Brian Houston does not have any upcoming events.
As a leader, Pastor Brian Houston is highly regarded for his boldness, innovation and vision. As a church pastor, he is well respected for his passion for the cause of Christ and the local church, and for preaching messages that change mindsets and ministries. But at HOME - Hillsong Church – Pastor Brian is most loved for being the 'dad of the house' - a pastor who loves God and people, and a speaker of life who declares, 'The best is yet to come'.
WHEN THE WORLD SAYS YOU CAN'T, GOD SAYS YOU CAN!
Founder and Global Senior Pastor of Hillsong Church shows how, with God's power, you can believe and achieve a life that exceeds every earthly expectation. When you surrender control and follow God's guidance, you become empowered and equipped to do the impossible, reach higher, and go further than you could have ever imagined—giving Him all the glory.
By building our lives around godly principles, surrounding ourselves with wisdom, and living for a cause greater than ourselves, we can effectively live out God's purposes and have an enduring impact. As Brian Houston has personally discovered, when we depend on Jesus as the source for our identity, our lives can be a lasting legacy that maximize the gifts He has given each of us as unique individuals. There is More is for anyone who wants to be challenged to live with expectancy. Ephesians 3 is a call to release the immeasurable potential within you and the exceeding, abundant, and above plans and purposes of a Holy God that are beyond your greatest imagination.


text:
The 497 Greenwich Street penthouse in Soho
The 497 Greenwich Street penthouse is located in a residential Soho district in Manhattan, New York City, USA. The modern residence of over 325 square meters costs 12 million dollars and looks even more than luxury. First of all, this penthouse has a fascinating view overlooking the Hudson River and the Statue of Liberty from the height. And huge windows fastened with steel beams give you an opportunity to enjoy the beauty of the landscape from the almost every part of the house. Secondly, unusual finishing looks really magnificent. Also, designers have created an excellent combination of different textures with a predominance of bright colors and also created a style where each area of the house is unique. The original design of the interior impresses you with its lightness. This is a very luxury and comfortable apartment at the same time. Designers complied with owners' wish and set the audio equipment and home appliances of the highest quality that fit well into the design of the project. However, in contrast to the bright living area, the bedroom design with darker shades is more relaxing.
And, of course, there is the open-air terrace on the roof, gym, swimming pool, sauna, high-end home theater and all other stuff that should be in the residence of 12 million dollars.
© Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman © Douglas Elliman
© Douglas Elliman
Luxury penthouse for $ 14 million in New York The luxury house for $ 8.3 million in Old Naples, USA Penthouse in New York The charming village house in Indianapolis, Indiana, USA is displayed for sale for $ 2.5 million The luxury penthouse Elysium in Grace Tower, Vancouver, Canada The Interior Design: Bleecker Street Loft A Stylish House In British Columbian Mountains for $8.5 Million The excellent glass house for $ 14.9 million in Malibu, USA
Home » Apartments Design Ideas » The 497 Greenwich Street penthouse in Soho
Previous The art studio on a tree by Rockefeller Partners Architects
Next The Bolton house from Naturehumaine architecture firm in Quebec

redpajama_set_name: RedPajamaCommonCrawl


With guaranteed snow from mid-November until the beginning of May the Obertauern ski area is ideal for passionate skiers, snowboarders, families and alpine sports enthusiasts. Well-prepared ski slopes, modern lifts and a wealth of cosy ski huts make skiing great fun. Obertauern is also a sports eldorado for non-skiers – snowshoe walks, cross country trails, toboggan runs etc. ensure lots of variety.
In the vicinity of our hotel you'll find everything you need for a perfect ski holiday in Obertauern. The sports shop, Intersport Frühstückl offering ski rental and other services is situated just 100 m away from our Spa hotel Schneider ****Superior. Take advantage of the discounts you receive as our hotel guest. Thanks to the current weather forecast you'll be well prepared for your winter sports activities in Obertauern.


Ibn-Racin was the name of the Berber tribe that established here with the Muslim invasion. After some time it became a Roman town. The town was called Santa María de Oriente when it was Christian…The ancient history of Albarracín begins like this, a history where protection and defense have been the premise. Today, as you walk its narrow streets and breathe the calmness, it is difficult to imagine that its location was not imagined just for pleasure but as a shelter against danger.
The amazing old part of town is just for pedestrians and very high up. If you go up on the afternoon you will see that the passageways under the arches, the stairs, the doors of the mansions, seem as if they were inhabited by mysterious shadows. Albarracín oozes an aristocratic air; you can see it in its studded doors, in the iron door knockers, in its balconies made with carved wood, in its spectacular work with wrought iron. In one of the winding streets there are little squares that open over the abysm from where the houses seem to hang. The landscape is extremely beautiful. For the construction of houses, they have traditionally used plaster instead of stone, and the color of its walls has the same color of the ravines underneath them.
But even if you have heard a lot about it, it still impacts you on your arrival.
You can get to Albarracín through the same road but with two different visions. One, when you arrive from Teruel, with the A512, and after a curve you suddenly find yourself on its lower part. Then, the town gives you the incredible image of a city, with infinite tones of orange and pink, embedded in the rocks, and you can't stop looking at it.
Known as "the most beautiful town in Spain", Albarracín is located 30 minutes away from Teruel. Located southeast of the lower part of Aragón, the town of Albarracín owes its name to the Muslim conquerors Aben Razin, the name of an Arab family. To walk its streets is to really go back in time all the way to the Middle Ages.


text:
\section{Introduction}
Bounds on transport coefficients are an important tool to quantify
the strength of correlations in quantum many-body systems. If one
can identify a theoretical value for a minimal electrical conductivity
or viscosity, then one can judge
how strongly-interacting a system is. A highly influential bound for
momentum conserving scattering of quantum fluids was proposed by Kovtun,
Son, and Starinets\cite{Kovtun2005} (KSS) for the ratio of the shear viscosity
and entropy density
\begin{equation}
\eta/s\geq\frac{\hbar}{4\pi k_{{\rm B}}}.\label{eq:KSS}
\end{equation}
It is obeyed in systems like the quark gluon plasma\cite{Schaefer2014}
or cold atoms in the unitary scattering limit\cite{Thomas2009}. Graphene
at charge neutrality is another example that is expected to be close
to this bound\cite{Mueller2009}. Within the Boltzmann transport theory
one finds that a bound for $\eta/s$ can be related to the
ratio $l_{{\rm mfp}}/\lambda$ of the mean-free path $l_{{\rm mfp}}$ and the mean distance $\lambda$ between carriers. However, Eq.(\ref{eq:KSS})
is valid even for systems that cannot be described in terms of the
quasi-classical Boltzmann theory. Indeed, the bound is saturated for
quantum field theories in the strong coupling limit as was shown in
Ref.\cite{Kovtun2005} using the holographic duality of conformal
field theory and gravity in anti-de-Sitter spacetime\cite{Maldacena1998,Witten1998,Gubser1998}.

Limiting bounds for the charge transport like the electrical conductivity
are somewhat more subtle. A much discussed example is the Mott-Ioffe-Regel
limit\cite{Ioffe1960,Mott1972,Gurvitch1981} that corresponds to a
threshold value of the electrical resistivity when $l_{{\rm mfp}}/\lambda\sim{\cal O}\left(1\right)$.
While some systems clearly show a saturation of the resistivity once
$\lambda/l_{{\rm mfp}}$ reaches unity, materials like the cuprate
or iron-based superconductors violate this limit\cite{Emery1995}.
For a detailed discussion of correlated materials that obey or systematically
violate the Mott-Ioffe-Regel bound, see Ref.\cite{Hussey2004}. \textcolor{black}{Transport
	properties in quantum critical systems were argued under certain circumstances
	to be governed by a Planckian relaxation rate $\hbar\tau^{-1}\approx k_\text{B}T$\cite{Zaanen2004,Sachdev2011},
	which would also limit the electrical conductivity
	at quantum critical points.}\textcolor{red}{{} }A bound on charge transport
that is less restrictive and theoretically better justified than the
Mott-Ioffe-Regel limit was proposed in Ref.\cite{Hartnoll2015}.
It constrains the value of the charge diffusivity as determined by
the Einstein relation:
\begin{equation}
D_{c}=\frac{\sigma}{\chi_{c}}\geq C_{D}\frac{\hbar v^{2}}{k_{{\rm B}}T},\label{eq:diffusivity}
\end{equation}
with $C_{D}$ is a numerical coefficient of order unity. Here $v$
is a characteristic velocity of the problem, $\sigma$ is the electrical
conductivity, and $\chi_{c}=\partial\rho/\partial\mu$ the charge
susceptibility with particle density $\rho$ and chemical potential
$\mu$. The latter is related to the charge compressibility since
$\chi_{c}=-\frac{\rho^{2}}{V}\frac{\partial V}{\partial p}$. If $v^{2}\chi_{c}$ stays constant as $T\rightarrow0$, the electrical
resistivity cannot vanish slower than linearly in $T$\cite{Hartnoll2015}.
Ref.\cite{Blake2016,Blake2016b} proposed  the butterfly velocity $v=v_{\mathrm{B}}$ as the characteristic velocity.
$v_{\mathrm{B}}$ follows from the analysis of out-of-time-order (OTOC) correlations
$C\left(\mathbf{x},t\right)=-\left\langle \left[A\left(\mathbf{x},t\right),B\left(\mathbf{0},0\right)\right]^{2}\right\rangle $
that are discussed in the context of chaos and information scrambling\cite{Shenker2014,Shenker2015,Roberts2015,Roberts2016,Maldacena2016}.
It can be obtained from the long-distance behavior, e.g. via
\begin{equation}
C\left(\mathbf{x},t\right)\sim e^{2\lambda_\text{L}\left(t-\frac{\mathbf{\left|x\right|}}{v_{\mathrm{B}}}\right)}.
\end{equation}

The scrambling rate $\lambda_\text{L}$ that enters the OTOC is also subject
to the bound $\lambda_\text{L}\geq2\pi k_\text{B}T/\hbar$\cite{Maldacena2016}.
While the interpretation of $\lambda_\text{L}$ and its relation to transport
and thermalization rates is not always correct\cite{Blake:2017qgd,Klug2018,Davison:2018ofp,Davison2018,1710.00921,1706.00019},
the butterfly velocity seems to yield a natural scale for the characteristic
velocity of a system, even if no clear quasiparticle description is
available.  A caveat applies when a symmetry of the system is weakly broken and triggers a sound-to-diffusion crossover: in this case, the resulting diffusivity is more naturally expressed in terms of the sound velocity and the gap \cite{Davison2015b,Davison:2018ofp,Grozdanov:2018fic}.

The focus of this paper is the investigation of anisotropic systems,
where the conductivity tensor $\sigma_{\alpha\beta}$ and the viscosity
tensor $\eta_{\alpha\beta\gamma\delta}$ exhibit a more complex structure
with potentially different temperature dependencies for distinct tensor
elements \cite{Cook2019,Link2018}. The anisotropy that we consider is most naturally expressed
in terms of the relation between characteristic energies and momenta
along different directions. For a system with two space dimensions, it holds then that:
\begin{eqnarray}
\omega & \sim & \left|k_{x}\right|^{z/\phi}\nonumber \\
\omega & \sim & \left|k_{y}\right|^{z}\label{eq:def phi 1}
\end{eqnarray}
with dynamical exponent $z$. We characterize the anisotropy
in terms of the exponent $\phi$ that relates typical momenta along
the two directions according to
\begin{equation}
\left|k_{x}\right|\sim\left|k_{y}\right|^{\phi}.\label{eq:def phi 2}
\end{equation}
{A single particle dispersion that is consistent with such scaling
would be $\varepsilon\left(\mathbf{k}\right)\sim\left|k_{x}\right|^{z/\phi}+a\left|k_{y}\right|^{z}$
that corresponds to a system at a Lifshitz point\cite{Lifshitz1942,Dzyaloshinski1964,Goshen1974,Hornreich1975,PhysRevB.62.12338,SHPOT2001340,Shpot_2005,0802.2434}.}
However, our conclusions do not require the existence of well defined
quasiparticles with this dispersion relation.

{Anisotropic systems, that obey scaling behavior of a Lifshitz transition
were recently shown to violate the viscosity bound\cite{Rebhan2012,Jain2015, Ge:2014aza,Ge2017,Link2018,1805.01470, Ge:2014aza,1205.1797,1306.1404,Pedraza2018}.}
In Ref.\cite{Link2018} a model of anisotropic Dirac fermions that emerged from two ordinary Dirac cones was analyzed as an explicit condensed matter realization\cite{Isobe2016}. Within a quasiparticle
description of the transport processes and a Boltzmann equation approach,
the conductivity anisotropy was found to diverge: one direction is
metallic and another one insulating. Based on the quasiparticle transport
theory, a modified bound was conjectured, that involves not just the
viscosity tensor elements $\text{\ensuremath{\eta_{\alpha\beta\alpha\beta}}}$
and the entropy density $s\left(T\right)$, but also the conductivities\cite{Link2018}:
\begin{equation}
\frac{\eta_{\alpha\beta\alpha\beta}}{s}\frac{\sigma_{\beta\beta}}{\sigma_{\alpha\alpha}}\geq\frac{\hbar}{4\pi k_{{\rm B}}}.\label{eq:bound}
\end{equation}
Here, no summation over repeated indices is implied.\\
Other tensor elements like $\eta_{\alpha\beta\beta\alpha}$ continue to
obey Eq.(\ref{eq:KSS}). The origin for this combined viscosity-conductivity
bound is the different scaling behavior of the typical velocities
$v_{\alpha}$ for different directions. { Candidate materials with Lifshitz transitions are the organic conductor $\alpha-(\text{BEDT-TTF}_2)\text{I}_3$ under pressure\cite{Julia27}, and the heterostructure of the $5/3 \text{TiO}_2/\text{VO}_2$ supercell\cite{Julia28,Julia29}. Moreover, the surface modes of topological crystalline insulators with unpinned surface Dirac cones\cite{Julia30} and quadratic double Weyl fermions\cite{Julia31} are expected to exhibit such a behavior}. The analysis of Ref.\cite{Link2018}
was based on the Boltzmann equation and did not allow to explicitly
analyze a model that satisfies this bound or determine the precise
numerical coefficient in Eq.(\ref{eq:bound}), i.e. the factor ${1}/{4\pi}$.
This can only be done within a formalism that addresses transport
in strongly-coupled non-quasi-particle many-body systems. In the same
context it is of interest to address the related question of whether
the diffusivity bound, Eq.(\ref{eq:diffusivity}), is also modified
for anisotropic systems.

In this paper we perform a holographic analysis of anisotropic transport,
exploiting the duality between strongly coupled quantum field theories
in $d+1$ dimensions and gravity theories in one additional dimension\cite{Maldacena1998}.
{The calculation is based on an Einstein-Maxwell-dilaton (EMD) action, where the anisotropy is generated by massless scalars, linear in  the boundary spatial coordinates. {See Refs.\cite{Rebhan2012,Jain2015,Ge:2014aza,1306.1404,Jeong2018,Donos:2014uba,Ge2017,Mateos:2011ix,Mateos:2011tv,1205.1797,Pedraza2018,Donos2014,1202.4436} for previous studies of these holographic systems.} } As a consequence, the scalars also break translations,
momentum is not conserved and the viscosity cannot be interpreted as a hydrodynamic coefficient.
\\
{It is well known that in such holographic frameworks the KSS bound is violated\cite{Rebhan2012,Jain2015, Ge:2014aza,Ge2017,1805.01470, Ge:2014aza,1205.1797,1306.1404,Pedraza2018,Hartnoll2016,Alberte:2016xja,Ciobanu:2017fef,Burikham2016,Ling_2016,Ling_2017}.} We
will choose a geometry where the momentum is conserved along one of
the spatial directions, say the $\beta$-direction. Thus, the stress tensor
elements $T_{\alpha \beta}$ serves as currents of the conserved momentum density along the direction $\beta$.
Consequently, the viscosity elements $\eta_{\alpha \beta \gamma \beta}$ maintain
their meaning as hydrodynamic coefficients, for all $\alpha$ and $\gamma$.
In particular the model obeys
\begin{equation}
\frac{\eta_{\alpha\beta\alpha\beta}}{s} =
\frac{\hbar}{4\pi k_\text{B}} \frac{\sigma_{\alpha\alpha}}{\sigma_{\beta\beta}}.
\label{eq:at the bound anisotr.}
\end{equation}
The generalized bound Eq.\eqref{eq:at the bound anisotr.} has to be understood as a relation between hydrodynamic coefficients. Moreover, the combination $\frac{\eta_{\alpha\beta\alpha\beta}}{s} \frac{\sigma_{\beta\beta}}{\sigma_{\alpha\alpha}}$ serves as an indicator of strong coupling behavior in anisotropic systems.
In Fig.\ref{temperature_behavior} we show typical temperature dependencies for these
transport coefficients for a specific value of the crossover exponent $\phi$ that characterizes the anisotropy.

In addition, we determine the anisotropic butterfly velocity $v_{B,\alpha}$ ({see Refs.\cite{1805.01470,Jeong2018,Pedraza2018,1610.02669,Blake:2017qgd,1710.05765,1708.07243,1811.06949} for previous studies})
and the compressibility, and obtain for the anisotropic diffusivity
the generalization of Eq.(\ref{eq:diffusivity})
\begin{equation}
D_{c,\alpha} =	\frac{d_\text{eff}-\theta}{\Delta_\chi} \frac{\hbar v_{B,\alpha}^{2}}{2\pi k_{{\rm B}}T},
\label{eq:Diffusivity aniso}
\end{equation}
where $d_\text{eff}$ is the effective spatial dimensionality -- see Eq.(\ref{deff}) below, $\theta$ the hyperscaling violating exponent,  and $\Delta_\chi$ the scaling dimension of the charge susceptibility. Thus, the bound of Eq.(\ref{eq:diffusivity}) can be generalized to anisotropic systems. In distinction to the viscosity bound, the anisotropy only changes the universal coefficient that now depends on the exponents $\phi,z,$ and $\theta$. Furthermore, (\ref{eq:Diffusivity aniso}) recovers the limit of isotropic charge neutral theories\cite{Blake2016}. In Ref.\cite{Jeong2018}, the thermal diffusivity was computed in anisotropic setups and also found to obey a relation similar to \eqref{eq:Diffusivity aniso}. See Ref.\cite{1805.01470} for an alternative proposal to \eqref{eq:Diffusivity aniso} at an anisotropic QCP.

Before we present the theories that yield these results, we
give some general scaling arguments, assuming charge and momentum conservation. This analysis motivates us to consider
the appropriate combinations of transport quantities that enter Eq.(\ref{eq:bound})
and Eq.(\ref{eq:Diffusivity aniso}). The scaling analysis is then
followed by a holographic analysis of the combined viscosity-conductivity
bound, the charge susceptibility, and the butterfly velocity within
an anisotropic gravity theory.
\begin{figure}[t]
	\centering
	\includegraphics[scale=0.31]{boundsyxyx}
	\caption{
		Main panel: temperature dependence of the $\eta/s$ tensor. In the anisotropic case the KSS bound (orange) can be parametrically violated (green line). Here, $T_0$ is a temperature scale below which the anisotropy effects are dominant. The conductivity ratio might constitute a new lower bound when rotations are broken (green).
		Inset: temperature dependence of the conductivity tensor elements $\sigma_{xx}$
		and $\sigma_{yy}$.
		$\phi$ is the crossover exponent that characterizes the anisotropy between the different spatial
		directions $k_{x}\sim k_{y}^{1/\phi}$. Once $\phi\protect\neq0$
		one element of the conductivity of a two-dimensional system must be
		insulating and the other must be metallic.
	}
	\label{temperature_behavior}
\end{figure}
\section{Scaling arguments}

\label{scaling_arguments} We consider the scaling behavior of transport
coefficients in anisotropic systems near a quantum critical Lifshitz point.
As we will see, scaling arguments can be efficiently used to make
statements about transport bounds. Once a combination of physical observables has scaling dimension zero, it naturally
approaches a universal value in the limit $T,\mu,\omega\cdots\rightarrow0$,
that corresponds to an underlying quantum critical state. If one can
argue, usually based on an analysis of conservation laws, that
this value is neither zero nor infinity, it should be some dimensionless
number times the natural unit of the observable. In other words, this combination should be insensitive to irrelevant deformations of the quantum critical point. As an example we
consider the electrical conductivity at zero density. For isotropic systems its scaling
dimension is $d-2$, a result that follows from single-parameter scaling
and charge conservation. Thus the conductivity of a zero density two-dimensional
system is expected to reach a universal value in units of the natural
scale $e^{2}/h$. Under the same conditions, both the viscosity and
the entropy density have scale dimension $d$ such that their ratio
has scaling dimension zero. Then $\eta/s$ should approach a universal
value times $\hbar/k_{{\rm B}}$ which yields the correct physical
unit. This observation helps to rationalize a result like Eq.(\ref{eq:KSS}).
As an aside, these scaling considerations also offer a natural explanation
why the bound Eq.(\ref{eq:KSS}), while applicable, is not very relevant
for Fermi liquids. Here, the existence of a large Fermi surface gives
rise to hyper-scaling violating exponents \cite{Huijse:2011ef}. If
one performs the appropriate scaling near the Fermi surface\cite{Shankar1994},
then it seems more natural to use $\eta s^{2}$ as the natural bound,
a quantity that approaches a constant value as $T\rightarrow0$.

The conclusions of this section require that scaling relations are
valid, i.e that the system under consideration behaves critical and
is below its upper critical dimension. In the remainder of this section
we assume that this is the case. To be specific, we analyze a $d$-dimensional
system and allow for one direction to be governed by a characteristic
length scale with a different scaling dimension $\phi\neq1$
than the other spatial directions, see Eqs.(\ref{eq:def phi 1},\ref{eq:def phi 2})
above. In addition, the temporal direction is characterized by a dynamic
scaling exponent $z$. Let us then consider a physical observable $O\left(\mathbf{k},\omega\right)$.
By assumption the observable obeys the scaling relation
\begin{equation}
O\left(k_{\perp},\mathbf{k}_{\parallel},\omega\right)=b^{-\Delta_{O}}O\left(b^{\phi}k_{\perp},b\mathbf{k}_{\parallel},b^{z}\omega\right).\label{eq:scaling law}
\end{equation}
Here $\Delta_{O}$ is the scaling dimension of the observable. The
$d$-dimensional momentum vector $\mathbf{k}=\left(k_{\perp},\mathbf{k}_{\parallel}\right)$
consists of one component $k_{\perp}$ that is governed by the exponent
$\phi$ and a $d-1$ dimensional component $\mathbf{k}_{\parallel}$.
In the subsequent holographic analysis we focus on a system with two
spatial coordinates and use the notation $k_{\perp}=k_{x}$ and $k_{\parallel}=k_{y}$.
While the scaling analysis presented here cannot determine the values
of the exponents, it allows for rather general conclusions once those
exponents are known. For an explicit model with nontrivial exponents
$z$ and $\phi$, see Ref.\cite{Link2018}.

\subsection{Scaling of thermodynamic quantities}

We begin our discussion of scaling laws with thermodynamic quantities.
For the free-energy density of the system holds the following scaling
law:
\begin{equation}
F\left(T,\mu\right)=b^{-d_{{\rm eff}}-z}F\left(b^{z}T,b^{z}\mu\right),
\end{equation}
with effective dimension
\begin{equation}
d_{{\rm eff}}=d-1+\phi.
\label{deff}
\end{equation}
As an energy density, $F$ should scale like unit energy per unit
volume. To obtain its scaling dimension it is then easiest to start
from the usual result $d+z$ for isotropic systems\cite{Sachdev2011}
and replace $d$ by $d_{{\rm eff}}$. This takes into account the
different weight of the directions $\mathbf{k}_{\parallel}$ and $k_{\perp}$.
With $s=-\partial F/\partial T$ and $\rho=\partial F/\partial\mu$
we obtain immediately the scaling dimensions
\begin{equation}
\Delta_{s}=\Delta_{\rho}=d_{{\rm eff}}
\end{equation}
for the entropy density $s$ and particle density $\rho$, respectively.  Away from zero density, the relation $\Delta_{\rho}=d_{{\rm eff}}$ generally does not hold \cite{Gouter_2014_mom_diss,Hartnoll2015-1,Davison:2018ofp}.
The second derivative of the free energy with respect to the chemical
potential yields charge susceptibility
\begin{equation}
\chi^{\left(\rho\right)}\left(T,\mu\right)=b^{-\Delta_{\chi}}\chi^{\left(\rho\right)}\left(b^{z}T,b^{z}\mu\right)\label{eq:chi scaling}
\end{equation}
with $\Delta_{\chi}=d_{{\rm eff}}-z$. We can now use these thermodynamic
relations to determine the scaling behavior of the conductivity and viscosity. To
do so is possible because of the restrictions that follow from charge and momentum
conservation.

\subsection{Scaling of transport coefficients}

The conductivity is determined via a Kubo formula from the current-current
correlation function, e.g.
\begin{equation}
{\rm Re}\,\sigma_{\alpha\beta}\left(\omega\right)=\frac{{\rm Im}\,\Pi_{\alpha\beta}\left(\omega\right)}{\omega}.
\end{equation}
At zero density, the system has a finite d.c. conductivity. $\Pi_{\alpha\beta}\left(\omega\right)$
is the Fourier transform of the retarded current-current correlation
function $\Pi_{\alpha\beta}\left(t\right)=-i\theta\left(t\right)\left\langle \left[j_{\alpha}\left(t\right),j_{\beta}\right]\right\rangle $.
In order to exploit the implications of charge conservation we use
the continuity equation
\begin{equation}
\partial_{t}\rho+\partial_{\alpha}j_{\alpha}=0
\end{equation}
and obtain the well known relation between the longitudinal conductivity
$\sigma_{\alpha\alpha}\left(\omega\right)$ and the density-density
correlation $\chi^{\left(\rho\right)}\left(\mathbf{k},\omega\right)$
\begin{equation}
\sigma_{\alpha\alpha}\left(\omega\right)=\lim_{\mathbf{k}\rightarrow0}\frac{\omega}{k_{\alpha}^{2}}\chi^{\left(\rho\right)}\left(\mathbf{k},\omega\right).
\end{equation}
Here $\chi^{\left(\rho\right)}\left(\mathbf{k},\omega\right)$ is
the temporal Fourier transform of $\chi^{\left(\rho\right)}\left(\mathbf{k},t\right)=-i\theta\left(t\right)\left\langle \left[\rho\left(\mathbf{k},t\right),\rho\left(-\mathbf{k},0\right)\right]\right\rangle $,
where $\rho\left(\mathbf{k},t\right)$ is the spatial Fourier transform
of the density $\rho\left(\mathbf{x},t\right)$. Since $\chi^{\left(\rho\right)}=\lim_{\mathbf{k}\rightarrow0}\chi^{\left(\rho\right)}\left(\mathbf{k},\omega=0\right)$,
the scaling dimension of $\chi^{\left(\rho\right)}$ is also $\Delta_{\chi}$,
given below Eq.(\ref{eq:chi scaling}). Thus we find
\begin{eqnarray}
\Delta_{\sigma,\parallel} & = & \Delta_{\chi}+z-2=d_{{\rm eff}}-2,\nonumber \\
\Delta_{\sigma,\perp} & = & \Delta_{\chi}+z-2\phi=d_{{\rm eff}}-2\phi,
\end{eqnarray}
for the conductivities along the two directions. This yields for the
conductivities:
\begin{eqnarray}
\sigma_{\parallel}\left(T,\omega\right) & = & b^{3-\phi-d}\sigma_{\parallel}\left(b^{z}T,b^{z}\omega\right),\nonumber \\
\sigma_{\perp}\left(T,\omega\right) & = & b^{\phi+1-d}\sigma_{\perp}\left(b^{z}T,b^{z}\omega\right).
\end{eqnarray}
If we return to the isotropic limit, where $\phi=1$, both components
of the conductivity behave the same with usual conductivity scaling
dimension $d-2$. Interestingly, in the anisotropic case, this continues to be the
dimension of the geometric mean $\sqrt{\sigma_{\parallel}\sigma_{\perp}}$.
Distinct scaling exponents for the tensor elements imply a different
temperature dependency of the conductivity for different directions.
Thus, a more insulating behavior along one direction will force the
other direction to be more metallic. For a two-dimensional system,
one direction will have to be insulating and the other then has to
be metallic as long as $\phi\neq1$. Finally, the ratio $\sigma_{\parallel}/\sigma_{\perp}$
of the conductivity is governed by $\Delta_{\sigma,\parallel}-\Delta_{\sigma,\perp}=2\left(\phi-1\right)$,
i.e.
\begin{equation}
\frac{\sigma_{\parallel}\left(T\right)}{\sigma_{\perp}\left(T\right)}=b^{-2\left(\phi-1\right)}\frac{\sigma_{\parallel}\left(b^{z}T\right)}{\sigma_{\perp}\left(b^{z}T\right)}.\label{eq:sigma ratio}
\end{equation}

We can perform a similar analysis for the viscosity tensor. It is given by a different Kubo formula
\begin{equation}
{\rm Re}\,\eta_{\alpha\beta\gamma\delta}\left(\omega\right)=\frac{{\rm Im}\,\Pi_{\alpha\beta\gamma\delta}\left(\omega\right)}{\omega},
\label{viscosity_scaling_section}
\end{equation}
with $\Pi_{\alpha\beta\gamma\delta}\left(\omega\right)$ the Fourier
transform of the retarded stress-tensor correlation function $\Pi_{\alpha\beta\gamma\delta}\left(t\right)=-i\theta\left(t\right)\left\langle \left[T_{\alpha\beta}\left(t\right),T_{\gamma\delta}\right]\right\rangle $.
Momentum conservation gives rise to the continuity equation for the
momentum density $g_{\alpha}$:
\begin{equation}
\partial_{t}g_{\beta}+\partial_{\alpha}T_{\alpha\beta}=0.
\end{equation}
We are considering a system without rotation invariance. In this case
it is important to keep track of the order of the tensor indices as
$T_{\alpha\beta}$ cannot be brought into a symmetric form\cite{Link2018b}. From the continuity equation
for the momentum follows for the viscosity
\begin{equation}
\eta_{\alpha\beta\gamma\delta}\left(\omega\right)=\lim_{\mathbf{k}\rightarrow0}\frac{\omega}{k_{\alpha}k_{\gamma}}\chi_{\beta\delta}^{\left(g\right)}\left(\mathbf{k},\omega\right),
\end{equation}
with momentum-density correlation function $\chi_{\beta\delta}^{\left(g\right)}\left(\mathbf{k},\omega\right)$,
i.e. the Fourier transform of $\chi_{\beta\delta}^{\left(g\right)}\left(\mathbf{k},t\right)=-i\theta\left(t\right)\left\langle \left[g_{\beta}\left(\mathbf{k},t\right),g_{\delta}\left(-\mathbf{k},0\right)\right]\right\rangle $.
Thus, we only need to know the scaling dimension of $\chi_{\beta\delta}^{\left(g\right)}$
to determine the behavior of the viscosity. The easiest way to obtain
this scaling dimension is to realize that under a boost operation,
a velocity field is thermodynamically conjugate to the momentum density.
A velocity has scaling dimension $z-1$ for the directions along $\mathbf{k}_{\parallel}$
and $z-\phi$ for $k_{\perp}$. To capture all the options we write
this as $z-\varphi_{\alpha}$ where $\varphi_{\alpha}=1$ for all
directions but along $k_{\perp}$ where we have $\varphi_{\alpha}=\phi$.
Thus, it holds
\begin{equation}
\chi_{\beta\delta}^{\left(g\right)}\left(k_{\perp},\mathbf{k}_{\parallel},\omega\right)=b^{-\Delta_{g,\beta\delta}}\chi_{\beta\delta}^{\left(g\right)}\left(b^{\phi}k_{\perp},b\mathbf{k}_{\parallel},b^{z}\omega\right).
\end{equation}
with $\Delta_{g,\beta\delta}=d_{{\rm eff}}-z+\varphi_{\beta}+\varphi_{\delta}$.
In the Appendix \ref{Alternative_viscosity} we obtain the same behavior from an analysis of strain
generators, following Refs.\cite{Link2018b,Bradlyn2012}. Using $\Delta_{g,\beta\delta}$
allows us to determine the scaling behavior of the viscosity tensor
\begin{equation}
\eta_{\alpha\beta\gamma\delta}\left(T\right)=b^{-\Delta_{\eta,\alpha\beta\gamma\delta}}\eta_{\alpha\beta\gamma\delta}\left(b^{z}T\right)
\end{equation}
with
\begin{eqnarray}
\Delta_{\eta,\alpha\beta\gamma\delta} & = & \Delta_{g,\beta\delta}+z-\varphi_{\alpha}-\varphi_{\gamma}\nonumber \\
& = & d_{{\rm eff}}-\varphi_{\alpha}+\varphi_{\beta}-\varphi_{\gamma}+\varphi_{\delta}.\label{eq: viscosity dimension}
\end{eqnarray}
For isotropic systems, this gives the well known result that the scaling
dimension of the viscosity is $d$, i.e. the same as for the entropy
or particle density. For an anisotropic system the scaling dimensions
of the viscosity and the entropy density can still be the same. This
is the case whenever $\varphi_{\alpha}+\varphi_{\gamma}=\varphi_{\beta}+\varphi_{\delta}$.
Examples are $\eta_{\perp\perp\perp\perp}$, $\eta_{\perp\perp cd}$,
$\eta_{ab\perp\perp}$$,$$\eta_{a\perp\perp d}$, or $\eta_{\perp bc\perp}$,
where $a$,$b$ etc. stand for components of $\mathbf{k}_{\parallel}.$

However, the scaling dimension of the viscosity can also be different
from the one of the entropy density. This is the case for
\begin{eqnarray}
\eta_{a\perp c\perp}\left(T\right) & = & b^{-\left(d-3+3\phi\right)}\eta_{a\perp c\perp}\left(b^{z}T\right)\nonumber \\
\eta_{\perp b\perp d}\left(T\right) & = & b^{-\left(d+1-\phi\right)}\eta_{\perp b\perp d}\left(b^{z}T\right).
\end{eqnarray}
If we now take the ratio of the viscosity to entropy density, we find
\begin{eqnarray}
\frac{\eta_{a\perp c\perp}\left(T\right)}{s\left(T\right)} & = & b^{-2\left(\phi-1\right)}\frac{\eta_{a\perp c\perp}\left(b^{z}T\right)}{s\left(b^{z}T\right)},\nonumber \\
\frac{\eta_{\perp b\perp d}\left(T\right)}{s\left(T\right)} & = & b^{2\left(\phi-1\right)}\frac{\eta_{\perp b\perp d}\left(b^{z}T\right)}{s\left(b^{z}T\right)}.\label{eq:eta ratio}
\end{eqnarray}
Thus, for $\phi\neq1$ there is always one tensor element of the viscosity,
where $\eta_{\alpha\beta\gamma\delta}/s$ diverges as $T\rightarrow0$
and another one that vanishes. The latter will then obviously violate
any bound for the ratio of a viscosity to entropy density. In Ref.\cite{Link2018}
it was shown that precisely these tensor elements turn out to be important
for the hydrodynamic Poiseuille flow of anisotropic fluids.

The origin of unconventional scaling of both the conductivities and the
viscosities is geometric, i.e. rooted in the anisotropic scaling of
spatial coordinates at the Lifshitz point. If one combines Eqs.(\ref{eq:sigma ratio})
and (\ref{eq:eta ratio}), it is straightforward to see that the combinations
that enter Eq.(\ref{eq:bound}) always have scaling dimension zero.
While it certainly does not offer a proof of Eq.(\ref{eq:bound})
this is necessary for such quantity to approach a universal, constant
low-temperature value.

Finally we comment on the scaling behavior of the diffusivity bound,
Eq.(\ref{eq:Diffusivity aniso}). To check whether this bound even
makes sense for an anisotropic system, we consider the quantity
\begin{equation}
X_{\alpha}=k_{{\rm B}}TD_{c,\alpha}/\hbar v_{\alpha}^{2}
\end{equation}
where $v_{\alpha}$ is the characteristic velocity along the $\alpha$-th
direction and $D_{c,\alpha}=\sigma_{\alpha\alpha}/\chi_{c}$ the diffusivity
along this direction. It obviously holds
\begin{equation}
\Delta_{X_{\alpha}}=z+\Delta_{\sigma,\alpha}-\Delta_{\chi}-2\left(z-\varphi_{\alpha}\right)
\end{equation}
where we used again that a velocity scales as $z-\varphi_{\alpha}$.
If we now insert our above results, it follows
\begin{equation}
\Delta_{X_{\parallel}}=\Delta_{X_{\perp}}=0.
\end{equation}
This implies that $X_{\alpha}$ should approach a universal constant
times $\hbar/k_{{\rm B}}$. Thus, we expect Eq.(\ref{eq:diffusivity})
to be valid even for anisotropic systems, which yields Eq.(\ref{eq:Diffusivity aniso}).
In this sense is this bound even more general than the original viscosity
bound of Eq.(\ref{eq:KSS}).

\section{Holographic Derivation of the viscosity-conductivity Bound}

The correspondence between gravity theories and quantum field theories,
as it occurs in the anti-de Sitter space/conformal field theory
duality\cite{Maldacena1998,Witten1998,Gubser1998}, is a powerful
tool to analyze the universal properties of strongly-coupled field theories.
In what follows we analyze an anisotropic bulk geometry in order to
determine the relationships between distinct transport coefficients
of anisotropic quantum many-body problems in the strong-coupling limit.
To this end we use the membrane paradigm\cite{Thorne1986} to express
boundary theory transport coefficients in terms of geometric quantities
at the horizon\cite{Iqbal2009}. To be specific, we consider a system
of two space dimensions, i.e. with $D=2+1$ space-time coordinates
at the boundary. The relation between the generating functional of
the quantum field theory and the gravity action for imaginary time is given by\cite{Witten1998,Gubser:1998bc}
\begin{equation}
\left\langle e^{-\int d^{3}x\Phi_{0}{ O}}\right\rangle =\left.e^{-S\left[\Phi\right]}\right|_{\Phi\left(r\rightarrow\infty\right)=\Phi_{0}},
\end{equation}
where ${ O}$ is an operator of the field theory, $\Phi_{0}$
a conjugate source, $\Phi$ the dual field, and $S$ a gravitational action in
the $D+1$ dimensional bulk, with additional coordinate $r$. Here
we chose a system of coordinates where the boundary lies at
$r=\infty$. Following Ref.\cite{Iqbal2009}, retarded Green's functions
of the field theory can be obtained from
\begin{equation}
\left\langle { O}\left(\mathbf{x},t\right)\right\rangle _{\Phi_{0}}=\lim_{r\rightarrow\infty}\Pi\left(r,\mathbf{x},t\right),
\label{radial_momentum}
\end{equation}
where $\Pi$ is the canonical momentum conjugate to $\Phi$, as follows
from the gravitational version of the Hamilton-Jacobi formalism. This
finally allows for the determination of retarded Green's functions
$
G\left(\mathbf{x},t\right)=-i\theta\left(t\right)\left\langle \left[{ O}\left(\mathbf{x},t\right),{ O}\left(\mathbf{0},0\right)\right]\right\rangle .
$
For the Fourier transform with respect to momentum and frequency follows
\begin{equation}
G\left(\mathbf{k},\omega\right)=-\lim_{r\rightarrow\infty}\frac{\Pi\left(r,\mathbf{k},\omega\right)}{\Phi\left(r,\mathbf{k},\omega\right)}.
\end{equation}
Causality is preserved if one considers $\Phi$ that satisfies in-falling
boundary conditions at the black hole horizon\cite{Son2002,Herzog2003}.
The related transport coefficient is given by $-\lim_{\omega,\mathbf{k}\to0}\frac{1}{\omega}{\rm ImG\left(\mathbf{k},\omega\right)}.$

\begin{figure}[t]
	\centering \includegraphics[width=3.4in]{RGflow} \caption{Cartoon of an AdS black hole geometry in ($D$+1)-spacetime dimensions.
		The extra coordinate $r$ is between $r=\infty$ and $r=r_{+}$ where the
		boundary and the horizon of the AdS are located, respectively. The
		evolution along $r$ is dual to the RG-flow of the dual $D$-spacetime
		dimensional QFT\cite{McGreevy}. Furthermore, in
		order to underline how the spatial anisotropy becomes higher as one
		approaches the IR region, we have depicted how an ellipse is distorted
		in the spatial directions as $r$ decreases.
	}
	\label{fig:example}
\end{figure}

Anisotropic, static bulk geometries cannot come from a pure gravitational
action. Thus, we need to couple gravity with {axial} {gauge} fields or
massless scalar fields. See also Ref.\cite{1202.4436,1310.6725,1306.1404} for generalities on anisotropic studies in holography.

We start from the Einstein-Maxwell-Dilaton
action
	\begin{equation}
	S=\int d^{3+1}x\sqrt{-g}\left(R+{\cal L}_{\text{M}}\right),
	\label{EMD_model}
	\end{equation}
with Lagrangian
\begin{equation}
{\cal L}_{\text{M}}=
-
\frac{1}{2}\left(\nabla\varphi\right)^{2}-V\left(\varphi\right)
-\frac{Y\left(\varphi\right)}{2}\left(\nabla\psi\right)^{2}
-\frac{Z\left(\varphi\right)}{4}F^{2}.
\label{matter_lagrangian}
\end{equation}
$\varphi$ is referred to as the dilaton. It is a scalar field which enters the action modifying
all the couplings involved. $V\left(\varphi\right)$ is its own potential.
We include the dilaton as it will {allow} us to consider anisotropic geometries that arise near the horizon of near-extremal black holes. In the absence of the dilaton
{field, the model} reduces to the usual $\text{AdS}_4$ system with electromagnetic
field, i.e $V\left(0\right)=2\Lambda$ with cosmological constant
$\Lambda=-3/\ell^{2}$, $Z\left(0\right)=1$ and $Y(0)=0$. $\ell$ is the radius of
curvature of the AdS space.
As we will shortly see, by considering a bulk profile that depends linearly on one of the boundary spatial coordinates, the massless scalar $\psi$ will break the rotation and translation symmetries of the dual field theory. {For related work on this family of holographic models, see Refs.\cite{Mateos:2011ix,Mateos:2011tv,Donos:2013eha,Andrade2014,Rebhan2012, Pedraza2018,Ge2017,Ge:2014aza,Jain2015,Ling_2016,Ling_2017,Amoretti2016_chas,Amoretti2018_CDW,Andrade2014,Davison2015b,Donos2014,Hartnoll2016} }
\\
$F^{2}$ is
the Maxwell Lagrangian with $F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$
the usual field tensor with vector potential $A_{\mu}$. This term
is needed to implement a $U\left(1\right)$ global symmetry in the
boundary theory and to determine the electrical conductivity.

We summarize the field equations of motion that follow from Eq.\eqref{EMD_model} varying the action with respect to the fields $g_{\mu\nu},A_\mu,\psi,\varphi$.

Varying the metric, we obtain the Einstein equations
\begin{equation}
\label{eq:Einstein}
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=-\frac{1}{\sqrt{-g}} \frac{ \delta (\sqrt{-g} \mathcal{L}_\text{M})}{\delta g^{\mu\nu}}.
\end{equation}
The variation of the gauge field yields the Maxwell equations
\begin{equation}
\partial_{\mu}(\sqrt{-g}Z(\varphi)F^{\mu\nu})=0.
\label{gaugeDG}
\end{equation}
Notice that both scalar fields are neutral such that the Maxwell equations are bulk conservation equations for the two-form $F$. Ultimately, this will let us evaluate the boundary charge current at the black hole horizon. Finally, the wave equations for the two scalars are:
\begin{eqnarray}
&\partial_\mu (\sg\,Y(\varphi)\partial^\mu\psi)=0,
\label{axions_EOM}
\\
&\partial_\mu(\sg\,\partial^\mu \varphi)={\partial_\varphi} V_\text{eff},
\label{dialton_EOM}
\end{eqnarray}
where
\begin{equation}
\frac{V_\text{eff} }{\sg}=
{V(\varphi)+\frac{Y(\varphi)}{2}(\partial\psi)^2+\frac{Z(\varphi)}{4}F^2}.
\label{Veff}
\end{equation}
In the absence of external perturbations, we use the following ansatz
\begin{eqnarray}
&& ds^2 = - g_{tt}(r)dt^2 + g_{rr}(r)dr^2+ \sum_\alpha g_{\alpha\alpha}(r)dx_\alpha^2  \nonumber\\
&&\varphi=\varphi(r), \quad A=A_t(r) dt, \quad \psi = a y,
\label{eq:metric}
\end{eqnarray}
where $a$ is real and $\alpha=\{x,y\}$. The Ansatz for $\psi$ is consistent with the field equations and
preserves the homogeneity of the other fields. Indeed, $\psi$ back-reacts on the equations of motion only through gradients so that all dependence on $y$ drops out of the field equations. However, translations along the $y$-direction are broken and momentum is dissipated at a strength set by $a$. On the other hand, momentum along $x$ direction is conserved which allows us to perform a hydrodynamic analysis of the viscosity tensor elements $\eta_{\alpha x\beta x}$.
The metric in (\ref{eq:metric}) describes anisotropic bulk geometries since, in general, $g_{xx}\left(r\right)\neq g_{yy}\left(r\right)$. The coefficient $a$ determines the temperature scale $T_0$ below which the anisotropy effects are large. Setting $a=0$ restores both rotations and translations in the dual field theory.

In its more general formulation, the holographic correspondence maps the RG-flow of the dual (strongly coupled) field theory to the evolution along the radial direction\cite{McGreevy} -- see Fig.\ref{fig:example}.
The near boundary region
captures the UV of the dual field theory, while the near
horizon region describes the IR.
In the UV ($r\to\infty$) the geometry is assumed to be asymptotically $\text{AdS}_4$ :
\begin{equation}
ds^2= - r^2 dt^2 + \frac{dr^2}{r^2} + r^2(dx^2+dy^2)+\dots
\label{AdS_metric}
\end{equation}
where the dots denote subleading terms as $r\to+\infty$.
This requires
\begin{equation}
\begin{split}
V_\text{UV}\equiv V(0) = -6,
&\quad
Y_\text{UV}\equiv Y(0)= 0,		\\
Z_\text{UV}\equiv &Z(0) = 1
\end{split}
\end{equation}
with the dilaton vanishing like $\varphi= \varphi_{s} r^{\Delta_\varphi-3}+\varphi_{v} r^{-\Delta_\varphi}+\dots$ coming from the near boundary expansion of Eq.(\ref{dialton_EOM}). {$\Delta_\varphi<3$ is the largest solution of $M^2=\Delta_\varphi(\Delta_\varphi-3)$,  $M$ being the mass of the field. The dilaton field is thus dual to a relevant deformation of the UV CFT, with source $\varphi_s$ and vacuum expectation value $\varphi_v$\cite{McGreevy}.

	From this discussion, we also see that the bulk field $\psi$ sources a marginal deformation of the UV CFT.
	Similarly, $A_{t}=\mu-\rho r^{-1}+\dots$  with chemical potential $\mu$ and charge density $\rho$. In the following we analyze the charge neutral case $A_t=0$.

	Since both scalars are dual to relevant/marginal deformations of the UV CFT, we expect the system to be able to flow to a non-trivial quantum critical phase. This IR endpoint of the RG flow is represented in the bulk by a power law geometry, which arises in the near horizon region at very low temperatures compared to the sources of the UV CFT. To find such geometries, we assume that the dilaton runs logarithmically in the IR ($r\to0$) $\varphi=2\kappa \log(r)$ and that the scalar potentials take the following form\cite{Charmousis:2010zz,Gouter_2014_mom_diss}
	\begin{equation}
	V_\text{IR} = -V_0 e^{\delta \, \varphi},
	\quad
	Y_\text{IR} = e^{\lambda \varphi},
	\quad
	Z_\text{IR} = \, e^{\zeta \varphi}\,.
	\label{coupling_IR}
	\end{equation}
	The critical scaling of the previous section is holographically realized by a Lifshitz geometry of the form
	\begin{equation}
	ds^2 = r^\theta \left( -\frac{dt^2}{r^{2z}} +L^2 \frac{dr^2}{r^2} + \frac{dx^2}{r^{2\phi}} + \frac{dy^2}{r^2} \right),
	\label{IR_scaling_model}
	\end{equation}
	which is covariant under the scale transformation
	$
	(t,r,x,y) \to (b^{-z} t,b^{-1}{r}, b^{-\phi} x,b^{-1} y),
	$
	up to a conformal factor $ds^2\to b^{-\theta}ds^2$. Therefore, $\phi$ and $z$ coincide with the anisotropic and dynamical exponents, and $\theta$ quantifies the violation of scale invariance in the metric \cite{Pedraza2018,Gouteraux:2011ce,Huijse:2011ef}.
	All the parameters involved are real and $V_0,\delta,L>0$.
	The explicit derivation of such a solution can be found in Appendix \ref{single_axion_model}, for the (marginally) relevant single axion case, which has $z=\phi\neq1$, the marginally double axion case, which has $z>1$, $\phi\neq1$, and the irrelevant single axion case, which has $z=1$, $\phi=1$ (and where rotations/translations along $x$ are only broken away from the IR endpoint through the irrelevant deformation).
	\\
	A finite temperature can be introduced via the emblackening factor
	\begin{equation}
	f(r)= 1-\left({\frac{r}{r_+}}\right)^{\delta_0},
	\label{blackening_factor}
	\end{equation}
	where $r_+$ denotes the location of the event horizon and $\delta_0=1+\phi+z-\theta$.
	The Hawking temperature is
	\begin{equation}
	4\pi T ={\frac{|\delta_0|}{L}} r_+^{-z}
	\end{equation}
	and satisfies $T\to b^z T$, consistently with the scaling analysis.
	The fact that scaling stops at a finite value of the flow is reflected
	in the event horizon at finite $r_+$.
	The entropy density follows from the area of the horizon $s =4\pi r_+^{\phi+1}$.

	Thus, with an appropriate choice of $V\left(\varphi\right)$,
	and $Y\left(\varphi\right)$ we can ``engineer'' a holographic
	dual that generates a desired crossover exponent $\phi$. Without
	more constructive statements about the field theory-gravity dual,
	it is not possible to determine the values of $\phi$ for a given
	quantum field theory. However, we can make statements about a number
	of physical observables for a given value of $\phi$.

	\subsection{Analysis of the conductivity}
	In this section we review the results of Ref.\cite{Iqbal2009,Donos2014,Donos:2014uba} to express the electric conductivities in terms of IR quantities. In particular, we calculate the d.c. conductivity along the $\alpha$-direction
	\begin{equation}
	\sigma_{\alpha\alpha} =
	\lim_{\substack{\omega\to0\\r\to\infty}} \text{Im}\frac{j^\alpha(r,\omega)}{\omega A_\alpha(r,\omega)},
	\label{electric_conductivity}
	\end{equation}
	working directly at zero frequency and switching on a constant and small electric field. $A_\alpha$ is the fluctuation respect to which we linearize the gauge equations, and $j^\alpha$ is the associated canonical momentum.

	Within the homogeneous ansatz (\ref{eq:metric}), Maxwell equations assume the form
	\begin{equation}
	\partial_{r}(\sqrt{-g}Z(\varphi)F^{\mu r})=0.
	\label{maxwell_conserving}
	\end{equation}
	The quantity in brackets coincides with the conjugate
	momentum of the gauge field $j^{\mu}=\delta S/\delta\left(\partial_{r}A_{\mu}\right)$.
	From the holographic dictionary (\ref{radial_momentum}) follows that the boundary value $j^{\mu}\left(r=\infty \right)$ of this quantity is the electric current density of the dual field theory. From (\ref{maxwell_conserving}) $j^\mu$ is radially conserved, i.e. $\partial_{r}j^{\mu}=0$. Thus, we can
	determine the current at the boundary from the behavior of $j^{\mu}$
	at the horizon
	\begin{equation}
	j^{\mu}\left(r=\infty,\mathbf{x},t\right)=\lim_{r\rightarrow r_{+}}j^{\mu}\left(r,\mathbf{x},t\right).
	\end{equation}
	In the absence of external fields, the only non zero component of
	$j^{\mu}$ is the temporal one $	j^{t}=\sqrt{-g}Z\left(\varphi\right)F^{tr
		\label{charge_density_gauge_field}}
	$
	which corresponds to the charge density $\rho$ of the field theory. In the following we focus on the charge neutral case $\rho=0$.
	\\
	In order to determine the conductivity, we add a small electric field
	$E_{\alpha}=F_{\alpha t}$ in the $\alpha$-direction, e.g. via
	\begin{equation}
	A_{\alpha}^{{\rm ext}}=-E_{\alpha}t.
	\end{equation}
	This electric field will polarize the system and therefore induce
	small corrections to the metric and matter fields. We parametrize
	those corrections via
	\begin{eqnarray}
	&&A_{\alpha}  =  -E_{\alpha}t+\delta A_{\alpha}\left(r\right),\nonumber \\
	&&	g_{t\alpha}  =  \delta g_{t\alpha}\left(r\right),\nonumber \\
	&&	g_{r\alpha}  =  g_{\alpha\alpha}\left(r\right)\delta h_{r\alpha}\left(r\right),
	\end{eqnarray}
	and $	\psi  =  a y+\delta\psi \left(r\right)$ if $\alpha=y$.

	All terms $\delta A_{\alpha}$ etc. are assumed to be of first order
	in the electric field. They can be related to each other through a
	perturbative solution of the field equations.
	The above ansatz yields at first order and for zero density $\rho=0$:
	\begin{equation}
	j^{\alpha}=
	-\frac{\sqrt{-g}Z\left(\varphi\right)}{g_{rr}g_{\alpha\alpha}}\partial_{r}\delta A_{\alpha}.
	\end{equation}
	This result further simplifies our analysis as we only need to determine
	$\delta A_{\alpha}$. To this end we perform a transformation to a
	set of coordinates that is free of singularities at the horizon. This
	is accomplished by the Eddington-Finkelstein (EF)
	coordinates\cite{Eddington1924,Finkelstein1958} $t'=t+r_\star(r)$, where $dr_\star=dr/\gamma(r)$ is the tortoise coordinate, $\gamma(r)=\sqrt{g_{tt}(r)/g_{rr}(r)}$.
	In these variables holds that near the horizon
	\begin{equation}
	A_{\alpha}=-E_{\alpha}t'+{E_{\alpha}} r_\star(r)+\delta A_{\alpha}.
	\end{equation}
	If we now demand regularity of $ A_{\alpha}$ in the EF coordinates
	it follows for the leading, singular contribution:
	\begin{equation}
	\delta A_{\alpha}\left(r\rightarrow r_+\right)=-E_{\alpha} r_\star(r).
	\end{equation}
	It is now straightforward to determine the conductivities
	\begin{equation}
	\sigma_{\alpha\alpha}=\lim_{r\rightarrow r_+}\frac{j^{\alpha}}{E_{\alpha}}=\sqrt{\frac{g_{\overline{\alpha}\overline{\alpha}}}{g_{\alpha\alpha}}}Z\left(\varphi\right)\bigg|_{r_+},\label{eq:conduct holo}
	\end{equation}
	where $\overline{x}=y$ and $\overline{y}=x$.

	\subsection{Analysis of the viscosity}
	In order to compute the correlation function (\ref{viscosity_scaling_section}), we act on the bulk-metric field which is dual to the boundary stress tensor\cite{Kovtun2005}. To get the shear viscosity components, we  switch on small off-diagonal fluctuations of the spatial sector
	\begin{equation}
	ds^2 \mapsto ds^2 + e^{-i \omega t} \delta h_{xy}(r) dxdy.
	\end{equation}
	In the following we linearize Einstein equations with respect to the one-index-up parametrization $h_\alpha^\beta = g^{\beta	\beta} \delta h_{\alpha\beta}$ and compute the viscosity through
	\begin{equation}
	\eta_{\alpha\beta\alpha\beta}= \lim_{\substack{\omega\to0\\r\to\infty}} \frac{1}{\omega}\, \text{Im}\frac{\Pi_{\beta}^\alpha(r,\omega)}{ {h}_\alpha^\beta(r,\omega)},
	\label{viscosity_first_step_nodiss}
	\end{equation}
	where $\Pi_\beta^\alpha$ is the associated radial momentum\footnote{Strictly speaking, the most natural notation in Eq.\eqref{electric_conductivity} and \eqref{viscosity_first_step_nodiss} would have been $\sigma^{\alpha\alpha}$ and $\eta^{\alpha\,\alpha}_{\,\,\beta\,\,\beta}$. Since however the boundary is flat, we could choose the notation with all indices down in order to be consistent with section \ref{scaling_arguments}.}. Since the model is anisotropic, there will be two fluctuations satisfying different equations of motion.

	We start with the simpler case to review the standard derivation of the viscosity, and consider $\delta h_{xy} = g_{xx} h^{x}_{y}$. The Einstein equations (\ref{eq:Einstein}) yield
	\begin{equation}
	\partial_\mu\left( \frac{\sg}{\mathcal{N}} \partial^\mu h^{x}_y \right) =0,
	\end{equation}
	which describes the dynamics of a massless scalar with radial dependent coupling $\mathcal{N}(r) = g_{yy}(r) g^{xx}(r)$.
	The canonical momentum is
	\begin{equation}
	\Pi^y_x  = \frac{\sg}{\mathcal{N}}\partial^r  h^x_y,
	\end{equation}
	satisfying ${\mathcal{N}} \partial_r \Pi^y_x = - \omega^2 {\sg} h^x_y$.
	In the low frequency limit, i.e. $\omega\to0$ keeping ${\omega} h^x_y$ and $\Pi$ fixed\cite{Iqbal2009}, both the fluctuation and the momentum are radially conserved allowing to perform a near horizon limit in Eq.(\ref{viscosity_first_step_nodiss}). Here the fluctuation satisfies the in-falling conditions
	\begin{equation}
	h^x_y(r,\omega) \to {h}_0(r)\, e^{-i\omega r_\star(r)}.
	\label{horizon_condition_fluct_nodiss}
	\end{equation}
	$h_0$ is the real solution to the frequency independent wave equation, which asymptotes to a constant at the boundary and is regular at the horizon. Due to the radial conservation $h_0(r)=\text{const}\equiv1$.
	We then obtain
	\begin{equation}
	\frac{\eta_{yxyx}}{s} =  \frac{1}{4\pi} \frac{g_{xx}}{g_{yy}}\bigg|_{r_+},
	\label{viscosity_conserving}
	\end{equation}
	which reproduces the bound of Eq.(\ref{eq:KSS}) in the isotropic
	limit $g_{xx}=g_{yy}$. These results, together
	with our findings of Eq.(\ref{eq:conduct holo}) for the conductivities
	immediately yield the expression Eq.(\ref{eq:at the bound anisotr.})
	given in the introduction. This is one of the key results of this
	paper.

	For the $y$-index-up parametrization we find
	\begin{equation}
	\partial_\mu	\left({{\sg}{\mathcal{N}}\,  \partial^\mu h^{y}_x} \right)
	=
	{\sg}{\mathcal{N}}\, m^2h^{y}_x
	\label{EOMdiss}
	\end{equation}
	with radial mass $m^2(r)=  a^2Y(\varphi) g^{yy}(r)$ arising due to the breaking of translations along $y$. As before, we define the conjugate momentum via
	\begin{equation}
	\Pi^x_y  = {{\sg}{\mathcal{N}}\,  \partial^r h^{y}_x},
	\label{hamilton_eqs_fluctuations_diss}
	\end{equation}
	with $\partial_r \Pi^x_y ={\sg} \mathcal{N}  ( m^2-\omega^2 g^{tt}) h^{y}_x$.
	The non vanishing mass makes the evolution along $r$ non trivial even at zero frequency.
	However, from the equations follows that $\text{Im}\left[\Pi^x_y \,{h}^{y\star}_x\right]$ is radially conserved\cite{Jain2015}, ${h}^\star$ denoting the complex conjugate fluctuation. In particular we can switch to the near horizon limit in the numerator
	\begin{equation}
	\eta_{xyxy} = \lim_{\omega\to0} \frac{ \lim_{r\to r_+} \text{Im}\left[\Pi^x_y \,{h}^{y\star }_x\right]}{ \lim_{r\to \infty} \omega|h^{y}_x|^2}.
	\end{equation}
	Using the in-falling conditions in the numerator we obtain
	\begin{equation}
	\frac{\eta_{xyxy}}{s} =  \frac{1}{4\pi} \frac{g_{yy}}{g_{xx}}\bigg|_{r_+}
	h_0^2(r_+).
	\label{viscosity_diss}
	\end{equation}
	$h_0(r_+)$ denotes the horizon value assumed by $h^y_x(r)$.
	A similar result obtains in isotropic backgrounds with momentum relaxation\cite{Hartnoll2016,Burikham2016,Alberte:2016xja}.
	$h_0(r_+)$ originates from the simultaneous breaking of rotations and translations along $y$ caused by the massless scalar. Since it has a non-trivial radial evolution, we expect that it will differ from unity as temperature decreases, i.e. as the system flows away from the UV AdS$_4$.

	We can also discuss the temperature dependence of $h_0(r_+)$ at low temperatures. First, we discuss the case where the massless scalar $\psi$ vanishes faster than other bulk fields towards the extremal horizon. Then, the IR endpoint enjoys rotation and translation symmetries, which are broken only through an irrelevant deformation sourced by $\psi$. We expect that $h_0(r_+)$ goes to a constant typically less than unity, as in Ref.\cite{Hartnoll2016}.

	Alternatively, the translation/rotation breaking field $\psi$ can source a marginal deformation at $T=0$. In this case, there is no notion of momentum, although of course we can still compute the response to shear strain using the Kubo formula. But then the object we are computing does not have the interpretation of a shear viscosity.
	Its temperature dependence follows from an asymptotic analysis near the boundary of the IR region and yields:
	\begin{equation}
	\frac{\sigma_{yy}}{\sigma_{xx}}\frac{\eta_{xyxy}}{s} \sim   T^{\frac{\delta_0-2(\phi-1)}{z} \left(-1+\sqrt{1+
			\left(\frac{2 a L}{\delta_0-2(\phi-1)}\right)^2
		} \right) }.
	\end{equation}
	The sign of the exponent is not fixed, hence the tensor element can vanish or diverge -- for details on the parameter range see Appendix \ref{single_axion_model}.
	This result is still valid when two axions are taken into account  \eqref{scaling_solution}. The isotropic limit of this last case is consistent with Ref.\cite{Ling_2016,Ling_2017} at charge neutrality.

	In any case, the viscosity-conductivity bound stated through the scaling analysis is holographically realized at least for one of the $\eta/s$-tensor elements.

	\section{Holographic Derivation of the charge-diffusivity Bound}
	The charge diffusivity in the $\alpha$-direction is determined by the electrical conductivity and the charge susceptibility via the Einstein relation $D_{\text{c},\alpha}=\sigma_{\alpha\alpha}/\chi_\text{c}$. In section \ref{scaling_arguments} we demonstrated that the
	combination
	\begin{equation}
	X_{\alpha}=\frac{k_{{\rm B}}TD_{c,\alpha}}{\hbar v_{\alpha}^{2}}
	\end{equation}
	has scaling dimension zero, which suggests that it approaches at low
	temperatures a universal value. In the subsequent sections we will
	use our result Eq.(\ref{eq:conduct holo}) for the conductivity, obtained through the holographic approach and determine, within the same theory, the charge susceptibility
	and the butterfly velocity of the system.  Without loss of generality we set $\zeta=0$, as
	in the charge neutral case the exponent of $Z_\text{IR} = e^{\zeta \varphi}$ is not constrained -- see Appendix \ref{single_axion_model}.
	We then obtain the result that
	\begin{equation}
	X_{\alpha}=\frac{1}{2\pi}\frac{1+\phi-\theta}{1+\phi-z}
	\end{equation}
	independent	on the space direction $\alpha$ leads to Eq.(\ref{eq:Diffusivity aniso}).
	\subsection{Analysis of the diffusivity}
	An important ingredient for the bound on the diffusivity in Eq.(\ref{eq:diffusivity})
	is the isothermal charge susceptibility $\chi_{c}=(\partial\rho/\partial\mu)_{T}$. In order to derive the correspondent holographic relation, we formally solve Maxwell equations (\ref{maxwell_conserving}):
	\begin{equation}
	A_t(r) = A_t (r_+) +\rho \int_{r_+}^r  \frac{dr}{\sg Z(\varphi) g^{rr}g^{tt}}\,.
	\end{equation}
	As mentioned, $A_t$ yields the chemical potential near the boundary and vanishes at the horizon, therefore
	\begin{equation}
	\chi_\text{c}^{-1} =  \int_{r_+}^{\infty} \frac{dr}{\sg Z(\varphi) g^{rr}g^{tt}},
	\label{inverse_compressibility}
	\end{equation}
	see also Ref.\cite{Iqbal2009}. Due to the non locality of the above formula, the integral can only be worked out by explicitly solving the RG flow from the boundary to the horizon.
	Keeping in mind that $r_+\propto T^{-{1}/{z}}$, we observe that the near horizon geometry contribution scales as $T^{-{\Delta_\chi}/{z}}$. Within a low temperature analysis, this is the dominant term if ${\Delta_\chi}/{z}>0$ and the charge diffusion is uniquely controlled by the IR physics, in accord with the isotropic analysis of Ref.\cite{Blake2016,Blake2016b}. In this case we obtain
	\begin{equation}
	\chi^{-1}_\text{c} = -\frac{L}{\Delta_\chi}\frac{r^{\Delta_\chi}}{Z(\varphi)}\bigg|_{r_+}.
	\end{equation}
	We can alternatively Taylor-expand the integrand $i(r)$ of (\ref{inverse_compressibility}) near the horizon. From the IR scaling behavior follows the recursion rule
	\begin{equation}
	i^{(n)} (r)= \frac{(-1)^n}{r^n} \left[\prod_{k=1}^{n} (k-\Delta_\chi)\right] i(r),
	\end{equation}
	$i^{(n)}(r)$ denotes the $n$-th radial derivative of $i(r)$. Plugging this expression into the Taylor expansion we find
	\begin{equation}
	i(r) = i(r_+) \sum_{n=0}^{\infty} {{n-\Delta_\chi}\choose{n}} \left({1-\frac{r}{r_+}}\right)^n
	\end{equation}
	Performing the binomial series we obtain $
	i(r) = i(r_+) ({{r}/{r_+}})^{\Delta_\chi-1}$, which yields the same result of the previous analysis.
	\smallskip\\
	The susceptibility together with the holographic conductivities (\ref{eq:conduct holo}) yields the diffusion constants
	\begin{equation}
	D_{\text{c},\alpha}
	= -\frac{L}{\Delta_\chi} \frac{r^{\theta-z}}{g_{\alpha\alpha}(r)}\bigg|_{r_+}.
	\label{diffusivity}
	\end{equation}
	The above results are still valid in the $\zeta\neq0$ case.

	\subsection{Analysis of the butterfly velocity in anisotropic systems}
	Following Ref.\cite{Shenker2014}, we determine the butterfly velocity for an anisotropic holographic system using a shock-wave analysis. As mentioned in the introduction, the butterfly velocity can be thought of as the velocity of growth of out-of-time-order correlation functions of local operators.
	Holographically, it can be calculated from the back-reaction of the
	metric due to a massless particle falling towards the horizon of the black
	hole. The velocity of growth of this back-reaction can then be identified
	as the butterfly velocity.

	For the subsequent analysis it is convenient to use Kruskal-coordinates
	\begin{equation}
	uv  = - e^{\gamma'(r_+) r_\star(r)}, \quad
	u/v = - e^{-\gamma'(r_+) t},
	\end{equation}
	where $\gamma'$ denotes the radial derivative.
	$uv=0$ and $uv=-1$ correspond to the horizon and to the boundary respectively -- see Fig.\ref{Kruskal}. The anisotropic metric (\ref{eq:metric}) takes the form
	\begin{equation}
	ds^{2}  =  -g_{uv}\left(uv\right)du dv\nonumber  + \sum_\alpha g_{\alpha\alpha}\left(uv\right)dx_\alpha^{2}.
	\end{equation}
	Next we perturb the system by adding $\delta T_{uu}\propto Ee^{2\pi Tt_{w}}\delta(u)\delta(x)\delta(y)$
	to the holographic stress-energy tensor, which represents a particle of energy $E$ released at the left boundary at time $t_{w}$ in the past and propagating
	towards the $u=0$ horizon\cite{Shenker2014,Sfetsos1995}. The perturbed
	metric can then be expressed in the following shock-wave form
	\begin{eqnarray}
	ds^{2} & = & -g_{uv}\left(uv\right)du dv+g_{uv}\left(uv\right)h(x,y)du^{2}\nonumber \\
	& + & \sum_\alpha g_{\alpha\alpha}\left(uv\right)dx_\alpha^{2}.
	\end{eqnarray}
	The equation of motion for $h\left(x,y\right)$ follows from Einstein equations at near the $u=0$ horizon:
	\begin{equation}
	\left(
	\sum_\alpha
	\frac{\partial_\alpha^2}{c_\alpha^{2}}
	-m_{h}^{2}\right)h(x,y)=b\delta(x)\delta(y),
	\label{shockwave_eq}
	\end{equation}
	with $c_\alpha = \sqrt{g_{\alpha\alpha}(0)}$, $ b\propto Ee^{2\pi Tt_{w}}/ {g_{uv}(0)}$ and mass
	\begin{equation}
	m_{h}^{2}=
	{\frac{1}{g_{uv}}}\frac{\partial \log(g_{xx} g_{yy})}{\partial (uv)}\Bigg|_{u=0}.
	\end{equation}
	\begin{figure}[t]
		\includegraphics[scale=0.31]{KruskalCoordinates}
		\caption{Black hole AdS space in Kruskal coordinates. The $(x,y)$-space,
			which is attached at each point, is not shown. The boundary (left/right
			edge) is located at $uv=-1$, the horizon (diagonal lines) is located
			at $uv=0$ and the singularity (upper/lower edge) is located at $uv=1$.
			Dashed lines represent surfaces of constant $u$. Time $t$ is flowing
			upwards in the right wedge and downwards in the left wedge.}
		\label{Kruskal}
	\end{figure}
\\
	Eq.(\ref{shockwave_eq}) is consistent with the isotropic case of Ref.\cite{Blake2016b}. The solution can be expressed in terms of the 0th modified Bessel function of the second kind $\mathcal{K}_{0}$ as
	\begin{equation}
	h(x,y)\propto-\frac{b\, c_{x}c_{y}}{2\pi}\mathcal{K}_{0}
	(m_h\varrho),
	\end{equation}
	where  $\varrho^2={c_x^2 x^{2}+c_y^2 y^{2}}$.
	At large values of $\varrho$, i.e. at large spatial distances, this gives
	\begin{equation}
	h(x,y)\propto \frac{1}{{\sqrt{\varrho}}}
	\exp \left[{2\pi T\left({ t_{w}-\frac{m_{h}}{2\pi T} \varrho }\right)}\right].
	\end{equation}
	From the exponent we can extract the direction-averaged scale for the velocity
	\begin{equation}
	\bar{v}_{\mathrm{B}} =
	\frac{2\pi T}{m_h}.
	\end{equation}
	In order to switch to the original system of coordinates, we use the identity $uv \,g_{uv}(uv)= g_{tt}(r)/\partial_r g_{tt}(r_+)^2$ and obtain
	\begin{equation}
	\bar{v}_{\mathrm{B}}^2
	= - {\frac{2\pi T L}{d_\text{eff}-\theta}}r_+^{\theta-z}.
	\end{equation}
	To determine the butterfly velocity along $x$,
	we consider the case where $y=0$ and we move in the $x$-direction.
	This gives
	\begin{equation}
	\frac{\varrho}{\bar{v}_{\mathrm{B}}}=
	\frac{c_x|x|}{\bar{v}_{\mathrm{B}}}
	\equiv\frac{|x|}{v_{\mathrm{B},x}}.
	\end{equation}
	It then follows for the velocity along the $\alpha$-direction
	\begin{equation}
	v_{\mathrm{B},\alpha}=\frac{\bar{v}_{\mathrm{B}}}{\sqrt{g_{\alpha\alpha}(r_+)}},
	\end{equation}
	in accord with Ref.\cite{Jeong2018,Pedraza2018,1805.01470,1610.02669,1710.05765,Blake:2017qgd}.
    This result violates the upper bound of the isotropic case pointed out in Ref.\cite{1612.00082}, consistently with Ref.\cite{1708.07243,1811.06949}

	Considering the ratio between the diffusion constant (\ref{diffusivity}) and the square
	butterfly velocity we finally obtain
	\begin{equation}
	\frac{D_{\text{c},\alpha}}{v_{\text{B},\alpha}^{2}}=
	\frac{d_\text{eff}-\theta}{\Delta_\chi}\frac{1}{2\pi T},
	\end{equation}
	which yields the result Eq.(\ref{eq:Diffusivity aniso}) for the diffusivity bound, another key result of this paper.

	\section{Conclusions \label{sec:ccl}}
	In this paper we analyzed transport coefficients at a quantum Lifshitz point in the strong coupling limit, using scaling arguments and exploiting the duality between quantum field theories and gravity theories. We  have focused on particle-hole symmetric theories at charge neutrality which admit a gravitational dual description. We have shown that bounds on transport coefficients of the isotropic case can be generalized to the anisotropic one.
	\\
	We analyzed the behavior of several observables after a spacetime dilatation, emphasizing that the scale dimensionless ones must approach a constant value for low temperatures. It turned out that some elements of the  $\eta/s$-tensor have a nonzero dimension while the diffusivity still exhibits the scaling of the rotational invariant case. In order to address the former, we included the electric transport, multiplying the ratio by a specific combination of conductivities such that the dimension of the resulting quantity is zero.
	\medskip\\
	 Within the Einstein-Maxwell-dilaton model considered, translational symmetry is broken along the $y$ direction by a massless scalar in the bulk with a bulk profile linear in $y$. Thus, the $x$-component of the momentum
		is still conserved and $T_{\alpha x}$ continues to be the current
		of a conserved quantity. Therefore, the viscosity tensor elements $\eta_{\alpha x\beta x}$
		maintain their meaning as hydrodynamic transport coefficients. Since
		we can find solutions of the field equations that yield either $\phi<1$
		or $\phi>1$, we can always construct an anisotropic geometry that
		violates the isotropic viscosity bound for at least one tensor element, while fulfilling the generalized bound given in Eq.\eqref{eq:at the bound anisotr.}.}

	In the direction where translations are broken, momentum relaxes at a rate $1/\tau_\text{mr}$. Provided $1/\tau_\text{mr}\ll\Lambda$, where $\Lambda$ is a UV cutoff, there is a range of intermediate times $1/\tau_\text{mr}\lesssim t\ll\Lambda$ where momentum is approximately conserved. In this regime, the viscosity can be defined from the shear Kubo formula, yet is still found to violate the viscosity-to-entropy-density-ratio bound. Alternatively, the diffusivity of transverse momentum can be considered, and has been reported\cite{Ciobanu:2017fef} to obey a bound of the kind \eqref{eq:diffusivity}.

	Differently from the other quantities, the diffusivity is not solely given by data on the horizon and is expressed through an integral over the radial direction. Although we do not have the full expression of bulk fields, we have derived a near horizon formula for the compressibility, and could relate the diffusion constant to the horizon data in a simple fashion. Indeed, the near IR geometry dominates at low temperature\cite{Blake2016,Davison:2018ofp,Davison2018}.
	On the other hand we have calculated the butterfly velocities by moving to the Kruskal system of coordinates and using a generalization of the shock-wave technique. We have computed the proportionality factor between the diffusivity to the square butterfly velocity ratio and the inverse temperature, finding that it can be expressed in terms of the critical exponents $z,\phi$, and $\theta$.
	\\
	Eventually, both the viscosity and the diffusivity bounds could be analyzed within higher-derivative gravitational backgrounds\cite{Cremonini:2011iq,Baggioli2017} to take into account finite coupling effects. Moreover, particle-hole symmetry breaking could be taken into account as well.
	\\
	Thus, we conclude that the transport properties of a strongly-interacting many-body system near a quantum Lifshitz point can be efficiently described using holographic methods and requires a generalization of the viscosity bound obtained in isotropic theories.
	\begin{acknowledgments}
		We are grateful to A. Amoretti, S. A. Hartnoll, E. I. Kiselev, N. Maggiore, N. Magnoli, B. N. Narozhny, and K. Schalm
		for stimulating discussions.
		BG has been supported during this work by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 758759).
	\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
You are here: Parliament home page > Parliamentary business > Publications and Records > Hansard > Commons Debates > Daily Hansard - Debate
30 Nov 2006 : Column 1326
I was privileged to attend the world AIDS conference this year in Toronto and was appalled to hear first-hand of the problems facing Africa and other countries, including India, where 5.2 million are suffering with HIV. At the conference I had the opportunity to meet many groups and I applaud the work of the voluntary sector and professionals, who do fantastic work in education and by ensuring that drugs are getting through to the community, particularly rural communities, encouraging people to come for testing, and working with other agencies in countries that have a high prevalence of HIV/AIDS or in their own communities, especially universities, helping to remove the stigma.
I welcome the support of Bill and Melinda Gates and Bill Clinton with their amazing foundation. They do so much to help get drugs to the right people and to raise the profile of the issue. I welcome the fact that the Department for International Development—I am delighted to see the Secretary of State in his place, which shows how significant he thinks today's debate is—has given £1.5 billion since 1997, trying to strengthen health systems in the developing world and over the next three years has committed to a further £1.5 billion.
I welcome where the money is to be spent, but I urge the Secretary of State to look at other areas where money is also vital. Will he target much of the money at his disposal at helping to provide vital drugs that are needed to combat the disease? Cheap drugs, such as Neviraprine, which costs as little as £6, prevents HIV from being passed from mother to baby. I applaud this week's articles in Metro that pinpoint Ethiopia, where the drug has helped to provide protection for the babies of pregnant women.
I urge the Secretary of State to ensure that in the coming years Government funding is prioritised so that vital drugs such as these are provided to patients. We must also look at other drugs that could make such a difference to people's lives. Microbicide cream could make a huge difference to prevent the spread of AIDS. I welcome the development of recent years and we are now in the final stages of that drug being tested. For women across the world, particularly in Africa, it will provide independence and empower them to make positive choices to ensure their well-being.
Will the Secretary of State also work with multinational businesses in this country so that all introduce positive work practices for their work force in countries that suffer from the pandemic? Some companies, such as Virgin, SAB Miller and De Beers, are already showing a lead, but we must make sure that all companies that operate within those countries provide a lead.
Mr. Andrew Mitchell (Sutton Coldfield) (Con) rose—
Mr. Deputy Speaker (Sir Alan Haselhurst): Order. It is not the convention of the House for the Opposition Front-Bench spokesman to intervene in a half-hour Adjournment debate at the end of the day, but I am sure that the House appreciates the importance that he attaches to the occasion by his presence.
Mr. Evans: I, too, am delighted to see my hon. Friend the shadow Minister in his place. We have had talks about this issue and I know how deeply he feels about the problem, so I am grateful for his presence here today.
Clearly, I could say much more about the issue, but I want others to speak. When I attended the Commonwealth Parliamentary Association conference in Nigeria recently I visited a small township just outside Abuja called Lugbe where I met Sister Cecilia and others who do fantastic work on the ground in a Catholic mission. I praise the work of the Catholic Church and other Churches. She introduced me to a lady called Pauline who lost her husband last year to AIDS. She herself was infected and destined to die. She lived in a typical humble setting and many thought that she would die soon. The drugs reached her and when I talked with her she was smiling and grateful, but more than anything else she displayed a vision of hope. Two of her sons beamed as they played around their mother. Had she died, the children would have been known as a cadet family, with the eldest brother looking after the youngest and all helping one another. That would have been pitiful. At least now the children have hope and the mother has a hope of seeing her children grow up. In this case tragedy was averted and I wish Pauline and her family the best of health in the coming years, but not everyone is so fortunate.
On world AIDS day 8,000 people will die of AIDS and, more chillingly, there will be 12,000 new infections. That is the scale of the problem we now face.
Mr. Neil Gerrard (Walthamstow) (Lab): I will be brief so that I leave enough time for the Minister to respond. I am grateful to the hon. Member for Ribble Valley (Mr. Evans) for suggesting that he would welcome the presence of colleagues in this debate. This is not a party political issue and never will be.
I want briefly to make three points, two on what is happening internationally and one on what is happening in the UK. The theme of world AIDS day this year is accountability. Organisations outside Parliament are asking us to be accountable for the promises that were made at Gleneagles and other international conferences, particularly on universal access to treatments by 2010. We have made real progress in the past few years. Not long ago, many people were saying that we will never be able to deliver anti-retrovirals in developing countries. Although we did not meet the target of 3 million people by 2005, the "3 by 5" campaign made a step change in the approach to the delivery of drugs.
There are two specific issues apart from the obvious one of finance. The first is that of drugs pricing. At the moment, there are significant differences in prices for second line therapies. That will become increasingly important over the next few years. Some of the first line therapies that are now available quite cheaply will not be as effective, and we will have to try to secure cheaper access to second line therapies. Connected with that is the TRIPS—trade-related aspects of intellectual property rights— agreement and the way in which that might affect the supply of generics. The waiver was agreed at Doha, but no developing country has yet taken advantage of it. I hope that through DFID we can actively help developing countries to do so.
The second issue is that of the pattern of the epidemic—that is, the growing danger of major epidemics in eastern Europe and in India and China. If we get things right, we might stop those epidemics
becoming as generalised as they have become in many parts of sub-Saharan Africa. It is partly down to money. However, to a large extent, particularly in eastern Europe, it is not so much a question of what the UK does to provide financial aid but of what we can do in working with those countries on developing sensible policies, particularly those where intravenous drug use is driving the epidemic and there is still considerable resistance from many politicians to adopting policies of harm reduction. If we can have influence through our contacts with politicians in some of those east European countries, we might be able to make a significant difference in time to stop the further development of mass pandemics in eastern Europe and in Eurasia.
Finally, I turn to the situation in the UK. The hon. Member for Ribble Valley mentioned the statistics. We are still experiencing a growing number of infections, although the Health Protection Agency figures seem to show a bit of levelling off in 2005. I hope that that is a genuine trend and that we will start to see a reduction. All the evidence suggests that a major epidemic in the UK is still possible. Although the figures for infections are relatively small, the number of new sexually transmitted infections is evidence that a lot of people out there are indulging in pretty risky behaviour. There is still the potential for a significant increase in infection in the UK, particularly in marginalised populations where men have sex with men, and especially now in African communities. This is not the Secretary of State's direct responsibility, but there seems to be a contradiction between what we are doing internationally in talking about universal access to treatments and what we are doing domestically in denying treatments to some people in the UK who are infected. That does not make good sense on public health grounds.
I am deeply grateful for the leadership on this issue that has been shown by Ministers in our Department for International Development. We have had an important influence on what has been happening internationally, and I hope that we will continue to do so.
Mr. Jeremy Hunt (South-West Surrey) (Con): I start by thanking my hon. Friend the Member for Ribble Valley (Mr. Evans) for allowing me to make a brief contribution to the debate.
I would like to express the thanks of the whole House for the life and work of Father Angelo d'Agostino, a noted AIDS campaigner who died in Kenya last week and whom the Secretary of State has met. He was famous throughout the world for his work in founding the Nyumbani orphanage in Kitui, southern Kenya, and for his work in distributing anti-retroviral drugs in the slums of Nairobi. He will be greatly missed, but his work will continue.
Echoing the comments of the hon. Member for Walthamstow (Mr. Gerrard), I thank the Secretary of State for his personal commitment to the battle against AIDS. I would like to thank him not just for the achievement of Gleneagles and the universal access target, not just for persuading the UN to adopt interim country-level targets—the Secretary of State knows that I campaigned for them—but for putting his money where his mouth is. The fact is that DFID is the second biggest donor internationally in the battle against
AIDS. By doing that, he shows—and we show as a country—that we recognise that without progress in the battle against AIDS, there can be no progress in development at all in Africa.
I would be grateful if the Secretary of State dealt with three concerns about moves towards universal access. First, I have researched some figures that seem to indicate that the cost of distribution of anti-retroviral drugs is up to five times higher through the global fund as compared with the distribution through PEPFAR—the President's Emergency Plan for AIDS Relief. That is connected to the fact that PEPFAR is quite happy to distribute anti-retrovirals directly to non-governmental organisations, whereas the global fund tends to prefer to distribute through host country Governments. That is obviously a great concern and the effectiveness of the global fund will be incredibly important in this battle.
My second concern is the continued lack of availability of paediatric anti-retroviral drugs, which I know that the Secretary of State has looked into. My third concern is how we are going to meet the targets for universal access in conflict and post-conflict zones—in countries such as the Democratic Republic of the Congo, where there is little or no health infrastructure. I am not sure that we have a strategy for determining whether we can achieve that and, if so, how best to do it. It is a very important consideration.
I finish by urging the Secretary of State to show the same commitment to achieving the goals of Gleneagles as he showed in securing them. I am sure that he will show that commitment. The hopes of a whole generation of Africans and, indeed, the hopes of the entire House rest on his personal commitment.
The Secretary of State for International Development (Hilary Benn): I begin by congratulating the hon. Member for Ribble Valley (Mr. Evans) on securing this debate on the eve of world AIDS day, and on the passion with which he spoke. I join him in paying tribute to my hon. Friend the Member for Walthamstow (Mr. Gerrard), who does great work chairing the all-party AIDS group. The hon. Member for South-West Surrey (Mr. Hunt) played a really important part in securing progress on interim targets. He argued the case and got us thinking about it. He had a real impact. I also pay tribute to our absent colleague, my hon. Friend the Member for Calder Valley (Chris McCafferty), who does such a sterling job. I am also very pleased to see in his place the hon. Member for Sutton Coldfield (Mr. Mitchell).
I undertake to draw to the attention of my right hon. Friend the Secretary of State for Health the points that were made about funding in the UK. I really hope that the central message of this debate and world AIDS day this year will be not only about all the things that have already been discussed, but about the importance of tackling stigma and discrimination. That will be central if we are to turn the tide of the epidemic.
We have heard the statistics. When we remember that 15 million children have lost the love and care of those on whom they relied most—one or both of their parents—to AIDS; when we know that a child born in Zambia today has a 50 per cent. chance of dying of
AIDS in his or her lifetime; when we know that in Zimbabwe female life expectancy is now just 34 years; and when we see that the epidemic is actually growing fastest in eastern Europe and central Asia, with 90 per cent. of cases in Ukraine in Russia, we know how much of a challenge we have on our hands. Those are awful statistics, and behind every single one lies an individual. I join the hon. Member for South-West Surrey in acknowledging the exceptional work of Father Angelo d'Agostino, who he rightly praised. I also acknowledge the efforts of the civil society organisations and foundations that were rightly praised by the hon. Member for Ribble Valley.
We are making some progress. We and others worked jolly hard to get the commitments last year at Gleneagles, and at the United Nations in June all Governments agreed to work towards achieving universal access to treatment, care and support by 2010. One way of measuring the progress is to acknowledge that there are now 10 times as many people on anti-retroviral treatment in sub-Saharan Africa as there were three years ago. That involves 1 million people. My hon. Friend the Member for Walthamstow and other colleagues were right to point out that, five years ago, that would have seemed like a dream. I remember that my very first visit to Africa was to Malawi, where people were discussing how they might provide anti-retroviral treatment to just some of the population. It shows what we can do collectively when we put our minds to it. However, in Africa we are reaching only a quarter of those who need treatment, and an even smaller proportion in Asia. So, it is progress, but is it enough? No, it is not: we need to do a lot more.
The other thing that we agreed at the UN in June was that countries should set out plans with ambitious interim targets. Forty-four countries have submitted targets, and 20 have submitted fully costed plans. What those countries need to do is very clear and simple, in one sense. They need to build their health infrastructure, and that involves building clinics, ending user fees, employing doctors and nurses, doing the tests and buying anti-retroviral drugs. Those are all things that hon. Members have mentioned tonight. The countries in question have their own funds, but they need our support alongside that. The UK is pushing hard for donors to get their act together—that is why we worked so hard on the "three ones"—so that the help that we give comes in a form that the countries can make best use of.
The commitments on aid made at Gleneagles were important, as is replenishing the global fund. Hon. Members have generously referred to the funding that we have made available, including the £100 million going to the global fund each year. I will look into the point raised by the hon. Member for South-West Surrey about the cost of distribution under the global fund, because we need to ensure that every single pound that we commit gets to work to make a difference on the ground.
The hon. Member for Ribble Valley was right to highlight the importance of preventing mother-to-child transmission. I am pleased to tell him that a further £25 million of DFID support in Zimbabwe will provide more services to prevent transmission from pregnant women to children. We also support similar efforts in several others countries, including Malawi, Mozambique
and Zambia. It is also essential, as the hon. Member for South-West Surrey said, that we work in conflict and post-conflict countries, where the Governments are weak. Zimbabwe is one example, and Burma is another, of countries where we work with the UN, non-governmental organisations and faith-based organisations. That helps to build the infrastructure that they need.
We also need to take more steps to bring down the price of drugs. The hon. Member for Ribble Valley was absolutely right about microbicides, and that is why we have already invested £50 million in their development. If that comes off, it will be a really significant step forward, not least because it will put some control into women's hands. Many women have very little control over what happens to their lives.
I share the concern that has been expressed about medicines for children. That is why, in September, we helped to found UNITAID, the new international drug purchasing facility. That predictable, long-term funding aims to lower drug prices, and to get more people on to treatment. UNITAID's first board meeting in October agreed $36 million to fund anti-retroviral treatment for up to 100,000 children in 2007 and double that number in 2008, and approved $70 million to expand second-line therapy—where first-line anti-retroviral treatment does not work—to 100,000 patients.
I warmly welcome the Clinton Foundation's announcement at the UNITAID board meeting this morning of cheaper treatments that will reach 100,000 children. Cipla and Rambaxy are reducing the price of their paediatric anti-retrovirals to $60 a year, which represents a 45 per cent. reduction. That is the good news from this morning. We have made a 20-year commitment to UNITAID, as hon. Members are aware. We are also helping countries to use the flexibilities under the TRIPS—trade-related aspects of intellectual property rights—agreement. They are there, but people have to work their way through them to deal with this public health emergency.
The other point that I want to come to is the fact that although treatment is the priority for keeping people alive today, if we are to achieve an AIDS-free generation—that is what every single one of us wants—prevention is the key for tomorrow and the day after. That will not happen unless prevention is directed particularly at those who are most at risk: young people, women and young girls, men who have sex with men, sex workers and injecting drug users.
We know that women make up almost two thirds of the people living with HIV in sub-Saharan Africa. In some African countries, young women are almost three times more likely to be HIV-infected than men of the same age. Why? It is because of gender discrimination, social restriction, violence, fear of violence and lack of financial security. Women sell themselves for food or money because that is how they keep themselves alive. That makes women much more vulnerable to HIV.
Next Section Index Home Page


text:
DeLand är en stad (city) i Volusia County i delstaten Florida i USA. Enligt United States Census Bureau har staden en folkmängd på 27 041 invånare (2011) och en landarea på 45,6 km². DeLand är huvudort i Volusia County.

Kända personer
 Terence Trent D'Arby, artist
 Jacob deGrom, basebollspelare
 Bridgette Gordon, basketspelare
 Chipper Jones, basebollspelare
 Luke Scott, basebollspelare
 Maurice Starr, musiker och producent

Referenser

Externa länkar

 State & County QuickFacts, US Census Bureau.

redpajama_set_name: RedPajamaWikipedia


text:
No peace for the dead in 'illegal tombs'
Authorities in Fuzhou have ordered the demolition of hundreds of burial sites built without permission. Photo from South China Morning Post
By He Huifeng
Authorities in southeast China have ordered the demolition of hundreds of illegally built tombs as President Xi Jinping's campaign to combat illicit and environmentally harmful construction projects continues to gather momentum.
The municipal committee of the Communist Party in Fuzhou, capital of Fujian province, ordered governments at all levels to remove all such graves built in the city and return the sites to their natural state.
The announcement came after Xinhua on Friday reported that a huge number of illegal burial sites had been developed in mountainous areas and that many of them had damaged the environment and posed a threat to local drinking water sources.
The report highlighted a development on Nanyang Mountain in the Jiangtian township of Fuzhou, where it said there were more than 100 luxury tombs under construction. In the worst cases, as much as 100 square metres (1,076 square feet) of forest had been felled to make room for them, while in other areas whole reservoirs had been surrounded by unapproved grave sites, it said.
In the months since Xi issued his latest directive to clamp down on the illegal construction of luxury villas and tombs that damage the environment, officials across the country have been racing to comply. Many of those who failed to act have been sacked or even jailed.
Many officials have been sacked or even jailed for failing to stop illegal developments. Photo from South China Morning Post
Meanwhile, in the northeastern province of Heilongjiang on Sunday, authorities oversaw the demolition of a villa development covering more than 19 hectares on Zhangguangcai Mountain. The move came after China National Radio reported on the unauthorised project, which was accused of encroaching on the natural environment.
As well as luxury homes, the Gardens of Cao development featured three formal gardens and a museum built in the Huizhou architectural style.
While local authorities had granted its developers permission to develop a "scenic spot" for the public on 2.8 hectares in 2006, the company continued to extend it, adding the villas, gardens and museum, the report said.
In a separate case, Zhao Zhengyong, a former party chief of Shaanxi in northwest China, this month became the latest official from the province to be detained on corruption charges related to a villa project developed without permission in the Qinling mountains.
It was in Shaanxi in 2014 that Xi issued his first directive to provincial leaders to combat the illegal construction of luxury properties in the mountain range, which supports not only a diverse range of plant life but also an endangered giant panda population.


text:
Q: Jackrabbit Performance goes down on saving node I'm Using jackrabbit to store assets(contains text + image files).
Here is My workspace structure.
       root
        |
      node1
  //////|\\\\\\
      nodes(each node has 5-7 child for file type)
      //|\\
    file type
////////|\\\\\\\
around 25-75 child file nodes(versionable) in each parent node

I have also some referenceable node (50-100) which are referenced by many nodes.

I have more than 55000 files. I'm Using File System Datastore to store file.
<DataStore class="org.apache.jackrabbit.core.data.FileDataStore">
    <param name="path" value="${rep.home}/repository/datastore"/>
    <param name="minRecordLength" value="100"/>
</DataStore>

I imported all file in repository in structure i explain above.
Now when i'm saving session after adding node it takes too long time.
5-6 minute for storing 60-70 nodes.
Write operation takes long time. reading is taking normal time.
I'm using MSSQL database. And Jackrabbit implemented in java with spring framework.
So any possible solution for increase performance for adding new node.
Thanks

redpajama_set_name: RedPajamaStackExchange