Physics Letters B 747 (2015) 395–399
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Non-abelian (hyperscaling violating) Lifshitz black holes in general
dimensions
Xing-Hui Feng
∗
, Wei-Jian Geng
Department of Physics, Beijing Normal University, Beijing 100875, China
a r t i c l e i n f o a b s t r a c t
Article history:
Received
6 March 2015
Received
in revised form 1 June 2015
Accepted
11 June 2015
Available
online 16 June 2015
Editor:
M. Cveti
ˇ
c
We consider Einstein gravities coupled to a cosmological constant and multiple SU(2) Yang–Mills fields in
general dimensions and find that the theories admit colored Lifshitz solutions with dynamic exponents
z > 1. We also introduce a Maxwell field and construct exact electric charged black holes that asymptote
to the z = D − 1 colored Lifshitz spacetimes and analyze their thermodynamical first law. Furthermore,
we introduce a dilaton to the system and construct Lifshitz spacetimes with hyperscaling violations.
After turning on the Maxwell field, we obtain a class of hyperscaling violating Lifshitz black holes when
θ =
2
D−2
[z − (D − 1)].
© 2015 Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Holographic technique provides a powerful and wide-ranging
tool to study strongly-coupled field theories by embedding them
in the boundaries of some gravitational backgrounds such as the
anti-de Sitter (AdS) spacetimes. In condensed matter physics the
gauge/gravity duality has been applied successfully to – among nu-
merous
other systems – quantum critical points exhibiting Lifshitz
[1] and Schrödinger [2] symmetry. Lifshitz quantum critical points
are invariant under the scaling symmetry
t → λ
z
t, x
i
→ λx
i
, (1)
where z is the dynamical critical exponent. In [1], a candidate
gravity dual for Lifshitz fixed points was proposed, with metric
ds
2
=
2
−
r
2z
dt
2
+
dr
2
r
2
+r
2
dx
i
dx
i
.
(2)
This is the so-called Lifshitz spacetimes. It is invariance under (1)
provided
if one scales r → r/λ, where r is the coordinates of the
extra dimension. When z = 1, the metric reduces to the usual AdS
metric in Poincaré coordinates with AdS radius .
Recently,
non-relativistic backgrounds with scale symmetry as a
conformal isometry – but not an isometry – were also put forward
as holographic duals to quantum systems exhibiting hyperscaling
*
Corresponding author.
E-mail
addresses: xhfengp@gmail.com (X.-H. Feng), vergilgeng@gmail.com
(W.-J. Geng).
violation [3–6]. The hyperscaling violating Lifshitz metric is of the
form
ds
2
=r
θ
−
r
2z
dt
2
+
dr
2
r
2
+r
2
dx
i
dx
i
,
(3)
where θ is the hyperscaling parameter. When θ = 0, it reduces to
the Lifshitz spacetime.
Lifshitz
and hyperscaling violating Lifshitz backgrounds have
been discussed in a number of recent papers, including [7–35] and
the references therein. Especially, some exact charged black holes
in SU(2)-colored Lifshitz spacetimes were obtained in four and five
dimensions [7,8]. In this theory, Lifshitz spacetimes themselves can
be colored in Einstein–Yang–Mills gravity.
The
main work of this paper is to generalize the program in
[7,8]. In these two papers, the authors considered Einstein grav-
ity
with a cosmological constant minimally coupled to one SU(2)
Yang–Mills field. The self-interaction of the Yang–Mills field allows
one to find a class of colored Lifshitz vacua. In this construction,
the dimension of the Euclidean space x
i
is related to the number
of SU(2) generators. Since the group SU(2) has three generators,
it turns out that there are two possible non-trivial choices of the
gauge potentials, giving rise to four and five spacetime dimen-
sions [7,8].
In order to consider non-abelian Lifshitz spacetimes in
higher dimensions, we need to consider larger groups. The sim-
plest
way is to generalize the single SU(2) Yang–Mills field to
multiple SU(2) fields. We shall construct such theories in Section 2.
Not so surprisingly, we find that the theories admit colored Lifshitz
vacua with the scaling exponents z > 1in higher dimensions. Fur-
thermore,
we find that introducing a Maxwell field allows us to
http://dx.doi.org/10.1016/j.physletb.2015.06.030
0370-2693/
© 2015 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
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