Diffusion and butterfly velocity at finite density
Received: April
Di usion and butter y velocity at nite density
Chao Niu 0 1 2 3 4
KeunYoung Kim 0 1 2 3 4
di usion. At 0 1 2 3 4
regime ( =T 0 1 2 3 4
0 describing the coupled
1 Gwangju 61005 , Korea
2 School of Physics and Chemistry, Gwangju Institute of Science and Technology
3 Open Access , c The Authors
4 exactly, we nd that in the incoherent
We study di usion and butter y velocity (vB) in two holographic models, linear axion and axiondilaton model, with a momentum relaxation parameter ( ) at nite density or chemical potential ( ). Axiondilaton model is particularly interesting since it shows linearT resistivity, which may have something to do with the universal bound of nite density, there are two di usion constants D di usion of charge and energy. By computing D ArXiv ePrint: 1704.00947
nite; density; Holography and condensed matter physics (AdS/CMT); Gaugegravity cor

1) D+ is identi ed with the charge di usion constant (Dc)
and D
at very small density, D
are `maximally' mixed in the sense that D+(D ) is identi ed
regime De
and . However, Dc
C ~vB2=kBT where C
2=16 2T 2 so, in general, C+
respondence
Contents
1 Introduction Methods 2 3
Di usion constants
Butter y velocity
Linear axion model
Axiondilaton model
Zero density
Finite density
Conclusion
Thermodynamics and transport coe cients
Introduction
heavy fermions etc.), resistivity ( ) is linear in temperature (T )
so called Homes' law1 have been observed [2, 3].
While such interesting phenomena in
holographic methods.
s(T = 0) = C DC(Tc)Tc ;
where C is a material independent universal number.
so called `Planckian' time scale [3, 10]
If Dc saturates to the bound and v2 is temperature independent,
1=T hence linearT
At nite density, there are two di usion constants D
describing the coupled di usion
D+ + D
relaxation time ( P ) [11]:
= Dc , where
is the charge susceptibility. The Einstein relation with (1.4)
is the thermoelectric
susceptibility. If the charge density is zero, since
= 0, D
are decoupled and D+
and D
constant (De) respectively.
In [11], it was proposed the di usion constants are bounded as
2The conductivities may be diagonalized as in [12].
di usivity [14].
inhomogeneous SYK model [26].
perature limit. Unlike the previous studies, we rst study D
at nite density without
between D
on the GubserRocha model. We
nd that there are two branches of classical solutions.
both at zero and nite density. In section 5, we conclude.
models, it has been also observed in condensed matter systems [14, 16{18].
a dynamical transition to an manybody localization phase.
be written as
where C
B T
2 T D v
2 T D
which will be computed and displayed in section 3 and 4.
1 and =
is chemical
is governed by di usion of charge and energy [11].
Di usion constants
From (1.6) and (1.7) two di usion constants are computed as
+ M ;
( ; ; c ) and conductivities ( ; ; ).
First, thermodynamic susceptibilities are de ned as
@T 2 = T
G =
thermodynamics.
S =
1 X2 ( I ( )@ I )
with the ansatz
ds2 =
U dt2 +
+ V1dx2 + V2dy2 ;
A = adt ;
I = I Iixi ;
Z( )s
4 V1
12 1( ) r=rh
12 1( ) r=rh
12 1( )s r=rh
h[W (t; ~x); V (0; 0)]2i ;
i denotes thermal average. The function
general, C(t; ~x) takes the following form
conductivity, is
Butter y velocity
or the following average of the commutator squared:
C(t; ~x) = e L(t t vj~xBj ) +
\butter y e ect cone". Outside of the cone C(t; ~x)
1, even if the operators V and W
chaos spatially propagates through the system.
2 kBT =~ = 2 = P ;
terms of the characteristic parameters in quantum chaos
& vB2 P & 2 vB2= L :
with an infrared geometry
ds2d+2 =
U (r)dt2 +
+ V (r)d~x2d :
L =
B2 =
V 0(rh)
the bound (2.13), the butter y velocity is not.
Linear axion model
S =
1 F 2
The second term is nothing but
with the negative cosmological constant,
a system at
last term, two massless scalar
elds of the form
I =
Iixi =
Iixi are introduced to
relaxation e ect so makes conductivity
nite [27]. An advantage of this ansatz for massless
I , a classical
solution of the action (3.1) is
ds2 =
f (r)dt2 +
f (r) =
A =
I =
we set L = 1.
is interpreted
as the chemical potential of the boundary eld theory,
T =
f 0(rh)
so rh is expressed in terms of T; and :
The entropy density is
rh =
s = 4 rh2 =
= rh =
B2 =
4 T + p6 2 + 16 2T 2 + 3 2 :
4 T + p6 2 + 16 2T 2 + 3 2 2
4 T + p6 2 + 16 2T 2 + 3 2 :
6 T
4 T + p6 2 + 16 2T 2 + 3 2
The butter y velocity (2.16) is
from the metric (3.2) and (3.4).
4 T + p6 2 + 16 2T 2 + 3 2
6 2 + 16 2T 2 + 6 2 !
1 + p6 2 + 16 2T 2 + 3 2
c = T
c = c
9p6 2 + 16 2T 2 + 3 2
9(6 2 + 6 2 + 4 T (4 T + p6 2 + 16 2T 2 + 3 2))
so the electrical, thermal, thermoelectric conductivities are
= 1 +
2 =
4 T + p6 2 + 16 2T 2 + 3 2 2
4 T + p6 2 + 16 2T 2 + 3 2 ;
here for convenience,
D+ =
B2 = p
4 T
3p6 2 + 16 2p6T 2
4 T + p6 2 + 16 2T 2 + 3 2 2
c1 =
c2 =
+ M ;
M =
The analytic formulas of D
can be obtained by plugging (3.8), (3.9), (3.11), and (3.12)
into (3.14). Because the
nal expressions are complicated and not very illuminating we
2 D . The green curve means vB2 (3.7). As =T increases both 2 T D and vB2 go to zero, but all of them behave as 1=( =T ). Thus, 2 T D =vB2 saturate the nite lower bound as
shown in
gure 1(b).
1 and =
1) can be read also from
the analytic expression of D
and vB2 at large :
= 0;
I = 1;
Z = 1;
V =
I =
(a) Di usion constants (2 T D ) and the butter y velocity (vB2)
1 and =
=T = 0:1; 1; 5 from
which yield
C+ =
2 T D+ = 2 +
2 T D = 1 + 2
We nd that C
Indeed, in this regime, D+ and D
can be identi ed with Dc and De respectively because
bounds C+ = 2 and C
identify Dc
= and Dc
gure 2, where the
term is important and for
e ect is `maximal' in the sense D
is identi ed with Dc and D+ is identi ed with De,
T M = p
(a) Di usion constants: 2 T Dc and 2 T De
2 ▲ ▲ ▲ ▲▼▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
2 ▲ ▲▼▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
▼ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
mixing term, the solid curves in gure 1.
which implies that the mixing term may be small if
=T is large even for small
=T . In
and energy di usions are negligible so D+ and D
may be identi ed with Dc and De
respectively. (Because
=T 6 5 in
gure 2, =4 T
5 is the incoherent regime.)
1). To see it more clearly let us consider a di erent case
1 and
which yield the following expansions:
2 T
2 T
B2 =
D+ =
C+ =
2 T D+ =
2 T D
= 1 + p
2 T T
Thus C+ is not universal while C
in a class of holographic models that ow to AdS2
xed points in the infrared. Our
; (3.21)
D+ =
B2 =
M =
energy di usion is expanded as
of our solution is as follows.
and D
= De and
C+ =
2 T Dc =
2 T De = 1 + p
and , which
and a function of
1) it becomes universal.
i.e. C+ = 2.
Axiondilaton model
linear axion model in the previous section.
The action is
S =
1 X2 (@ I )
in low frequency approximation.
{ 11 {
which belongs to (2.7) and is reduced to (3.1) if
h(r)dt2 + dx2 + dy2 +
(Q + rh)3
(Q + r)3
I =
2(Q + rh)2
r2g(r)h(r)
2(Q + rh)2
Q + rh
Q + r
g(r) =
ds2 =
h(r) = 1
A =
r2g(r)
2(Q + r)2
3Q(Q + rh)
log(g(r)) ;
where rh > 0 and rh >
here we set L = 1.
Thermodynamics and transport coe cients
chemical potential and charge density are
T =
s =
rh2g(rh)
rh2g(rh)h(rh) 0
prh(6(Q + rh)2
8 (Q + rh)3=2
= 4 prh(Q + rh)3=2 ;
3Q(Q + rh) 1
= (Q + rh) 3Q(Q + rh) 1
2(Q + rh)2
2(Q + rh)2
t~ = t rh ;
x~ = x rh ;
y~ = y rh ;
~ =
Q~ =
Consequently, we may de ne the eld theory variables as:
T~ =
~ =
~ =
{ 12 {
To nd a possible range of Q~, we impose physical condition T
Without loss of generality, we can take
0. All these inequalities imply
4 T = 0
>><0 < 4 T
1 < Q~
Q~. They correspond to the green region in
gure 3. The boundary of the green region is
nothing but the condition for
gure 3: the red curve
temperature [29]
G =
(Q + rh)3
G(Q~ < 0). At zero density, there is no positive Q~
To compute the di usion constants D
we rst need to compute thermodynamic
sus(4.6) and entropy
and . In principle, rh and Q can be expressed in terms of T and
from (4.3) and (4.5)
but their analytic expressions are very complicated except for
ities read
and from (2.16) the butter y velocity is
= e (rh) +
B2 =
Q + 4rh
Q + rh
{ 13 {
(a) =T = 0:1
(b) =T = 5
G(Q~ < 0) is shown. The positive Q~ solution is always
Zero density
Let us rst consider a neutral case, i.e.
For Q~ = 0, the dilaton eld
vanishes and the solution (4.2) is reduced to (3.2) with
and susceptibilities are given as
4 T + p6 2 + 16 2T 2 ;
c =
4 T + p6 2 + 16 2T 2 2
9p6 2 + 16 2T 2
= 0 and
7The energy di usion constant was also computed in [46].
Dc =
De =
4 T + p6 2 + 16 2T 2 p6 2 + 16 2T 2
De. The green curve displays vB2.
Q~ =
The butter y velocity is
B2 =
6 T
4 T + p6 2 + 16 2T 2
2 T Dc = 2 ;
2 T De = 1 +
2 T
(a) Q~ = 0
(b) Q~ = 1 + p~
4 T + p6 2 + 16 2T 2 :
{ 15 {
For Q~ =
1 + p , there is a nontrivial
coe cients and susceptibilities are8
= 0 and
= 2 ;
Dc =
De =
2 T 4 T ;
B2 =
for =T
2 T Dc =
2 T De =
16 2T 2 for =T
for =T
c = 4 2 2
The butter y velocity is
Q~ = 0, Dc; De
For Q~ =
1 + ~=p2, Dc
Finite density
1 + ~=p2
1 + ~=p2, we nd
1. For
T = and v2
1=T and De
T = 2 while vB2
The horizon position rh can be expressed in terms of T and
by eliminating Q in (4.3)
and (4.5). However, unlike the previous cases, if
{ 16 {
2 T ; v
2 T De
{ 17 {
(a) Di usion constants (2 T D ) and the butter y velocity (vB2)
1 and =
limit, which we
positive Q branch, rh=
! 0 and Q=
are interested in for the universal bound, can be read o
analytically. For large
in the
= 2 ;
Therefore, the charge and energy di usion constants are
4 T
Together with the butter y velocity at large
we have
axion term.
2 T D+
8The electric conductivity was also computed in [47].
(a) Di usion constants: 2 T Dc and 2 T De
=T =
results with the mixing term, the solid curves in gure 5.
Thus we nd that the bounds for D
at zero density (4.22) still hold at nite density. The
correction by
may be identi ed
for Dc and De in
term, where we simply identify Dc
and Dc
is identi ed with Dc and D+ is identi ed with De.
di usion (1.5) is not realized. Even though
realized because
is temperatureindependent in this model.
Conclusion
cases, the axion eld is of the form
I =
Iixi and plays a role of momentum relaxation,
where large
{ 18 {
Linear axion
Axiondilaton
p3=2
1 and =
a ground state by comparing their grand potentials.
There are two di usion constants D
describing the coupled di usion of charge and
We have showed the exact relation between D
and (Dc; De) in
gure 2 and
suppressed so D+ and D
can be identi ed with Dc and De respectively. However, in the
decoupled and D
is `maximal' in the sense that D
is identi ed with Dc and D+ is identi ed with De, which
ten as
2 T
where C
agree to the values at zero density and C
is universal independently of density. In [20],
6 1 at low temperature limit.
Because the IR geometry of the axion model is AdS2
= 1 can be anticipated
Rd so the
Rd) studied
in [20]. However, our result C
Rd, which can be
allows i.e. 1=2 < C
The charge di usion constant is written as
2 T
{ 19 {
2 T : Thus, if there is more relevant velocity scale than the butter y velocity and if it is temperatureindependent, the relation between the universality of charge di usion and
di usion by the following observations.
susceptibilities,
and , the full
di usion constant in the incoherent regime is the ratio of
to c , it can be written only
butter y velocity is robust [49].
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