\section{\@startsection{section}{1}{\z@
                                   {-3.5ex \@plus -1ex \@minus -.2ex
                                   {2.3ex \@plus.2ex
                                   {\normalfont\normalsize\bfseries}}
\def\subsection{\@startsection{subsection}{2}{\z@
                                     {-3.25ex\@plus -1ex \@minus -.2ex
                                     {1.5ex \@plus .2ex
                                     {\normalfont\normalsize\bfseries\itshape}}
\def\@seccntformat#1{\csname the#1\endcsname.~~}
\long\def\@makecaption#1#2
  \vskip\abovecaptionskip
  \sbox\@tempboxa{\small#1. #2
  \ifdim \wd\@tempboxa >0.9\hsize
  {\leftskip=0.05\hsize\rightskip=0.05\hsize\relax\small
    #1. #2\par}
  \else
    \global \@minipagefalse
    \hb@xt@\hsize{\hfil\box\@tempboxa\hfil
  \fi
  \vskip\belowcaptionskip}
\def\Appendix{


\section{Introduction and Summary}\label{intro}

Black holes have strange thermodynamic behaviour. Unlike other thermodynamic objects its entropy is proportional
to {\it area} (of the event horizon) instead of its volume.
\begin{equation}
S= {A_{d-1} \over 4 G_{d+1}},
\end{equation}
where $A_{d-1}$ is $d-1$ dimensional horizon (spatial)area and $G_{d+1}$ is Newton's constant in $d+1$ dimension.
Therefore all the microscopic details of a black hole are encoded in one less dimension.
It was first observed by Bekenstein for two derivative gravity. Later Wald has prescribed a generalized covariant formula for non-extremal black hole entropy in presence any higher derivative terms. Though the area law does not hold in higher derivative gravity but to calculate entropy using Wald's formula we only require the horizon values of metric and other fields forming the black holes. In the extremal limit one has to be careful using Wald's formula since the bifurcate killing horizon does not exist in this case. In \cite{sen} Sen has discovered an elegant method to calculate the entropy for a wide class of extremal black holes. Again, only the near horizon values of the fields are required to calculate entropy.

Black holes in asymptotically AdS space is more interesting. AdS/CFT predicts that the thermodynamic and hydrodynamic behaviour of strongly coupled boundary plasma are captured by black holes in one higher dimension.
The entropy of a black hole in
AdS space is related to the thermodynamic entropy of the boundary
gauge theory at a finite temperature (which is the same as the
Hawking temperature of the black hole). Thus there are two
apparently different holographic descriptions of the entropy. One in
terms of the horizon and the other in terms of the boundary
gauge theory \cite{dg}. In a rough sense, the latter is the UV description from
the microscopic (gauge theory) point of view, while the former is the
IR description. One might therefore expect some kind of RG flow to relate the
two.

Like entropy, the hydrodynamic quantities of boundary conformal plasma can be evaluated holographically. For example the retarded Green's function for tensor mode (for example $h_{xy}$) can be calculated using Kubo formula,
\begin{equation}
G^R_{xx,xy} = P - i  \eta \omega + \eta \tau_{\pi} \omega^2 + {\kappa \over 2}[(p - 2)\omega^2 + q^2]
+ {\cal O}(\omega^2),
\end{equation}
where, $p$ is spatial dimension and $p \geq 3$ and
\begin{eqnarray}\displaystyle
P &:& pressure,\\
\eta &:& shear \ viscosity \ coefficient,\\
\tau_{\pi} &:& relaxation \ time \ for \ shear \ viscous \ tensor,
\end{eqnarray}
and $\omega$ is frequency and $q$ is spatial momentum.

As we have seen that it is possible to find the the entropy of boundary field theory, only specifying the complete near horizon geometry of its dual black hole spacetime, one can also ask in the same spirit, whether
different hydrodynamics coefficients of the boundary plasma can be found from the knowledge of bulk near-horizon geometry.
There are enough hints \cite{thorne,pw1,sonsh,liu} which seems to point to an affirmative answer to this question. In \cite{liu} it has been shown that in the low
frequency limit ($\omega \rightarrow 0$) the shear viscosity coefficient $\eta$ is related to transport
coefficient of membrane fluid, which in turn is given by the effective coupling constant of graviton
($h_{xy}$) evaluated on the horizon. In \cite{cai1,ns1} the prescription has been generalized for any higher
derivative gravity\footnote{Recently in \cite{paulos} a Wald like formula for shear viscosity has been proposed.}.
All these approaches indicate that there can be an "IR" description of transport coefficients $\eta$. However, it is not
quite clear how other transport coefficients like $\tau_{\pi}$ etc. which appear in next order in $\omega$ can be
calculated only in terms of horizon data. The derivation of \cite{liu,ns1} was strictly valid only in
$\omega \rightarrow 0$ limit.

Although, all the above references state that that one can possibly analyse the transport coefficient $\eta$ of the boundary plasma completely in terms of horizon data, but still the full bulk solutions were used in the actual computation. In this paper, we
observe that only with the knowledge of near horizon geometry of a black hole spacetime one
can easily calculate the thermodynamic and hydrodynamic quantities, namely entropy and shear viscosity of boundary
fluid and their ratio. One does not need to know the full analytic solutions of Einstein equations. This observation
is helpful when bulk Lagrangian is very complicated. For example when gravity is coupled to various matter fields
(gauge field, scalars etc.) in non-trivial way, it may not be always possible to find an analytic solution.
But if a black hole solution exists then it is possible to find the near
horizon geometry, i.e. how the metric and other fields behave in near horizon limit. The method is very simple.
We do not need to solve any differential equation to find the near horizon geometry. First we write the field equations.
Then take a suitable near horizon ansatz for different fields. In general different components of the metric goes like $g_{tt} \sim a_1 (r-r_h) + a_2 (r-r_h)^2+ \cdots, \ \ g_{rr}\sim {b_1 \over r-r_h}(1+ b_2 (r-r_h)+\cdots$)(for non-extremal case), scalar fields behaves as $\varphi \sim \varphi_h + \varphi_1 (r-r_h)+\cdots$ and gauge fields goes as $A_t = q(r-r_h)+\cdots$. Substituting these in the field equations we find the coefficients
consistently order by order in $(r-r_h)$. Once we find the near-horizon structure of the spacetime we find entropy of the system using Wald's formula (for non extremal black hole) or entropy function formalism (for extremal black hole). In either case we only need to calculate the horizon values of some quantities. To calculate shear viscosity we adopt the method proposed in \cite{liu,cai1,ns1}. We write the effective action for transverse graviton in black hole near-horizon region and find the coupling constant. For non-extremal black hole usually (in presence of higher derivative terms) the effective coupling depends on radial coordinate and we need to take $r\rightarrow r_h$ limit at the end to find the horizon value of the coupling constant. For extremal black hole it turns out to be constant.

In case of non-extremal black holes the near horizon geometry is not completely determined always.
For example we consider dilaton field coupled to cosmological constant and higher derivative terms in sec.
\ref{neutralisation} and dilaton field coupled to gauge field non trivially in sec. \ref{dilcharge}. We found
that the horizon values of the dilaton field ($\varphi_h$) is not determined. Other coefficients of metric, gauge fields and dilaton depend on $\varphi_h$.
Therefore the ratio $\eta/s$ depends on the horizon values of the dilaton field. Where as in extremal case the near-horizon geometry is completely fixed in terms of black hole charges and hence $\eta/s$ is also completely determined in  terms of the charges of black hole.

Our paper is organised as follows. In section \ref{nonex} we study the near-horizon geometry of different non-extremal black holes and find $\eta/s$ for dual conformal plasma in presence of global charges. In particular we studied $F^4$ terms in Gauss-Bonnet Gravity (sec. \ref{f4}), non-trivial dilaton coupled to Gauss-Bonnet term (sec. \ref{neutralisation}) and Maxwell term (sec. \ref{dilcharge}). We study near horizon geometries of extremal black holes in presence of different higher derivative terms and find $\eta/s$ of boundary field theory at zero temperature (sec. \ref{ex}). Finally in sec. \ref{lif} we consider near-horizon geometry of black holes in asymptotically Lifshitz spacetime and find $\eta/s$ for its dual plasma at Lifshitz fixed point.

\section{Non Extremal Black Holes}\label{nonex}

Now, we will consider theories with some matter coupled to metric. Here, we find $\eta/s $ for these theories by computing the horizon value of graviton's effective coupling. In \cite{ns1}, we proved that for generic higher-derivative term added to the Einstein-Hilbert action, one can actually write an effective action (which gives the same equation of motion as the original one) in canonical form. The canonical form of the action was required to conclude that the transport coefficients (in particular shear-viscosity coefficient) of the boundary plasma is related to the effective coupling of the transverse graviton. Hence, to justify that even in theories with arbitrary matter content, the effective coupling can still be related to the transport coefficient of the dual boundary plasma, we need to prove that the action would maintain its canonical form. This issue has been well studied in \cite{kovtun}. In presence of generic matter in the action, one can still show that the transverse graviton satisfies the equation of a massless scalar field. Therefore, the action evaluated on shell would always have the canonical form.

\subsection{$F^4$ Terms In Gauss-Bonnet Gravity}\label{f4}

As a first example we consider the following Lagrangian.
\begin{eqnarray}\displaystyle
{\cal I}&=&{1\over 16 \pi G_5} \int d^5 x \sqrt{-g} \Big{[} R + 12 + g \left( R^2 -4 R_{ij}R^{ij}+R_{ijkl}R^{ijkl} \right) \nonumber \\
&& -{1\over 4} F_{ij}F^{ij}  + c_1 (F_{ij}F^{ij})^2 + c_2 F_{ij}F^{jk}F_{kl}F^{li} \Big{]}\ .
\end{eqnarray}

From the string theory point of view, the $F^4$ term is of order of $\alpha'$ as the Gauss-Bonnet term.
In fact, in general one can also include other four derivative terms in the action\cite{myerscharge,cremonini} which arise in five dimensional gauge supergravity. However for simplicity we consider the above action. In \cite{cai1}, the authors have calculated shear viscosity to entropy density ratio using full bulk solution. Since the complete bulk solution is very complicated in presence of Gauss-Bonnet term it turns out to be difficult to express the ratio completely in terms of black hole charges. Here we study the near-horizon geometry of this black hole spacetime which is in fact very simple and compute shear viscosity, entropy and their ratio. We are able to express them in terms of black hole charges. Our expressions are exact in $g$.

We consider the following ansatz for metric and gauge field,
\begin{eqnarray}\displaystyle
ds^2&=& -g_{tt}(r) dt^2 + {\sigma^2(r)\over g_{tt}(r)}dr^2 + r^2 d\vec{x}^2\\
A(r)&=&A_t(r) \ dt
\end{eqnarray}
where,
\begin{eqnarray}\displaystyle
g_{tt}(r)&=&a_1(r-r_h)+ a_2(r-r_h)^2 \cdots,  \nonumber \\
\sigma(r)&=&\sigma_0 + \sigma_1(r-r_h)+ \cdots,  \nonumber \\
A_t(r)&=& q \ \sigma_0(r-r_h)+ \cdots.
\end{eqnarray}
The coefficient $\sigma_0$ can be set to one by rescaling the time and the parameter $q$ is related to the physical charge of the system.
Substituting this ansatz in field equations one can solve for $a_1$, $a_2$, $\sigma_1$ etc. order by order in ($r-r_h$). It turns out that only the coefficient $a_1$ appears in entropy, shear viscosity and their ratio.
\begin{equation}
a_1=-{r_h (6 c_1 q^4 + 3 c_2 q^4 + q^2 - 96)\over 24} \ .
\end{equation}

Once we know the near horizon geometry one can use Wald's formula to find the entropy of the system. It is given by (s in entropy density),
\begin{equation}
s={r_h^3 \over 4 G_5}\ .
\end{equation}
Using the prescription given in \cite{liu,cai1,ns1} we calculate shear viscosity coefficient. It turns out to be,
\begin{equation}
\eta = \frac{r_h^3}{16 \pi  G_5}\left( 1 - {2 g a_1\over r_h} \right)\ .
\end{equation}
Therefore the ratio $\eta/s$ is given by,
\begin{eqnarray}\displaystyle
{\eta \over s} &=&\frac{1}{4 \pi }\left( 1 - {2 g a_1\over r_h} \right) \nonumber \\
&=&\frac{1}{4 \pi }\left( 1 + { g}\ {6 c_1 q^4 + 3 c_2 q^4 + q^2 - 96\over 12}\right)\ .
\end{eqnarray}

Temperature can also be calculated with near horizon data. It turns out to be,
\begin{equation}
T={a_1 \over 4 \pi}
\end{equation}
Therefore one can express $\eta/s$ in terms of black hole temperature,
\begin{equation}
{\eta \over s} =\frac{1}{4 \pi }\left( 1- {8 \pi g \over r_h} {T}\right)\ .
\end{equation}
This expression matches with \cite{cai1}. However here we are able to express $\eta/s$ in terms of charge parameter also. In the limit $T\rightarrow 0$ (which is extremal limit), the ratio $\eta/s$ approaches its universal value.

\subsection{Non-trivial Dilaton}\label{neutralisation}

So far we have considered Maxwell-Einstein-Hilbert action in presence of four derivative terms. In this section we consider dilaton field which is non-minimally coupled to gravity and find the effects of non-trivial dilaton on $\eta/s$.
\begin{equation}
{\cal I}= {1 \over 16 \pi G_{5}}\int d^5x \sqrt{- g} \left [ R -{1 \over 2}\partial_{\mu}\varphi \partial^{\mu}\varphi
- 2 \Lambda \,\,e^{\tau \varphi}+{g}e^{- \gamma \varphi}\,\, {\cal L}_{(GB)}\right ]
\end{equation}
with,
\begin{equation}
{\cal L}_{(GB)}= R^2+ R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma} -4 R^{\mu\nu}R_{\mu\nu}
\end{equation}
where, $\gamma$ and $\tau$ are constants and $\varphi$ is
the dilaton field. One can get this action by transforming the string frame action with
a non-trivial dilaton to the Einstein frame. Though from string theory point of view the values of $\tau$ and $\gamma$ are fixed but here we keep them as some arbitrary parameter.  In \cite{got} it has been shown that there exists an asymptotically AdS black hole solution for some suitable range of parameters. However here we examine the near horizon geometry of this black hole spacetime and calculate $\eta$, $s$ and their ratio. It turns out that the ratio (also $s$ and $\eta$) depends on the horizon values of the dilaton field. The horizon value of the dilaton field is not some arbitrary parameter of the theory it is fixed in terms of the charges of the black hole (in this case only mass). Our results are exact in $g$.

The near horizon ansatz for this system is given by,
\begin{eqnarray}\displaystyle
ds^2&=& -B(r) e^{-2 \delta(r)} dt^2+{dr^2 \over B(r)}+ r^2 d\vec{x}^2,\nonumber \\
\delta(r)&=& \delta_0+\delta_1 (r-r_h)+ \delta_2 (r-r_h)^2+\cdots,\nonumber \\
B(r)&=& b_1 (r-r_h)+b_2(r-r_h)^2+\cdots,\nonumber \\
\varphi(r)&=& \varphi_h+ \alpha_1 (r-r_h)+\alpha_2(r-r_h)^2+\cdots\ .
\end{eqnarray}
Here, $\varphi_h$ is the constant horizon value of the dilaton field. We substitute these ansatz in the equation of motion
and get the following values of the coefficients (we present the expressions of those coefficients which appear in $\eta/s$).
\begin{eqnarray}\displaystyle
\alpha_1 &=&-{8 e^{-(\gamma-\tau)}g \gamma \Lambda + 3 \tau \over r_h},\nonumber   \\
\delta_1&=& - {r_h \alpha_1^2 \over 6}, \nonumber \\
b_1&=&-{2 \over 3}e^{\tau \phi_h}r_h \Lambda .
\end{eqnarray}

Once we know the near horizon geometry we calculate $s$ and $\eta$ as before and get,
\begin{equation}
s= {r_h^3 \over 16 \pi G_5},
\end{equation}
and
\begin{equation}
\eta=
r_h^3\frac{e^{-2 \gamma  \varphi_h} \left(32 g^2 \gamma ^2
   \Lambda ^2 e^{2 \tau  \varphi_h}+4 g \Lambda  (3
   \gamma  \tau +1) e^{\varphi_h (\gamma +\tau )}+3
   e^{2 \gamma  \varphi_h}\right)}{48 \pi  G_5}\ .
\end{equation}
Therefore the ration turns out to be,
\begin{eqnarray}\displaystyle
{\eta\over s}&=&\frac{1}{4 \pi }+
\frac{8 g^2 \gamma ^2 \Lambda ^2 e^{2 \tau  \varphi_h-2
   \gamma  \varphi_h}}{3 \pi }+\frac{g \gamma  \Lambda
   \tau  e^{(\tau - \gamma)
   \varphi_h}}{\pi }+\frac{g \Lambda  e^{(\tau - \gamma) \varphi_h}}{3 \pi
   } \nonumber \\
&=& {1\over 4 \pi} \left [ 1-8 g e^{-(\gamma-\tau)\varphi_h} \left( 1+ 3 \gamma \tau - 48 \gamma^2 g
e^{-(\gamma-\tau)\varphi_h} \right) \right ]\ , \ \ \ \Lambda = -6\ .
\end{eqnarray}
The answer has been obtained in \cite{cai2}.

\subsection{Charged Black Holes with Dilaton}\label{dilcharge}

The last non extremal black hole we consider is electrically charged black hole in presence of non-trivial dilaton field in presence generic four derivative terms. The action is given by \cite{gm},
\begin{eqnarray}\displaystyle
{\cal I}={1 \over 16 \pi G_5}\int d^Dx \sqrt{-g}[ R &-& {1 \over D-2} \partial_{\mu}\phi
\partial^{\mu}\varphi - V(\varphi) - {1 \over 4} e^{{- g_{0} \varphi}} F_{\mu\nu}F^{\mu\nu}\\
&-& {1 \over 2(D-2)!} e^{- g_{0} \varphi} F_{\mu_1\mu_2....\mu_{D-2}}F^{\mu_1\mu_2....
\mu_{D-2}}+ g {\cal L}_{(4)}]
\end{eqnarray}
with,
\begin{equation}
{\cal L}_{(4)} = \beta_1 R^2+ \beta_2 R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}+\beta_3 R^{\mu\nu}R_{\mu\nu}
\end{equation}
It has been shown in \cite{ptw} that when $V(\varphi)=|\Lambda|$ (positive cosmological constant) there exists not black hole solution. However there exists an asymptotically AdS spherical black hole solution when, $V(\varphi)=-|\Lambda|$ with a maximum single horizon.
The asymptotic value of the dilaton field is fixed in terms of charges of the black hole. Therefore it is not an independent hair. In \cite{ptw} authors found a numerical black hole solution with two parameters namely position of the horizon $r_h$ and horizon value of the dilaton field $\varphi_h$. Here we find near-horizon geometry of the black hole spacetime and calculate $\eta/s$. The ratio depends on the horizon value of the dilaton. The results are valid up to order $g$.

We will work in five dimension and only with electrically charge gauge fields. We also treat the four derivative terms perturbatively. The leading (in $g\rightarrow 0$ limit) near-horizon ansatz is
\begin{eqnarray}\displaystyle
ds^2&=& -A(r) \sigma(r)^2 dt^2 +{dr^2 \over A(r)}+ r^2 dX^2, \nonumber \\
A(r)&=&\alpha_1 (r-r_h)+\alpha_2 (r-r_h)^2 +\cdots, \nonumber \\
\sigma(r)&=& \sigma_0+\sigma_1 (r-r_h)+\sigma_2 (r-r_h)^2 +\cdots ,\nonumber \\
\varphi(r)&=&\varphi_h+\varphi_1 (r-r_h)+\varphi_2 (r-r_h)^2 +\cdots ,\nonumber \\
A_t(r)&=&Q(r-r_h) + q_2(r-r_h)^2 + \cdots\ .
\end{eqnarray}
The near-horizon behaviour of the fields as follows
\begin{equation}
\alpha_1={r_h \over 24}\lb8 \Lambda - e^{-g_0 \varphi_h}Q^2\right),\ \ \ \ \ \ \ \sigma_1 = {3 g_0^2 Q^4
\sigma_0 \over 2 r_h (Q^2-8 e^{-g_0 \varphi_h}Q^2)}
\end{equation}
and
\begin{equation}
 \alpha_2= \frac{e^{-g_0 \varphi_h} \left(\left(9 g_0^2+14\right) Q^4-128 Q^2 \Lambda
e^{g_0 \varphi_h}+128 \Lambda ^2 e^{2 g_0
   \varphi_h}\right)}{96 \left(Q^2-8 \Lambda e^{g_0 \varphi_h}\right)}   .
\end{equation}

The entropy and shear viscosity is given by,
\begin{equation}
s={r_h^3 \over 4 G_5} \left( 1 + {e^{-g_0 \varphi_h}(-Q^2(\beta_1 +7\beta_2+2\beta_3)
-8 e^{g_0 \varphi_h}(5\beta_1 -\beta_2+\beta_3)\Lambda )g \over 12}\right) + {\cal O}(g^2)
\end{equation}
\begin{equation}
\eta={r_h^3 \over 16 \pi G_5} \left( 1 + {e^{-g_0 \varphi_h}(-Q^2(\beta_1 +6\beta_2+2\beta_3)
-8 e^{g_0 \varphi_h}(5\beta_1+\beta_3)\Lambda )g \over 12}\right)\ + {\cal O}(g^2).
\end{equation}
Therefore the ratio is given by,
\begin{equation}
{\eta\over s}={1 \over 4\pi}\left [ 1+ g \beta_2 {e^{-g_{0}\varphi_h} Q^2 - 8 \Lambda \over 12} \right ]\ + {\cal O}(g^2).
\end{equation}
The black hole temperature can also be calculated from near-horizon geometry.
\begin{equation}
T\sim e^{-g_{0}\varphi_h} Q^2 - 8 \Lambda\ .
\end{equation}
In $T\rightarrow 0$ limit the the horizon value of the dilaton is also determined in terms of black hole charges and the ratio approaches its universal value.

\section{Extremal Black Holes}\label{ex}

We will first briefly discuss how charge black holes can arise in five
dimensions in the context of string theory.  A consistent truncation
of ${\cal N}=8$, $D=5$ gauged supergravity with $SO(6)$ Yang-Mills
gauge group, which can be obtained by $S^5$ reduction of type IIB
supergravity, gives rise to ${\cal N}=2$, $D=5$ gauge supergravity
with $U(1)^3$ gauge group.
The same theory can also be obtained by compactifying eleven
dimensional supergravity, low energy theory of M theory, on a
Calabi-Yau three folds. The bosonic part of the action of ${\cal
  N}=2$, $D=5$ gauged supergravity is given by,
\begin{eqnarray}\displaystyle\label{gsacn}
{\cal I} &=& {1 \over 16 \pi G_{5}}\int d^5x \sqrt{- g} \left [ R
- {1 \over 4}X_I^2 F^I_{\mu\nu}F^{\mu\nu I}-{1 \over 2}X_I^{-2}
\partial_{\mu} X_I \partial^{\mu}X_I -V \right ]  \nonumber \\
&& + C.S \,\,\, terms
\end{eqnarray}
where, $X^I$'s are three real scalar fields, subject to the constraint
$X^1 X^2 X^3 =1$. $F^I$'s, which are field strengths of three Abelian gauge
fields (I,J=1,2,3), and the scalar potential $V(X)$ is given by,
\begin{eqnarray}\displaystyle
F_{\mu\nu}^I &=& 2 \partial_{[\mu}A^I_{\nu]},\,\,\,\,\ V= - 4 \gamma^2 \Sigma
_{I=1}^{3}X_I,\,\,\,\ X_1X_2X_3=1\ .
\end{eqnarray}

Solving the equations of motion one gets a black hole solution with three $U(1)^3$ charges. Using this equations of motions one can calculate entropy and shear viscosity of boundary plasma and their ratio. It turns out that the ratio saturates the famous {\bf KSS} bound \cite{sonr}.
We will consider higher derivative corrections to this action in next two subsections. Calculation of entropy and shear viscosity for non-extremal black hole for three independent non-zero charges seems to be very difficult. One can consider different limits, for example three equal charges (which is same as Maxwell-Einstein-Hilbert action) and calculate different thermodynamic and hydrodynamic coefficients.
However we will concentrate only on the extremal limit of this black hole (for three independent charges). In the extremal limit the near horizon geometry is universal and given by \cite{morales}.  We calculate corrections to entropy for higher derivative terms using entropy function formalism. Evaluating the effective action for transverse graviton in this near horizon geometry we also compute higher derivative correction to shear viscosity coefficient of zero temperature boundary plasma in presence of global $U(1)^3$ charges.

\subsection{Four Derivative Corrections to R-Charged Black Holes}

In this section we consider a generic four derivative correction to action (\ref{gsacn}).
\begin{eqnarray}\displaystyle
{\cal I} &=& {1 \over 16 \pi G_{5}}\int d^5x \sqrt{- g} \left [ R
- {1 \over 4}X_I^2 F^I_{\mu\nu}F^{\mu\nu I}-{1 \over 2}X_I^{-2}
\partial_{\mu} X_I \partial^{\mu}X_I -V + g \,\, {\cal L}_{(4)}\right ]  \nonumber \\
&& \ \ \ \ \ \ +  C.S \,\,\, terms
\end{eqnarray}
and
\begin{equation}
 {\cal L}_{(4)}= \beta_1 R^2
+ \beta_2 R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}+\beta_3 R^{\mu\nu}R_{\mu\nu}\ .
\end{equation}
The extremal Black hole
has $AdS_2$ part in there near horizon geometry. The full near-horizon ansatz is as follows,
\begin{eqnarray}\displaystyle\label{exnh}
ds^2 &=& v_1 (- r^2 dt^2 + {dr^2 \over r^2}) + v_2 d\vec{x}^2 \nonumber \\
X_I&=& u_I, \,\,\,\ A^I= - e_I r dt,\,\,\,\ F^I_{tr}=e_I\ .
\end{eqnarray}
We will not explain the entropy function formalism here, as it is very well studied in literature. We will just present the results here.
Following \cite{morales} we choose three parameters $\mu_1, \mu_2$ and $\mu_3$ and parametrize $u_I's$ and $v_2$ as,
\begin{equation}
u_I = {\mu_I \over (\Pi \mu_J)^{1/3}}, \,\,\,\,\ v_2= (\Pi \mu_J)^{1/3}\ .
\end{equation}
Then the other parameters are given by (in terms of $\mu_I's$),
\begin{eqnarray}\displaystyle
v_1&=&{(\Pi \mu_J)^{1/3} \over 4 \gamma^2 \Sigma \mu_J}, \nonumber \\
e_I &=& e_I^0+ g e_I^g, \,\,\,\,\,\,\,\,\,\  q^I=q^I_0+ g q^I_g,\nonumber \\
e_I^0&=& {\sqrt{{\Pi \mu_J \Sigma_{J \ne I}\mu_J \over 2 \gamma^2 (\Sigma\mu_J)^2}} \over \mu_I}, \,\,\,\
e_I^g= - {4 (\Pi \mu_J)^{2/3} (2 \beta_1+ 2\beta_2+ \beta_3)  \over 3 \mu_I \sqrt{{\Pi \mu_J
\Sigma_{J \ne I}\mu_J \over 2 \gamma^2 (\Sigma\mu_J)^2}}},
\end{eqnarray}
and
\begin{equation}\label{qi}
q^I_0 = {2 V_3 \gamma^2 \Sigma \mu_J \mu_I^2 \over (\Pi \mu_J)^{1/2}} e_I^0 ,\,\,\
q^I_g = {2 V_3 \gamma^2 \Sigma \mu_J
\mu_I^2 \over (\Pi \mu_J)^{1/2}} e_I^g\ .
\end{equation}
Here, $\Sigma \mu_J= (\mu_1+\mu_2+\mu_3)$
and $\Pi \mu_J=\mu_1\mu_2\mu_3 $. Repeated indices in $q^I$ are $not$ summed over.

After finding higher derivative correction to the near horizon geometry we compute the entropy of the extremal Black hole,
\begin{equation}
s={\sqrt{\Pi \mu_J} \over 4 G_5}\left [ 1 - 8 g \gamma^2 (2 \beta_1+2\beta_2+\beta_3)
{\Sigma \mu_J \over (\Pi \mu_J)^{1/3}}\right ]\ + {\cal O}(g^2).
\end{equation}

Similarly we can also find the shear-viscosity coefficient from gravitons effective coupling.
It is given as,
\begin{equation}
\eta={\sqrt{\Pi \mu_J} \over 16 \pi G_5}\left [ 1 - 8 g \gamma^2 (2 \beta_1+2\beta_2+\beta_3)
{\Sigma \mu_J \over (\Pi \mu_J)^{1/3}}\right ]\ + {\cal O}(g^2).
\end{equation}
In order to obtain the expressions in terms of black hole charges one has to invert the equation (\ref{qi}) and write $\mu_I's$ in terms of $q_I's$.
The shear viscosity to entropy density ration is given by,
\begin{equation}
{\eta\over s}= {1 \over 4 \pi} + {\cal O}(g^2).
\end{equation}
Thus we see that in the extremal limit the correction saturates the {\bf KSS} bound for any generic four derivative terms in the action.

\subsection{Six Derivative Correction R-Charged Black Holes}

 In this section, we consider generic six derivative terms to the five dimensional gauge sugra Lagrangian
and determine the $\eta/s$ ratio. There are eight
possible six derivatives terms constructed our of curvature tensors listed below, and a generic six derivative
correction term is constructed by taking arbitrary linear combinations of them. They are as follows,
 \begin{eqnarray}\displaystyle
 T_1&=& R^{\mu\nu\sigma\kappa}R_{\sigma\kappa\rho\tau}R^{\rho\tau}_{\mu\nu}, \,\,\,\,\
 T_2= R^{\mu}_{\nu\rho\sigma}R^{\rho}_{\tau\mu\lambda}R^{\sigma\tau\nu\lambda},\nonumber \\
 T_3&=&R^{\mu\nu\sigma\kappa}R_{\sigma\kappa\nu\rho}R^{\rho}_{\mu}, \,\,\,\,\
T_4= R R^{\mu\nu\sigma\kappa}R_{\sigma\kappa\mu\nu}, \nonumber \\
 T_5&=&R^{\mu\nu\sigma\kappa}R_{\mu\sigma}R_{\nu\kappa}, \,\,\,\,\,\
T_6=R^{\mu}_{\nu}R^{\nu}_{\sigma}R^{\sigma}_{\mu}, \nonumber \\
 T_7&=& R R^{\mu}_{\nu}R^{\nu}_{\mu}, \,\,\,\,\,\,\ T_8= R^3, \nonumber \\
 {\cal L}_{(6)}&=& \alpha_1 T_1 +\alpha_2 T_2 +\alpha_3 T_3 +\alpha_4 T_4
+\alpha_5 T_5 +\alpha_6 T_6 +\alpha_7 T_7 +\alpha_8 T_8
 \end{eqnarray}

Now, we consider an action with this generic six derivative correction to (\ref{gsacn}),
\begin{eqnarray}\displaystyle
{\cal I} &=& {1 \over 16 \pi G_{5}}\int d^5x \sqrt{- g} \left [ R - {1 \over 4}X_I^2 F^I_{\mu\nu}F^{\mu\nu I}-{1 \over 2}X_I^{-2}
\partial_{\mu} X_I \partial^{\mu}X_I -V + g \,\, {\cal L}_{(6)}\right ]  \nonumber \\
&& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + C.S \,\,\, terms \ .
\end{eqnarray}
In \cite{ns2}, we considered this correction (even with the lower order GB term) to Einstein-Hilbert action. Using
field redefinition, we were able to show that few terms would not occur in the final expression of $\eta/s$.
This is no longer true for this action\footnote{It is easy to check that the arguments given in reference \cite{ns2} no longer holds for all coefficients.}. The near-horizon geometry is given by (\ref{exnh}). The solution turns out to be,
\begin{equation}
u_I = {\mu_I \over (\Pi\mu_J)^{1/3}}, \,\,\,\,\ v_2= (\Pi\mu_J)^{1/3}, \ \ \ v_1 = v_1^0+ g v_1^g, \,\,\,\,\,\,\,\
v_1^0= {v_2 \over 4 \gamma^2 \Sigma \mu_J}, \,\,\,\,\ v_1^g= -{\alpha \over v_1^0}
\end{equation}
where,
\begin{equation}
e_I=e_I^0+ g e_I^g , \,\,\,\,\,\,\,\ q^I=q^I_0+ g q^I_g\ ,
\end{equation}
\begin{equation}
e_I^0 = {\sqrt{{\Pi \mu_J \Sigma_{J \ne I}\mu_J \over 2 \gamma^2 (\Sigma\mu_J)^2}} \over \mu_I} \ , \ \ \
e_I^g= {8 \gamma^2 \alpha (\Pi \mu_J)^{1/3} {(\Sigma \mu_J+ 3 \mu_I)
\over 3 \mu_I \sqrt{{\Pi \mu_J \Sigma_{J \ne I}\mu_J \over 2 \gamma^2
(\Sigma\mu_J)^2}}}}\ ,
\end{equation}
\begin{equation}
q^I_0= {V_3\mu_I^2 e_I^0 \over 2 \sqrt{v_2}v_1^0}\ , \ \ \ \ \ \ \ \
q^I_g= {V_3 \mu_I^2  \over 2 \sqrt{v_2}}
(e_I^g - {v_1^g \over (v_1^0)^2})
\end{equation}
and $\alpha$ is defined to be,
\begin{equation}
\alpha=4 \alpha_1 - 2 \alpha_3+4 \alpha_4 +\alpha_5+\alpha_6+2 \alpha_7+4 \alpha_8 \ .
\end{equation}

Now, using entropy function formalism, it is easy to find out the corrected entropy (density) for the extremal
black holes. It result is,
\begin{equation}
s={\sqrt{\Pi \mu_J} \over 4 G_5}\left [ 1 + 48 g \gamma^4 \alpha {(\Sigma \mu_J)^2 \over (\Pi \mu_J)^{2/3}}\right ] + {\cal O}(g^2)\ .
\end{equation}

Now, following the effective action technique, we can compute the shear-viscosity coefficient for these
extremal black holes. The coefficient and the $\eta/s$ ratio are as follows,
\begin{equation}
\eta={\sqrt{\Pi \mu_J} \over 16 \pi G_5}\left [ 1 - 16 g
\gamma^4 (3\alpha_2+4 \alpha_3-12 \alpha_4-2 \alpha_5-3 \alpha_6-6 \alpha_7-12 \alpha_8)
{(\Sigma \mu_J)^2 \over (\Pi \mu_J)^{2/3}}\right ] + {\cal O}(g^2)\ .
\end{equation}
Finally $\eta/s$ is given by,
\begin{equation}
{\eta\over s}={1 \over 4 \pi}\left [ 1 -16 g \gamma^4 (12\alpha_1+3 \alpha_2-2 \alpha_3+\alpha_5)
{(\Sigma \mu_J)^2 \over (\Pi \mu_J)^{2/3}}\right ] + {\cal O}(g^2)\ .
\end{equation}
The ratio approaches to the universal bound for pure Lovelock gravity.

\section{Black Holes in Asymptotically Lifshitz Spacetime}\label{lif}

In this section we study the hydrodynamic behaviour of boundary gauge theory plasma at Lifshitz like fixed point. The dual gravity
theory of a Lifshitz like fixed point was first proposed by Kachru et.al. \cite{kachru}. They presented a four dimensional bulk
action in presence of one and two form fields. Although it is not possible to embed this action
in any string theory \cite{taka} but it is a good holographic phenomenological model to study the behaviour of boundary field theory
at Lifshitz like fixed point. The action is given by,
\begin{equation}
{\cal I}={1\over 16 \pi G_4}\int d4^x \sqrt{-g} \left( R - 2 \Lambda -{1\over 4} F_{\mu\nu}F^{\mu\nu}-{c^2 \over 2}{\cal A}_{\mu}{\cal A}^{\mu} + g {\cal L}_{(4)}\right)\ .
\end{equation}
where,
\begin{equation}
F=d{\cal A}
\end{equation}
and
\begin{equation}
c={\sqrt{2 z} \over L},\ \ \ \ \ \Lambda=-{z^2+z+4 \over 2 L^2} \ .
\end{equation}
There can be order $g$ corrections to $c$ and $\Lambda$ in presence of Gauss-Bonnet term \cite{anindya}. But they will not affect $\eta/s$ calculation at order $g$.
This action is related to \cite{kachru} up to some field re-definition.

The finite temperature solution (black hole solution) (numerically) for this kind of action was proposed in
\cite{ulf,mann} for $z=2$ where $z$ is the critical exponent. Later in \cite{bertoldi} authors have found a numerical black hole
solution for any values of $z$. A complete analytic black hole solution for
this kind of phenomenological action is not yet known (see \cite{taylor} for a different model). We study the near horizon
structure of this black hole and compute entropy, shear viscosity and their ratio of the dual field theory at Lifshitz
fixed point. Our results are valid up to order $g$.

We consider the black hole solution of the following form.
\begin{eqnarray}\displaystyle
ds^2 &=&-e^{2 A(r)}dt^2 + e^{2 \sigma(r)}dr^2 + r^2 (dx_1^2 + dx_2^2)\ , \nonumber \\
{\cal A}&=&e^{G(r)}dt\ .
\end{eqnarray}
Substitute this ansatz for metric and gauge fields in the action (in $g\rightarrow limit$) and find equations of
motion for $A(r)$,
$\sigma(r)$ and $G(g)$. Then we consider the following near-horizon ansatz for the fields
\begin{eqnarray}\displaystyle
&&A(r)=\ln\left(r^z L\left(a_0(r-r_h)^{\frac12}+a_0a_1(r-r_h)^{\frac32}+\cdots\right)\right),\nonumber \\
&&\sigma(r)= \ln\left(\frac{L}{r}\left(c_0(r-r_h)^{-\frac12}+c_1(r-r_h)^{\frac12}+\cdots\right)\right), \nonumber
\\ &&G(r)=\ln\left(\frac{L^2r^z}{z}\sqrt{\frac{2z(z-1)}{L^2}}\left(a_0g_0(r-r_h)+a_0g_1(r-r_h)^2+\cdots\right)\right).
\end{eqnarray}
and substituting them into the equations of motion we solve for different coefficients order by order and find (see \cite{bertoldi} also),
\begin{eqnarray}\displaystyle \label{c0equ}
c_0 = \frac{\sqrt{(4z+g_0^2 r_h(z-1))} r_h^{\frac12}}{\sqrt{2 z}\sqrt{{{(z^2+z+4)}}}} \ ,
\end{eqnarray}
\begin{equation}
a_1=\frac{2 g_0^2 r_h \left(z^3+2 z^2+z-4\right)+g_0^4 r_h^2
   (z-1)^2-8 z^2 \left(z^2+z+4\right)}{8 r_h z
   \left(z^2+z+4\right)}\ ,
\end{equation}
\begin{equation}
c_1=\frac{-2 g_0^4 r_h^2 (z-1)^2 \left(z^2-11 z+4\right)+48 g_0^2 r_h
   (z-1) z^2+3 g_0^6 r_h^3 (z-1)^3+32 z^2 \left(z^2+z+4\right)}{8
   \sqrt{2} z^2 \left(z^2+z+4\right)^{3/2} \sqrt{\frac{r_h
   \left(g_0^2 r_h (z-1)+4 z\right)}{z}}}\ ,
\end{equation}
\begin{equation}
g_1=\frac{g_0 \left(g_0^4 r_h^2 (z-1)^2+8 g_0^2 r_h (z-1) z-4 z
   \left(z^3+2 z^2+z+4\right)\right)}{4 r_h z
   \left(z^2+z+4\right)}\ .
\end{equation}

Having found out the leading near-horizon geometry we find entropy and shear viscosity of dual gauge theory
at strong coupling as,
\begin{equation}
\eta=\frac{r_h^2}{16 \pi  G_5}-\frac{g \left(z^2+z+4\right) \left(g_0^2 r_h (z-1) \left(2 \beta _1+2 \beta _2+\beta _3\right)+2 z \left(4 \beta _1+\beta
   _3\right)\right)}{8 \pi  G_5 L^2 \left(g_0^2 r_h (z-1)+4 z\right)}r_h^2 + {\cal O}(g^2)
\end{equation}
and
\begin{equation}
s=\frac{r_h^2}{4 G_5}-\frac{g \left(z^2+z+4\right) \left(g_0^2 r_h (z-1) \left(2 \beta _1+2 \beta _2+\beta _3\right)+2 z \left(4 \beta _1+\beta
   _3\right)\right)}{2 G_5 L^2 \left(g_0^2 r_h (z-1)+4 z\right)}r_h^2 + {\cal O}(g^2)\ .
\end{equation}

The shear viscosity to entropy density ratio is given by,
\begin{equation}
{\eta\over s}={1\over 4 \pi} + {\cal O}(g^2)+ {\cal O}(g^2)\ .
\end{equation}
In \cite{kats} the ratio $\eta/s$ has been computed for any generic four derivative terms in any dimension for asymptotically AdS spacetime. It turns out that in four dimension the ratio saturates the universal bound. Here also we get the same result since the near horizon geometry is same for both the cases.

When we are preparing this manuscript some interesting papers appear \cite{cai3,sayan,bindu}.\\

\noindent
{\bf Acknowledgement}

We would like to that D. Astefanesei, A. Bagchi, R. Gopakumar, R. Gupta, D. Jatkar, Prem Kumar, A. Mukhopadhyaya and A. Sen for useful discussion.
This work of NB is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is financially supported by the Netherlands Organisation for Scientific Research (NWO).
SD would like to acknowledge the hospitality of Harish-Chandra Research Institute, Allahabad, India where some part
of this work was completed.


\section{Introduction}

Complex oxide solids are known to exhibit structural phase
transitions~\cite{citeulike:5298220}. These transitions are relevant for
applications, as material properties below and above the transition typically
differ, but are also interesting in their own right, regarding the fundamentals
of solid phase behavior. Consequently, these transitions have received
considerable attention, including increasingly, by means of computer simulation.
An example is the study of \olcite{citeulike:13839098}, where a simple particle
model is proposed featuring a displacive structural phase transition. We shall,
in what follows, refer to this model as the BVST model (after the first letter
of each author's surname) and focus exclusively on its {\it equilibrium} phase
behavior.

Despite the apparent simplicity of the BVST model, its equilibrium phase
behavior is far from trivial. Indeed, the authors already announce the
possibility of a Kosterlitz-Thouless (KT) transition, citing
\olcite{joseRenormalizationVorticesSymmetrybreaking1977}. Further inspection of
\olcite{joseRenormalizationVorticesSymmetrybreaking1977}, as well as following
up on valuable comments made by an anonymous referee of an earlier draft of this
manuscript~\cite{vinkCriticalBehaviorDisplacive2018a}, suggest that the BVST
model could be in the universality class of the two-dimensional $q=6$ clock
model. If this is the case, the model should, in fact, undergo two separate
phase transitions, and, consequently, support three thermodynamically distinct
phases~\cite{joseRenormalizationVorticesSymmetrybreaking1977}. The purpose of
the present paper is to verify, via Monte Carlo (MC) simulations and finite-size
scaling, whether this scenario applies.

As it turns out, our simulations strikingly show that the BVST model is a member
of the $q=6$ clock model universality class. The existence of two phase
transitions, implying three phases, is clearly visible. In addition, critical
exponents obtained using finite-size scaling, are consistent with those of the
$q=6$ clock model. In what follows, we will present the analysis leading to
these results. The outline is as follows: We first describe the BVST
model~\cite{citeulike:13839098}, followed by a brief summary of the $q=6$ state
clock model. We then present our MC data, followed, in Section~IV, by a
discussion and summary. The details of the MC methods used in this work are
provided in the Appendix.

\section{Models}

\subsection{BVST model}

The BVST model~\cite{citeulike:13839098} provides a simple description of a
material exhibiting a structural phase transition. It qualitatively resembles a
system whereby, during the transition, one of the particle species remains fixed
(as do, e.g.~Ba atoms in the case of $\rm BaTiO_3$~\cite{citeulike:5298220}).
The fixed species is assumed to provide an underlying lattice structure, as well
as to give rise to a (static) external field acting on the mobile species. It is
assumed there is only one type of mobile particle species, and the total number
of these particles is denoted~$N$. In addition, there is a pair interaction
between the mobile species, described in the form of permanent anharmonic bonds.
The total energy of the system is thus given by $E = \sum_{[ij]} u_{ij} +
\sum_{i=1}^N h_i$, where $[ij]$ is a sum over bonded pairs of particles $i$ and
$j$, $u_{ij}$ the corresponding bond energy, and $h_i$ the external field acting
on particle~$i$.

The underlying crystal structure is assumed to be a hexagonal lattice, i.e.~the
model is purely two-dimensional. The nearest neighbor distance between lattice
points is denoted~$a$. To each lattice position $\vec{R}_i$, a single particle
is assigned $(i=1,\ldots,N)$. The {\it displacement} of particle $i$ from
its lattice position $\vec{R}_i$ is denoted $\vec{r}_i=(x_i,y_i)$. During the
simulations, the particle displacements $\vec{r}_i$ are allowed to fluctuate,
but the lattice positions $\vec{R}_i$ remain fixed.

Each particle is bonded to its six nearest neighbors by means of an anharmonic
spring. The spring energy is given by $u_{ij} = b_2 (d_{ij}-a)^2 + b_4
(d_{ij}-a)^4$ with $d_{ij} = |\vec{R}_j + \vec{r}_j - \vec{R}_i - \vec{r}_i|$
the distance between the two particles. The bonds ($3N$ in total) are assigned
once at the start of the simulation. During the simulations, there is no
breaking of bonds, nor the formation of new bonds.

The external field acting on particle $i$ is defined in terms of its
displacement $\vec{r}_i$ as follows:
\begin{equation*}
 \frac{h_i}{\epsilon} = 1 + \left( \frac{r_i}{a_0} \right)^4 -
 2 \left[
  (3 \cos\theta_i - 4\cos^3 \theta_i) \frac{r_i}{a_0}
 \right]^2 \quad,
\end{equation*}
with $r_i=\sqrt{x_i^2+y_i^2}$ and $\cos\theta_i=x_i/r_i$. Due to the external
field $h_i$, each lattice position $\vec{R}_i$ is \ahum{surrounded} by six local
energy minima, at coordinates $\vec{R}_i + a_0 (\cos \pi \lambda/3, \sin \pi
\lambda/3)$, with $\lambda=1,2,3,4,5,6$. A single particle can thus minimize its
field energy by selecting one of the surrounding minima. In order to
simultaneously minimize the bond energy additionally requires that all particles
choose the same $\lambda$, which leads to the ground state of the model, where
the total energy $E=0$. Hence, upon lowering the temperature, one expects
ordering to occur, whereby the particles collectively choose the same value of
$\lambda$, reminiscent of a displacive structural phase transition. As will be
shown later, the transition to the ordered (low-temperature) state proceeds via
two phase transitions.

In what follows, $b_2 = 28.32U/a^2$, $b_4=784.35U/a^4$, $\epsilon=0.2U$, and
$a_0/a=0.05$, which are the values of the original
reference~\cite{citeulike:13839098}. The lattice constant $a$ will be our unit
of length, and temperature will be expressed in units of $U/k_B$, with $k_B$ the
Boltzmann constant. We use rectangular $L_x \times L_y$ simulation cells with
periodic boundary conditions in both dimensions. To prepare the initial
hexagonal lattice, we take as unit cell a $l_x \times l_y$ rectangle, with
$l_x=a$ and $l_y=\sqrt{3}a$. The unit cell contains two lattice sites, at
coordinates $(0,0)$ and $(l_x/2,l_y/2)$. This unit cell is then replicated $2n$
times in the $x$-direction, and $n$ times in the $y$-direction, with integer
$n$. Consequently, $N = 4n^2$, $L_x=2an$, $L_y=\sqrt{3} an$, and it is ensured
that, irrespective of $n$, all our simulation cells have the same aspect ratio
$L_x/L_y$.

\subsection{$\bf q=6$ clock model}

The (two-dimensional) $q$-state clock model considers a two-dimensional space
lattice (e.g.~square, hexagonal) of sites $i=1,\ldots,N$. To each lattice
site~$i$, a discretized direction is attached, expressed via the angle $\theta_i
= 2\pi n_i/q$, with integer $n_i=1,\ldots,q$. The energy $E = -J \sum_{[ij]}
\cos(\theta_i-\theta_j)$, coupling parameter $J>0$, and sum over pairs of
nearest neighbors. The case $q=2$ is the Ising model; $q \to \infty$ the
XY-model. For $q=6$, the clock model features two phase
transitions~\cite{joseRenormalizationVorticesSymmetrybreaking1977}, at
temperatures $T_1$ and $T_2$, respectively (we set $T_1<T_2$ in what follows).
The model thus supports three, thermodynamically distinct, phases. The phases are
characterized by the decay of the angular correlation function, $G(r) = \avg{
\sum_{r_{ij}=r} \cos(\theta_i - \theta_j)} / N_r$, where the sum is over all
pairs of sites $i-j$ separated by a distance $r_{ij}=r$, $N_r$ the number of
such pairs, and $\avg{\cdot}$ a thermal average.

In the high-temperature phase, $T>T_2$, the correlations decay exponentially to
zero, $G(r) \propto e^{-r/\xi}$, $\xi$ being the correlation length. The phase
is disordered: There is no alignment of the angular directions over distances
beyond $\sim \xi$, implying that the order parameter $\Delta=0$. Provided the
simulation box dimensions $L_x,L_y > \xi$, one expects only negligible
finite-size effects in simulation data.

In the low-temperature phase, $T<T_1$, there is long-ranged order, with a
macroscopic fraction of the site orientations \ahum{pointing} in the same
direction (which can be any one of the $q$ possibilities). Consequently, the
order parameter $\Delta>0$. The correlation function still decays exponentially,
but to a finite value, $G(r)-G_\infty \propto e^{-r/\xi}$, with $G_\infty>0$.
Provided $L_x,L_y > \xi$, finite-size effects in simulation data are again
negligible.

The intermediate phase, $T_1<T<T_2$, is a {\it critical} phase, where the
correlations decay as a power law, $G(r) \propto r^{-\eta}$, implying that $\xi$
is infinite. Power law decay of correlations also implies that, in the
thermodynamic limit, the susceptibility $\chi$ is infinite, and the order
parameter $\Delta=0$. Since $L_x,L_y \ll \xi$ is now unavoidable, finite-size
effects in simulation data are strong!

We emphasize that the correlation length $\xi$, the plateau value $G_\infty$,
and the exponent $\eta$ are functions of temperature. According to theoretical
predictions, $\eta(T_2) \equiv \eta_2 =1/4$, $\eta(T_1) \equiv \eta_1
=4/q^2=1/9$~\cite{joseRenormalizationVorticesSymmetrybreaking1977}. Simulation
estimates~\cite{challaCriticalBehaviorSixstate1986,
tomitaProbabilitychangingClusterAlgorithm2002} are close to these values, though
not in perfect agreement.

The consensus is (but do note discussions in
Refs.~\cite{lapilliUniversalityAwayCritical2006, hwangSixstateClockModel2009,
baekCommentSixstateClock2010}) that both transitions are of the
KT-type~\cite{kosterlitz.thouless:1973, kosterlitz:1974}, implying exponential
growth of $\xi$ upon approach of the critical phase:
\begin{equation}
\label{eq:kt}
\xi(T)  \propto \begin{cases}
 e^{a_1 t_1^{-1/2}}, \quad t_1 \equiv \frac{T_1-T}{T_1} & (0 < t_1 \ll 1), \\
 \infty & (T_1 \leq T \leq T_2), \\
 e^{a_2 t_2^{-1/2}}, \quad t_2 \equiv \frac{T-T_2}{T_2} & (0 < t_2 \ll 1). \\
\end{cases}
\end{equation}
For the XY-model at finite temperature, only the transition at $T_2$ remains,
for which $a_{XY} \sim 1.5$~\cite{kosterlitz:1974}. Computer
simulations~\cite{challaCriticalBehaviorSixstate1986} show that this value is
compatible with the $q=6$ clock model as well, for {\it both} transitions:
$a_1=a_2 \sim 1.54$.

The specific heat $c_V$ of the $q=6$ clock model always remains finite. However,
the variation of $c_V$ with temperature does reveal two maxima, one occurring
close to $T_1$, the other closer to $T_2$. This property is convenient in
simulations to provide first evidence of two phase
transitions~\cite{challaCriticalBehaviorSixstate1986}.

\section{Results}

In what follows, moderate system sizes $L \equiv L_x = 30-80$ are used to study
the BVST model, see Appendix for details about the MC methods. This approach is
in line with \olcite{orkoulas.panagiotopoulos.ea:2000}, where it was noted that
moderate system sizes, in combination with finite-size scaling methods, can
yield a very adequate description of the phase behavior in the thermodynamic
limit.

\subsection{Specific heat}

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig1}
\caption{\label{fig1} Specific heat $c_V$ of the BVST model, versus the
temperature $T$, for various system sizes $L$. The data reveal
two maxima, consistent with the two-transition scenario of the $q=6$ clock
model. Note that finite-size effects in the peak heights are weak, indicating
that $c_V$ does not diverge at any of the transitions.}
\end{center}
\end{figure}

In \fig{fig1}, we plot the specific heat per particle of the BVST model, $c_V =
(\avg{E^2} - \avg{E}^2)/(NT^2)$, versus the temperature $T$, for various system
sizes $L$. Consistent with the $q=6$ clock model, the data strikingly reveal two
maxima, corresponding to two phase transitions. In addition, finite-size effects
in $c_V$ are small. The absence of a strong increase of the peak heights with
$L$ indicates that $c_V$ does not diverge at any of the transitions, consistent
with the $q=6$ clock model. The reader is encouraged to compare our \fig{fig1}
to specific heat MC data of the $q=6$ clock
model~\cite{tobochnikPropertiesQStateClock1982,
challaCriticalBehaviorSixstate1986}, which look very similar.

\subsection{Susceptibility}

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig2}

\caption{\label{fig2} Susceptibility $\chi$ of the BVST model, versus the
temperature $T$, for various system sizes $L$. The data reveal
negligible size effects at low and high temperature, while $\chi$ increases
strongly with $L$ at intermediate temperatures (note the vertical logarithmic
scale). This supports the two-transition scenario of the $q=6$ clock model.}

\end{center}
\end{figure}

For a given set of particle displacements, in line with
\olcite{citeulike:13839098}, we use the vector sum, $M = | \sum_{i=1}^N
\vec{r}_i | / a_0$, to quantify the degree of order. In the perfectly ordered
ground state $M/N=1$, since here all the particles have selected the same
minimum, while $M/N<1$ when the ordering is not perfect. In the language of
vector spin models, $M$ is analogous to the absolute value of the total
magnetization, commonly used in studies of the $q=6$ clock
model~\cite{tobochnikPropertiesQStateClock1982,
challaCriticalBehaviorSixstate1986}. Note, however, that this is not the only
possible choice, for example, an angular order parameter could be used
also~\cite{baekTrueQuasilongrangeOrder2009}. We now define the susceptibility,
$\chi = \left( \avg{M^2} - \avg{M}^2 \right)/ (NT)$, which we plot in \fig{fig2}
as function of temperature, for various system sizes. The key observations are
that finite-size effects are negligible in the low- and high-temperature
regimes, while they are very strong in the intermediate regime, roughly
corresponding to the temperature range spanned by the specific heat maxima. The
pronounced increase of $\chi$ with $L$ at intermediate temperatures is
consistent with a critical phase, where $\chi \to \infty$ in the thermodynamic
limit $L \to \infty$. Hence, \fig{fig2} supports the two-transition scenario of
the $q=6$ clock model, with non-critical phases at low and high temperature,
separated by a critical phase (although we still need to check the nature of the
order in each of the phases).

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig3}

\caption{\label{fig3} Finite-size scaling analysis of the temperatures (a)
$T_2(L)$, and (b) $T_1(L)$, assuming the transitions are of the KT type. The
intercept of the lines yields the transition temperature of the thermodynamic
limit when all three parameters are fitted. The square symbols indicate the
transition temperature obtained for two-parameter fits, where $a_1=a_2=a_{XY}$
was imposed, using all system sizes (closed symbol), and system sizes $L=70,80$
(open symbol).}

\end{center}
\end{figure}

\subsection{Phase transition temperatures}

To determine the transition temperatures, we perform a finite size scaling
analysis. In a system of size $L$, approaching the lower transition at $T_1$
from below, \fig{fig2} shows that the slope $d\chi/dT$ initially increases, then
levels off, i.e.~$d\chi/dT$ reaches a local maximum. Let $T_1(L)$ be the
temperature at the maximum. Similarly, approaching the upper transition at $T_2$
from above, $d\chi/dT$ reaches a local minimum, defining $T_2(L)$. In the limit
$L \to \infty$, these finite-size estimators converge to the transition
temperature of the thermodynamic limit: $\lim_{L \to \infty} T_i(L)=T_i$
($i=1,2$).

Assuming the $q=6$ clock model scenario, the transitions at $T_1,T_2$ are both
of the KT type, with $\xi$ given by \eq{eq:kt}. This implies scaling relations
$T_1(L) = T_1 - a_1^2 / \ln^2(f_1 L)$ and $T_2(L) = T_2 + a_2^2 / \ln^2(f_2 L)$,
with $f_i$ constants of order unity, and $a_i$ the coefficients of
\eq{eq:kt}~\cite{chungEssentialFinitesizeEffect1999}. In \fig{fig3}(a), we fit
our $T_2(L)$ estimates to the expected scaling form. The fit captures the data
well, using $T_2 \approx 0.060$, $a_2 \approx 1.3$, $f_2 \approx 1.0$.
\fig{fig3}(b) shows the corresponding fit to our $T_1(L)$ data, where $T_1
\approx 0.056$, $a_1 \approx 1.3$, $f_1 \approx 2.3$ yielded the best fit. It is
gratifying that both fits yield the same value for $a_i$, although the expected
KT value is somewhat higher. If we repeat the procedure setting $a_1=a_2=a_{XY}
\sim 1.5$, and only fit $f_i,T_i$, we obtain good fits also. The closed square
symbols in \fig{fig3} indicate the transition temperatures obtained in this way,
with $f_1 \approx 3.7$, $f_2 \approx 1.3$. Finally, again setting
$a_1=a_2=a_{XY} \sim 1.5$, but this time only using data for $L=70,80$, i.e.~our
largest systems, the open square symbols are obtained. From this analysis, we
conclude $T_1 = 0.0560 \pm 0.0006$, and $T_2=0.0600 \pm 0.0006$; the vertical
height of the shaded bands in \fig{fig3} indicates the corresponding ranges.

\subsection{Critical exponents}

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig4}

\caption{\label{fig4} Application of the method of
Loison~\cite{loisonBinderCumulantKosterlitzThouless1999} to obtain the critical
exponent $\eta_2$ of the high-temperature transition. For this analysis, only
data in the high-temperature phase, $0 < t_2 < 0.1$, was used, with $t_2$
defined in \eq{eq:kt}. Substituting the accepted value $\eta_2=1/4$ an excellent
collapse of the data is observed!}

\end{center}
\end{figure}

Next, we turn to measuring the critical exponent $\eta_2$, i.e.~the value of
$\eta$ at the high-temperature transition. The latter is most conveniently
obtained using the method of
Loison~\cite{loisonBinderCumulantKosterlitzThouless1999}; see also
\olcite{baekTrueQuasilongrangeOrder2009} where this method is applied to the
$q=8$ clock model. Here, one varies $T$, and plots the scaled susceptibility,
$\chi L^{-(d-\eta_2)}$, with $d=2$ the spatial dimension, as function of the
Binder cumulant, $U_4 = 1 - \avg{M^4} / (3\avg{M^2}^2)$. Provided the correct
value of $\eta_2$ is used, the data for different system sizes $L$ should
collapse onto a single curve. The result is shown in \fig{fig4}, where the
accepted value $\eta_2=1/4$ was substituted, using our largest set of system
sizes $L=70,80$. The quality of the collapse is quite remarkable! If we repeat
the analysis using all our available system sizes, a somewhat larger exponent,
$\eta_2 \sim 0.3$, is obtained. In \olcite{baekTrueQuasilongrangeOrder2009}, it
is mentioned that deviations are likely due to logarithmic size corrections,
which still are strong in small system sizes. Hence, we believe that
$\eta_2=1/4$ obtained for our largest systems is the most reliable estimate,
fully consistent with the $q=6$ clock model.

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig5}

\caption{\label{fig5} Finite-size scaling analysis of the order parameter
$\Delta$ in the vicinity of the low-temperature transition. (a) $\Delta
L^{\eta_1/2}$ versus $T$ for three different system sizes $L$, where
$\eta_1=1/9$ was used. Note that curves for different $L$ intersect. (b)
Intersection temperatures $T_X$ for pairs of system sizes, plotted versus
$1/\ln^2 L^\star$. The shaded region marks the lower portion of the $T_1$
estimate range of \fig{fig3}(b). As $L^\star$ increases, $T_X$ approaches this
range.}

\end{center}
\end{figure}

For the low-temperature transition, Loison's method is not directly applicable,
and so, to obtain $\eta_1$, we follow a different route, based on the order
parameter, $\Delta = \avg{M}/N$. In the critical intermediate phase, i.e.~for
temperatures $T_1 \leq T \leq T_2$, the order parameter vanishes, following
finite-size scaling, as $\Delta \propto L^{-\eta(T)/2}$, where the exponent
$\eta(T)$ is an increasing function of $T$. For $T<T_1$, i.e.~in the ordered
phase, $\Delta$ is finite, since here there is long-ranged order. Consider now
the quantity $\Delta L^{\eta_1/2}$, with $\eta_1=1/9$ being the expected
theoretical value of $\eta(T)$ at the lower transition temperature $T_1$. This
quantity should diverge with $L$ for $T<T_1$ (since here $\Delta$ is finite),
remain constant at $T_1$, and decay to zero above $T_1$ (since $\eta(T)$
increases with $T$). Hence, plotting $\Delta L^{\eta_1/2}$ versus $T$, for
different system sizes $L$, the data for different $L$ are expected to intersect
at $T=T_1$. The result is shown in \fig{fig5}(a), which clearly reveals
intersections! For a pair of system sizes, $(L_i,L_j)$, we now define $T_X$ as
the temperature where the corresponding curves intersect. \fig{fig5}(b) shows
$T_X$ versus $1/\ln^2(L^\star)$, $L^\star \equiv (L_i+L_j)/2$, for all available
pairs ($L=30,40,50,60,70,80$, i.e.~a total of 15 pairs). As $L^\star$ increases,
there is a clear trend for $T_X$ to increase as well, approaching values that
are consistent with \fig{fig3}(b). Hence, while \fig{fig5} does not constitute a
direct measurement of $\eta_1$, it does show that the BVST model is {\it
consistent} with the theoretically expected $q=6$ clock model value
$\eta_1=1/9$.

\subsection{Correlation functions}

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig6}

\caption{\label{fig6} Correlation functions, $G(r)$, of the BVST model
obtained at $T=0.04; 0.045$ (top two curves), $T=0.056; 0.06$ (center two
curves), $T=0.075; 0.1$ (lower two curves). The three sets of curves represent,
from top to bottom, the low-temperature phase with long-ranged order, the
critical intermediate phase with power-law decay of correlations, and the
high-temperature phase with short-range order.}

\end{center}
\end{figure}

Finally, we still present the correlation function $G(r)$ [\fig{fig6}]. For
$T<T_1$, one indeed finds that $G(r)$ decays to a finite plateau value,
consistent with long-ranged order, and a finite correlation length $\xi$. For
$T>T_2$, $G(r)$ quickly decays to zero, consistent with short-ranged order, and
finite $\xi$. In the intermediate phase, $G(r)$ decays slowly, but there is no
sign of $G(r)$ saturating to a finite value, consistent with a critical phase
where $\xi$ is infinite. Hence, also the dependence of $G(r)$ with temperature
is consistent with the $q=6$ clock model.

\section{Conclusion}

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig7}

\caption{\label{fig7} Shape of the nearest-neighbor pair interaction for the
BVST model, the clock model, and the generalized clock model using $p=2.8$. In
this plot, the potential minimum of each interaction was shifted to zero,
followed by a scaling of the potential to have the maximum at unity.}

\end{center}
\end{figure}

In conclusion, we have presented MC data of the BVST model, showing that it
belongs to the universality class of the two-dimensional $q=6$ clock model. The
hallmark features are the presence of two KT transitions with, consequently,
three thermodynamically distinct phases. Our data are consistent with the
theoretical $q=6$ clock model critical exponents $\eta_1=1/9$ and $\eta_2=1/4$,
of the lower and higher transition, respectively. Our estimates of the
transition temperatures, $T_1$ and $T_2$, are somewhat below $T_{\rm c} \approx 0.075$
reported in \olcite{citeulike:13839098}. However, since the focus of
\olcite{citeulike:13839098} was on frictional behavior, no finite-size scaling
study was performed, which we believe explains the deviation.

The reason the BVST model is in the $q=6$ clock model universality class is due
to the external field $h_i$ {\it and} the bond potential $u_{ij}$. For each
particle, the field gives rise to six local energy minima for this particle,
corresponding to the \ahum{clock} states. It is important to note that, already
at temperatures $T \sim T_2$, the external field is \ahum{confining} each
particle to its set of minima. Hence, even though each particle in the BVST
model has two degrees of freedom, $\vec{r}_i=(x_i,y_i)$, the field effectively
reduces this to a set of six possible angles, $\theta_i=\pi\lambda/3$,
$|\vec{r}_i|=a_0$, $\lambda=1,\ldots,6$, just as in the clock model. If we now
plot the BVST bond energy, $u_{ij}$, as a function of the angular difference
$\theta_{ij}$ between two nearest-neighboring particles, the bell-shaped curve
\ahum{BVST} of \fig{fig7} is obtained, which qualitatively resembles the
\ahum{cosine} shape of the pair potential of the clock model. Hence, in the BVST
model, both the number of states and the shape of the pair potential correspond
to the $q=6$ clock model, which explains why the universality classes are the
same.

We still comment on the possibility of a single transition. Our data indicate
that the difference in transition temperatures, $T_2-T_1$, is rather small.
Hence, a critical reader might argue that, in the limit $L \to \infty$, the two
transitions could well merge into one. However, based on what is known about
clock models and their generalizations, this scenario is unlikely. The
above-mentioned merging of transitions is known to happen in {\it generalized}
clock models, where the nearest-neighbor interaction $u_{ij} \sim 1 -
\cos^{2p^2} (\theta_{ij}/2)$, which reduces to the standard clock model when
$p=1$. The two transitions merge into a single transition when the
nearest-neighbor interaction becomes sufficiently \ahum{sharp and narrow},
implying a large enough value of $p$. For the $q=8$ clock model, the
corresponding value $p>2.8$~\cite{baekTrueQuasilongrangeOrder2009}. However, as
one can see in \fig{fig7}, the BVST pair potential is far removed from the
\ahum{sharp and narrow} shape required to bring about such merging (on the
contrary, the BVST model rather resembles $p<1$). As a side remark, if the
transitions were to merge, we should expect two-dimensional $q=6$ state Potts
behavior~\cite{baekTrueQuasilongrangeOrder2009}. The latter has a first-order
transition~\cite{revmodphys.54.235}, for which neither our data, nor that of
\olcite{citeulike:13839098}, show any evidence.

Finally, we discuss what might be expected in $d=3$ dimensions. In $d=3$, the
clock model has a single second-order phase transition, for all values of
$q$~\cite{scholtenCriticalBehaviorQstate1993a}. For $q=6$, the critical
exponents are consistent with those of the $d=3$
XY-model~\cite{miyashitaNatureOrderedPhase1997}. Hence, the two-transition
scenario of $d=2$ does not survive in $d=3$. We emphasize that, in the $d=3$
clock/XY models, only the lattice space is three-dimensional, the angular
degrees of freedom remain two-dimensional. A more realistic description of a
displacive transition in $d=3$ should likely use three-dimensional displacement
vectors. For an fcc-material, a possible generalization of the BVST model could
be an fcc-lattice, where, to each lattice site, 12 minima are assigned, each one
displaced a small distance in the direction of one of the nearest neighbors.
Such a model could easily be simulated using Monte Carlo methods, but we are not
aware of such simulations having been carried out.

\acknowledgments

We acknowledge support by the German research foundation (SFB-1073, TP A01). We
are additionally indebted to an anonymous referee for bringing to our attention
the two-transition scenario of the six-state clock model, and for pointing out
its possible connection to the BVST model.

\bibliographystyle{clear}