0$. The existence of the massive modes implies that the lower bound of the viscosity/entropy ratio is violated~\cite{Hartnoll-ml-2016tri}. (See also~\cite{Alberte-ml-2016xja,Liu-ml-2016njg}.) However, in this case, the massive function $m(r)$ diverges on the boundary as $r\rightarrow \infty$. Consequently, it can be easily verified that we have $\phi_2\sim \exp[\pm \sqrt{-1/(24 g^2 q^2\lambda)}\, r^2]$ as $r\rightarrow \infty$. This implies that $\phi_2$ should be truncated out by the boundary condition. Here, we develop a new method to compute the shear viscosity, which we find particularly convenient when higher-derive curvature terms are involved. We consider the radially conserved current constructed in~\cite{Liu-ml-2017kml} for general gravity theories, namely \bea {\cal J}^{x_1} &=& \sqrt{-g} J^{r x_1}\,,\label{jx1} \eea where \be J_{}^{\mu\nu} = 2 \fft{\partial L} {\partial R_{\mu\nu\rho\sigma}} \nabla_\rho\xi_\sigma + 4 \xi_\rho \nabla_\sigma \fft{\partial L} {\partial R_{\mu\nu\rho\sigma}} + \xi^\rho A_\rho F^{\mu\nu}\,.\label{J2grav} \ee For our particular example, the Killing vector is given by $\xi=\partial_{x_2}$. For the static background, the current ${\cal J}^{x_1}$ vanishes identically; it gives non-trivial contribution at the linear order once the perturbation $\Psi(t,r)$ is turned on. We find \be 16\pi {\cal J}_{\rm lin}^{x_1} = g^2 r^2 f\, \psi' + \fft{6\lambda q^2}{r^4} f\, \Big(r^2 f \psi''' + 2r (r f' - f) \psi'' - 6 (rf' + 2 f - 2g^2 r^2) \psi' \Big)\,. \ee It can then be easily verified that the radial conservation law $\partial_r {\cal J}_{\rm lin}^{x_1}=0$ gives precisely the linear equation~(\ref{shearlinear}). In appendix~\ref{sec:rccbst}, we show that this radially conserved current matches with the boundary stress tensor $T^{x_1x_2}$, demonstrating that this is indeed the bulk dual of the holographic shear viscosity. The horizon boundary conditions for $\psi$ must be that $\Psi(t,r)$ is ingoing, implying that \be \psi = \varsigma\, \fft{ \log (r-r_0)} {f'(r_0)} + \cdots\,. \ee We can now read off the radially conserved current for the $\phi_1$ of~(\ref{phi1phi2}) using the horizon data. We have \be J_{\rm lin}^{x_1} (\phi_1)= g^2 r_0^2 \varsigma\,,\qquad\Rightarrow\qquad \eta = \fft{1}{g^2} \fft{\partial}{\partial \varsigma} J_{\rm lin}^{x_1}(\phi_1)= \fft{1}{16\pi}r_0^2\,. \ee Thus the shear viscosity/entropy ratio is exactly given by~(\ref{visbound}). The quasi-topological term ${\cal Q}^\3$ does not alter the viscosity/entropy ratio. It is of interest to note that the radially conserved current for the $\phi_2$ mode of~(\ref{phi1phi2}) vanishes identically, which implies that the corresponding shear viscosity would vanish. This is of course an irrelevant result since the mode is inconsistent with the boundary condition. Finally, we would like to comment that the radially-conserved method we adopt in this subsection is particularly convenient to use for calculating the holographic shear viscosity in theories with higher-order curvature terms, owing to the general radially-conserved current formula obtained in~\cite{Liu-ml-2017kml}. The conserved current implies that the detail of the perturbation function $\psi$ is unimportant, except that it should satisfy the ingoing boundary condition on the horizon and it falls sufficiently fast at the asymptotic infinity so that the linearized equation is well approximated. In fact the $\phi_2$ mode was ruled out precisely because it diverges asymptotically, giving rise simply to~(\ref{visbound}). ]]>

0 \qquad \hbox{or}\qquad \lambda \le -\fft{r_0}{12f_2 f_1}\,.\label{lambdaregion} \ee Note that in the context of holographic shear viscosity discussed earlier, the coupling constant has to be negative for the stable TT modes. When the second inequality is saturated, we have $\tilde m=0$. The butterfly velocity is defined as \be v_B = \fft{2 \pi T}{m} \,. \ee Thus the corresponding butterfly velocities associated with the $m_0$ and $\tilde m$ modes are given~by \be v_0^2 = \fft{f_1}{4 g^2 r_0} \,, \qquad \tilde v^2 = \fft{v_0^2}{1 + \fft{r_0}{12\lambda f_1 f_2}} \,. \ee The quantity of $v_0$ is the standard butterfly velocity and can be expressed in terms of thermodynamical variables~\cite{Feng-ml-2017wvc} \be v_0^2 = \fft12 \fft{TS}{V_{\rm th} P}\,. \ee Thus we see that for positive $\lambda$, we have $\tilde v^2\le v_0^2$. For some appropriate negative value of $\lambda$, we can have $\tilde v^2 \rightarrow \infty$, which may potentially violate the conjectured holographic diffusivity bounds~\cite{elebound,thebound,Blake-ml-2016wvh,Blake-ml-2016sud}. We shall discuss this in the next subsection. Additional butterfly velocity in Schwarzschild AdS planar black hole of quadratically-extended Ricci gravity was also obtained in~\cite{Alishahiha-ml-2016cjk}. However there is an important difference. The mass of the new massive mode in~\cite{Alishahiha-ml-2016cjk} is solely determined by the coupling constants of the theory, and if it vanishes, it is zero for all solutions. In our case, $\tilde m$ in~(\ref{m0mtilde}) depends not only the coupling constant $\lambda$, but also the integration constant of the solution, such as mass and charges. Thus in our case, the vanishing of $\tilde m$ depends on the specific property of a solution, provided that $\lambda$ is negative. This provides a mechanism of breaking the diffusivity bounds, which we shall discuss in the next subsection. ]]>

C_\pm$. However, as was explained in sections~\ref{sec:viscosity} and~\ref{sec:butterfly}, the coupling constant $\lambda$ can be negative, and should be negative in the discussion of holographic shear viscosity. The bounds can then be violated for negative $\lambda$, for which case the butterfly velocity $\tilde v^2$ is unbound above, leading to a possible infinitesimal $\tilde C_\pm\rightarrow 0^+$. Although it appears that $\tilde C_\pm$ in~(\ref{allcs}) can be negative for negative $\lambda$. This will not happen since we require~(\ref{lambdaregion}) so that $\tilde v^2$ be non-negative, but possible to be unbound above. Note also that in our model, the violations of diffusivity bounds occur simultaneously for both the diffusion constants $D_c$ and $D_e$. Violations of charge diffusivity bound was previously obtained~\cite{Baggioli-ml-2016oqk,Baggioli-ml-2016oju,Gouteraux-ml-2016wxj,Baggioli-ml-2016pia}, but not for the energy diffusivity bound. It is worth pointing out that higher derivative gravities are liable to violate causality~\cite{Camanho-ml-2014apa}. However, it was demonstrated~\cite{Li-ml-2017txk} recently that linearized quasi-topological Ricci-polynomial gravities, (which all our examples are,) preserve the causality of the type discussed in~\cite{Camanho-ml-2014apa}, at least for perturbations on Einstein metrics. The metrics we have discussed in this section, on the other hand, are beyond Einstein, and the linearized perturbation for the butterfly velocity involves higher derivatives and hence beyond the scope discussed in~\cite{Camanho-ml-2014apa} which considered generic two-derivative perturbations. However, it is important to note that the extra mode that gives $\tilde v \rightarrow \infty$, depends on the integration constant of the solution, which is different from typical higher-derivative gravities. Nevertheless, the causality constraint in our case of violating the diffusivity bound requires further investigation. ]]>