Physics Letters B 759 (2016) 410–416
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
On anisotropic black branes with Lifshitz scaling
Dibakar Roychowdhury
Indian Institute of Technology, Department of Physics, Kanpur 208016, Uttar Pradesh, India
a r t i c l e i n f o a b s t r a c t
Article history:
Received
28 March 2016
Accepted
6 June 2016
Available
online 8 June 2016
Editor:
N. Lambert
In this paper, based on the method of scalar perturbations, we construct the anisotropic charged Lifshitz
background perturbatively up to leading order in the anisotropy. We perform our analysis both in the
extremal as well as in the non-extremal limit. Finally, we probe the so called superfluid phase of the
boundary theory and explore the effects of anisotropy on the superconducting condensate.
© 2016 The Author(s). 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. Overview and motivation
For the past couple of years, the holographic constructions of
anisotropic black branes in an asymptotically anti-de Sitter (AdS)
space time has been an active field of investigation due to its var-
ious
remarkable implications that range between the physics of
QCD to various exotic phases in usual condensed matter applica-
tions
[1–23]. Motivated by these analyses, recently the authors in
[24] have constructed four dimensional gravitational theories (with
AdS
4
asymptotic) those are homogeneous but do not preserve the
usual SO(2) rotational invariance along the two dimensional spatial
hypersurface. In particular, motivated by the analysis performed in
[1,2], the authors in their analysis [24] have deformed the usual
charged AdS
4
black brane configuration by a relevant scalar de-
formation
that depends linearly on one of the spatial directions
of the brane. Such deformations turned out to be the true source
of anisotropy in the sense that they generate the anisotropic en-
ergy
momentum tensor in which, T
x
x
= T
y
y
. In their analysis
[24], the authors have considered the examples from Einstein–
Maxwell-Dilaton
theory and Einstein–Maxwell-Dilaton–Axion the-
ory
with SL(2, R) symmetry where these scalar deformations could
be thought of as perturbations either due to the dilation or that of
the axion profile in the bulk.
Motivated
by these analyses, the purpose of the present exer-
cise
is to extend the notion of anisotropy from AdS black branes to
black brane configurations with asymptotic Lifshitz scaling [25,26]
that
does not seem to have been addressed in the literature so
far. Before we proceed further, two issues are extremely important
to understand. First of all, it is important to realize that grav-
itational
constructions over (asymptotic) Lifshitz-like geometries
(those are dual descriptions of Quantum Field Theories at Lifshitz
E-mail addresses: dibakarphys@gmail.com, dibakarr@iitk.ac.in.
fixed points) play crucial role in various condensed matter appli-
cations,
1
for example, the hydrodynamic description of quantum
(non-Fermi) liquids near the quantum critical point [28] and in
particular explaining the low temperature anomalous behavior of
the specific heat in heavy Fermion compounds like CeCu
5.9
Au
0.1
for which the value of the dynamic critical exponent turns out
to be, z ≈ 2.7 [29]. It is in fact quite evident by now that holo-
graphic
models with anisotropic (Lifshitz-like) scaling are the basic
building blocks for theories consisting of de-confined compressible
fermionic excitation distributed over hidden Fermi surfaces [30,31].
All the above discussions indeed suggest that holographic descrip-
tions
of Lifshitz-like geometries play extremely important role in
various condensed matter applications where the usual perturba-
tive
approach to field theories does not work well.
The
second important issue turns out to be the role of
anisotropy in holographic theories comprising (asymptotic) Lifshitz
isometry group. In order to address this issue, we need to un-
derstand
the following: In usual condensed matter systems, due
to the presence of background lattice and/or crystal structures
one could in principle imagine the breaking of rotational sym-
metry
associated with the Fermi surfaces. Therefore, under these
circumstances, holographic constructions (those are homogeneous
but not isotropic) with Lifshitz asymptotic might play a crucial role
in order to probe heavy Fermion compounds near its (anisotropic)
Fermi surfaces. The present analysis could be regarded as the first
step towards this direction.
The
organization of the paper is the following: In Section 2,
we review the essential features of unperturbed (original) charged
1
Particularly, in this context it is noteworthy to mention that in nature there
exist several ferromagnetic materials like, MnSi and some unconventional cuprate
superconductors as well as iron pnictides, electronic nematics like Sr
3
Ru
2
O
7
which
exhibit Lifshitz-type anisotropic scaling (namely, t → λ
z
t and x → λx, z > 1) near
its quantum critical point [27].
http://dx.doi.org/10.1016/j.physletb.2016.06.008
0370-2693/
© 2016 The Author(s). 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
.