JHEP02(2020)132
For black branes, we impose the following ansatz:
ds
2
= −
U(r)
L
2
e
2W
0
(r)
dt
2
+
r
2
L
2
e
2W
1
(r)
dx
2
+
r
2
L
2
dy
2
+
L
2
dr
2
U(r)
,
z(r, x) = R(r)e
ikx
. (2.4)
This configuration is called Q-lattice which is useful to see the modulation effect in holo-
graphic study [22]. Taking this ansatz, the sources and the vacuum expectation val-
ues(VEVs) of the oprators in (2.1) become trigonometric functions of x.
The equations of motion are given by second order differential equations for W
1
(r) and
R(r) and first order differential equations for U (r) and W
0
(r). Furthermore we take the
following scaling:
r = r
h
˜r , U(r) = r
2
h
u(˜r) , R(r) = ρ(˜r) ,
W
0
(r) = w
0
(˜r) , W
1
(r) = w
1
(˜r) , k =
r
h
L
2
˜
k , (2.5)
where r
h
is the location of the horizon. Then the regularity conditions at the horizon is
given as follows:
u(1) = 0 ,
ρ
0
(1) = −
ρ(1)e
−2w
1
(1)
˜
k
2
ρ(1)
2
+ 1
+ 2
ρ(1)
2
− 1
e
2w
1
(1)
ρ(1)
2
− 3
,
w
0
1
(1) = −
2
˜
k
2
ρ(1)
2
e
−2w
1
(1)
ρ(1)
4
− 4ρ(1)
2
+ 3
. (2.6)
The detailed explanation to obtain these conditions is provided in appendix B. Therefore, a
hairy black brane solution is parametrized by r
h
, ρ(1), w
0
(1), w
1
(1) and
˜
k. However w
0
(1)
and w
1
(1) do not play any crucial role in this system. A choice of these values changes
w
0
(∞) and w
1
(∞) which are compensated by a trivial coordinate transformation:
t → e
W
0
(∞)
t , x → e
W
1
(∞)
x , k → e
−W
1
(∞)
k. (2.7)
Thus w
0
(1) and w
1
(1) can be fixed by suitable values and the relevant parameters of the
black brane are r
h
, ρ(1) and
˜
k. We plot a numerical solution in figure 1. For k = 0 case,
the numerical solution is reduced to that in [31].
The temperature and entropy density of the black branes are given by
T =
U
0
(r
h
)
4πL
2
e
W
0
(r
h
)−W
0
(∞)
=
r
h
4πL
2
u
0
(1)e
w
0
(1)−w
0
(∞)
=
r
h
4πL
2
e
w
0
(1)−w
0
(∞)
3−ρ(1)
2
1−ρ(1)
2
, (2.8)
s =
r
2
h
4GL
2
e
W
1
(r
h
)−W
1
(∞)
=
r
2
h
4GL
2
e
w
1
(1)−w
1
(∞)
. (2.9)
On the other hand, we need holographic renormalization [32–34] to evaluate various phys-
ical quantities in the dual field theory. See [12, 35] for closer applications to this work.
Such quantities are determined by the asymptotic behavior of the fields. One can find
– 4 –
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