6.4 Probes as deformations and gapless excitations in complex systems
The dynamical regime in which brane effective actions hold is particularly suitable to describe physical
systems made of several interacting subsystems in which one of them has a much smaller number of degrees
of freedom. Assume the larger subsystems allow an approximate description in terms of a supergravity
background. Then, focusing on the dynamics of this smaller subsector, while keeping the dynamics of
the larger subsystems frozen, corresponds to probing the supergravity background with the
effective action describing the smaller subsystem. This conceptual framework is illustrated in
Figure 9.
This set-up occurs when the brane degrees of freedom are responsible for either breaking the symmetries
of the larger system or describing an interesting isolated set of massless degrees of freedom whose
interactions among themselves and with the background one is interested in studying. In the following, I
very briefly describe how the first approach was used to introduce flavour in the AdS/CFT correspondence,
and how the second one can be used to study physics reminiscent of certain phenomena in
condensed-matter systems.
Probing deformations of the AdS/CFT:
Deforming the original AdS/CFT allows one to come up with set-ups with less or no supersymmetry.
Whenever there is a small number of degrees of freedom responsible for the dynamics (typically D-branes),
one may approximate the latter by the effective actions described in this review. This provides a
reliable and analytical tool for describing the strongly-coupled behaviour of the deformed gauge
theory.
As an example, consider the addition of flavour in the standard AdS/CFT. It was argued in [333] that
this could be achieved by adding
D7-branes to a background of
D3-branes. The D7-branes give rise
to
fundamental hypermultiplets arising from the lightest modes of the 3-7 and 7-3 strings, in the brane
array
The mass of these dynamical quarks is given by
, where
is the distance between the D3-
and the D7-branes in the 89-plane. If
the D3-branes may be replaced (in the appropriate
decoupling limit) by an AdS5 × S5 geometry, as in the standard AdS/CFT argument, whereas if, in
addition,
then the back-reaction of the D7-branes on this geometry may be neglected. Thus, one is
left, in the gravity description, with
D7-brane probes in AdS5 × S5. In the particular case of
,
one can use the effective action described before. This specific set-up was used in [348] to study the
linearised fluctuation equations for the different excitations on the D7-probe describing different scalar and
vector excitations to get analytical expressions for the spectrum of mesons in
SYM, at strong
coupling.
This logic can be extended to non-supersymmetric
scenarios.
For example, using the string theory realisation of four-dimensional QCD with
colours and
flavours discussed in [499
]. The latter involves
D6-brane probes in the supergravity background dual
to
D4-branes compactified on a circle with supersymmetry-breaking boundary conditions and
in the limit in which all the resulting Kaluza–Klein modes decouple. For
and for
massless quarks, spontaneous chiral symmetry breaking by a quark condensate was exhibited
in [349] by working on the D6-brane effective action in the near horizon geometry of the
D4-branes.
Similar considerations apply at finite temperature by using appropriate black-hole backgrounds [499] in
the relevant probe action calculations. This allows one to study phase transitions associated with the
thermodynamic properties of the probe degrees of freedom as a function of the probe location. This can be
done in different theories, with flavour [379], and for different ensembles [343, 378].
The amount of literature in this topic is enormous. I refer the reader to the reviews on the use of
gauge-gravity duality to understand hot QCD and heavy ion collisions [137] and meson spectroscopy [207],
and references therein. These explain the tools developed to apply the AdS/CFT correspondence in these
set-ups.
Condensed matter and strange metallic behaviour:
There has been a lot of work in using the AdS/CFT
framework in condensed matter applications. The reader is encouraged to read some of the
excellent reviews on the subject [283, 296, 385, 284, 285], and references therein. My goal in
these paragraphs is to emphasise the use of IR probe branes to extract dynamical information
about certain observables in quantum field theories in a state of finite charge density at low
temperatures.
Before describing the string theory set-ups, it is worth attempting to explain why any AdS/CFT
application may be able to capture any relevant physics for condensed matter systems. Consider the
standard Fermi liquid theory, describing, among others, the conduction of electrons in regular metals. This
theory is an example of an IR free fixed point, independent of the UV electron interactions, describing the
lowest energy fermionic excitations taking place at the Fermi surface
. Despite its success, there is
experimental evidence for the existence of different “states of matter”, which are not described by this
effective field theory. This could be explained by additional gapless bosonic excitations, perhaps arising as
collective modes of the UV electrons. For them to be massless, the system must either be tuned
to a quantum critical point or there must exist a kinematical constraint leading to a critical
phase.
One interesting possibility involving this mechanism consists on the emergence of gauge fields
(“photons”) at the onset of such critical phases. For example, 2 + 1 Maxwell theory in the presence of a
Fermi surface (chemical potential
)
is supposed to describe at energies below
, the interactions between gapless bosons (photons) with the
fermionic excitations of the Fermi surfaces. The one-loop correction to the classical photon propagator at
low energy
and momenta
is
Due to the presence of the chemical potential, this result manifestly breaks Lorentz invariance, but there
exists a non-trivial IR scaling symmetry (Lifshitz scale invariance)
with dynamical exponent
, replacing the UV scaling
. Since these
systems are believed to be strongly interacting, it is an extremely challenging theoretical task to
provide a proper explanation for them. It is this strongly-coupled character and the knowledge of
the relevant symmetris that suggest one search for similar behaviour in “holographic dual”
descriptions.
The general set-up, based on the discussions appearing, among others, in [334
, 398
, 286
], is as
follows. One considers a small set of charged degrees of freedom, provided by the probe “flavour”
brane, interacting among themselves and with a larger set of neutral quantum critical degrees of
freedom having Lifshitz scale invariance with dynamical critical exponent
. As in previous
applications, the latter is replaced by a gravitational holographic dual with Lifshitz asymptotics [324]
where
will play the role of the holographic radial direction. Turning on non-trivial temperature
corresponds to considering black holes having the above asymptotics [162, 370, 102, 34]
where the function
depends on the specific solution and characterises the thermal nature of the
system.
In practice, one embeds the probe “flavour” brane in the spacetime holographic dual, which
may include some non-trivial cycle wrapping in internal dimensions when embedded in string
theory, and turns on some non-trivial electric
and magnetic fluxes
on the brane
At low energies and in a quantum critical system, the only available scales are external, i.e., given by
temperature
, electric and magnetic fields
and the density of charge carriers
. Solving the
classical equations of motion for the world volume gauge field, allows one to integrate
, whose
constant behaviour at infinity, i.e., at
in the above coordinate system, defines the chemical
potential
of the system. Working in an ensemble of fixed charge carrier density
, which is
determined by computing the variation of the action with respect to
, the free energy density
is given by
where
stands for the volume of the non-compact 2-space spanned by
and
is the
on-shell Dp-brane action. As in any thermodynamic system, observables such as specific heat or magnetic
susceptibility can be computed from Eq. (469) by taking appropriate partial derivatives. Additionally,
transport observables, such as DC, AC or DC Hall conductivities can also be computed and studied as a
function of the background, probe embedding and the different constants controlling the world volume
gauge field (468).
More than the specific physics, which is nicely described in [334, 398, 286], what is important to stress,
once more, is that using the appropriate backgrounds, exciting the relevant degrees of freedom and
considering the adequate boundary conditions make the methods described in this review an extremely
powerful tool to learn about physics in regimes of parameters that would otherwise be very difficult to
handle, both analytically and conceptually.