Physics Letters B 774 (2017) 569–574
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Physics Letters B
www.elsevier.com/locate/physletb
Origin of the Drude peak and of zero sound in probe brane holography
Chi-Fang Chen
∗
, Andrew Lucas
∗
Department of Physics, Stanford University, Stanford, CA 94305, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received
10 September 2017
Accepted
10 October 2017
Available
online 16 October 2017
Editor:
M. Cveti
ˇ
c
At zero temperature, the charge current operator appears to be conserved, within linear response, in
certain holographic probe brane models of strange metals. At small but finite temperature, we analytically
show that the weak non-conservation of this current leads to both a collective “zero sound” mode and
a Drude peak in the electrical conductivity. This simultaneously resolves two outstanding puzzles about
probe brane theories. The nonlinear dynamics of the current operator itself appears qualitatively different.
© 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
.
1. Introduction
One of the earliest applications of the AdS/CFT correspondence
to condensed matter physics was to study holographic “probe
branes” at finite density [1,2]. The holographic dual of such models
is (in the simplest cases) widely believed to be N = 2 supersym-
metric
fundamental matter (analogous to quarks), localized on a
defect within the N = 4 supersymmetric Yang–Mills plasma [3].
Because electrical transport in strongly interacting quantum sys-
tems
remains a challenging problem in condensed matter, much of
the work on these probe brane models focuses on the transport of
the conserved U(1) baryon number. Unfortunately, the probe limit,
where the background plasma is unaffected by the dynamics of the
baryon matter, leads to certain simplifying features of transport
that are absent in more “realistic” holographic models for strange
metals [4,5].
It
is still important to understand the transport properties of
the probe brane models, however; they remain rare examples of
solvable interacting quantum systems in higher dimensions. And
at low temperature, the behavior of probe branes has remained
rather mysterious for almost a decade. Firstly, at low temperatures
one often finds a collective, propagating “sound mode” [6–10].
This cannot be ordinary sound, as the energy-momentum tensor is
dominated by the ‘decoupled’ N =4 plasma; it also cannot be (su-
perfluid)
second sound as the U(1) symmetry is not broken. From
this line of thought, [6] subsequently concluded that this propagat-
ing
mode was analogous to zero sound: the sloshing of the Fermi
*
Corresponding authors.
E-mail
addresses: chifangc@stanford.edu (C.-F. Chen), ajlucas@stanford.edu
(A. Lucas).
surface in a Fermi liquid at low temperature [11]. While there is no
strong evidence for a well-defined baryonic Fermi surface, save for
finite momentum spectral weight [12], we will follow the literature
and call this propagating mode zero sound. Furthermore, when the
zero sound waves are present, the low frequency electrical conduc-
tivity
σ (ω) has an apparent Drude peak at low temperature [13].
Such a Drude peak would normally be associated with approxi-
mate
conservation of momentum [14,5], but as we have already
mentioned, that cannot be the case in probe brane models.
We
demonstrate below that within linear response in probe
brane holography, the charge current operator itself appears to be
conserved at zero temperature when the dynamical critical ex-
ponent
z of the background plasma obeys z < 2. This emergent
conservation law, and the resulting “hydrodynamics”, is responsible
for the Drude singularity in the electrical conductivity, as well as
the propagation of the zero sound mode. At small but finite tem-
perature,
the charge current decays at a rate ∼ T
2/z
; this decay is
responsible for both the breakdown of zero sound modes as well
as the broadening of the Drude peak. We also emphasize that this
low temperature hydrodynamics is distinct from the high temper-
ature
hydrodynamics of probe branes, which is conventional, and
describes a single diffusive mode for charge.
This
mechanism is analogous to the behavior of electrical con-
ductivity
[5,14,15] and ordinary sound waves [16] in a normal fluid
with weak momentum relaxation; such similarity was qualitatively
observed before [17]. Here, of course, the conserved momentum
is replaced by the charge current operator itself. Unlike the hy-
drodynamics
of ordinary fluids, we find that the nonlinear hydro-
dynamics
of the current operator is often ill-posed: the gradient
expansion always fails at low enough temperatures. Thus, as we
resolve the mysteries of the zero sound modes and the Drude con-
ductivity
which arise within linear response, our work also calls
https://doi.org/10.1016/j.physletb.2017.10.023
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
.