
ant theory with reduced symmetry is a Lifshitz field theory (LFT), which is generally classified
by the dynamical critical exponent denoted by z. In particular, the conformal symmetry is
restored at z=1. A LFT shows a specific dispersion relation with ω ∼ pz . The same scale sym-
metry can be realized in a Lifshitz geometry. For example, the metric of a three-dimensional
Lifshitz geometry is given by
dt2 dx2 du2
ds2 = − 2z + 2 + 2 , (1)
u u u
where u indicates a radial coordinate. The above metric is invariant under the scale transfor-
mation:
u → λu, t → λz t and x → λx, (2)
From the point of view of symmetry, this Lifshitz geometry can be considered as the dual of
a two-dimensional LFT. In the AdS/CFT context, the radial coordinate of the AdS space is
identified with the energy scale probing the dual CFT. This identification is useful to connect
the RG flow of the condensed matter system to the dual geometry [57–60]. Especially, the
u → 0 limit corresponds to the UV region. The identification of the radial coordinate of the
Lifshitz geometry with the RG scale of the dual LFT entails that the Lifshitz geometry appears
in the IR regime after deforming an AdS space with a specific relevant operator, as will be seen.
Now, let us discuss the holographic entanglement entropy of such scale invariant theories.
As explained before, the scale invariant theories have the well defined dual geometries. In such
dual geometries the entanglement entropy can be holographically calculated by applying the
Ryu-Takayanagi formula. In the Lifshitz geometry the holographic entanglement entropy is
expressed as
1√
Z l/2
1
SE = dx 1 + u02 , (3)
4G −l/2 u
where l denotes the size of a subsystem and the prime means a derivative with respect to x.
Note that this formula accounts for the ground state entanglement entropy. Another point we
should note is that the above entanglement entropy does not depend on the dynamical critical
exponent. This is because we do not consider the time evolution of the entanglement entropy.
When time is fixed, the minimal surface representing the holographic entanglement entropy
lives in a spatial section of the dual geometry and the resulting entanglement entropy becomes
3independent of the time component of the metric. Since the dual geometries of a CFT and
a LFT are distinguishable only in the time direction, their entanglement entropies have no
distinction without considering the time evolution of the entanglement entropy. On the other
hand, if we take into account the time evolution of the entanglement entropy, we must exploit
the covariant formalism instead of the Ryu-Takayanagi formula. Since the covariant formulation
usually includes information about the time component of the metric, it can distinguish a LFT
from a CFT. In this work, we focus only on the time independent case where the scaling
behavior of the holographic entanglement entropy is governed by the scale transformation of
spatial coordinates, u → λu and x → λx. One can easily see that the above holographic
entanglement entropy is invariant under this spatial scale transformation.
Through an explicit calculation, it can be shown that the minimal surface describes a
p
geodesic curve represented as u = l2 /4 − x2 . Substituting it into the Ryu-Takayanagi for-
mula, the resulting entanglement entropy reproduces the known two-dimensional CFT result
c l
SE ∼ log + const, (4)
3
3R
where c = 2G and indicate the central charge and a UV cutoff (or the lattice spacing),
respectively. This result can also be understood by the previous spatial scale symmetry. Since
the dimensionful parameters scaled under the spatial scaling are and l, only possible scale
invariant combination is given by l/. After the dimension counting, the allowed terms are the
result in (4) and terms having the form of (/l)α with a positive power (α > 0). Since the
UV cutoff is taken to be → 0, the terms ∼ (/l)α automatically vanish. Note that above
the constant term is not uniquely determined because of the regularization scheme dependence.
Anyway, the lesson from the scale symmetry is that the entanglement entropy of a scale invariant
two-dimensional theory has no nontrivial dependence on l except the logarithmic term.
Although information about the dynamical critical exponent is not involved in the ground
state entanglement entropy, it is not the case for the entanglement entropy of the excited state.
The entanglement entropy change between the ground and excited states can be described
by a relative entropy which is independent a regularization scheme and generally leads to
the entanglement entropy bound [15]. Especially, when the entanglement entropy bound is
saturated, it gives rise to the thermodynamics-like law, ∆E = TE ∆SE . In this case, the
increased entanglement entropy is directly related to the excitation energy, so that the scaling
behavior of the time direction becomes important. For a LFT, the entanglement temperature
TE crucially relies on the dynamical critical exponent
1
TE ∼ z . (5)
l
In order to understand z-dependence of the entanglement temperature, we need to recall the
scaling behavior in (2). The entanglement entropy is invariant under the scale transformation
4as mentioned before, and the energy scales as E → λ−z E. Thus, the entanglement temperature
must scale as TE → λ−z TE . Since the above thermodynamics-like law is regularization scheme
independent, the scaling behavior of the entanglement temperature should be explained by the
remaining parameter, l. As a result, the result in (5) naturally appears because of the scale
invariance of a LFT.
Explained above, the scale invariance is indispensable in the study of critical behavior. Now,
let us deform a UV CFT with a specific relevant operator. Denoting its source (or coupling
contant) as µ and its conformal dimension as ∆s , the modified entanglement entropy near the
UV fixed point has the following form
c l
log + SD µl∆s ,
SE = (6)
3
where SD denotes the contribution caused by the deformation. The nontrivial l dependence
of the last term usually breaks the scale symmetry and causes a nontrivial RG flow. In the
holographic entanglement entropy context, when is fixed, 1/l can be regarded as an RG scale.
However, due to the ambiguity associated with the regularization scheme, it is not clear what
value one should assign to the UV cutoff and how it is related to the traditional regularization
scheme of a QFT, (although SE relies on the regularization scheme, its derivative does not).
However, in the case of a condensed matter system such ambiguities do not exist as there is
very well-defined UV cutoff namely the lattice spacing. Furthermore, as evidenced from Eq. 7,
the RG flow of the entanglement entropy in terms of the subsystem size λ is impervious to the
UV cut-off.
dSE c dSD (µl∆s )
= + . (7)
d log l 3 d log l
There exists another interesting and physical interpretation for this RG flow. Let us de-
compose the entanglement entropy into
cA
log + SL µl∆s ,
SE ≡ − (8)
6
with
cA
SL µl∆s = log l + SD µl∆s .
(9)
6
Here the first term including log represents a short distance correlation across the entangling
surface which gives rise to the area law, A = 2. On the other hand, SL is independent of the
UV cutoff , which corresponds to the shortest length scale, and can be interpreted as the long
range quantum correlation between the subsystem and its complement. For a two-dimensional
LFT, the RG flow of the entanglement entropy entropy can be well described by the long range
5correlation defined above due to the following relation
dSE dSL
= . (10)
d log l d log l
In this case, the entanglement entropy SL caused by the long range correlation is regularization
scheme independent up to a constant term which does not affect the RG flow. When regarding
a higher dimensional theory, however, we must be careful [14]. Since the power-law divergence
of a higher dimensional theory prohibits us from decomposing the entanglement entropy naively
into short and long range quantum correlations, an appropriate renormalization procedure is
required before extracting a long range quantum correlation [61].
When the RG flow of the deformed QFT meets a critical point at a certain IR energy scale,
what happens? Generally the scaling symmetry is restored at such a critical point, so the QFT
again becomes a scale invariant theory. Due to this fact, we can expect that the entanglement
entropy is again described by (4) at the IR fixed point. In the next section, we will show that
the disordered source causes a nontrivial RG flow from a UV CFT to a LFT at an IR fixed point
and that the corresponding entanglement entropy has the same form at those two fixed points.
Related to this fact, another interesting question is whether the entanglement entropy can show
a universal behavior near the IR fixed point similar to the second order phase transition. In the
following section, we will show that the disorder effect near the IR fixed point behaves as ∼ l−δ
in which the exponent does not crucially depend on the strength of the disordered source.
3 The entanglement entropy of the randomly disordered
system
Recently, significant progress was made in understanding the disordered system holographically.
It has been shown that when a two-dimensional CFT is deformed by a randomly disordered
fluctuation, its contribution is finite even in the IR as well as UV regimes [28]. The dual
geometry of the randomly disordered system has been analytically constructed by using the
Poincaré-Lindstedt resummation technique. Although it is a perturbative geometric solution,
it opens a new window to study an IR critical behavior holographically. Interestingly, the dual
geometry modified by the disordered fluctuation remains as an AdS space in the UV limit, while
in the IR limit it continuously changes into a Lifshitz geometry with a nontrivial dynamical
critical exponent. This geometry implies that the dual QFT runs from a CFT to a LFT along
the RG flow. Since the dual geometry connects the UV and IR fixed points smoothly, it provides
a good playground to study the RG flow caused by a randomly disordered fluctuation. In this
section, after briefly reviewing the dual geometry of a randomly disordered system [28], we will
study how the entanglement entropy of the dual QFT changes along the RG flow [60–63] .
6The starting point to describe a disordered fluctuation holographically is a (2+1)-dimensional
Einstein-scalar gravity described by the following action
√
Z
1 3 2 a 2m 2
S= d x −g R + 2 − 2∂a φ∂ φ − 2 φ , (11)
16πG L L
where L is an AdS radius and the scalar field corresponds to the source of disorder. From now
on, we will set L = 1 for simplicity. When m = −3/4, the scalar field near the asymptotic AdS
boundary can be expanded into
φ = u1/2 φ1 (x) + u3/2 φ2 (x) + · · · , (12)
and its dual scalar operator saturates the Harris criterion. In this case, the randomly disordered
source can be represented as (see [28] for the details)
N
X −1
φ1 (x) = v An cos (kn x + γn ) , (13)
n=1
where kn = n∆k with ∆k = k0 /N and γn denotes a random phase uniformly distributed in
(0, 2π). Here k0 indicates the highest mode of the disorder and the disorder amplitude An is
given by
p
An = 2 S(kn ) ∆k , (14)
where the function S(kn ) controls the correlations of the noise. The random average denoted
by h· · ·iR is defined in the N → ∞ limit and accounts for the local Gaussian noise
hφ1 (x)iR = 0 and hφ1 (x)φ1 (y)iR = v 2 δ(x − y). (15)
Regarding the gravitational backreaction of the scalar field, the resulting dual geometry of
the randomly disordered system becomes up to v 2 order [28]
1 A(u, x) 2
ds2 = − dt + B(u, x)dx2
+ du2
, (16)
u2 F (u)β(v)
with
h i 2k u
0
A(u, x) = 1 + v 2 e−2k0 u (1 + 2k0 u) {log(2k0 u) + γ} − 2k0 u − log p ,
1 + k02 u2
B(u) = 1 + v 2 e−2k0 u + γ − 1 ,
F (u) = 1 + k02 u2 ,
v2
β(v) = , (17)
2
7where γ is the Euler constant, γ ≈ 0.577. Even when v is order one, the gravitational backreac-
tion of the disordered source is still regular in the entire range of u. Interestingly, this geometry
smoothly changes from an AdS geometry in the UV regime (u → 0) into a Lifshitz one in the
IR regime (u → ∞) with the dynamical critical exponent
v2
z =1+ . (18)
2
Although higher order corrections of v are also regular [28], hereafter we concentrate on the
order v 2 correction.
3.1 In the UV regime
In order to study the disorder effect, let us investigate the holographic entanglement entropy.
Since the perturbative expansion with respect to v is applicable in the entire regime of the dual
geometry, here we focus on the leading correction of v 2 order. By using Ryu and Takayangi’s
formula [11, 12], the entanglement entropy can be evaluated by calculating the area of the
minimal surface extended to the dual geometry, whose boundary have to coincide with the
entangling surface dividing a total system into two parts. In order to clarify the entangling
surface, we take a subsystem defined in the following interval
l l
− ≤x≤ . (19)
2 2
The configuration of the minimal surface can be expressed by u as a function of x. Using this
parameterization, the induced metric on the minimal surface reduces to
1
ds2in = B(u) + u02 dx2 ,
2
(20)
u
where the prime indicates a derivative with respect to x. Then, the entanglement entropy is
governed by the following action
Z l/2
1 1p
SE = dx B(u) + u02 . (21)
4G −l/2 u
In order to obtain a unique minimal surface, we should impose two boundary conditions which
fix the integral constants of the second order differential equation. Natural boundary conditions
are u(l/2) = 0 and u0 (0) = 0. Here the first constraint is required because the entangling surface
must be located at the boundary u = 0, while the second is needed to find a smooth minimal
surface at the turning point, x = 0.
Before studying the disorder effect in the IR regime, we first investigate the entanglement
entropy in the UV regime satisfying k0 l 1. Above the action is invariant under x → −x, so
8that we can further reduce the action into
Z l/2−x∗
1 1p
SE = dx B(u) + u02 . (22)
2G 0 u
where x∗ is introduced to denote a UV cutoff in the x-direction. Due to the existence of the
small expansion parameter v in the entire regime, u can be expanded to be
u(x) = u0 (x) + v 2 u2 (x) + O v 4 .
(23)
Then, u0 (x) and u2 (x) satisfy the following equations of motion
0 = u0 u000 + u02
0 + 1, (24)
0 = e2k0 u0 (γ + 2u00 u02 + u2 u000 + u0 u002 − 1) − 2k0 u0 2 u000 − k0 u0 + 1. (25)
The boundary conditions discussed above reduces to ui = 0 at x = ±l/2 and u0i = 0 at x = 0.
The solution of the first equation, u0 , describes the geodesic curve in a pure AdS3 space [11]
1√ 2
u0 (x) = l − 4x2 , (26)
2
which automatically satisfies two required boundary conditions. The second equation governs
the deformation of the minimal surface caused by the disorder. It does not allow an analytic
solution, so that a numerical study is needed to figure out its effect in the IR regime. However,
in the UV regime the existence of another small expansion parameter still enables us to look
into the entanglement entropy analytically but perturbatively. In the UV regime where the
highest momentum of the disorder is much smaller than the energy scale observing the dual
QFT, k0 l is small and u2 can be further decomposed into
k0 l
u21 (x) + O k02 l2 .
u2 (x) = u20 (x) + (27)
2
The first solution satisfying the equation of motion is given by
c1 l + 2c2 x − γx2
u20 (x) = √ . (28)
l2 − 4x2
The smoothness at x = 0 fixes c2 = 0 and the other boundary condition, u(l/2) = 0, gives rise
to c1 = γl/4. Using this result, u21 (x) satisfying the equation of motion reads
√
[(γ − 3)l2 + 2(3 − 2γ)x2 ] l2 − 4x2 + 6c3 l2 + 12c4 lx − 3l2 x tan−1 √l22x
−4x2
u21 (x) = √ . (29)
6l l2 − 4x2
Imposing again two required boundary conditions, we finally obtain c4 = 0 and c3 = πl/8.
9Now, let us introduce a UV cutoff, , in the u-direction which is associated with the UV
cutoff, x∗ , introduced previously. Using the perturbative solutions we found, they are related
by
h π i 2
k0 l + γ v 2 + O 3 .
x∗ = 1 − (30)
8 l
Substituting these results into (21) and integrating it perturbatively, the holographic entangle-
ment entropy can be written as
c
SE = − log + SL (l, v, k0 ), (31)
3
with
c c cv 2 √
SL (l, v, k0 ) = log l + 1 + 2γv 2 − 2 log 2 − 32 2 − 3π k0 l, (32)
3 12 288
where we used the relation, c = 3G/2L. Here the first part including the UV divergence can be
regarded as the short distance correlation across the entangling surface (two boundary points),
while SL (l, v, k0 ) corresponds to a long range quantum correlation between the subsystem and
its complement. In the UV regime the result shows that the disorder corrections reduce the
long range quantum correlation and that its effect linearly increases along with the RG flow.
3.2 In the IR regime
In order to understand the disorder effect in the IR regime, we need to know the exact minimal
surface configuration extended into the deep interior of the dual geometry. In the interior
corresponding the IR regime of the dual field theory, the expansion parameter used in the
previous section is not small, so that we can not apply the previous expansion any more. In
this section, therefore, we will investigate the disorder effect in the IR regime numerically.
Noting that the minimal surface action in (21) does not depend explicitly on x, we can find a
conserved quantity
B
H=− √ . (33)
u B + u02
At the turning point satisfying u0 = 0 at x = 0, it reduces to
√
B0
H=− , (34)
u0
where u0 and B0 indicate the turning point and the value of B on it. Comparing these two
results, we can determine the subsystem size and entanglement entropy in terms of u0
Z u0 √
u B0
l(u0 , v, k0 ) = 2 du √ p 2 ,
0 B Bu0 − B0 u2
10Z u0
√
c u0 B
SE (u0 , v, k0 ) = du p 2 , (35)
3 u Bu0 − B0 u2
where a UV cutoff in the second integral is introduced for the regularization. Note that unlike
the previous section where l was used as a free parameter, above l is determined by three free
parameters, u0 , v and k0 . Since c appears as an overall constant in the entanglement entropy
formula, the exact value of c is not important to understand the qualitative behavior of the
entanglement entropy.
Now, let us concentrate on the long range quantum correlation in order to look into IR
physics
c
SL = log l + SD . (36)
3
Here, the first term represents the common dependence on the subsystem size, while the second
term corresponds to the disorder effect, which can be obtained from the numerical data of SE
c l
SD ≡ SE − log . (37)
3
From now on, we take a specific UV cutoff, = 10−10 , for the numerical calculation. If changing
the UV cutoff, the value of SE must be changed. However, the qualitative behavior described
by the RG flow is not affected as mentioned before, so that we use SD and SL to describe the
physical properties. Fig. 1(a) shows that the value of SD in the large l limit converges into a
certain constant. This fact becomes clear in Fig. 1(b) where the slope of SD also approaches
to zero. These results strongly indicate that a new scale invariant theory arises at the IR
critical point (l → ∞). The change of the long range quantum correlation along the RG flow
is depicted in Fig. 2(a). It shows that the long range quantum correlation at the IR fixed point
is proportional to log l as a two-dimensinoal scale invariant theory should do.
Due to the existence of the scale invariance at the IR critical point, it would be interesting to
check whether the IR entanglement entropy can show a universal critical behavior represented
as a critical exponent. Assuming that SD is given by a polynomial of l, v and k0 , then near the
IR critical point it can be expanded into
SD = S∞ + C l−δ + · · · . (38)
Above the ellipsis means higher order corrections, O (l−σ ) with σ > δ, and can be ignored in
the l → ∞ limit. In order to check the above expected form, we fit numerical data of SD
with (38). Fig. 2(b) shows in the large l limit that numerical data of SD can be well fitted
by (38) with S∞ = −0.00284, C = 0.825 and δ = 2.036 for k0 = 0.1, v = 0.2 and c = 1.
In addition, the negative value of SD implies that the disorder effect reduces the long range
quantum correlation.
11(a) (b)
0 in terms of l, for v = 0.1 (dotted), 0.2 (dashed), and
Figure 1: The numerical results for SD and SD
0.3 (solid). These two results show that SD in the large l limit approaches to a certain constant value
(See also Fig. 3(b)).
In order to study how the disorder deformation affects the long range correlation behavior
and what the critical behavior of the entanglement entropy is near the IR fixed point, we extract
information about S∞ , SL , C and δ depending on v in Fig. 3, where we use l = 200, c = 1 and
k0 varies from 0.1 to 0.5. The values of S∞ in Fig. 3(a) and C in Fig. 3(c) are given by well-
behaved continuous functions. When the amplitude of the disorder v increases S∞ decreases,
while C increases. In general, δ in (38) is given by a function of the intrinsic parameters of the
theory, k0 and v. However, the resulting δ in Fig. 3(d) shows an almost constant value with a
small deviation for v > 0.3. In this case, the small deviation seems to be caused by a numerical
error. If so, the result in Fig. 3(d) indicates that the first correction to the IR entanglement
entropy is suppressed by a critical exponent δ which is independent of v.
There are several remarks before closing this section. First, the critical exponent δ seems
to be universal as it depends only very weakly on the amplitude of the disordered fluctuation
which is one of the intrinsic parameters of the UV microscopic field theory. Second, this result
is obtained by taking into account just v 2 order disorder deformation, so that if one further
considers higher order corrections like v 4 , the critical exponent we found can be modified. Lastly,
although we obtained a critical exponent of the entanglement entropy in a small v region, it is
not clear whether such a well-defined critical exponent for the long range entanglement entropy
always appears in the IR critical point, for example, even in the large v limit. When we applied
our numerical program to the case with a large value of v, our program crashed similar to the
previous work in [28]. Therefore, it remains as an interesting and important issue to check
whether there still exists a universal critical exponent in the IR entanglement entropy even for
a large v. We hope to report some more results on this issue in future works.
12d SL
d log l
0.334
0.333
0.332
0.331
0.330
0.329
0.328
0 50 100 150 200
l
(a) (b)
Figure 2: (a) The change of the long range quantum correlation SL relying on the subsystem size l
for k0 = 0.1, v = 0.2 and c = 1, (b) numerical data of SD depending on l (solid curve) and fitted by
(38) (dashed line).
4 Mutual information
In order to understand the IR physics of the randomly disordered system, let us further inves-
tigate the mutual information. The mutual information is defined as [24, 64–68]
I(A, B) = SE (A) + SE (B) − SE (A ∪ B). (39)
By definition the mutual information has no UV cutoff dependence, so it can measure the
long range quantum correlation. For more details, let us define an operator OA which is in a
subsystem A. Then, a two point correlation function between two operators living in different
subsystems, A and B, can be denoted by
C(OA , OB ) = hOA OB i − hOA i hOB i . (40)
Interestingly, it was known that these two concepts are related to each other [68–70]
(hOA OB i − hOA i hOB i)2
I(A, B) ≥ ≥ 0. (41)
2|OA |2 |OB |2
This relation shows that the mutual information is larger than the square of the two point
correlation function. Thus, the mutual information plays a role as the upper bound of a two
point correlation function. In addition, the last inequality implies a non-negativity of the mutual
information. From this relation, we can see that the vanishing mutual information indicates the
absence of the quantum correlation between two disjoint subsystems. Due to this reason, the
13SL
S∞
0.000
0.05 0.10 0.15 0.20 0.25 0.30
v 1.766
-0.002 1.764
k 0 =0.1, 0.2, 0.3, 0.4, 0.5 1.762 k 0 =0.1, 0.2, 0.3, 0.4, 0.5
-0.004
1.760
-0.006
-0.008
1.758
0.05 0.10 0.15 0.20 0.25 0.30
v
(a) S∞ (v) is independent of k0 (b) SL (v) is independent of k0
C δ
2.0 2.2
k 0 =0.1 k 0 =0.2 k 0 =0.3 k 0 =0.1 k 0 =0.2 k 0 =0.3
1.5 2.1
k 0 =0.4 k 0 =0.5 k 0 =0.4 k 0 =0.5
1.0 2.0
0.05 0.10 0.15 0.20 0.25 0.30
v
0.5
1.9
0.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30
v 1.8
(c) C(v) for different values of k0 (d) δ(v) for different values of k0
Figure 3: Coefficients of the fitting function(38) with l = 200, c = 1 and k0 varies from 0.1 to 0.5.
S∞ < 0 (a) but SL > 0 (b), which implies that the disorder weakens the long range correlation. (b)
and (c) show continuous functions, whereas (d) gives an almost constant δ around 2, which may be
interpreted as a critical exponent of the entanglement entropy near the IR fixed point. The small
jumps of δ around v = 0.15 seems to be a numerical error for k0 > 0.3.
14mutual information was used as an indicator of the phase transition in the holographic setup
[71–73].
From the dual geometry of the randomly disordered system, we can easily derive the mutual
information which in the UV limit (l ∼ h 1) is written as the following perturbative form
c l2 c √
I(A, B) = log + 32 2 − 3π k0 v 2 h + · · · , (42)
3 h(2l + h) 144
where we set two subsystems to have the same size l and h indicates the distance between two
subsystems. Then, the critical distance when the phase transition occurs is given by
√ √
√ 64( 2 − 1) − 3(2 − 2)π 2 2
v k0 l + O l3 .
hc = ( 2 − 1)l + (43)
192
Below the critical distance (h < hc ), the mutual information becomes positive. This implies
that the two subsystems have nontrivial quantum correlations. In other words, the union of
the subsystems has a lower entanglement entropy than the entanglement entropy sum of the
subsystems. At the critical distance (h = hc ), the mutual information vanishes (see Fig. 4) and
the quantum correlation between the subsystems also disappear. Above the critical distance
(h > hc ), since there is no quantum correlation between the subsystem, the entanglement
entropy of the union of the subsystem, SE (A ∪ B), must be the same as the entanglement
entropy sum of each subsystem, SE (A) + SE (B). For the CFT without the disorder (v = 0),
the critical distance is linearly proportional to the subsystem size. On the other hand, the
existence of the disorder increases the mutual information and the critical distance slightly in
the UV regime. For a CFT, the critical distance is linearly proportional to the subsystem size in
the entire RG scale. In the UV regime of the randomly disordered system, the similar behavior
also happens with small higher order corrections (see (43)). This is because the UV regime of
the randomly disordered system can be regarded as a small deformation of the UV CFT.
In Fig. 4(a), we depict the mutual information relying on h at a given subsystem size and
plot how the critical distance depends on the subsystem size in Fig. 4(b). Interestingly, the
numerical result in Fig. 4(b) shows a linearly increasing behavior, which can be well fitted by
hc ∼ 0.41 l − 0.004, (44)
where we take c = 1, v = 0.1, k0 = 0.1, and δ = 2.034. Can we understand the critical
distance relying on the subsystem size linearly even in the IR regime? In order to discuss the
mutual information in the IR limit, let us first take into account the fitting function in (38)
which explains the IR entanglement entropy of the disordered system well. Using this fitting
function, the mutual information in the IR limit (l ∼ h 1) can be represented as
l2
c 2 1 1
I(A, B) = log +C δ − δ − + ··· , (45)
3 h(2l + h) l h (2l + h)δ
15I(A,B) hc
100
1.5 l=50 l=100 l=200 80
60
1.0
40
0.5
20
0.0
20 40 60 80
h
100
0
0 50 100 150
l
200
(a) (b)
Figure 4: (a) The mutual information in terms of the internal distance h and (b) the critical distance
hc in terms of l between two subsystems where we have taken k = 0.1 and v = 0.1. In (b), the critical
distance has a small deviation from the CFT result (the detail is depicted in Fig. 5).
where we set the subsystem sizes to be l. In this result, the UV cutoff dependence and the con-
stant part, S∞ , of the entanglement entropy do not appear because they are exactly cancelled.
Since these terms are associated with the short distance correlation of the quantum entan-
glement, the mutual information can be regarded as the measure observing the long range
correlation, as mentioned before.
The mutual information in (45) has a critical distance of hc where the mutual information
vanishes. Above this critical distance, the long range quantum correlation between two disjoint
systems disappears. In the IR region, the second term in (45) is relatively small. Thus, the
critical distance satisfying I(A, B) = 0 can be expressed as
hc = al + blγ + · · · , (46)
where γ < 1. The leading term linearly proportional to l is required to make the logarithmic
term of the mutual information in (45) vanish. After substituting this ansatz into (45) and
expanding it, we can determine the critical distance as
√ √
√ 3 ( 2 + 1)δ + ( 2 − 1)δ − 2 C 1
hc = ( 2 − 1)l − √ + ··· . (47)
2 2c lδ−1
√
Noting that 2 − 1 ≈ 0.41 and γ = −(δ − 1). Similar to the UV case in (43), the critical
distance is still linearly proportional to the subsystem size with a small negative correction. In
Fig. 5, we depict the deviation of the critical distance from the CFT case with ∆hc = 0. In
the limits having a small and large subsystem sizes, the analytic forms of the critical distance
in (43) and (47) are well matched to the numerical data in Fig. 5. This result shows that the
16critical distance is almost linearly proportional to the subsystem size with a small corrections
caused by the disordered source.
Now, let us consider the mutual information in the IR region with a large l. The short range
correlation can be represented by a small distance between two subsystems (h 1 l). In
this case, the mutual information reduces to
c l2 n c 2
o
I(A, B) = log + S∞ − 1 + 2γv − 2 log 2
3 h(2l + h) 12
cv 2 √
+C 2l−δ − (2l + h)−δ + 32 2 − 3π k0 h + · · ·
288
c l c
1 + 2γv 2 − 2 log 2 .
≈ log + S∞ − (48)
3 2h 12
Note that the leading contribution to the mutual information is independent of the parameters
describing the disorder. Since S∞ decreases as v increases in Fig. 3, the above result indicates
that the short range quantum correlation decreases as the amplitude of the disorder becomes
large. When h increases, the mutual information decreases logarithmically. For the long range
correlation with l & h 1, the h dependence of the mutual information at a given l reduces to
2 1 1
I(A, B) ∼ ICF T + C δ − δ − , (49)
l h (2l + h)δ
where ICF T indicates the mutual information of the undeformed CFT. For CFT, the long range
mutual information also decreases logarithmically by − log [h(2l + h)]. The second term above
shows the effect of the disordered source in the IR limit.
5 Discussion
In this work, we have holographically investigated the IR entanglement entropy of a two-
dimensional CFT deformed by a disordered source. In the AdS/CFT contexts, the dual geome-
try of it has been constructed in [28]. There are several noticeable points in this dual geometry.
The dual scalar field of a disordered source has a finite gravitational backreaction in the entire
regime, so that the perturbative expansion in terms of the strength of the disorder is possible
even in the IR regime. Since the disorder is relevant, the UV CFT flows to another theory in
the IR regime. Intriguingly, the disorder considered here allows a Lifshitz fixed point in the IR
regime. This Lifshitz fixed point is usually classified by the dynamical critical exponent, which
in this model is determined by the strength of the disorder. In this work, we have take account
of the quantum entanglement entropy and its RG flow in order to figure out quantum aspects
of the IR physics. To do so, we have decomposed the entanglement entropy into two parts,
short distance and long range correlations. The short distance correlation occurs due to the
17Δhc
0.005
0.000
50 100 150 200
l
-0.005
-0.010
-0.015
√
Figure 5: (a) The numerical result for relative critical distance ∆hc ≡ hc − ( 2 − 1)l in terms of the
internal distance l (Blue curve). The purple dashed line is from the analytical result of UV limit (43),
and the red dashed line is from the analytical result of IR limit (47).
entanglement across the entangling surface, so it is usually associated with the UV divergence
of the entanglement entropy. On the other hand, the UV finite term accounts for the long
range quantum correlation between the subsystem and its complement. As a consequence, the
quantum IR physics can be well described by the long range correlation which is the main point
of interest in this study.
At UV and IR fixed points, the theory becomes scale invariant. Due to this scale invariance,
the entanglement entropy at those critical points shows a simple logarithmic behavior relying
on the subsystem size. In the entanglement entropy context, the RG flow can be parameterized
by the subsystem size. Using it, the entanglement entropy near the UV fixed point decreases
linearly along the RG flow. Near the IR fixed point still decreases but slowly along the RG
flow. In this case, the rate of decrease approaches to zero as the subsystem size goes to the
infinity (or the IR limit). In this case, the long range correlation can be parameterized by the
subsystem size with a certain critical exponent. Through the numerical analysis, intriguingly,
we found that the critical exponent of the long range correlation near the IR fixed point is
universal. In other words, the critical exponent is very weakly dependent on the strength of
the disorder similar to the critical exponent appearing in the second order phase transition. It
would be interesting to study the IR behavior of the long range correlation for other models
and to find the similar universal critical exponent. We hope to report more results in future
works.
18Acknowledgement
RN would like to thank APCTP for their hospitality during his visit. This work was
supported by the Korea Ministry of Education, Science and Technology, Gyeongsangbuk-Do
and Pohang City. RN was partially supported by National Research Foundation (NRF) funded
by MSIP of Korea (Grant No. 2015R1C1A1A01052411). CP was also supported by Basic
Science Research Program through the National Research Foundation of Korea funded by the
Ministry of Education (NRF-2016R1D1A1B03932371).
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