
Physics Letters B 749 (2015) 376–382
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Condensation for non-relativistic matter in Ho
ˇ
rava–Lifshitz gravity
Jiliang Jing
a,b,∗
, Songbai Chen
a,b
, Qiyuan Pan
a,b
a
Department of Physics, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Synergetic Innovation
Center for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan 410081, PR China
b
State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
a r t i c l e i n f o a b s t r a c t
Article history:
Received
25 June 2015
Received
in revised form 1 August 2015
Accepted
3 August 2015
Available
online 6 August 2015
Editor:
N. Lambert
Keywords:
Holographic
superconductors
Non-relativistic
matter
Ho
ˇ
ra
va–Lifshitz gravity
We study condensation for non-relativistic matter in a Ho
ˇ
rava–Lifshitz black hole without the condition
of the detailed balance. We show that, for the fixed non-relativistic parameter α
2
(or the detailed balance
parameter ), it is easier for the scalar hair to form as the parameter (or α
2
) becomes larger, but the
condensation is not affected by the non-relativistic parameter β
2
. We also find that the ratio of the
gap frequency in conductivity to the critical temperature decreases with the increase of and α
2
, but
increases with the increase of β
2
. The ratio can reduce to the Horowitz–Roberts relation ω
g
/T
c
≈ 8
obtained
in the Einstein gravity and Cai’s result ω
g
/T
c
≈ 13 found in a Ho
ˇ
rava–Lifshitz gravity with the
condition of the detailed balance for the relativistic matter. Especially, we note that the ratio can arrive
at the value of the BCS theory ω
g
/T
c
≈ 3.5by taking proper values of the parameters.
© 2015 The Authors. 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
The AdS/CFT correspondence [1–3] relates a weak coupling
gravity theory in an anti-de Sitter space to a strong coupling con-
formal
field theory in one less dimensions. Recently it has been
applied to condensed matter physics and in particular to super-
conductivity
[4,5]. In the pioneering papers Gubser [4,5] suggested
that near the horizon of a charged black hole there is in opera-
tion
a geometrical mechanism parameterized by a charged scalar
field of breaking a local U(1) gauge symmetry. Then, the gravita-
tional
dual of the transition from normal to superconducting states
in the boundary theory was constructed. This dual consists of a
system with a black hole and a charged scalar field, in which the
black hole admits scalar hair at temperature lower than a critical
temperature, but does not possess scalar hair at higher tempera-
tures [6].
In this system a scalar condensate can take place through
the coupling of the scalar field with the Maxwell field of the back-
ground.
Much attention has been focused on the application of
AdS/CFT correspondence to condensed matter physics since then
[7–19].
Ho
ˇ
ra
va [20,21] proposed a new class of quantum gravity. The
key property of this theory is the three-dimensional general covari-
ance
and time re-parameterization invariance. It is this anisotropic
rescaling that makes Ho
ˇ
rava’s theory power-counting renormaliz-
able.
Therefore, many authors pay their attention to this gravity
*
Corresponding author.
E-mail
address: jljing@hunnu.edu.cn (J. Jing).
theory and its cosmological and astrophysical applications, and
found many interesting results [22–35]. These investigations im-
ply
that there exists the distinct difference between the Ho
ˇ
rava–
Lifshitz
theory and Einstein’s gravity.
In
the Ho
ˇ
rava–Lifshitz gravity, Kiritsis and Kofinas [36], Kimp-
ton
and Padilla [37] proposed the non-relativistic matter. They
constructed the most general action of matter coupled to gravity
with the foliation-preserving diffeomorphism. The action obeys the
usual power-counting renormalizability conditions used in Ho
ˇ
rava–
Lifshitz
gravity and assuming the temporal derivatives are as in the
relativistic theory.
Recently, in order to see what difference will appear for the
holographic superconductivity in the Ho
ˇ
rava–Lifshitz theory, com-
paring
with the case of the relativistic general relativity, Cai et
al. [38] studied the phase transition of planar black holes in the
Ho
ˇ
rava–Lifshitz gravity with the condition of the detailed balance
in which the metric function is described by f (r) = x
2
−
√
c
0
x.
They argued that the holographic superconductivity is a robust
phenomenon associated with asymptotic AdS black holes. And they
also got a relation connecting the gap frequency in conductivity
with the critical temperature, which is given by
ω
g
T
c
≈ 13, with the
accuracy more than 93% for a range of scalar masses. More re-
cently,
Lin, Abdalla and Wang [39] generalized the investigation
to the holographic superconductors related to the non-relativistic
matter in the Schwarzschild black hole in the low energy limit of
Ho
ˇ
rava–Lifshitz spacetime.
Note that the Ho
ˇ
rava–Lifshitz black hole without the condition
of the detailed balance has rich physics [40–42], i.e., changing the
http://dx.doi.org/10.1016/j.physletb.2015.08.009
0370-2693/
© 2015 The Authors. 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
.