JHEP03(2018)056
Recently there has been a particular focus on studying diffusion of heat and charge in
the context of holography. This stems, in part, f r om the suggestion that diffusive processes
may be a key to understanding universal aspects of transport in incoherent metals [
13].
Furt her m ore , it was also suggested in [
13] that there m i ght be lower bounds on diffusion
constants by analogy with bounds on shear viscosity associated wi t h diffusion of momen-
tum [
14]. A key idea is to write D ∼ v
2
τ, where D is suitabl e diffusi on constant and v, τ
are characteristic velocities and time scales of the system, and it was suggested in [
13] that
τ should be the ‘Planckian time scale’ τ = ~/(k
B
T ) [
15, 16]. An interesti ng subsequent
development was the suggestion that v should be identified with the butterfly velocity, v
B
,
extracted from out of time order cor r e l ator s [
17, 18] and used as a measure of the onset of
quantum chaos.
While there has been a r ange of interesti ng holographic results in this dir ec ti on , includ-
ing [
19–32], wit h an appreciation that it is the thermal diffusion should be rel at e d to v
B
, it
is fair to say that within hol ograp hy a sharp global picture has yet to emerge.
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Almost all
of the holographic stud y in this area has b e e n in the setting of sp e c i fic types of ‘homoge-
neous’ hologr aphi c lattices [
41, 42], which maintain a translationally invariant metric. In
these cases it is straightforward to extract v
B
by studying a sh ock wave entering the black
hole horizon [
43, 44] (see also [45]). A notabl e exception is [19] who studied holograp hi c
lattices in one spatial dimension, but working in a hydrodynamic, high temperat ur e li mi t
of the background hol ogr aphi c lattice. We hope that the present work, which illuminates
universal aspects of diffusion for arbitrary spatial modulation in holography, will be useful
in further developments.
In a di ff er e nt direction, our derivation of the dispersion relation lead s to a general
connection between thermodynamic instabilities and dynamic instabilities. Some tim e ago,
building on [
46, 47], it was shown in a spe cific holographic context with a translationally
invariant horizon, that thermodynamic instability implies an imaginary speed of sound,
leading to unstable quasi-normal modes and dynamical instability
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[52]. For general spatial
modulation within holography, any sound modes will onl y appear on scal e s much smaller
than the scale of the modulation and he nc e this wi l l not be a universal channel to deduce
dynamical instability from thermodynamic instability. Instead, the diffusion modes do
provide such a channel. Specifically, in the presence of spatial modulation, we can deduce
the following result. If the heat and charge susceptibility matrix has a negative e i genvalue,
then the system is thermody n ami cal l y unstable and then the dispersion relati on implies
that t he re is at least one mixed diffusion mode, involving heat and charge, living in the
upper half plane which will necessarily lead to a dynamic al instability.
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It is striking that a relation of the f or m D ∼ v
2
B
τ has also appeared in a variety of other non-holographic
contexts, including [
22, 33–40], with τ ∼ λ
−1
L
where λ
L
is the Lyapunov exponent [35].
4
Some recent discussion of both hydrodynamic and n o n -hydrodynamic modes and the connect io n with
instabilities in a translationally invariant setting, appeared in [
48–51].
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