#### Conjecture on the butterfly velocity across a quantum phase transition

Received: May
Conjecture on the butter y velocity across a quantum
Matteo Baggioli 0 1 2 4
Bikash Padhi 0 1 2 3
Philip W. Phillips 0 1 2 3
Chandan Setty 0 1 2 3
0 1110 W. Green Street, Urbana, IL 61801 , U.S.A
1 71003 Heraklion , Greece
2 Department of Physics, University of Crete
3 Department of Physics and Institute for Condensed Matter Theory, University of Illinois , USA
4 Crete Center for Theoretical Physics, Institute for Theoretical and Computational Physics
We study an anisotropic holographic bottom-up model displaying a quantum phase transition (QPT) between a topologically trivial insulator and a non-trivial Weyl semimetal phase. We analyze the properties of quantum chaos in the quantum critical region. We do not nd any universal property of the Butter y velocity across the QPT. In particular it turns out to be either maximized or minimized at the quantum critical point depending on the direction of propagation. We observe that instead of the butter y velocity, it is the dimensionless information screening length that is always maximized at a quantum critical point. We argue that the null-energy condition (NEC) is the underlying reason for the upper bound, which now is just a simple combination of the number of spatial dimensions and the anisotropic scaling parameter.
Holography and condensed matter physics (AdS/CMT); AdS-CFT Corre-
1 Introduction 2
The holographic model 3 4
2.1
2.2
2.3
Weyl semimetals
Holographic Weyl semimetal
Anomalous transport
Quantum chaos & universality
Conclusion
A The holographic background B Butter y velocities in anisotropic backgrounds 1 4
h [V(~x; t) ; W(0; 0)]2 i
e L (t t
j~xj=vB);
(1.1)
where V
; W are two local Hermitian operators,
L the Lyapunov exponent, t is the so
called scrambling time and
is just the thermal timescale. The appearance of the
butter y velocity in this correlation function motivates it as the relevant velocity for de ning
bounded transport [2]. The monotonic growth of the Lyapunov exponent at a quantum
critical point and its subsequent decrease away from the critical point determined by some
non-thermal coupling constant, g is sketched in
gure 1. See also [3] for an exploration
of the connection between quantum chaos and thermal phase transitions. Moreover, this
behavior is believed to hold also at nite but low temperature inside the quantum critical
region. Preliminary studies connected to the proposal of [
1
] and to the onset of quantum
chaos across a QPT have been already performed within the holographic bottom-up
framework in [4]. Nevertheless a complete study, beyond simple models, is still lacking. A full
analysis of this problem appears to be in order in view of the recent experiments where the
OTOC has been measured using Lochsmidt echo sequences [5] and NMR techniques [6].
{ 1 {
line) at the quantum critical point g = gc. It is not clear, and indeed the purpose of our investigation,
if the butter y velocity vB display a similar behaviour or not.
The aim of this paper is indeed to understand the onset of quantum chaos across
a quantum phase transition in more complicated holographic models displaying a
quantum phase transition. In particular, we will perform our computations in the holographic
bottom-up model introduced in [7, 8] which exhibits a QPT between a trivial insulating
state and a Weyl semimetal. The particular new wrinkle we bring to bear on the transition
is the presence of anisotropy. In other words, the rotational group SO(3) is broken to the
SO(2) subgroup by an explicit source in the theory. As a consequence of the underlying
anisotropy, we can de ne two butter y velocities that we will denote as v and v , where
denotes the direction(s) of the anisotropy, while ? in the direction perpendicular to it.
Throughout the paper, we use this notation for all such directional quantities.
?
The results of our paper show that while the perpendicular velocity v
? displays a
behavior similar to that in [
1
], the parallel one v
does not. In particular, the butter y
velocity along the anisotropic direction will not display a maximum at the critical point
g = gc but rather a minimum. We pinpoint as the origin of this violation, the presence of
anisotropy itself.1 Interestingly, the bound on the viscosity is also violated in an anisotropic
system [13, 14] and by a strong magnetic eld [15, 16]. Here the mechanism leading to the
violation of the bound are very analogous, that is explicit breaking of the SO(3), leading
to spatial-anisotropy. Note that [17] spontaneous breaking of rotation symmetry, despite
leading to a non-universal value for =s, does not provide a violation of the KSS bound.
We expect this to be case for butter y velocity as well. Here
is the viscosity and s is the
entropy density.
1E ects of anisotropy on the butter y velocity were previously investigated in [9{12].
{ 2 {
To understand if any universal statement can be made about the butter y velocity,
especially in the presence of anisotropy, we identify a quantity related to the spatial-spread
of information which is insensitive to the breaking of the SO(D) symmetry, where D is the
number of space dimensions. We do so by computing the OTOC holographically. Given
an anisotropic bulk spacetime of the form
ds2 =
gtt(r) dt2 + grr(r) dr2 + h?(r) d~x2? + + h (r) d~x2 ;
where we denote by
the D
anisotropic directions and with ? the D
? remaining
directions.
The butter y velocities can be computed for this background as (
v
=
L=M ; where
L = 2 =
is the Lyapunov exponent and all the quantities are
computed at the horizon. The parameter M(?; ) controls the screening of the information
=?; )
spreading in the (?; ) directions,
and it clearly depends on the warp factor h . As a consequence, the butter y velocity can
not represent a good and universal quantity in the presence of anisotropy. Contrastly, we
can de ne a dimensionless quantity controlling the screening of information through
(t; x )
e L t M jxij;
2
M 2
h (r0)
:
(1.2)
(1.3)
(1.4)
(1.5)
The factor h (r0) is indeed the reason why we see dissimilar result from [
1
] in an anisotropic
setup. The important point is that our new physical parameter
has no spatial dependence
and hence, it is completely insensitive to any anisotropy present in the system. Our proposal
is to consider the dimensionless information screening length L, which can be de ned as
L
1= . Our claim can be rephrased as the dimensionless information screening length
L, which can be de ned via the OTOC, is always maximum at the quantum critical point.
Moreover, for a theory passing through a Lifshitz-like critical point, given the number of
spatial directions D
? which scales similarly as time and the number of the directions D
which has an anisotropic scaling, 0, the conjecture regarding L can be restated as
2L
1
D
? +
We will later see that such a bound can be justi ed from NEC and in our model this is
saturated at the quantum critical point g = gc. In a similar spirit, [18] points out a bound
on the butter y velocity for an isotropic space with di erent warp factors appearing along
the r, t directions, gtt(r); grr(r). Since in our case gtt(r)grr(r) = 1, we always saturate
their bound.
The paper is organized as follows. In section 2, we present the holographic model and
its main, and known, transport features. In section 3, we study the onset of quantum chaos
in the model and in particular the butter y velocity and the related conjectured bound.
Conclusions are reached in section 4. In appendices A and B we provide more technical
details about our computations.
{ 3 {
We begin by reviewing the holographic model of [7, 8] which exhibits a QPT from a
topologically non-trivial Weyl semimetal to a trivial insulating phase. Although the boundary
theory exhibiting this topological transition in eq. (2.1) is a free theory, the holographic
bulk theory strictly describes a strongly correlated system. The hope is that they share the
same set of symmetries, thereby capturing the essential properties of the phase transition,
if not all the details of the transport pertaining to interacting physics. Note that this is
a phase transition in a certain topological invariant (such as Chern number) and not in
the symmetries; thus, one can not probe it through the free energy density as it never
depends on any topological term in the action. The order parameter is represented by the
anomalous Hall conductance, AHE which is zero in the trivial gapped phase and nite in
HJEP07(218)49
the Weyl semimetal phase.
2.1
Weyl semimetals
Weyl semimetals are a class of three dimensional topological materials characterized by
(point) singularities in the Brillouin zone (BZ) at which the band gap is zero. This peculiar
property gives rise to exotic transport phenomena (see [20] for a comprehensive review).
Quasiparticle excitations near such band-touching points, also called Weyl nodes, can be
described by (left- or right-handed) Weyl spinors. In a time-reversal symmetry broken
insulator, the left- and right- Weyl nodes are separated in the BZ which can be controlled
by a chiral or axial gauge potential, ~b. It is the interplay of this axial eld and the (chiral)
mass of the spinor, M , that gives rise to di erent phases (see gure 1). Deep in the
semimetal phase, b (
node-separation equal to [21] be = b (1
j~bj) is much larger and M simply renormalizes it causing a reduced
M 2)1=2, where M
M=b. On the other hand, for
a larger M , renormalization by a weaker b reduces the gap to Me = b (M 2
1)1=2. Thus,
the semimetal-insulator phase transition occurs at Mc
capturing this physics is [22]
O(1). The continuum description
L =
eB=
5
~ ~b + M ) :
Here the slash denotes contraction by Dirac gamma matrices,
. The matrix
i 0 1 2 3 allows one to project the Dirac spinors,
, into the chiral sectors,
5
(1
) . B is the electromagnetic gauge potential; without loss of generality [23], we
choose the axial gauge potential to be ~b = b e^z. The axial symmetry, however, is anomalous
can be seen [24] in the response of the axial current, J~5
~
b
e
(dJ5 6= 0), leading to a non-conservation of the number of particles of given chirality. This
dA~, that is the anomalous
Hall conductance, AHE
be . This clearly vanishes in the insulating phase, that is for
su ciently large M . The mass term and the axial term act as relevant deformations. Thus,
with increasing M , the theory moves from UV to IR thereby traversing through a xed
point at Mc.
(2.1)
5 =
L;R =
{ 4 {
Now we turn to the holographic model of the above phase transition. The bulk action takes
the form ( xing 2G2N = L = 1, where GN is Newton's constant, and L the AdS radius):
S =
Z
d5x p
g R + 12
1
4
F 2
1
4 F52 +
3
A
F 5
F 5
+ 3 F
F
(D
) (D
)
V ( )
The bulk elds are an electromagnetic vector U(1) gauge eld B with elds strength
iqA
, and the scalar potential is chosen to be V ( ) = m2j j2 + 2 j j4.
Since the phase of the scalar eld is not a dynamical variable, with out loss of generality
we assume it to be real. The mass of the
eld, m = p
(
d) controls the scaling
dimension,
, of the boundary operator corresponding to . Throughout the paper, we
will use d as the space-time dimension of the boundary eld theory, occasionally denoting
the boundary spatial dimension as D = d
1. From the mass deformation in eq. (2.1)
and the above relation, it is clear that one needs to choose m2 =
3 (see [19] for di erent
choices of m2 and [25, 26] for further studies of the model), such that the dual operator
has conformal dimension
= 3. Note that this imaginary mass is perfectly allowed within
AdS/CFT since it is with in the Breitenlohner-Freedman (BF) bound, m2
d2=4. The
UV boundary conditions for the vector and scalar eld are chosen to be
(2.2)
(2.3)
(2.4)
(2.5)
Although not necessary for computing the butter y velocity, we will rst discuss the
behavior of zero-temperature solutions for understanding the various low-temperature
limits. For nite temperature, we assume the presence of a black hole horizon at r = r0 such
that f (r0) = 0. For the zero temperature background there is a Poincare horizon at r0 = 0,
{ 5 {
lim r
r!1
= M ;
r!1
lim Az = b ;
where both M and b represent a source for the corresponding dual operators. The
parameter b can be thought as an axial magnetic eld that explicitly breaks the rotational SO(3)
symmetry of the boundary to the SO(2) subgroup. From
gure 2, one can see that this
controls the e ective separation between Weyl nodes. On the contrary, the source M for
the scalar eld is simply introducing the mass scale required by the physics of the problem.
Note the presence of two more (bulk) free parameters in the problem; the quartic coupling,
, controls the location of the quantum critical point (QCP) by changing the depth of the
e ective potential of , and the charge q relates to the mixing between the operators dual
to
and A . Following [7, 8] we x these parameters to q = 1, and
= 1=10, which xes
Mc to 0:744. The generic solution of the system is given by the following ansatz
dr2
f (r)
ds2 =
the bands cross) separated by 2~be in momentum. The band structure on the (top) right is that
of a topologically trivial insulator with an explicit band-gap 2Me . At the QCP, the two Dirac
cones merge together, giving rise to a Lifshitz
xed point (black dot in the bottom
gure) with a
scaling anisotropy along the same direction as ~b. In the holographic picture, away from the QCP,
the theory
ows to two di erent types of (deep IR) near-horizon geometries, AdS5 (Weyl semimetal
phase) or domain wall-AdS5 (trivial insulator phase). The
gure shares some resemblances with
those in [7, 19].
and f (r) = g(r). There are tree types of solutions at zero temperature | (i) insulating
background (for M > Mc), (ii) critical background (for M = Mc), and (iii) semimetal
background (for M < Mc). These solutions can be obtained by solving the equations of
motion, the details of which we discuss in the appendix A. We quote the results here (up
to leading order near the IR).
Insulating background. | Similar to a zero-temperature superconductor, the near-horizon
geometry of a topologically trivial insulator is an AdS5 domain-wall
f (r) =
1 +
r2 ;
h(r) = r2 ;
Az(r) = a1r 1 ;
(r) =
+ 1r 2 :
(2.6)
r 3
3
8
Here a1 is xed to 1 and
1 is treated as a shooting parameter. Exponents 1;2 can be
expressed as functions of (m; ; q), and are (2:69; 0:29) for our choice of parameters. Thus,
the near-horizon value of Az is always zero, and that of
is p3=
(for
= 1=10, it is
(r0) ' 5:477).
parametrized by 0
,
Critical background. | This solution is exact and displays an anistropic Lifshitz-like scaling
f (r) = f0r2 ;
h(r) = h0r2 0 ;
Az(r) = r 0 ;
(r) = 0 :
(2.7)
The scaling anisotropy is explicitely induced by the source of the axial gauge eld A ,
hence is along the direction of ~b. The parameters (f0; h0; 0; 0) are determined by
x{ 6 {
ing (m; ; q). For the parameter choice mentioned previously, we have (f0; h0; 0; 0) '
(1:468; 0:344; 0:407; 0:947). From the zero-temperature equations of motion, it can be shown
that 0 =
the semimetal phase, which is simply AdS5
f (r) = r2 = h(r) ; Az(r) = a1 +
a
dependence is hidden in higher order terms. Note in this case, the near horizon
solution of Az is nite; a1, however, (r0) vanishes. Figures 8 and 9 of appendix A provides
the full A(r) and (r) functions for various values of M . The apparent deviations of A(r0)
and
(r0) from the IR asymptotes described above owes to the fact that we obtain the
solutions for a small but nite temperature up to order O(T ), where T
treat M and T as the free parameters in the theory to control the phase transition.
T =b. We will
2.3
Anomalous transport
As mentioned before, the order parameter for the QPT is the anomalous Hall conductivity.
The DC, limit of all the conductivities can be extracted from (for both zero and
nite
temperatures) horizon data as follows
AHE = 8
Az(r0) ;
? = ph(r0) ;
=
g(r0)
ph(r0)
:
(2.9)
Here
is just a short hand for zz and `?' refers to the conductivity matrix elements,
xx; yy, and should not be mistaken for the transverse conductivity. In gures 3 and 4,
we plot the above conductivities as functions of M , for various temperatures T . We discuss
them individually, starting from their zero-temperature behavior. In order not to sacri ce
numerical stability, we con ne our lowest temperature value to T = 0:005 and treat it as
zero temperature.
diag '
Note that AHE
Az(r0), and from the discussion of the zero-temperature solutions,
we see
AHE is nite only for M < Mc. A more physical picture could be that since in the
IR, the axial gauge eld is completely screened [27], there are no degrees of freedom that
could be coupled to it and hence, it can not be probed any further. As the temperature
is increased, the sharp phase transition slowly becomes a cross-over. At zero-temperature,
the onset of the semimetal phase is well tted by
AHE /
Mc
M
0:21. For M = 0
(or, M = 0), the near-horizon geometry is the deformed AdS5 background of eq. (2.8).
With our choice of normalizations, for low temperatures, g(r0) = h(r0) =
2T 2 and hence,
T , which clearly vanishes at T = 0. The subscript `diag' collectively refers to all
the diagonal components of the conductivity matrix, xx; yy; zz. There are two features
of diag of interest. First, for vanishing b (or, M
1) the near-horizon geometry is the
domain-wall AdS5 geometry of eq. (2.6), which makes diag ' c T , where c < 1 and
{ 7 {
xy) as a function of the dimensionless
mass parameter M for temperatures T = 0:1; 0:05; 0:005 (from green to orange). Note for a very
low temperature the conductivity sharply drops to zero at a critical value, Mc
0:74. This marks
the semimetal-insulator topological phase transition.
0.75
0.5
0.25
0
0
0.5
1
¯
M
1.5
2
0.5
1.5
2
1
¯
M
M for temperatures T = 0:1; 0:05; 0:005 (from green to orange).
independent of temperature. This is due to the fact that it is a phase transition between
a semimetal-insulator transition and some degrees of freedom are now gapped out in the
trivial phase. The reason why the conductivity is still nite in the insulating phase can be
understood by computing the ratio of the gapped to un-gapped degrees of freedom [19],
which eventually becomes a statement about the geometry or more precisely about the
holographic a-theorem [28]. This ratio can be made to vanish by controlling m2 and .
Second, and the most relevant for our discussion, is the fact that at the critical point, there
{ 8 {
ε
4πη 0.5
s
0.75
0.25
0
0
1
¯
M
0.5
1.5
2
1
¯
M
(Left) The anisotropy parameter "0 evaluated at the horizon for various T =
0:1; 0:05; 0:005 (from green to orange).
(Right) Viscosity to entropy ratio, 4
k=s along the
anisotropic direction for T = 0:005; 0:05; 0:1. The viscosity is given in terms of the horizon data as
k = g2(r0)=ph(r0) [29]. The violation of the KSS bound is evident. On the contrary the ratio
along the isotropic direction saturates exactly the KSS bound 1=4
and it is not shown here.
are strong divergences at zero temperature. This can be attributed to the anisotropy of the
critical point. For convenience, we de ne the ratio "0 at the horizon (also see gure 5a),
as the measure of spatial anisotropy along the z direction at the horizon. More precisely,
from the expressions of the diag in eq. (2.9), one can see that the ratio of the two at zero
temperature becomes
"0
h(r0)
g(r0)
1
? =
h(r0)
g(r0)
r
0
2( 0 1)
;
2
HJEP07(218)49
(2.10)
(2.11)
which clearly diverges at the quantum critical point Mc. Another way of achieving the same
conclusion is to analyze the AC conductivities [30]. From there, or simply from eq. (2.11),
we can indeed conclude that
?
=
later see that this ratio "0 plays a key role in the behaviour of the butter y velocity. In
some sense, such a result is not surprising [11, 12] since in theories with anisotropic scalings,
one also observes a violation of the KSS bound [13, 14, 31]. As shown in [29], in the model
we consider, the viscosity along the anisotropic direction
violates the KSS bound (see
gure 5b). It is important to note that the ratio between the ? quantities and their
relatives is always xed by the anisotropic parameter de ned previously,
!2( 0 1), which blows up at the DC limit. We will
? =
? = 1 + "0 :
(2.12)
We will next see that this will still be true for the butter y velocities vB2 and will ultimately
be responsible for the violation of the maximization hypothesis. We show the behavior of
the anisotropy parameter "0 is a function of M in gure 5a. As already discussed, the
{ 9 {
anisotropy parameter is peaked around the quantum critical point and it blows up at
T = 0 following eq. (2.11).
3
Quantum chaos & universality
In this section, we compute the butter y velocity for the above holographic model.
After obtaining a general expression of vB in terms of the near-horizon data for a given
background, we (numerically) solve it near the quantum phase transition. Consider an
anisotropic black brane metric
dr2
f (r)
(not to be confused with viscosity) counts the number of di erent warp factors,
h( )(r), present in the
= f~x( )g sub-manifold of the above background; thus, D =
P d , where d = dim( ). The growth of the commutator in eq. (1.1) can be studied in
holography by perturbing a black hole with a localized operator V(~x; t) [32, 33]. After a
su ciently long time, (t > tr =
) the backreaction of this perturbation grows enormously,
giving rise to a shockwave pro le, (~x; t), spreading at a speed vB. Before the perturbation
has been completely scrambled (t < ts + j~xj=vB), the OTOC behaves as
appendix B we solve the shock-pro le for the above background and obtain the butter y
(~x; t)2. In
velocities for an anisotropic AdS background. Note that in an anisotropic background, the
velocity of the shockwave-front will depend on the spatial sector
, and the full pro le
(~x; t) can be approximated as a product of the shock-pro le of each sector. Doing so,
we obtain
Note that 1= de nes a theory-dependent, dimensionless IR length-scale in the problem,
a screen length over which the shock-pro le (exponentially) decays, see eq. (B.14). This
quantity plays an important role in our discussion and below we analyze this further.
An alternative way to express this is through the following near-horizon quantities |
surface gravity,
= 2
T , and the area density of the r-slices, which relates the horizon
with the entropy density of our dual QFT. We de ne the density of an r-slice which is
simply proportional to the area of the spatial surface, A2(r)
h(d )(r). Thus is
2 =
log A
r=r0
:
(3.2)
(3.3)
For the holographic model considered in the previous section, we have one anisotropic
direction z, that is, two butter y velocities. The velocity along the z-axis is denoted v
?
and that on the xy-plane is denoted v . Now we use eq. (3.2) to obtain the butter y
velocities for the background in eq. (2.5). Since this a holographic theory, the Lyapunov
exponent naturally saturates the Maldacena bound [
34
], L = 2 = . In the unit of ~ =
1 = kB, the maximal Lyapunov exponent is equal to surface gravity, L = ; however, to
avoid ambiguity relating the source of the thermal factor, we continue distinguishing them
and write
v
? =
2 =
2
pg1
g2 +
g1
;
h2
2h1
v =
Here we have used the near-horizon expansion of the metric functions, g(r) = g1 +g2(r r0)
and h(r) = h1 + h2(r
r0) discussed in appendix A, which involves Az(r0)
Az1 and
1. Also, we have set the horizon radius to r0 = 1. As discussed in the previous
section, the boundary theory is described by two dimensionless parameters, (M ; T ). In
turn, this xes two near-horizon quantities, ( 1; Az1). All other IR variables are functions
of (M ; T ), through ( 1; Az1). In gure 6 we numerically obtain the behavior of the butter y
velocities. Although, as noted in [35], there is a characteristic behavior of vBs near the
critical point; however, there is a clear departure from the result of [
1
] since the velocity
along the anisotropic direction seems to attain a local minimum around the critical point,
instead of a local maximum. The apparent inability of v to attain a maximum can be
traced back to the anisotropic scaling. As before, this can be seen from the ratio,
2
v
v
?2 =
h(r0)
g(r0)
= 1 + "0 :
(3.4)
(3.5)
(3.6)
?
Since we observe nite v2 at g = gc, the divergence of this ratio at the critical point causes
v to vanish. In other words, it is the length scale appearing in the formula of the butter y
velocity that sources the deviation from the maximization behavior. Hence, modulo this
B
length scale, v( ) maximizes only when
is minimized. Hence, if we consider the
dimensionless information screening length L
1= instead, perhaps a universal statement can
be made irrespective of the anisotropic scaling of the QPT. In this regard, we conjecture
that L, and not the butter y velocity vB, maximizes across a quantum phase transition.
Notice that in the isotropic case, the two statements are perfectly equivalent, and therefore
the previously conjectured bound holds. Before discussing this more generally, we analyze
the asymptotic limits of 2 in our system, using eq. (3.5) as a guide.
Firstly, at M = 0, since there is no perturbation, we have UV = p
of 6 is simply twice the spatial-dimension of the boundary CFT, 2D = 2 P d , which also
xes the butter y velocity of a d-dimensional Schwarzschild black hole background [37]. At
zero temperature, as M is increased, until one crosses Mc, there is no condensate, causing
2 to stay unchanged. At the critical point (using eq. (2.7) for the critical background)
6 ' 2:45. The factor
we have c = (4 + 2 0)1=2
observes no transition in
2
causes 2 to sharply decrease at the critical point. For an isotropic system ( 0 = 1), one
' 2:19. As discussed before, NEC forces 0 < 1. In turn this
. This sharp transition at the critical point for 0 6= 1 smears
out becoming a cross-over behavior at nite temperature. A
nal question is whether or
not 2 monotonically decreases after the transition or if it increases. The IR asymptotic
2We thank Viktor Jahnke for pointing this out. This bound was observed to be violated in [11, 12].
0.5
1.5
2
0.5
show the QCP. As one lowers the temperature the behavior of vBs near the critical point becomes
increasingly non-analytic.
Note the longitudinal (w.r.t. anisotropy direction) butter y velocity
behaves exactly opposite to its maximization observed in [
1
]. The vB values have been normalized
by their asymptotic values at M = 0, that is, 2=p6. This is obtained from eq. (3.5), and is equal
to the bound in [
36
], vB2 = (D + 1)=2D, which is clearly violated2 by v? at larger M .
value of 2, using the data of eq. (2.6), is
larger than c; in fact it is bound to be larger than
IR = (6 + 9=4 )1=2 ' 5:34. Clearly this is
UV as well since
is always positive.
At nite temperature this asymptotic value softens but stays larger than the critical value
for low enough temperature. We plot the behavior of
in gure 7 which conforms to our
inference and conforms to
c
UV
IR
or, L c
L UV
L IR :
(3.7)
Now, in the spirit of [19], we attempt to understand whether this conclusion remains
valid if the boundary operator assumes any other scaling dimension. This discussion is
con ned just to the insulating phase since the scalar deformation operator condenses only
for large M . In other words, when the second- or higher- order terms in
in eq. (3.5). We focus on the behavior of
at low temperature, and when M
2 are turned on
Mc
1, so
that we can simplify our treatment by using the scalar hair 1 as a perturbation parameter.
Also, since away from the critical point,
behaves analytically and monotonically so as to
the QPT, we rst consider m2 only. At this order, 2
6
establish our lower-bound conjecture, it su ces to justify that
starts increasing as one
enters slightly into the insulating phase. The coe cient of O( 21) term is simply the e ective
mass of the scalar hair, me2 = m2 + gzzq2Az2. Since at low temperature gzz = 1=h1 ! 0 at
m2 21=2, and only for m2 < 0
one has increasing . Recall [38] that the mass of a bulk scalar eld is xed by the scaling
dimension of the dual boundary operator as m2 =
(
d). The BF instability prevents
this mass from becoming smaller than mBF =
conjecture, m2 < 0 is true as long as
d2=3 (in this case, mBF =
4). For our
< d, or the perturbation is relevant. It should
0.44
0.41
L
T = 0:1; 0:05; 0:005 (from green to orange). In the IR limit it asymptotes to L IR and in the UV
this is L UV. The inset zooms into the behavior around the critical point. For low temperature L
maximizes around the critical temperature and reaches the maximum, L c = (4 + 2 0) 1=2. Since
NEC ensures 0 < 1, thus L c is always larger than L IR. In the text we argue for this maximum to
be a universal property.
be noted that this is a fundamental requirement in order to generate a QPT, since by
perturbing a UV with an irrelevant operator, one can never generate a non-trivial RG
ow
towards an IR
xed point. This is indeed the case as noted in the numerical studies of [19].
Thus, irrespective of the scaling dimension of the boundary deformation operator, one can
de ne a lower bound on the length scale of information scrambling, which is xed by the
CFTd. For a non-relativistic CFTd with a scaling anisotropy
0, along a D -dimensional
sub-space (D
= D
D?), the upper bound is (using eq. (3.3) for a generic background)
2L
1
D + ( 0
1) D
2Lc ;
(3.8)
and the equality is saturated exactly at the quantum critical point,3 g = gc as illustrated in
the gure above. Note that ultimately it is the NEC that restricts 0 to be less than one, and
3Since the anisotropic geometry turns out to be the critical geometry in the above model, the saturation
happens at the QCP leading to the violation of the maximization-result. However, a system exhibiting
such geometries in the UV or IR might saturate this bound away from the QCP. Thus, the signi cance of
the bound should not necessarily be attached to quantum criticality but rather should be seen more as a
universal feature of the near-horizon IR geometry. We thank Elias Kiritsis for discussing this issue.
hence, makes the critical value Lc larger as compared to any other asymptotic value. In the
case of isotropy, the maximum on the information screening length L becomes translated
to the maximum of the butter y velocity vB since vB
L L. Nevertheless, as we showed,
in the presence of anisotropy ( 0 6= 1), the statement about the butter y velocity does not
hold anymore and it has to be replaced by the behavior of the dimensionless information
screening length L.
4
Throughout this work, we studied the onset of quantum chaos on an anisotropic quantum
phase transition in a holographic bottom-up model. In particular, we focused on the
behavior of the butter y velocities in the quantum critical region and across the quantum
phase transition.
We observed a disagreement with the results proposed in [
1
].
More
precisely, the butter y velocity along the anisotropic direction does not develop a maximum
but rather a minimum at the quantum critical point. We reiterate the similarity of our
conclusions with the violation of the Kovtun-Son-Starinets (KSS) lower bound on the
viscosity to entropy density ratio [13, 14]. In either cases, the presence of the anisotropic
scaling, 0 seems to play an identical role. The viscosities have indeed been computed [29]
within the holographic model we considered and, as expected and already mentioned, the
=s ratio along the anisotropic direction violates the KSS bound, recall gure 5b.
As a remedy, we propose an improved conjecture which also holds in the presence
of anisotropy, and is stated in eq. (3.8). This involves a length scale, L, from the bulk
perspective which can be computed using eq. (3.3). For the boundary theory this may
be indirectly extracted by measuring the ballistic growth of a local perturbation through
the OTOC and combining this with the measurement of various transport properties such
as viscosity or conductivity along speci c anisotropic directions. This is needed since the
factors g(r0) or h(r0) can only be made relevant to the boundary theory through these
quantities, such as in eq. (2.9). In an anisotropic case, we observe L c
L UV
L IR;
however for the isotropic case we do not expect L to have a local maximum at the critical
point, that is L c = L UV. It would be interesting to understand the physics behind this L
more precisely, especially to see if the emergence of this length scale in a strongly correlated
theory can be better understood without making any reference to AdS/CFT.
Acknowledgments
We thank Panagiotis Betzios, Alessio Celi, Thomas Faulkner, Karl Landsteiner, Yan Liu,
Napat Poovuttikul, Valentina Giangreco Puletti, for useful discussions and comments about
this work. We thank Ben Craps, Dimitrios Giataganas, Viktor Jahnke and Elias Kiritsis
for valuable and constructive comments on the rst version of this paper. We are grateful
to Wei-Jia Li for reading a preliminary version of the draft. We acknowledge support from
Center for Emergent Superconductivity, a DOE Energy Frontier Research Center, Grant
No. DE-AC0298CH1088. We also thank the NSF DMR-1461952 for partial funding of
this project. MB is supported in part by the Advanced ERC grant SM-grav, No 669288.
MB would like to thank Marianna Siouti for the unconditional support. MB would like to
thank University of Iceland for the \warm" hospitality during the completion of this work
and Enartia Headquarters for the stimulating and creative environment that accompanied
the writing of this manuscript.
A
The holographic background
We discuss some more details about the gravitational background here and some aspects of
the pertaining numerics. We follow closely [8]. The equations of motions derived combining
the action in eq. (2.2) with our ansatz in eq. (2.5) are (note in order to be consistent with
HJEP07(218)49
the notations in Landsteiner et al. we have switched f ! u; g ! f ):
Here the primes denote derivative with respect to the radial-coordinate.
We want to
nnumerically integrate the system of equations (A.1) from the horizon r = r0 to the
boundary r = 1. In order to do so we rst try to
nd the asymptotic behavior of the
solutions near the IR boundary (horizon) and UV (conformal) boundary. Close to the UV
boundary, the bulk elds have the following leading order asymptotic expansion:
u = r2 + : : : ;
f = r2 + : : : ;
h = r2 + : : : ;
Az = b + : : : ;
=
+ : : : :
(A.2)
M
r
Note that we have rescaled the boundary values of the three di erent metric functions to
unity, such that the boundary eld theory depends only on the following free parameters,
T; b; M . The removal of the boundary values of the metric is achieved by invoking the
following (three) scaling symmetries
1. (x; y) ! a(x; y); f ! a 2f ;
2. z ! az; h ! a 2h; Az ! a 1Az;
3. r ! ar; (t; x; y; z) ! (t; x; y; z)=a; (u; f; h) ! a2(u; f; h); Az ! aAz .
Owing to there symmetries we only have two dimensionless scales, T and M , which control
the entire of the solution space. The near-horizon expansion up to O(r
r0) can be
1.0
0.8
at T = 0:05. The various colors (from blue to brown) are M = 0:66; 0:724; 0:736; 0:743; 0:757; 0:8.
The phase transition can be seen from the a large shift og the near-horizon values of the bulk elds
when M exceeds 0:744.
HJEP07(218)49
1.2
1.0
Left: the values of (Az1; 1) for the horizon shooting. Center: the value of 1 in function of M .
Right: the value of f1 in function of M .
written as
u ' 4 T (r
r0) + u2 (r
r0) ;
Az ' Az1 + Az2 (r
r0) ;
f ' f1 + f2 (r
r
' 1 +
2 (r
r0) ;
r0) :
h ' h1 + h2 (r
r0) ;
(A.3)
Here Az1 and 1 are the only free parameters, being controlled by the boundary data T
and M . From now onward, we also set the horizon radius to r0 = 1. In summary, while
the horizon data are (T; r0; f1; h1; Az1; 1), using the (three) scaling symmetries they get
reduced to (T; Az1; 1). At the conformal boundary they take the form of (T; M; b). We
can now use shooting to construct the numerical background on the 2D plane of (M ; T ).
An example of the bulk pro les for the Az(r) and (r) elds is shown in gure 8.
Butter y velocities in anisotropic backgrounds
Here we set up the shock wave equation in a generic anisotropic (in the spatial eld theory
directions) background with constant curvature. For this we closely follow the derivations
presented in [
2, 33, 39
]. Consider the following d-dimensional background with a black hole
counts the number of di erent warp factors, h( )(r), present in the
manifold of the above background. The treatment of Sfetsos con nes to
= f~x( )g
sub= 1, however,
here we are interested in the case when
> 1. The black hole (or black brane) horizon
is assumed to be located r0, such that f (r0) = 0 with non-vanishing a(r0) and b(r0). The
temperature of the black hole is, 4 T = 2
gravity. The background is assumed to be sourced by a stress tensor, T (0). For further
= f 0(r0)pa(r0) b(r0), here
is the surface
simpli cations we rst move to tortoise coordinate,
In the last line we've done a near-horizon expansion of r which is justi ed since r (r0)
blows up. Next we move to Kruskal coordinate by exponentiating the null coordinates of
t r space,
u = e2 T (r t) ; v = e2 T (r +t) =) r =
ln(uv) ; t =
1
4 T
1
4 T
ln
v
u
In this coordinate the horizon is at uv = 0 and the boundary is at uv =
1. The black
hole singularity is at uv = 1. The above relation can be used to express the background in
Kruskal coordinates
ds(20) = 2A(uv)dudv + X h( )(uv)d~x(2 ) ; 2A(uv) =
a((2r)fT()r2) e 4 T r :
ds(20) = a(r)f (r) dr2
dt2 +
r (r) =
Z r
dr0
r0 f (r0)pa(r0) b(r0)
;
1
4 T
ln
r
r0
r0
:
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.7)
We will need the following relations later, h0(0) = r0h0(r0), and using near-horizon
expansion of f (r) we have, 2A(0) = (2 rT0 )2 a(r0)f 0(r0) and 2A0(0) = (2 rT0 )2 (a(r)f 0(r))0jr0.
2
One can think of the above background is being generated from stress tensor T (0) by
using Einstein equation, G(0) = 8 T (0), where G(0) is the Einstein tensor corresponding to
ds(20) and
T (0) = Tu(0v) dudv + Tu(0u) du2 + Tv(v0) dv2 + T (0) d~x(2 ) + Tu(0) dudx :
(B.6)
Starting from eq. (B.5) we now obtain the butter y velocity. For that we perturb our
background with a point particle that is released from ~x = 0 at time tw in the past. The
particle is localized onn the u = 0 horizon but moves in the direction of v with light speed.
For late time, tw >
its energy density can be written as [40]
Tupu = E0e 2 tw (u) (~x)
v, we replace dv ! dv
perturbed metric
and the stress tensor is (along with T p)
We want to compute the backreaction of this stress tensor on our background. This can
be done perturbatively for a small energy density. One can start with an ansatz solution
that v gets shifted by
(~x) only for u > 0, v ! v +
(u) (~x). This new geometry is the
shockwave geometry and we want to solve for
(~x), that is the shockwave. By relabeling
(u) (~x)du. Plugging this in the above metric we obtain the
Since ds(21) doesn't generate nite Einstein tensor, G(u1v) = 0, we can demand (u)Tv(v0) =
0 = (u)G(v0v). There remains only one relevant Einstein equation that gives rise to the
shock wave equation (which is subject to the previous contstraint)
Or, X
A(0)
h( )(0) ( )
dim( ) h0( )(0) !
2h( )(0)
G(u1u) = 8 Tupu
(u) (~x)G(u0v) :
(~x) = 8 E0e 2 tw (~x) ;
=)
( )
where, M 2 = h( )(0) X dim( )
h0( )(0)
2A(0)h( )(0)
(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
(B.13)
(B.14)
HJEP07(218)49
( )
In the second last line, assuming linear order, we have divided the solution space into
di erent anisotropy sectors, labeled by . Clearly, for the isotropic case,
= 1 = , one
recovers the shock equations of [2, 33], with dim(
) = d
2. Also if the eld theory
living at a constant r; t-slice is curved then the shock front is no longer planar but depends
on the curvature of the spatial slice, thus its dynamics involves curved space Laplacian,
pg( ) g( ) @ , rather than the at space Laplacian used above. This a ects
the spatial-pro le of the shock but not its speed, that is the butter y velocity [
41
]. We
want to solve this equation, which is equivalent to solving the Green's function of the at
space Laplacian. At very long distance (x
M 1) the solution becomes
(~x( ); t)
e 2 (tw t) M j~x( )j
j~x( )j 2
d 3
:
Note that the factor 2 = is the Lyapunov exponent for Einstein gravity. Note that M 1
de nes the screening length-scale in the problem and
ter y velocity, as can be seen in the above equation, is a ratio of these two scales
L1 de nes the timescale. The
but
M(2 ) = h( )(r)b(r)f 0(r) X dim( ) h0( )(r)
4h( )(r) r0
:
(B.15)
Here vB( ) is the velocity corresponding to the shockwave propagating in the
subspace.
In de ning M( ) we have used the expression in eq. (B.13) and switched from Kruskal
coordinates to usual Schwarzschild coordinates using the identities discussed previously.
For simplicity, we set a(r) = b(r) = 1 and rewrite M(2 ) in terms of a dimensionless quantity
, such that
( )
h( )(r)
=
T
X dim(
) h0( )(r)
h( )(r) r0
:
(B.16)
Open Access.
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