Physics Letters B 768 (2017) 393–396
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Van der Waals-like behaviour of charged black holes and hysteresis
in the dual QFTs
Mariano Cadoni, Edgardo Franzin
∗
, Matteo Tuveri
Dipartimento di Fisica, Università di Cagliari & INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy
a r t i c l e i n f o a b s t r a c t
Article history:
Received
10 January 2017
Received
in revised form 22 February 2017
Accepted
27 February 2017
Available
online 4 March 2017
Editor:
M. Cveti
ˇ
c
Using the rules of the AdS/CFT correspondence, we compute the spherical analogue of the shear viscosity,
defined in terms of the retarded Green function for the stress-energy tensor for QFTs dual to five-
dimensional
charged black holes of general relativity with a negative cosmological constant. We show
that the ratio between this quantity and the entropy density,
˜
η/s, exhibits a temperature-dependent
hysteresis. We argue that this hysteretic behaviour can be explained by the Van der Waals-like character
of charged black holes, considered as thermodynamical systems. Under the critical charge, hysteresis
emerges owing to the presence of two stable states (small and large black holes) connected by a meta-
stable
region (intermediate black holes). A potential barrier prevents the equilibrium path between
the two stable states; the system evolution must occur through the meta-stable region, and a path-
dependence
of
˜
η/s is generated.
© 2017 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
.
The investigation of black brane configurations with holo-
graphic
duals has become an important source of information both
for the understanding of fundamental features of the gravitational
interaction and for the description of strongly coupled QFTs [1–5].
In this context, the hydrodynamic limit of holographic QFTs plays
a very important role, because it allows computing transport coef-
ficients,
like the shear viscosity to entropy density ratio η/s, in the
strongly coupled regime of the QFT. This has led to the proposal
of a fundamental bound η/s ≥ 1/4π, known as the KSS bound [1],
which found support both from string theory [2] and quark–gluon
plasma experimental data [6]. By now, it is well-known that the
KSS bound is violated by higher curvature terms in the Einstein–
Hilbert
action [7] or by breaking of translational or rotational
symmetry of the black brane background [8–16]. Typically, when
the KSS bound is violated, η/s exhibits a non-trivial dependence
on the temperature [17].
Until
now, these investigations have been restricted to pla-
nar
topologies in the bulk (black branes) and have not concerned
spherical topologies (black holes). The main obstruction to this
generalisation is the absence of the usual hydrodynamic limit for
QFTs dual to spherical black holes. Indeed, differently from the
black brane case, the spherical geometry of the horizon breaks the
*
Corresponding author.
E-mail
addresses: mariano.cadoni@ca.infn.it (M. Cadoni),
edgardo.franzin@ca.infn.it (E. Franzin), matteo.tuveri@ca.infn.it (M. Tuveri).
translational symmetry in the dual QFT preventing the existence
of conserved charges. However, it is still possible to define a rela-
tivistic
hydrodynamics in curved spacetimes without translational
symmetry as an expansion in the derivatives of the hydrodynamic
fields of the stress-energy tensor [18] and a related Kubo formula
for the shear viscosity.
The
hydrodynamic limit of a QFT living on a curved spacetime
can be defined in the same way as for a QFT in the plane. We just
consider the system at large relaxation times (small frequencies)
and large scales compared to the microscopic scale of the system.
When the latter is unknown, we can still give a thermal descrip-
tion
of the system and associate this microscopic scale with the
inverse of the temperature T . Thus, the hydrodynamic limit cor-
responds
to consider excitations of the system with wavelength
λ 1/T . In this limit, the macroscopic behaviour of the QFT liv-
ing
in a curved background is described by a stress-energy tensor,
which can be written as [18,19]
T
ab
=
(
+ P
)
u
a
u
b
+ Pg
ab
+
ab
, (1)
where and P are the energy density and the thermodynami-
cal
pressure and u
a
is the fluid velocity, usually considered in the
frame in which the fluid is at rest. The tensor
ab
contains all the
dissipative contributions to the stress-energy tensor. At first order
in the velocity expansion, it depends on the spacetime curvature κ,
the relaxation time τ
and the shear viscosity η.
The
previous considerations hold for a QFT in a generic curved
space. Working in the AdS/CFT framework, we can apply eq. (1) to
http://dx.doi.org/10.1016/j.physletb.2017.02.060
0370-2693/
© 2017 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
.