text:
\section{Introduction}

The magnetism is one of the most intriguing topics in various fields of physics. One of the reasons is that interesting subjects of the magnetism are closely related to a fundamental quantum degrees of freedom that distinguish kinds of particles, the particle's statistics and phases of the system. One of the significant phenomena of magnetic systems is the ferromagnetism given by the spontaneous symmetry breaking. In addition, the ferromagnetic system usually shows hysteresis curves, which are representative phenomena of the ferromagnetism. In this work, we will consider a simple model to build the ferromagnetism and the hysteresis curve using gauge/gravity duality \cite{Maldacena:1997re,Aharony:1999ti}. In earlier studies, magnetized systems have already attracted much attention and have been studied many times in the holographic approach, {\it e.g,}\cite{Hartnoll:2007ai,Hartnoll:2007ih,Albash:2008eh,Jensen:2011xb,Kim:2013wiz,Blake:2014yla,Seo:2015pug,Blake:2015epa,Kim:2015wba,Blake:2015hxa,Blake:2015ina,Cai:2015bsa,Cai:2015xpa,Cai:2015jta,Donos:2017mhp}.

The gauge/gravity duality is a very powerful tool to study strongly coupled field theory systems. The main advantage of the duality is that the generating functional of the strongly coupled field theory can be obtained by the on-shell action of the corresponding gravity theory \cite{Gubser:1998bc,Witten:1998qj}. This correspondence has been applied to various area of physics. One of the active applications is about the condensed matter theories, {\it e.g,}\cite{Hartnoll:2007ih,Herzog:2007ij,Herzog:2008he,Hartnoll:2008vx,Hartnoll:2008kx,Gubser:2009qm,Herzog:2009xv}.

More specifically, the gauge/gravity duality is a relation between a $d$-dimensional field theory living on the boundary and the corresponding $(d+1)$-dimensional bulk gravitational system with an appropriate boundary condition. Such a boundary condition tells us the prescription for the external sources and the non-normalizable modes of the field theory and the gravitational system, respectively. In this subject, we mainly consider a system with the sources given by the external magnetic field and chemical potential.
These sources induce the magnetization, charge density and vacuum expectation value(VEV) of a scalar operator. We take into account $(2+1)$-dimensional systems in the present work.

The $(2+1)$-dimensional spacetime we considered has a very interesting magnetic nature because the direction of magnetic field is always orthogonal to the $2$-dimensional space when the spacetime is embedded in higher dimensions. The magnetization is described as a real value, just as the magnetic field is. Therefore, the spontaneous magnetization is displayed by breaking $\mathbb{Z}_2$ symmetry. To realize this spontaneous symmetry breaking, we use the well-known hairy black brane solution with a real scalar field $\phi$. Such a real scalar field is dual to a scalar operator in the boundary theory. Since our final goal is to construct spontaneous magnetization and hysteresis curves, we introduce an interaction among the scalar operator, the magnetic field and the charge density operator. In bulk point of view, this interaction is achieved by adding an axionic term, $\int d^4 x W(\phi)\epsilon^{MNPQ}F_{MN}F_{PQ}$,\, to the action. And we consider a large class of model which can describe the spontaneous magnetization. We propose a general formula for the magnetization by using a `scaling symmetry' trick \cite{Banados:2005hm,Ahn:2015shg,Ahn:2015uza}. This formula will be confirmed in a specific model.

The proposed large class of models enable us to describe the spontaneous magnetization by a Landau-Ginzburg type effective theory. We will show later how to associate gravity calculation to the parameters in the potential of the effective theory. Furthermore we find the effective mass of elementary excitation due to the quantization of magnetic fluctuations in the broken phase. In addition to this, we also give a comment on various local and global excitation in the effective theory.

As we metioned, one of the characteristics of the ferromagnetic material is the hysteresis curve of magnetization. In order to construct the hysteresis curves, the interaction potential $W(\phi)$ plays an important role. This is because the interaction potential $W(\phi)$ should be an odd function to break the $\mathbb{Z}_2$ symmetry    changing the sign of the scalar field $\phi$. We will consider an odd function for a specific model. Also, in fact, the hysteresis is obtained by the slowly varying magnetic field. In the real physics, we must consider a time-dependent background.  Since, however, we assume a very slowly varying magnetic field, we ignore the effect from the time dependent dynamics.
See \cite{Evans:2010xs} for an explicit time-dependent consideration for another holographic model.

This paper is organized as follows: In section \ref{sec2}, we propose the general form of the magnetization using the `scaling symmetry' trick and introduce a simple model. Then we check that our general formula works in the specific model. In addition, we find the expression of the on-shell action dual to the free energy in the boundary field theory. In section \ref{sec3}, we investigate the spontaneous magnetization and the hysteresis curve. The Landau-Ginzburg type potential is also constructed.
In section \ref{sec5}, we discuss our result and introduce possible future directions. Several formulas related to the holographic renormalization are summarized in Appendix.

\section{Basic Formulation}\label{sec2}

In this section, we consider a large class of holographic models describing spontaneous magnetization. The magnetization is given by derivative of free energy with respect to external magnetic field $\mathcal{B}$ as follows\footnote{See \cite{Hartnoll:2007ih} for details.}:
\begin{align}\label{mag01}
\mathcal M = - \frac{\delta \Omega}{\delta \mathcal{B}}~~,
\end{align}
where $\Omega$ is the free energy of $d$-dimensional system. This is the most essential quantity to understand physics in various magnetic materials. The main goal of the present work is to study this quantity through holographic approach.

The fundamental relation of thermodynamics is the first law describing a constraint among variations of thermodynamic quantities. The gauge/gravity duality enables us to interpret this relation in the bulk geometry. $(d+1)$-dimensional black holes also have relations among their parameters by on-shell variation. These two relations have a similarity, which leads to the holographic principle and the gauge/gravity correspondence. Especially, we are interested in the magnetic variation. It consists of the external magnetic field variation and the magnetization. The form of 4-dimensional magnetic black brane solutions is simpler than those in the higher dimensions. Thus we will focus on the asymptotically $AdS_4$ black branes in this paper.

Furthermore, this first law of thermodynamics can be integrated to a relation among finite thermodynamic parameters. The resulting relation is consistent with the definition of free energy density by construction. One may write down the relation as follows:
\begin{align}\label{pressure}
\omega\equiv\Omega/\mathcal{V}
= \epsilon - s T  - \mu\, \tilde{Q} ~~,
\end{align}
where $\omega$ is the free energy density of the system. In addition, spatial volume, energy density, entropy density, temperature, chemical potential and charge density are denoted by $\mathcal{V}$, $\epsilon$, $s$, $T$, $\mu$, $\tilde Q$, respectively. Also, it is well known that the pressure $\mathcal P$ is same with $-\,\omega$ for homogeneous systems.

When the system has the magnetization $\mathcal M$ under the magnetic field $\mathcal B$, the pressure consists of the magnetic part and the other contribution as follows:
\begin{align}\label{P+Pm}
\mathcal P = \mathcal{P}_{others} + \mathcal M\,\mathcal{B}~~.
\end{align}
From this decomposition one can easily read off the magnetization $\mathcal{M}$. In the bulk geometry, the above relation (\ref{pressure}) corresponds to the so-called Smarr relation. It is the constraint among black hole parameters given by vanishing time-time component of the metric at the horizon. We will show this explicitly.

Recently the general form of the Smarr relation has been studied and was derived by applying a modified Noether theorem to a reduced action of gravitational systems \cite{Banados:2005hm,Hyun:2015tia,Ahn:2015uza,Ahn:2015shg,Hyun:2016isn,Hyun:2017nkb}. This is very useful even in a hairy black hole background. We will extend the method for hairy magnetic black branes.

\subsection{Einstein-Maxwell-Dilaton Model}

Let us start with a large class of holographic models to describe magnetized systems.
\begin{align}\label{Action00}
S = &\,\frac{1}{16\pi G}\int  d^4 x \sqrt{-g}  \left\{   R + \frac{6}{L^2}    -\frac{Z(\phi)}{4} F^2   - \frac{1}{  2} \left(\partial \phi\right)^2   - V(\phi)     \right\} \nonumber\\&\,+ \frac{\theta}{16\pi G}   \int  d^4 x W(\phi) \epsilon^{MNPQ} F_{MN} F_{PQ} ~~,
\end{align}
where the last term plays an important role for the spontaneous magnetization and the hysteresis curve. In 3+1 dimension, this term is linear in the magnetic field. So one may notice that there can exist a non-vanishing magnetization even in the absence of magnetic field. See (\ref{mag01}) for this speculation. Furthermore, turning on the magnetic field with even functions of $Z(\phi)$ and $V(\phi)$,
an odd function of $W(\phi)$ breaks a $\mathbb{Z}_2$ symmetry by changing the sign of the scalar field($\phi \to - \phi$). In this case, however, the Lagrangian is invariant under another transformation that simultaneously changes the sign of the magnetic field ($\mathcal{B}\to - \mathcal{B}$). Since the latter symmetry is essential to build the hysteresis curve, we will take into account the odd function $W(\phi)$ with even $Z(\phi)$ and $V(\phi)$ in a specific calculation. It will be clear in section \ref{sec3}.

Now, we will derive the Smarr relation of black branes in the gravity model using a `scaling symmetry' method in \cite{Banados:2005hm,Ahn:2015uza,Ahn:2015shg,Hyun:2016isn,Hyun:2017nkb}. In order to apply the method, we take the following metric ansatz:
\begin{align}\label{ansatz0}
&ds^2 = - U(r)e^{-\mathcal{W}(r)} d t^2  + \frac{r^2}{L^2} \left( dx^2+ dy^2 \right) + \frac{dr^2}{U(r)}~~,~~\\
&\phi=\phi(r)~~,~~A= A_{t}(r)d t + \frac{H}{2}(xdy-ydx)
\end{align}
We will use a convenient time coordinate $\tau = e^{-\mathcal{W}(\infty)/2}\, t$ to keep the boundary metric $\eta_{\mu\nu}$\,.
As a result, the coordinate system of dual field theory becomes $(\tau,x,y)$ whose temporal component of the metric and gauge field is given by
\begin{align}
g_{\tau\tau}(r) = - U(r) e^{-(\mathcal{W}(r)-\mathcal{W}(\infty)) } ~~,~~A_\tau(r) = A_t(r)e^{\mathcal{W}(\infty)/2}~~.
\end{align}
For simplicity, we define $\tilde{\mathcal{W}}(r)\equiv\mathcal{W}(r)-
\mathcal{W}(\infty)$. Then the Hawking temperature and entropy density are given by the following forms:
\begin{align}
T_H = \frac{U'(r_h)}{4\pi}e^{-\tilde{\mathcal W}(r_h)/2}~~,~~s = \frac{r_h^2}{4G L^2}~~,
\end{align}
where $r_h$ is the location of the horizon defined by $U(r_h)=0$.

Plugging our ansatz (\ref{ansatz0}) into the gravitational action (\ref{Action00}), we obtained the following reduced action:
\begin{align}
S = \frac{1}{16\pi G} \int d\tau dx dy \int dr  L_{red}~~,
\end{align}
where
\begin{align}\label{Lred}
 L_{red}= &
\frac{e^{-\tilde{\mathcal{W}}(r)/2}}{L^2
 \left(\frac{6 r^2}{L^2}-\frac{H^2 L^4 Z(\phi )}{2 r^2}-\frac{r^2 U(r) \phi'(r)^2}{2} -r^2 V({\phi}(r))-2 r U'(r)-2 U(r)\right) \nonumber\\
&+\frac{ r^2 Z(\phi ) e^{\frac{\tilde{\mathcal{W}}(r)}{2 }} A_{\tau }'(r){}^2}{2 L^2}-8 \,\theta  H\, W(\phi ) A_{\tau }'(r)~.
\end{align}
Here we dropped out total derivative terms. When $H=0$, the reduced action has a scaling symmetry. One can show that this reduced action without magnetic field is invariant under the following `scaling transformation' up to total derivatives. We defined the scaling transformation as follows:
\begin{align}
&\delta( e^{-\tilde{\mathcal{W}}/2} )= - \lambda \left( 3\,e^{-\tilde{\mathcal{W}}/2} + r\,(e^{-\tilde{\mathcal{W}}/2})' \right)~~,~~ \\
&\delta U = -\lambda\,(-2\, U + r\, U')~~,\\
&\delta A_t = -\lambda\,( 2\, A_\tau + r A_\tau' )~~,\\
&\delta \phi = -\lambda\,r\,\phi'~~,
\end{align}
where $\lambda$ is a small parameter. On the other hand, this symmetry can be broken by turning on the magnetic field $H$. So we considered the invariant part of the Lagrangian. And one can show that
\begin{align}\label{varL0}
\delta ( L_{red} - L_H ) = \lambda \left( - r ( L_{red} - L_H ) \right)',
\end{align}
where $L_H$ is the non-invariant part of the reduced Lagrangian (\ref{Lred}). It is defined as follows:
\begin{align}
L_H = -\left( 8\, \theta H W(\phi ) A_\tau'+\,e^{-\tilde{\mathcal W}/2}\frac{H^2 L^2 Z(\phi )}{2 r^2} \right)~~.
\end{align}
Using the equations of motion, $\delta L_{red}$ is given as follows:
\begin{align}\label{delLred}
\delta L_{red}= \left( \sum_i \delta \Psi_i \frac{\partial L_{red}}{\partial \Psi'_i }  \right)' ~~,
\end{align}
where $\Psi_i$ denotes a collective expression of the fields such as $\left\{e^{-\tilde{\mathcal{W}}/2}, U, A_\tau, \phi\right\}$. (\ref{varL0}) and (\ref{delLred}) can be incorporated to describe a partially conserved `charge'(PCC) $\mathcal C$.
\begin{align}\label{PCC}
\lambda\, \frac{d}{dr} \mathcal C  = \frac{1}{32\pi G} \left(\delta L_H + \lambda\,\left( r L_{H} \right)'\right)= -\frac{1}{32\pi G}\,\delta_H L_H ~~,
\end{align}
where $\mathcal C$ is defined by
\begin{align}
\mathcal C \equiv \frac{1}{32 \pi G\,\lambda}\left(\sum_i \delta \Psi_i \frac{\partial L_{red}}{\partial \Psi'_i } + \lambda\,r   L_{red}\right)~~.
\end{align}
and $\delta_H$ is the complementary transformation acting on $H$ to make the $L_{red}$ invariant under a fake `scaling transformation':
\begin{align}
\delta_H H = 2\,\lambda\, H~~.
\end{align}
The origin of the PCC equation (\ref{PCC}) is nothing but the Hamiltonian constraint along the radial direction with other equations of motion. Even though PCC does not introduce a new symmetry, it is very useful to describe thermodynamics of black holes.

Now, we can easily find the form of PCC as follows :
\begin{align}
\mathcal C
=-  A_\tau \mathcal{Q} +\frac{1}{32\pi G\,L^2} \,r\, e^{-\tilde{\mathcal{W}}/2} \left( 2 r\, U'+ r^2\,U\, {\phi'}^2 - 4\, U \right)~~,
\end{align}
where $\mathcal Q$ is the charge density of the black brane, which is  given by
\begin{align}\label{chargeQ}
\mathcal Q = \frac{1}{16\pi G\,L^2 } \left( r^2  e^{\tilde{\mathcal{W}}/2} Z(\phi)A_\tau'(r)  - 8\, \theta\, W(\phi) H\,L^2\right)~.
\end{align}
We may take an integration of PCC over the outer horizon region.
\begin{align}
\mathcal{C}(r) -\mathcal{C}(r_h) = -\frac{1}{32\pi G\,\lambda} \int_{r_h}^r dr' \delta_H L_H(r')~~,
\end{align}
where PCC at the horizon is given by $\mathcal{C}(r_h)=  s T_H  $ with adopting a regularity condition $A_{\tau}(r_h)=\,U(r_h)=0$.

Since the PCC is finite at the boundary of AdS space, it is legitimate to consider the PCC at $r=\infty$. Therefore, the integration over the outer region gives us the following relation:
\begin{align}
s T_H = \left(\mathcal{C}(\infty) + \frac{1}{32\pi\,G\,\lambda}\int_{r_H}^\infty dr'\delta_H L_H(r') \right)~~.
\end{align}
We will identify the temperature in the dual field theory $T$ with the Hawking temperature. In addition the asymptotic value of the PCC is given by
\begin{align}
\mathcal{C}(\infty) = - \mu\, \tilde{\mathcal{Q}} + \lim_{r\to\infty}\frac{1}{32\pi G\,L^2} \,r\, e^{-\tilde{\mathcal{W}}/2} \left( 2 r\, U'+ r^2\, {\phi'}^2 U - 4\, U \right)~,
\end{align}
where $\mu\equiv\frac{1}{\sqrt{16\pi G}}A_\tau(\infty)$ and $\tilde{Q}$ are the chemical potential and the charge density in the dual field theory and we also identified $\mathcal{Q}$ with $\frac{1}{\sqrt{16\pi G}}\tilde{\mathcal Q}$.

Since we are taking into account a homogeneous system, we may expect that the following relation
using the thermodynamic potential density($\omega = -\mathcal{P}$) in (\ref{pressure}):
\begin{align}
\epsilon + \mathcal{P} =& \,\mu\,\tilde{\mathcal Q} + s\, T_H \\
=&  \lim_{r\to\infty}\frac{1}{32\pi G L^2} \,r\, e^{-\tilde{\mathcal{W}}/2} \left( 2 r\, U'+ r^2\, {\phi'}^2 U - 4\, U \right) + \frac{1}{32\pi\,G\,\lambda}\int_{r_H}^\infty dr'\delta_H L_H(r') ~,   \nonumber
\end{align}
where $\epsilon$ is the internal energy given by one component of the holographic tensor $T_{\tau\tau}$. Furthermore the pressure $\mathcal P$ can be decomposed into the two parts: $\mathcal{P}_{others}+\mathcal{M}\,\mathcal{B}$, where we have used the magnetic field $\mathcal{B}\equiv \sqrt{L}\,H/\sqrt{16\pi\,G}$ as a field theory quantity. Since we don't consider sources except for the chemical potential and the magnetic field, $\mathcal{P}_{others}$ is simply given by the holographic pressure, $ T_{xx}$. The holographic energy-momentum tensor is usually obtained from asymptotic values of fields. Therefore we may claim that the magnetization density is given by
\begin{align}\label{MagnetizationGeneral}
\mathcal M=&\, \frac{1}{32\pi\,G\,\lambda\,\mathcal{B}}\int_{r_h}^\infty dr'\delta_H L_H(r') \nonumber\\=&\, - L^2\,\int_{r_h}^\infty dr\, e^{-\tilde{\mathcal{W}}/2}\left(  \frac{\mathcal{B} Z(\phi)}{L r^2}  + \frac{64 \mathcal{B} \theta^2 W^2(\phi)}{L r^2 Z(\phi)} +\frac{8\,\theta \tilde{\mathcal{Q}} W(\phi)}{\sqrt{L}r^2 Z(\phi)}   \right)~,
\end{align}
where the charge density $\tilde{\mathcal{Q}} = \sqrt{16\pi G} \mathcal{Q}$ in the dual field theory. By using an appropriate coordinate transformation, it turns out that this expression is same with formula for the magnetization in \cite{Kim:2015wba,Blake:2015ina,Donos:2017mhp}. In particular, the authors discussed $\theta(\phi) F\wedge F$ term in \cite{Donos:2017mhp}, where $\phi$ is a dilaton field which is not condensed. The authors of Ref. \cite{Donos:2017mhp} were focusing on the DC transport and did not say anything about spontaneous magnetization because they didn't consider scalar condensation. Nevertheless, formulas in the paper are very useful to study DC transport. We will mention a future direction related to this work in discussion section.

As we mentioned about the magnetization, one can notice that something interesting happens. Even though $\mathcal{B}=0$, the magnetization still exist if scalar field has finite expectation value.  This means that $\mathbb{Z}_2$ symmetry of the magnetization is spontaneously broken with the help of the hairy configuration of the scalar field. The resultant spontaneous magnetization is given as follows:
\begin{align}
\mathcal{M}_{f}\equiv\mathcal{M}_{\mathcal{B}=0} = -\, L^{3/2}\int_{r_h}^\infty dr\, e^{-\tilde{\mathcal{W}}/2}\left(  \frac{8\,\theta \tilde{\mathcal{Q}}  W(\phi)}{r^2 Z(\phi)}   \right)~~.
\end{align}
One can see that the sign of $\theta$ determines that of the magnetization. Under proper conditions, the spontaneous magnetization is accompanied by a hysteresis curve which is  a trajectory in $\left(\mathcal{B},\mathcal M\right)$-configuration space. In order to obtain such a curve, we have to consider black brane solutions with a real scalar hair in a numerical method. In the following sections, we will discuss the spontaneous magnetization and the corresponding hysteresis curves in a particular model. Also, we will check whether our expression (\ref{MagnetizationGeneral}) is valid or not in the model.

\subsection{ A Simple Model }\label{AsimpleModel}

In this subsection we consider a specific model which is relevant to a spontaneous magnetization and a hysteresis curve. We start with adopting $Z(\phi)=1$, $V(\phi) = - \frac{2}{L^2} \phi^2$ and $W(\phi) = - \phi^n$. As we discussed, we will consider only $n=1$ as a representative of odd $W(\phi)$ cases. Together with this choice, the real scalar field is supposed to be dual to a dimension 2 operator ($\Delta = 2$) for simplicity.
Then the total action consists of the bulk part and the boundary part as follows\footnote{ $ x^M =( x^\mu , r  ) = (\tau, x^i, r)=( \tau , x, y, r) $} :
\begin{align}\label{explicitModel}
S_{total}= & \,S_B + S_b~, \\
S_B = & \,\frac{1}{16\pi G}\int_{\mathcal M} d^4 x \sqrt{-g}\, \mathcal{L}_{bulk}\nonumber\\
= &\frac{1}{16\pi G}\int_{\mathcal M} d^4 x \sqrt{-g}  \left\{ R + \frac{6}{L^2}    -\frac{1}{4} F^2 - \frac{1}{2} \left(\partial \phi\right)^2    +\frac{1}{L^2}  \phi^2     \right\} \nonumber\\&- \frac{\theta}{16\pi G} \int_{\mathcal M} d^4 x \phi^n \epsilon^{MNPQ} F_{MN} F_{PQ}~, \\
S_{b}=&\,- \frac{1}{16\pi G}\int_{\partial  \mathcal{M} }d^3 x  \sqrt{-\gamma}  \left( 2 K +  \frac{4}{L} +  \frac{\phi^2}{2 L }   \right)~,
\end{align}
where $K$ is the Gibbons-Hawking term which is the tace of extrinsic curvature. The other terms in the boundary action $S_b$ are counter terms for holographic renormalization. The equations of motion for matter fields are given by
\begin{align}\label{FphiEq}
&\nabla_M  \left( F^{MN} +4 \,\theta\,  \phi^n     \frac{1}{ \sqrt{-g}}  \epsilon^{MNPQ} F_{PQ}  \right)=0  \\
& \left( \nabla^2 + \frac{2}{L^2} \right) \phi  - n\,\theta\, \phi^{n-1} \frac{1}{\sqrt{-g}}\epsilon^{MNPQ} F_{MN}F_{PQ} =0~~.
\end{align}
And the Einstein equation is
\begin{align}\label{EinEq}
R_{MN} - \frac{1}{2} g_{MN} \left(   R +\frac{6}{L^2} - \frac{1}{4} F^2 - \frac{1}{2} ( \partial \phi )^2  + \frac{1}{L^2} \phi^2 \right) - \frac{1}{2} F_{MP} {F_N}^P  -\frac{1}{2} \partial_M \phi \partial_N \phi =0 ~.
\end{align}
In addition, the magnetization formula (\ref{MagnetizationGeneral}) becomes
\begin{align}\label{MagExpl}
\mathcal M = - L\int_{r_h}^\infty dr e^{-\tilde{\mathcal{W}}/2}\left(  \frac{\mathcal{B}}{r^2}  + \frac{64\, \theta^2 \mathcal{B}\,\phi^{2n}}{r^2} -\frac{8\sqrt{L}\,\theta\,\tilde{\mathcal Q}\, \phi^n}{r^2}   \right)~~,
\end{align}
In the next subsection, this formula will be checked by holographic renormalization.

\subsection{Magnetization with Holographic Renormalization}\label{sub2.3}

Now, we will show that our proposal for magnetization (\ref{MagnetizationGeneral}) is correct in the explicit model (\ref{explicitModel}) using holographic renormalization. In order to show the validity, we first need to find the Euclidean on-shell action. We extend similar calculations for the on-shell action in \cite{Hartnoll:2008kx,Kim:2015dna}. The trace of the Einstein Equation (\ref{EinEq}) is given by
\begin{align}
R = 2 \left(\mathcal{L}_{\text{bulk}}+\frac{\theta  }{\sqrt{-g}} \phi^n F_{M N} F_{P Q} \epsilon ^{M N P Q}\right) + \frac{1}{2}F^2 + \frac{1}{2}(\partial \phi )^2
\end{align}
or
\begin{align}
& 2 g^{x x} G_{x x}+G^t{}_t+G^r{}_r + 2 R \\
&~~~~= 2 \left(\mathcal{L}_{bulk}+\frac{\theta }{\sqrt{-g}} \phi ^n F_{M N} F_{P Q} \epsilon ^{M N P Q} \right) + \frac{1}{2}F^2 + \frac{1}{2}(\partial \phi )^2 ~~, \nonumber
\end{align}
where $G_{MN}$ is the Einstein tensor. And one can find $G_{xx}$,
\begin{align}
G_{xx} = \frac{r^2}{2 L^2} \left( \mathcal{L}_{bulk} -R +\frac{\theta }{\sqrt{-g}} \phi ^n F_{M N} F_{P Q} \epsilon ^{M N P Q} \right) +\frac{L^2}{2r^2}H^2
\end{align}
from $xx$-component of the Einstein equation. Using the above three equations, the $\mathcal{L}_{bulk}$ is given by
\begin{align}
\mathcal{L}_{bulk}= -G^t{}_t-G^r{}_r-\frac{L^4}{r^4} H^2-\frac{\theta }{\sqrt{-g}}\phi ^n F_{M N} F_{P Q} \epsilon ^{M N P Q}~.
\end{align}
To get the Euclidean bulk action, we should take the wick rotation of $\tau$.  The bulk action is given by the following integration:\footnote{Here $\tau=-i\tau_E$ and $S_B^E+S_b^E = - i ( S_B+ S_b)$.}
\begin{align}
S^E_B =& - \int_0^{1/T_H} d\tau_E \int d^2 x \int_{r_h}^{\infty} dr\\
& \left( \frac{(-2r\, U\, e^{ \tilde{-\mathcal{W}}/2  }  )'}{16\pi G L^2} - \mathcal{B}L^2 e^{-\tilde{\mathcal{W}}/2} \left(\frac{\mathcal{B}}{r^2} -\frac{8 \theta \tilde{\mathcal{Q}} {\phi}^n}{r^2}+\frac{64 \theta ^2 \mathcal{B} {\phi}^{2 n}}{r^2} \right)  \right)\nonumber~,
\end{align}
where $\tau_E$ is the Euclidean time and one can notice that the proposed magnetization expression (\ref{MagExpl}) appears in the bulk on-shell action.

To finish the calculation of the total on-shell action, let us move on to the boundary part. The Euclidean boundary action is given by
\begin{align}
S_b^E &=\lim_{r\to\infty} \frac{1}{16\pi G} \int_0^{1/T_H} d\tau_E \int d^2 x  \sqrt{-\gamma}  \left( 2 K +  \frac{4}{L} +  \frac{\phi^2}{2 L }   \right) \\
&=\,\frac{1}{16\pi G L^2} \frac{1}{T_H}\int d^2 x\lim_{r\to\infty}\,r\, e^{-\tilde{\mathcal{W}}/2}\left(r \sqrt{U} \left(\frac{4}{L}+\frac{\phi^2}{2 L}\right)-r\, U' - U (4-r\, \tilde{\mathcal{W}}') \right)~~.
\end{align}
By plugging the asymptotic behavior of the fields in Appendix, finally we obtained the on-shell action as follows:
\begin{align}\label{E_on-shell}
S_B^E + S_b^E = \frac{\mathcal{V}}{T_H} \,\omega = - \frac{\mathcal{V}}{T_H} \left( \frac{\epsilon}{2} + \mathcal{M}\,\mathcal{B} \right),
\end{align}
where $\mathcal M$ is same with (\ref{MagExpl}). Also, one can notice that $\epsilon/2$ is equivalent to the internal pressure $ \epsilon/2= T_{xx} = 1/2\, T_{\tau\tau}$ which is derived in Appendix. This relation supports the fact that the proposed form of magnetization (\ref{MagnetizationGeneral}) is correct.

\section{Spontaneous Magnetization and Hysteresis Curve}\label{sec3}

In this section we analyze the spontaneous magnetization and hysteresis curve in the explicit model (\ref{explicitModel}) by numerical methods. For numerical calculations, we will take  $16\pi\, G = 1$, $L=1$ and $r_h=1$. In contrast to usual discussions in holographic models, solutions with lager free energy are also important in our discussion. We will explain the role of such unstable solutions in more details in the following subsections.

\subsection{Spontaneous Magnetization}

We concentrate on the case with vanishing magnetic field in this subsection. Then the equations of motion  are reduced to those without the magnetic field.
In high temperature, the AdS Reisner-N$\ddot{o}$rdstrom(RN) Black brane is the preferable solution dual to normal phase. On the contrary, the RN black brane becomes unstable and changes to a hairy black hole dual to an ordered phase in low temperature. The hairy scalar describes the VEV of a dimension 2 operator $\mathcal O$. As we discussed, this hairy scalar contributes to the magnetization (\ref{MagExpl}) even in the absence of the magnetic field. Therefore this holographic model can describe a phase transition from a diamagnetism phase to a ferromagnetism phase. Here the sign of the magnetization is determined by the sign of $\theta$ and we have to fix it with a suitable value. For an odd $n$, both signs of magnetization are possible with a given $\theta$. On the other hand the sign of magnetization is fixed for an even $n$. In the latter case the direction of magnetization is always same for both positive and negative scalar condensations. This is the reason why we concentrate on the $n=1$ case because we need the both signs of magnetization to construct magnetic hysteresis.

\begin{figure}[ht!]
    \subfigure[ ]
    {\includegraphics[width=6.5cm]{solspaceHzero.pdf}  }
    \hskip1cm
    \subfigure[ ]
    {\includegraphics[width=7.5cm]{FreeE01.pdf}  }\\

    \subfigure[ ]
    {\includegraphics[width=7cm]{TO.pdf}  }
   \hskip1cm
    \subfigure[ ]
    {\includegraphics[width=7cm]{TM.pdf}  }

    \caption{(a) The set of black brane solutions satisfying zero source condition (b) The  difference of free energy of RN black branes(red) and hair black branes(blue). (c) Temperature dependence of the absolute value of the condensation. (d) Temperature dependence of the absolute value of the magnetization, where we took $\theta= - 0.2$ for numerical calculation.
} \label{fig:Hzero}
\end{figure}

Also, if we allow only solutions without source of scalar condensation, i.e, $J_{\mathcal{O}}=0$ in (\ref{AsympS}), an instability of the RN black brane happens when $T_H=T_c$.
Figure \ref{fig:Hzero} (a) shows a set of black brane solutions satisfying the sourceless condition. Above this critical temperature, the only solution is that of zero condensation which is nothing but the RN black brane. On the other hand, two more kinds of hairy black brane solutions show up below $T_c$(solid curve connecting C and D). They describe positive and negative values of $\left<\mathcal{O}\right>$ naturally coming from the $\mathbb{Z}_2$($\phi \leftrightarrow -\phi$) symmetry in the action (\ref{explicitModel}).

Figure \ref{fig:Hzero} (b) presents free energy difference between RN black brane solutions and the hairy ones. Red solid line denotes the free energy of the RN black branes as a reference and blue line corresponds to the free energy of the hairy black branes. Below $T_c$, the free energy of the hairy solutions is lower than RN solutions. Therefore there is a second order phase transition from RN to hairy black brane solution at $T=T_c$.

We also plot the condensation diagram in Figure \ref{fig:Hzero} (c). Since the integrand of magnetization density is proportional to $\phi$ near critical temperature, the magnetization shows similar behavior as shown in Figure \ref{fig:Hzero} (d). In addition, the charge density is essential to produce the magnetization. Thus it turns out that the charge carrier also carries magnetic moment and there exists an interaction between the charge operator $\tilde{Q}$ and the dimension 2 operator $\mathcal O$ in the dual field theory.

The scalar condensation comes from the spontaneous symmetry breaking. In the dual gravity picture, this corresponds to advent of normalizable hairy black brane solutions. The holographic superconductor model admits a complex scalar hair which is associated with spontaneous breaking of continuous $U(1)$ symmetry \cite{Hartnoll:2008vx}. For a real scalar condensation and a real scalar hairy black brane, the associated symmetry is now $\mathbb{Z}_2$ symmetry. This phase transition is usually explained by a Landau-Ginzburg type theory whose effective field is a real scalar field. We denote this field by $\Phi$. Therefore, the potential of the Landau-Ginzburg model governs this phase transition. The typical shape of potential for the broken phase should also admit $\mathbb{Z}_2$ symmetry so it has two global minima. At these minima, the field has values, $\Phi=\pm\,\Phi_0$ and free energy of the system is given by the on-shell action of hairy black branes.

Now, we construct the Landau-Ginzburg type effective potentials using the data in Figure \ref{fig:Hzero}. One can naturally deduce that the free energy density of the RN black brane solution, such as B in Figure \ref{fig:Hzero}, corresponds to the unstable extremum due to $\mathbb{Z}_2$ symmetry and vanishing condensation. This is quite reasonable because the RN black branes are solutions of the dual gravity system and the extrema of the potentials are also solutions of the corresponding effective theory. In addition to this observation, we may assume that the condensation $\left< \mathcal{O} \right>$ and the magnetization $\mathcal{M}$ are very small near the critical temperature. Then the effective potential of free energy can be approximated by a polynomial of $\mathcal{O}$ or $\mathcal{M}$. The suggested form of the potential is given as follows
\begin{align}\label{V1}
\mathbb{V}(\Phi) = \omega_{RN} + \frac{\omega_{RN}-\omega_{HB}}{\Phi_0^4}  \Phi^2   \left(\Phi^2-2 \Phi_0^2\right) +  \mathcal{O}\left(\Phi^6 \right)~~,
\end{align}
where $\Phi$ is the effective field which in our case denotes the magnetization $\mathcal{M}$ or $\mathcal{O}$ and $\Phi_0$ denotes $\mathcal{M}$ or $\left<\mathcal{O}\right>$ of the hairy black branes, respectively. In addition, $\omega_{RN(HB)}$ is the free energy density of the RN black brane (Hairy black brane). Here, we interestingly point out that the parameters of the Landau-Ginzburg potential can be obtained by the dual geometry data.
We plot the effective potentials in Figure  \ref{fig:Cond}. The figure shows how the potential changes as the temperature decreases.

\begin{figure}[ht!]
\centering
    \subfigure[ ]
    {\includegraphics[width=7.8cm]{broken_M.pdf}  }
    \hskip0cm
    \subfigure[ ]
    {\includegraphics[width=7.8cm]{broken_phi.pdf}  }
    \hskip0.2cm

    \caption{The effective potential in terms of (a) the magnetization, (b) the condensation. The point B, C and D correspond to the same points in Figure \ref{fig:Hzero} (a) and (b).}
      \label{fig:Cond}
\end{figure}

 In addition, we may consider the fluctuation of $\Phi$ near the local minima. This fluctuation corresponds to an effective quanta in the broken phase. In this phase, the mass of quasi-particles governs major characteristic of the effective theory. We can easily obtain the mass of the quasi particle as follows:
\begin{align}
m_{eff}^2 =8\,\frac{\omega_{RN}-\omega_{HB}}{\Phi_0^2}.
\end{align}
Then, the low energy effective potential for the broken phase up to quartic order is as follows:
\begin{align}
\mathbb{V}_{BP} \sim \frac{1}{2}\omega_{HB} +\frac{1}{2} m^2_{eff}\delta\Phi^2 + \frac{\lambda_3}{3!} \delta\Phi^3  +  \frac{\lambda_4}{4!} \delta\Phi^4~~,
\end{align}
where $\delta\Phi$ is the fluctuation near $\Phi=\Phi_0$ and
\begin{align}
\lambda_3 = 4!\, \frac{\omega_{RN}-\omega_{HB}}{\Phi_0^3} ~~\text{and}~~ \lambda_4=4!\, \frac{\omega_{RN}-\omega_{HB}}{\Phi_0^4} \,.
\end{align}
These couplings describe the qubic and quartic interactions among the quasi particles. Of course, all the mass and couplings are also determined from data of the bulk geometry. When $\Phi=\mathcal{M}$, $\Phi$ describes the fluctuation of the magnetization. This quasi-particle can be interpreted as the magnon whose mass and self-couplings are given by
\begin{align}
m_{magnon}^2 = 8\,\frac{\omega_{RN}-\omega_{HB}}{\left<\mathcal{M}\right>^2 }~,\lambda_3 = 4!\, \frac{\omega_{RN}-\omega_{HB}}{\left<\mathcal{M}\right>^3} ~, \lambda_4=4!\, \frac{\omega_{RN}-\omega_{HB}}{\left<\mathcal{M}\right>^4}\,,
\end{align}
where $\left<\mathcal{M}\right>$ is obtained by the black brane corresponding D in Figure \ref{fig:Cond}.
For more clear understanding, we provide cartoons in Figure \ref{fig:elementaty}. In the figure, the length and direction of arrows depict the magnitude  and the sign of the magnetization. Figure \ref{fig:elementaty} (a) shows a fluctuation in the broken phase D in Figure \ref{fig:Hzero} (a). This excitation plays a qualitatively same role of a magnon in condensed matter physics. In the figure, the convex part corresponds to a magnon and the concave part describes an anti-magnon.

One more interesting  feature  of these potentials is that the free energy potential in Figure \ref{fig:Cond} implies existence of a kink or anti-kink solution connecting two vacua D and C, which is depicted in (b) of Figure \ref{fig:elementaty}. In more general cases, the magnetization   can have locally opposite sign in 2-dimensional space. See Figure \ref{fig:elementaty} (c) for such a configuration which describe the fragmentation of the magnetization. This plays an important role in the hysteresis curve.
To realize these complicated backgrounds, we have to solve partial differential equations, which is
 left as a future study. Later, we will give a comment on the role of these configurations in hysteresis curves.

\begin{figure}[ht!]
\centering
    \subfigure[ ]
    {\includegraphics[width=5cm]{qparticle.pdf}  }
    \hskip0.2cm
    \subfigure[ ]
    {\includegraphics[width=5cm]{kink.pdf}  }
    \hskip0.2cm
   \subfigure[ ]
    {\includegraphics[width=5cm]{fragmentation.pdf}  }
    \hskip0.2cm

    \caption{{\bf Cartoons for various excitations in broken phase :}
(a) shows a magnon(convex) and an anti-magnon(concave) and (b) depicts a magnetization kink configuration. In addition, (c) is an intersection of a locally opposite sign configuration. This kind of configurations can describe a fragmentation of magnetization.}
      \label{fig:elementaty}
\end{figure}

\subsection{Hysteresis Curve}\label{hysteresis}

In the previous subsections, we showed that a holographic model can describe the spontaneous magnetization and the hairy black brane solution is dual to a ferromagnetic phase. As we mentioned in the introduction, one of important properties of the ferromagnetism is the hysteresis curve. We now show how to construct the hysteresis curves. For this purpose,  we first find the hairy black hole solutions with the non-vanishing magnetic field using a numerical method. We impose the boundary condition $J_{\mathcal{O}}=0$ for the scalar field $\phi$ in (\ref{AsympS}) to simplify the problem. Using the solutions below the critical temperature, we display the  relation of external magnetic field and magnetization in Figure \ref{fig:hys01}: there exist three solutions  for small magnetic field. The solutions on (D-O-E) line have higher free energy than those on (B-$C_1$-D)  and (E-$C_2$-F) lines at the same magnetic field. So the former represent unstable extrema of the free energy potential, while the other two  describe the stable and metastable extrema of the potential. If the black brane solution changes following the local minimum of the potential under slowly varying magnetic field, such process can give rise to magnetic hysteresis. See Figure \ref{fig:MF}.

 \begin{figure}[ht!]
\centering
    {\includegraphics[width=10cm]{BMT01.pdf}  }
\caption{External magnetic field dependence of the magnetization.}
      \label{fig:hys01}
\end{figure}

\begin{figure}[ht!]
\centering
    \subfigure[ ]
    {\includegraphics[width=6.5cm]{MF01.pdf}  }
    \hskip1cm
        \subfigure[ ]
    {\includegraphics[width=6.5cm]{MF02.pdf}  }\\
     \subfigure[ ]
    {\includegraphics[width=6.5cm]{MF03.pdf}  }
    \hskip1cm
    \subfigure[ ]
    {\includegraphics[width=6.5cm]{MF04.pdf}  }
\caption{Magnetization dependence of the effective potential for given magnetic field corresponding to Figure \ref{fig:hys01}. The potentials (a) (b) (c) and (d) are given upto $\mathcal{M}^6$-\,order.}
      \label{fig:MF}
\end{figure}

Let us describe how the effective potential changes in more detail. We start with A in Figure \ref{fig:hys01} and increase the external magnetic field slowly. Until the magnetic field reaches B, there is a single minimum and it should be located at the global minimum of the effective potential, see Figure  \ref{fig:MF} (a). When the magnetic field becomes zero, there are three solutions. Two of them are at the minima   and the other one is at the local maximum(RN black brane), see Figure \ref{fig:MF} (b).  Further variation of the magnetic field gives rise to an asymmetric potential. If the thermal fluctuation and the changing rate of the magnetic field are very small with infinitely large volume, then the system can be placed on a metastable point in the potential. See Figure \ref{fig:MF} (c). When the magnetic field reaches D in Figure \ref{fig:hys01}, the local minimum where the solution is located and the local maximum coincides each other, Figure \ref{fig:MF} (d). Then, the solution cannot stay there and falls to the local minimum F which stands sudden jump of the magnetization and will stay there as the magnetic field increases further. If we start from large magnetic field and decrease it, then the process will be reversed. Our result describing magnetic hysteresis is reminiscent of a bistable model accompanied by the Barkhausen effect in condensed matter physics. See \cite{Bertotti1998} for a text book.

While generating the hysteresis curve, we need to pay attention to several things. First, the time-variation of magnetic field should be slow to prevent a phase transition from the local minimum to the global minimum. Otherwise, the time variation $\dot{\mathcal{B}}$ increases $\dot{\mathcal{M}}$ and the effective kinetic energy of $\mathcal{M}$. If the kinetic energy is large enough, then the system jumps over the potential barrier. To prevent this situation we assume that the changing rate of magnetic field is very small. Second, we need to care the thermal fluctuation of the system. In high temperature the thermal fluctuation becomes large, however we are investigating a broken phase from the real scalar hair. It is well known that real scalar hairs appear below very low temperature and so is our case\footnote{n our numerical calculation, the black brane solutions have $T_c=3.97\times 10^{-4}$ in the unit of $L=1$ and $16\pi\, G=1$.}. In fact, such an effect changes hysteresis curves in Figure \ref{fig:hys02} slightly, but it is not significant, so we ignore that effect in the present work. Finally, there exist many nontrivial configurations such as (b) and (c) in Figure \ref{fig:elementaty}. These configurations contribute to the fragmentation of magnetization. The actually measured hysteresis curves have shapes that change less dramatically than (a) in Figure \ref{fig:hys02}. The fragmentation must be considered to produce milder shapes. All the above issues are very interesting and important, but they are not the main topic of the present work. So we will investigate these issues in a future study \cite{KimSeo}.

\begin{figure}[ht!]
\centering
    \subfigure[ ]
    {\includegraphics[width=7cm]{HysT01.pdf}  }
    \hskip1cm
    \subfigure[ ]
    {\includegraphics[width=7cm]{HysT02.pdf}  }

    \caption{(a) Hysteresis curve of the magnetization. (b) Hysteresis curve of the charge density. }
      \label{fig:hys02}
\end{figure}

Therefore, the resulting process of the magnetization hysteresis is drawn in Figure \ref{fig:hys02} (a). Figure \ref{fig:hys02} (b) shows hysteresis behavior of the charge density for (a). The charge carrier density is decreasing for both direction and suddenly increase when the magnetization changes.  In this paper, we do not consider the time dependence of the falling process, so the magnetization is suddenly dropped at the transition point. In fact, this dynamics should be considered with time dependent background and it is actually governed by a non-equilbrium physics\footnote{An example using holography was discussed for the chiral phase transition in \cite{Evans:2010xs}. See Figure 2 in that paper.}.

In Figure \ref{fig:hys02}, non-zero values of the magnetization cause the Hall effect and it gives nontrivial Hall conductivity($\sigma_{xy}$) or Hall resistance($\rho_{xy}$). We expect that there also be hysteresis behavior to the Hall resistance. On the other hand,  the charge carrier density $\mathcal{Q}$ would be proportional to the longitudinal conductivity $\sigma_{xx}$. The results are qualitatively same to the transport on the surface state of the topological insulator. See Figure 3 and 4 in \cite{Bao2013}. In addition, Figure \ref{fig:hys03} shows temperature dependence of the hysteresis curve. As temperature decreases, the size of hysteresis curve increases. It is natural that the expectation value of the scalar is bigger at lower temperature and so is the magnetization.

\begin{figure}[ht!]
\centering
    {\includegraphics[width=6cm]{HysT03.pdf}  }

    \caption{Temperature dependence of the hysteresis curve with $T/T_C =$ 0.98(red), 0.96(blue), 0.93(black). }
      \label{fig:hys03}
\end{figure}

\section{Discussion}\label{sec5}

In this paper, we have studied the magnetic properties of the strongly interacting system using gauge/gravity duality. To do this, we first defined the magnetization of the boundary theory in terms of the bulk data using the `scaling symmetry' trick. Then, we applied it to a specific model and confirmed that the formula agreed with the magnetization obtained from the holographic renormalization and satisfied thermodynamics laws.

In our model, the magnetization comes from the condensation of the scalar field due to the spontaneous symmetry(discrete $\mathbb{Z}_2$) breaking. By introducing a coupling between magnetic field and the real scalar, we showed that a spontaneous magnetization occurs in our model. Using these gravity backgrounds admitted in our model, we constructed the Landau-Ginzburg type potential in terms of bulk data just below the critical temperature. In the case without a magnetic field, we provided the closed form of the potential (\ref{V1}) in terms of the on-shell actions and the expectation values which can be easily obtained by a simple numerical calculation. This formula gives us a systematic way to relate the black hole physics to the Landau-Ginzburg type effective theory. As a by-product, We obtained magnon mass and its self-couplings from this effective potential.

Also, we analyzed the field theory systems with the magnetic field dual to the magnetic hairy black branes. Using the black brane solutions, we found the corresponding Landau-Ginzburg potentials under varying the magnetic field. This fact tells us that the gravity theory seems to be a kind of Landau-Ginzburg model. Since it is difficult to quantize Landau-Ginzberg Models in a generic situation, one can see the reason why quantization of the gravity theory is so hard without any dual description. Our result provide another direct explanation for difficulty of quantum gravity.

The effective potential implies that the magnetization hysteresis can be generated by slowly varying magnetic field. In this hysteresis process, the time scale of the variation and the thermal fluctuations are very important. So we assume very slow change of the system and we argued that thermal fluctuations are very small due to the low temperature. More detailed discussion would be interesting as a future direction. In addition to the issues, our resulting hysteresis has rapidly varying shapes and the fragmentation can make the shapes more smoothly. However, we did not consider the fragmentation of magnetization because the corresponding black brane solution is not easy to obtain. Thus we leave it as another future study.

As a final comment, it is expected that the magnetization and the charge carrier density are proportional to the Hall resistivity and longitudinal conductivity, respectively. Therefore, there exists hysteresis behavior of these quantities~\cite{Bao2013,Chang2013,Barzola2012}  . A study on this subject will be reported soon \cite{KimSeo}.

\section*{Appdendix}

\subsection*{A. Holographic Renormalization}

In this section we provide expression of boundary tensors such as energy-momentum tensor, current and condensation of scalar operator dual to bulk scalar field $\phi$ in the subsection \ref{AsimpleModel}. In order to get these formulas using holographic renormalization, we introduce ADM decomposition without the shift vector. Thus the metric (\ref{ansatz0}) can be decomposed as follows:
\begin{align}
ds^2= \gamma_{\mu\nu} dx^\mu dx^\nu + N^2 dr^2~~,
\end{align}
where $\gamma_{\mu\nu}= \text{diag}\left( -U(r)\,e^{\tilde{\mathcal W}(r)},\frac{r^2}{L^2},\frac{r^2}{L^2} \right)$ and $N = 1/\sqrt{U(r)}$. Then the extrinsic curvature tensor and the Gibbons-Hawking term $K$ are given by $K_{\mu\nu}=- \frac{1}{2N}\gamma'_{{\mu}\nu}$ and $\gamma^{\mu\nu}K_{\mu\nu}$, respectively. A boundary tensor is usually given by variations of the total action $S_B + S_b$ with respect to corresponding sources. In this model, they are given by
\begin{align}\label{asympto01}
&\left<T_{\mu\nu}\right> = \frac{1}{16\pi G}\lim_{r\to\infty} 2 \frac{r}{L} \left[ K_{\mu\nu}- K \gamma_{\mu\nu} - \left( \frac{\phi^2}{4 L} +\frac{2}{L}  \right)\gamma_{\mu\nu}  \right]\\
&\left<J^\mu\right> =  \frac{-1}{\sqrt{16\pi G}}\lim_{r\to\infty} \frac{\sqrt{-g}}{\sqrt{-\gamma}}F^{r\mu} \\
&\left<\mathcal{O}\right> =\frac{L}{\sqrt{16\pi G}} \lim_{r\to\infty} \frac{1}{r} \left( -\sqrt{-g}\,\nabla^r \phi - \sqrt{-\gamma}\, \phi/L \right)
\end{align}

On the other hand, the asymptotic behavior of the bulk geometry is given by
\begin{align}
&\tilde{\mathcal{W}}(r) \sim 4\pi G \left(J_{\mathcal{O}}^2 \left(\frac{L}{r}\right)^2 + \frac{8\, L\, J_{\mathcal O}\tilde{\mathcal{O}}}{3}\left(\frac{L}{r}\right)^3+\ldots \right)\\
&U(r) \sim \left(\frac{r}{L}\right)^2 +4\pi G\left( J_{\mathcal{O}}^2 - 2 \, L\, \epsilon \left(\frac{L}{r}\right) +\ldots\right)\\
&\phi(r)\sim \sqrt{16\pi G}\left(J_\mathcal{O}\left(\frac{L}{r}\right) +   L \tilde{\mathcal{O}}\left(\frac{L}{r}\right)^2 + \ldots \right)~~,\label{AsympS}\\
&A_\tau (r) \sim \sqrt{16\pi G}\left(\mu - L\, \tilde{\mathcal{Q}} \left(\frac{L}{r}\right)+\ldots\right)~,
\end{align}
where we are considering $J_{\mathcal{O}}=0$ case since we don't want to introduce another source except for the chemical potential and the magnetic field\footnote{ One can easily extend the case with a non-vanishing $J_{\mathcal{O}}$.}. Then, we can obtain the boundary tensors as follows:
\begin{align}
\left<T_{\mu\nu}\right> = \left(\begin{array}{ccc}
\epsilon & 0 & 0 \\
0 & \epsilon/2 & 0 \\
0 & 0 & \epsilon/2
\end{array}\right)~,~\left<J^\mu\right> =( \tilde{\mathcal Q},0,0)~,~\left< \mathcal{O}\right> = \tilde{\mathcal O}\,.
\end{align}

\section*{Acknowledgments}
K.K. Kim thanks Koji Hashimoto and Seung-Hyun Chun for helpful comments. The work of K.K. Kim is supported by the faculty research fund of Sejong University in 2018. YS is supported by Basic Science Research Program through NRF grant No. NRF-2016R1D1A1B03931443. SJS is supported by Mid-career Researcher Program through the National Research Foundation of Korea grant No. NRF-2016R1A2B3007687.
The work of K.-Y. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT $\&$ Future Planning(NRF- 2017R1A2B4004810) and GIST Research Institute(GRI) grant funded by the GIST in 2019. We acknowledge the hospitality at APCTP where part of this work was done.

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This paper is the last of my three articles on this super important subject… How to get a mentor. In my first and second articles I looked at: Why should we have a mentor? What is a mentor? Who makes a good mentor? How do you approach mentors? What do you want from a mentor? And the importance of being prepared when you get on a phone or in a meeting with your mentor. This time, I will focus on mentor etiquette… the "Do's and Don'ts" that go along with having a mentor.
Firstly, here are a few of the don'ts to keep in mind. Don't put them on the spot by asking them to hire you, audition you, or read your screenplay. If they request it, that's fine. But don't put them on the spot by asking.
Don't have them be the one who works to generate the conversation. It's your job to be totally ready. And don't ever use their name without permission. Just because they might mention a show you would be great for and they mention that they know someone associated with that show, if you decide to get in touch with that person, and you want to use your mentor's name, it is imperative that you get their permission to do so. Remember, you are going to them for their wisdom, not their rolodex.
Here are a few items to remember as far as the do's are concerned. Always be prepared, really prepared, with your list questions as well as information on them and their company. Make sure you've done your homework. Make good use of their time. Keep your word. Call when you said you'd call. Show up at the meeting on time. Really respect their time and acknowledge them and thank them for their time and wisdom.
Remember that it's a business relationship. So respect the term of your arrangement. If you've agreed to have three meetings or phone calls over a month long period and you decide you want to continue the relationship then get permission to do so.
When I was shooting my YouTube series for my YouTube channel on this subject of getting a mentor, I interviewed film and video game producer, Ruby Lopez. Ruby has been a volunteer at Women in Film for many years and their mentor program is a year long, so they are very strict about mentor etiquette.
That's brilliant advice from both Ruby and Diane. They are giving you gold here, so be sure to use it. And remember, we all should have mentors in our lives and careers. As I mentioned in my first article, everyone loves and wants to contribute and there are thousands of people in our industry who have many years of experience and would be honored to share their knowledge and expertise with you. So take the information in all three articles and of course watch my YouTube channel and the Master Talent Teachers video series on this subject, and go out and get yourself a mentor or two. I promise it will be well worth it.


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Doc Hopper: This is Snake Walker. Tell 'em what you do, Snake.
Frog Killer: [removes his sunglasses] Kill frogs.
MacCready: If you can't handle it, get me somebody who can! Get me Bruce Lee!
Cobb: Bruce Lee is dead.
MacCready: Then get his brother!
— Blind Fury
An underling fails, so their employer hires the best man (or woman) to get the job done — usually played out by villainous characters, who suffer a bitter and humiliating defeat at the hands of the hero by proxy, when the hero defeats their champion. Enraged at their defeat, they do not engage the heroes face to face themselves, but use their financial means to acquire the toughest specialists money can buy, to settle the score. The expert may be Only in It for the Money, but some are actually interested in the prospect of a Worthy Opponent to test their mettle. If the expert is soon defeated or killed despite their expertise, he (or she) is The World's Expert on Getting Killed.
Subtrope of Hired Guns. Compare Cast the Expert and Sorting Algorithm of Evil. Not to be confused with Expert Consultant.
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Throughout Area 88, the communist rebels resort to hiring veteran air force pilots as mercenaries to take out the mercenaries recruited by the Royal Asranian Air Force. Among them include ex-American Navy pilot Patrick Reed, who was recruited to fly their MiG-23 Flogger fighters supplied by the USSR.
Chimitsu Sunazara is a mercenary sniper recruited by a black ops unit called SCHOOL in the Battle Royale arc of A Certain Magical Index after their previous sniper was killed by ITEM agents.
Most of the plots in Black Lagoon involve the hiring of bounty hunters and mercenaries to go after Lagoon Company or the VIPs they're supposed to protect/escort. It rarely goes well for the bad guys.
Mercenary Tao in Dragon Ball was hired to kill Goku after the Red Ribbon Army suffered several curb-stomp battles at the hands of him and his friends.
In Durarara!!, Vorona and Slon are hired in Japan as mercenaries in order to abduct Akane Awakusu and blame it on a hostile Yakuza group as a last resort by Yodogiri Jinnai.
Gauron is recruited by Amalgam in Full Metal Panic! in order to cause headaches for Mithril during their anti-terrorist operations in the Asia-Pacific region.
Ghost in the Shell: Stand Alone Complex
In the 1st GiG episode "Automated Capitalism", a Chinese Triad group recruits an underworld assassin named Fem to take out a known banker named Yokose Kanemoto. She was eventually arrested by Section 9 before she could carry out the hit.
In the middle of 2nd GiG, ex-JGSDF soldier Hideo Kuze is recruited by militant refugees in Dejima to help them fight off the JGSDF since their overt attacks are foiled by law enforcement, including Section 9.
This happens a lot to Duke Togo. He is, quite literally, the world's greatest assassin, so when someone needs someone else killed (and can afford the hefty fee), they know who to call. On more than one occasion, he's called in not even to kill someone, but just to provide his assessment.
In Jormungand, a team of assassin led by Dominic is hired by rogue CIA SAD agent Hex in order to assassinate Koko and her men in North Africa as a last resort.
In Patlabor, Schaft Enterprise, courtesy of Richard Wong aka Utsumi, recruits an Indian boy named Badrinath Harchand to take on the Ingrams of SV2 and secure as much combat data as possible after the failures of using pro-environment terrorists to take on them with Brockens.
This is a minor plot in Spriggan's "Legend of the Mask" arc when ex-Machiner's Platoon officer Dutch "Iron Arm" Metrisk is hired by Koichi Sasahara under the influence of Tezcatlipoca to battle Yu Ominae in Mexico after all previous attempts to stop him in Japan failed.
In YuYu Hakusho's Spirit Detective arc, Yukina was held prisoner in Gonzo Tarukane's mansion in the outskirts of Tokyo in order to avoid detection by law enforcement. Since his subordinates were not able to make Yukina cry to get her tears hardened as crystal gems, Tarukane arranged to make contact with the Toguro brothers, known in the underworld as the leaders of the Apparition Gang that specializes in anything requested by their clients, including the use of torture methods towards demons. They succeed.
Film — Live Action
In Blind Fury: The Big Bad exasperated by his minions' failures to deal with the Blind Weaponmaster, tells his henchman to get Bruce Lee if that's what it'll take to deal with him. When The Dragon points out Bruce Lee is dead, he demands his brother or someone capable of it. They settle for hiring an expensive Japanese swordsman of their own, who does end up giving Nick his toughest fight in the movie.
In Cats & Dogs: The Revenge of Kitty Galore, the mercenaries Angus and Duncan MacDougall attempt to assassinate Seamus on the boat returning from the prison.
In Diggstown Gabriel Caine bets John Gillon that boxer "Honey" Roy Palmer can defeat ten local boxers in 24 hours. Gillon does not trust the local amateurs to get the job done so for the final fight he brings in a Super Ringer "Hammerhead" Hagan, a professional boxer who has beaten Palmer before. He then tries to do it again but Caine actually paid off the new expert to throw the fight.
The Empire Strikes Back has Darth Vader recruiting bounty hunters, including Boba Fett, in order to capture Han Solo and lure in Luke Skywalker instead of relying on the Stormtroopers.
The Ghost and the Darkness. John Patterson prepares a cunning trap for the man-eating lion that's attacking his workers. Sir Robert Beaumont looks like he's going to praise it, only to scoff at how ridiculous the idea is and say he's calling in a professional lion hunter, Charles Remington.
Gorgeous. Jackie Chan vs. an expert kickboxer hired by the Big Bad that used to be his friend.
The main plot of Guns And Talks involves four assassins recruited by clients for various jobs in South Korea, which usually involves killing off someone they know.
Hard Times. As Chaney beats Gandil's bald prizefighter, Gandil hires Street, the top fighter from Chicago to square things out.
Done twice in Kung Fu Hustle by the Axe Gang: First they hired the blind musical assassins to get rid of the three kung fu masters (Tailor, Coolie and Donut) in the Pig Sty Alley after their mooks are soundly beaten by the three. The musicians manage to do it, but then they face the Landlord and Landlady who are secretly kung fu masters too, and they soundly beat the musicians. Then the Axe Gang hires The Beast, an infamous bloodthirsty kungfu master that has locked himself up in an asylum, to fight the couple.
Moonraker. The villain Drax orders The Dragon Chang to kill James Bond. Chang fails repeatedly and Bond finally kills him. Drax then hires Jaws to finish the job.
Done twice in The Muppet Movie. Doc Hopper hires Professor Krassman to brainwash Kermit. When he fails, Hopper hires Snake Walker to assassinate Kermit.
Ocean's Thirteen has the best-in-business of subverting casino computer defence systems guy. Upon hearing of what Ocean's band has their sights on, the invitee immediately gives up and talks about returning the front money for his services.
Once Upon a Time in China. The Saho thugs hire out Iron Shirt Yim, a down-and-out martial arts master, to combat Wong Fei Hung.
Several 1970s Shaw Brothers kung-fu films had the villain suffer a crushing defeat at the hands of The Hero, resulting in them hiring outside martial arts experts to do their fighting for them, usually foreign experts from other parts of China or Asia.
Shiri has Lee Bang-Hee brought out of obscurity by Park Mu-Young in Seoul with orders to assassinate key targets in the South Korean government in order to keep Operation Center busy and force the South Korean government to transport the CTX prototype explosives by the South Korean Army and hijack it and partially do it as payback for OP's interference with a hijacking operation his men executed years ago.
The Sting. After Hooker and his partners con one of Lonegan's numbers runners out of the money he was carrying, Lonegan puts out a hit on them. When two of his underlings fail to kill Hooker, Lonegan has the assassin Salino assigned to get rid of him.
In Way of the Dragon, Tang Lung faces off against Colt in the Colosseum, hired by the Italian Mafia in order to get rid of him after their men have failed to make him leave Rome.
Inverted in Blood Rites, where Harry calls in Kincaid, the expert mercenary he'd met in the previous book to help deal with a gang of Black Court vampires.
In The Day of the Jackal the conspirators fail to kill De Gaulle so for the next attempt they hire a professional assassin known only as "The Jackal". Ironically, the amount of money they have to steal to afford his services is the first sign to the police of what is going on.
In Fighters of Danwait, the Lo'ona Aeo start a massive hiring spree of demobilized human soldiers, pilots, and Space Navy officers after the end of the final Void War between humanity and the Bino Faata. Previously, they recruited pretty much anyone, although mostly civilians, but the massive influx of out-of-work war vets bolsters their mercenary force (used to protect their borders and shipping) by a large factor. The civilian hires are relegated to the Convoy, providing escort to Lo'ona Aeo merchant ships, while the experienced mercs are members of the Patrol and spend their time, well, patroling Lo'ona Aeo border sectors for hostile incursion. Ironically, their typical foes are the Dromi, who had worked for the Lo'ona Aeo in the same capacity for over a millennium and are now pissed off at being terminated in favor of the upstart humans.
The Executioner. After an open contract and their own mooks fail to kill Mack Bolan in the first three novels, the Mafia Commission vote to bring in the Talifero brothers, who run the modern version of Murder, Inc..
Virgil Cole/Everett Hitch: Whenver Cole and Hitch are hired, the opposing faction will often try to hire similarly skilled gunmen to counter them:
Resolution: After Wolfson hires Hitch and Cole, Eamon Malley hires Cato and Rose.
Blue-Eyed Devil: General Laird hires Chauncey Teagarden to take revenge after Cole kills Laird's son.
Appaloosa: Bragg hires the Shelton brothers after his own men prove inadequate to deal with Cole and Hitch.
In Yendi, the Jhereg crime boss Laris arranges for several attempts on the life of the protagonist Vlad Taltos, but they fail each time. He finally gets serious and hires the Sword and the Dagger of the Jhereg, two expert assassins, to take care of Vlad.
Live Action TV
The A-Team episode "Deadly Maneuvers" had a syndicate of crime lords pay Major Douglas Kyle to assemble a Dream Team of mercenaries to hunt down the A-Team.
On Believe Skouras does not believe that the FBI is capable of bringing Bo back to him so he hires a mercenary to find her. When the mercenary fails, Skouras enlists another powerful psychic to locate Bo and neutralize her protectors.
Buffy the Vampire Slayer. In "What's My Line, Part 1" Spike brings in a trio of assassin demons to kill Buffy.
Dalton: Uh, yes, but... The Order of Taraka, I mean... isn't that overkill?
Spike: No, I think it's just enough kill.
The Verrat in Grimm are always brought in by the antagonists due to Nick and the others putting a stop to their activities in the Greater Portland Area.
In the premiere of Halt and Catch Fire Joe and Gordon reverse engineer the IBM PC BIOS chip which puts their employer, Cardiff Electric in a lot of legal trouble. To avoid a massive lawsuit from IBM, the company has to design and build their own PC clone to prove that they are not infringing IBM's copyright. Gordon could do it but all current Cardiff employees are considered legally 'tainted' and the company has to hire an outsider to do the actual work. Joe then subverts the trope by hiring Cameron to work on the project. While she is portrayed as a very gifted engineer, she is still in college and has never worked in the computer industry.
On Justified Katherine Hale and Wynn Duffy plan to steal millions from fellow mobster Avery Markham. However, Markham is believed to keep his money in a safety deposit box in a local bank and neither Hale nor Duffy know much about robbing banks. To rectify this, they partner up with Boyd Crowder who used to rob banks for a living before he switched to other criminal endeavors. Boyd then discovers that the money is actually kept in the vault of a former bank building so they start looking for and expert who can help them break into the vault. Duffy hires the "Wiz", an explosives expert who knows how to blast his way into bank vaults. When that literally blows up in their faces, Boyd instead decides to dig his way into the vault through a nearby abandoned mine. Fully aware of the dangers such an operation could present, he hires Zachariah Randolph, an expert miner who knows how to work the old mines without getting yourself accidentally killed.
When Markham's mercenaries prove to be no match for the US Marshals, Markham brings in Boone, a Psycho for Hire who is a master gunslinger equal in skill to Marshal Raylan Givens.
In Kaizoku Sentai Gokaiger, Space Empire Zangyack recruits a bounty hunter named Kiaido in order to take down Captain Marvelous and the other Gokaigers after previous attempts to kill them failed. He proves to be a troublesome enemy that the Gokaigers needed help to deal with him.
Star Trek: Picard: Who better to oversee the rehabilitation of former Borg on a derelict Borg cube than Hugh, an ex-drone? He's the Executive Director of the Borg Reclamation Project.
In Gangsters 2: Vendetta it was possible to hire professional hitmen to deal with rival mob leaders. Their services always came with a hefty price tag, however.
In the mission "You Lucky Bastard!" of Mafia: The City of Lost Heaven, Don Salieri repeatedly sends Tommy and other family hitmen to assassinate Don Morello's brother—but after they fail three times in a row, he calls them off and hires a pair of freelance Professional Killers. These also fail, however, so Tommy kicks subtlety to the curb, chases the man down, kills all of his bodyguards, and just shoots him in the head.
Western Animation
In Avatar: The Last Airbender, Admiral Zhao requisitioned an elite unit called Yuyan Archers famed for their Improbable Aiming Skills. Surely enough, they managed to capture Aang and bring them to Zhao's base.
Most incarnations of G.I. Joe has Zartan recruited as a mercenary by the future Cobra Commander to assassinate Snake Eyes prior to his recruitment as a GIJoe commando.
Samurai Jack has Aku recruiting the Imakandi, bipedal lion people renowned for never failing a hunt, in the episode Jack and the Hunters. While Jack gives them a lengthy chase across the city, they eventually succeed in capturing him, but were so impressed by his skill they let him go instead of turning him over to Aku.
Teenage Mutant Ninja Turtles (2012) has the Shredder recruit a mercenary assassin named Tiger Claw in order to deal with the Turtles and their attempts to get in the Foot Clan's way.
Thundarr the Barbarian. In the episode "Battle of the Barbarians", an evil wizard hires another barbarian to help him defeat Thundarr.
The Venture Bros.: Played with when OSI seemingly calls in the assassins Herr Trigger, La Tueur, and Go-Fish to kill Brock Samson after terminating him as an agent. It's inverted because Hunter Gathers actually sent the assassins so Brock would kill them and thus eliminate competition for the Black Hearts.
Bumbling Henchmen Duo
The Muppet Movie
ImageSource/Puppet Shows
Kirk's Rock


text:
Q: Binding WPF DataGrid ItemsSource creates memory leak I'm simply creating a Observable collection in Background and binding to it.
All objects inherit from INotifyPropertyChanged.
but nevertheless the Memory consumption is continuously raising.
The following Objects instances are continuously raising
WeakReference
FrugalObjectList<WeakEventManager+Listener>
ConditionalWeakTable<TKey, TValue>+Entry<Object, Object>[]
WeakEventTable+EventKey
ConditionalWeakTable<Object, Object>
SingleItemList<WeakEventManager+Listener>
Object
Int32[]
WeakEventManager+ListenerList<NotifyCollectionChangedEventArgs>
WeakEventManager+ListenerList<PropertyChangedEventArgs>
HybridDictionary
ListDictionary
ListDictionary+DictionaryNode
WeakEventManager+ListenerList<EventArgs>
WeakEventManager+ListenerList<CurrentChangingEventArgs>
CollectionRecord

I'm using .net 4.5.2.
See also the following Screenshots:
MemoryConsumptionOverview
ClassesIncreasing
Attached the sample code
XAML Markup:

<Window x:Class="BindingDataGridTest.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        Title="MainWindow" Height="120" Width="200"
        DataContext="{Binding RelativeSource={RelativeSource Self}}">
    <Grid>
        <DataGrid ItemsSource="{Binding BindingVals, Mode=OneWay}" />
    </Grid>
</Window>

Code behind:
using System;
using System.Collections.ObjectModel;
using System.ComponentModel;
using System.Windows.Threading;

namespace BindingDataGridTest
{
    public partial class MainWindow : INotifyPropertyChanged
    {

        public ObservableCollection<Values> BindingVals { get; set; }

        public MainWindow()
        {
            BindingVals = new ObservableCollection<Values>();
            InitializeComponent();

            DispatcherTimer myTimer = new DispatcherTimer { Interval = new TimeSpan(20) };
            myTimer.Tick += CreateVals;
            myTimer.Start();

        }
        private void CreateVals(object sender, EventArgs e)
        {
            Values myMainval = new Values
            {
                MyVal = "1V" + new Random().Next()
            };

            BindingVals.Clear();
            BindingVals.Add(myMainval);

            GC.Collect();
        }

        public event PropertyChangedEventHandler PropertyChanged;
    }

    public class Values : INotifyPropertyChanged
    {
        public string MyVal { get; set; }

        public event PropertyChangedEventHandler PropertyChanged;

    }

}

A: It's caused by DispatcherTimer.
A DispatcherTimer will keep an object alive whenever the object's methods are bound to the timer.
https://msdn.microsoft.com/ru-ru/library/system.windows.threading.dispatchertimer(v=vs.110).aspx
Use System.Timers.Timer instead and Dispatcher separately.
    public MainWindow()
    {
        BindingVals = new ObservableCollection<Values>();
        InitializeComponent();

        System.Timers.Timer myTimer = new Timer {Interval = 20};
        myTimer.Elapsed += CreateVals;
        myTimer.Start();

    }

    private void CreateVals(object sender, EventArgs e)
    {
        Dispatcher.Invoke(() =>
        {
            Values myMainval = new Values
            {
                MyVal = "1V" + new Random().Next()
            };

            BindingVals.Clear();
            BindingVals.Add(myMainval);

            GC.Collect();
        });

    }

redpajama_set_name: RedPajamaStackExchange


text:
CTMQ > Museums > 186. Nathaniel B. Palmer House Museum
186. Nathaniel B. Palmer House Museum
The World in the Palmer of His Hand
Stonington (Google Maps location)
I've been known in these very pages to whine about the fact that there are so many historic house museums in Connecticut for me to visit. It's not even the visits themselves I whine about – it's the process of writing about them each in a unique and interesting way. It's all about you. I do everything for you.
(As if anyone reads these museum pages. Hey, let me maintain the fantasy.)
Really? That's what I fantasize about? People reading my pages about museums on CTMQ? Yeah, I'm getting old.
I'm happy to tell you, the thousands of you reading this, that the Palmer House is definitely NOT your run-of-the-mill historic house museum. Not at all. Why? Because it's not the repository of a town's historical artifacts, that's why. In Stonington, a beautiful and historically rich town to begin with, the historical society maintains two museums: This one and the Old Lighthouse Museum which is also cool because it includes an old lighthouse with all the usual town history stuff.
The two museums are just over a mile apart and if you visit Stonington on a nice day, you can handle the walk – in fact, I urge you do so. As far as maritime and quaint and cute and interesting and beautiful towns go, you can't do much better than Stonington. Alas, we visited during a mock-hurricane so walking wasn't in the cards for us.
In fact, merely visiting the museum wasn't in the cards for Damian… Which therefore meant it wasn't in the cards for Hoang either. Damian had had enough of an adventure for the day, so he stayed in the car with Hoang while I tried to make the docent understand my tour was to be quick and efficient.
I requested that before I knew how cool this place was though. So for the time I was touring the house, I kept peering out the windows facing our car to make sure Damian wasn't going berserker. Note: we all survived – Fake hurricane Earl and real Hurricane Damian. (I keep mentioning the fake hurricane – it rained a lot and the wind blew, but was never as bad as it could have been.)
I should also note that your paid admission to the Old Lighthouse Museum gets you in here as well. I'm sure you can do them separately, but why would you? You want the whole Stonington history experience, don't you? You should. Especially if you're reading this.
Nerd.
Before going inside, let's take a moment to recognize the house as the National Historic Landmark that it is. Of course, I'm completing the list of our 60 NHL's – here's the somewhat redundant NHL report on the Palmer House.
This 16-room Victorian mansion was built in 1852 by two brothers, Captains Nathaniel Brown Palmer and Alexander Smith Palmer. Majestically sited on a high rise of ground overlooking the upper end of Stonington harbor, "Pine Point" offered sweeping water views in all directions. From its octagonal cupola, the family could identify ships arriving from distant ports. Meticulous craftsmanship of the ornate woodwork testifies to the work of shipwrights.
Indeed. It is a really nice house. It's one of the rare old houses I wouldn't mind living in. Which makes this next sentence from the Stonington Historical Society especially insane:
Once threatened by demolition, this magnificent home was purchased by the Stonington Historical Society in 1994 and is now preserved in all its beauty.
Now I'm not one of those preservationist nuts that never wants to tear down anything, but c'mon… Who in the world wanted to demolish this place? That's just crazy. The house remained in the Palmer family for four generations – over 125 years
By now you may be asking yourself who this Captain Palmer fellow was.
Nathaniel Palmer, born in 1799, and his brother Alexander, born seven years later, went to sea at an early age. Nathaniel became a ship's master before his 19th birthday as did Alexander by the time he was 21. They were young men who ventured into the most desolate regions of the world in search of profits from sealing and whaling.
During a voyage to the South Atlantic in 1820-21, Nathaniel, captain of the 47-foot sloop Hero, pressed southward in search of new seal rookeries. On November 17, 1820 he sighted "land not yet laid down on my chart." He had discovered a place which would later become known as Palmer Land on the continent of Antarctica.
That is the info from the museum's website. It's nuanced and contains much truthiness. But there's more. "Sealing" sounds much nice than "During the 1810s the skins of Antarctic Ocean seals were highly valued as items for trade with China. As a skilled and fearless seal hunter…" which is what Palmer was. I don't care, as this was 1810, not 2010. Seal hunting was just a job.
But the implication that Palmer was the first person to discover Antarctica is just not true. Palmer and his men became the first Americans and the third group of people to discover the Antarctic Peninsula. Larger ships skippered by Fabian Gottlieb von Bellingshausen and Edward Bransfield had reported sighting land earlier in 1820.
That's not to say this cat wasn't important. Heck, his name is all over the place in Antarctica and Mercury. Mercury? Yes, Mercury.
Palmer Land, part of the Antarctic Peninsula, as well as the Palmer Archipelago, were named in his honor. The Antarctic science and research program operated by the U.S. government continues to recall Palmer's role in the exploration of the Antarctic area. Palmer Station, located in the seal islands that Palmer explored, the clipper ship N.B. Palmer and the Antarctic icebreaker RV Nathaniel B. Palmer are named after Captain Palmer. Hero Bay, in the South Shetland Islands, is named for Captain Palmer's sloop Hero, one of the vessels of the Pendleton sealing fleet from Stonington which visited the islands in 1820-21. Also named after Palmer's sloop is Hero Rupes, an escarpment which was discovered in the southern hemisphere of the planet Mercury in 1973.
But the whole Antarctica thing was really just a part of who Palmer was. He was a top-notch sailor. He helped design the famous clipper ships of the 19th century and made improvements on sailing technology all the time. Once he was done with sailing, he bought shares in some of the biggest and fastest clipper ships and enjoyed the profits.
Let's explore the house already. Memorabilia pertaining to Nathaniel's discovery of Antarctica and the Palmer brothers' adventurous lives, as well as other Stonington family portraits, furnishings and artifacts are on display.
The first part of my tour took in the house set up as it would have been back in the 1800's. My pictures are pretty terrible, but you get the gist: This is a really well-crafted house with lots of little woodworking details that only top craftsmen like the Palmers could incorporate.
I don't know who incorporated the perfectly situated monkey wallpaper, but he or she is a genius. (I was told this wallpaper isn't original.)
I really liked how the house tour was set up. It seemed purposeful that the first few rooms downstairs (parlor, living room) and upstairs (various bedrooms) were more about the historic period and the fashions and furniture of the time. Then, the next few rooms slowly incorporate the Palmer story and finally back downstairs, there is a full-fledged Palmer exhibit in a few different rooms. Well done.
And well done on the creepy doll. With its shrunken head and alabaster pallor on its hands, they certainly upped the creepy doll ante.
In the upstairs hallway I read a pretty cool little anecdote about how Admiral Byrd visited the Palmer house to study Palmer's records of his trips south. Byrd hung out up here for days on end, studying Palmer's papers. I just learned Byrd's middle name was Evelyn, which makes me happy.
You know what else makes me happy? Cupolas. The Palmer house's was actually open and the stairs up to it are nuts. The views are nice and since the house is built on a small little hill in this sea-level town, you can see out to the bay easily enough. The Palmers used to hang up here to see who was coming into town.
Back downstairs and to some of the pure-Captain Palmer stuff. I know we've talked about the little ship he sailed to Antarctica on, but I'm not sure you fully grasped has crazy it was to sail down there on it.
The Hero was built in Groton in 1800 and was only 47 feet long, 16 feet wide and less than 7 feet deep. It had no galley, no head, one deck and one mast. The cargo deck cleared the water by a mere one foot! Captain Palmer and his crew of four sailed almost 20,000 miles on mountainous and frigid seas on their expedition in 1820. A rendering of the Hero and the historical society's portrait of Captain Palmer were used for a USPS stamp.
Palmer's study has been meticulously recreated here at the museum as well, which I rather enjoyed. In it, the society shows some of Ernest Shackleford artifacts from his Antarctic expeditions – which of course occurred a hundred years after Palmer's.
Oh, and if you're (justifiably) wondering what's up with my pants in the pictures, remember that we drove through a hurricane (sorta) and that storm brought impossibly hot and humid air up from the tropics. I was sweating… a lot.
This was a museum, not a fashion show.
Captain Nathaniel Palmer House
Filed Under: Museums Tagged With: Creepy Dolls, Firsts Onlies and Oldests, National Historic Landmarks, Nautical History, New London County, Stonington, stonington historical society
Ellington's Town Trails


text:
Niger Delta Stakeholders Demand Board For NDDC
admin August 26, 2022 No Comments
Stakeholders in the Niger Delta region have appealed to President Muhammadu Buhari to approve a board rather than a sole administrator for the Niger Delta Development Commission (NDDC).
The stakeholders, under the auspices of the Niger Delta Movement for Development (NDMD), made the appeal in a statement signed by its Publicity Secretary, Mr Ogbakiri Yanga, on Thursday in Yenagoa.
The NDMD members said that they would only accept the current management team led by Dr. Efiong Akwa as the commission's last interim administrator.
The stakeholders said their call became necessary because they recently "observed with dismay another selfish plot to make the NDDC an appendage of the Ministry of Niger Delta Affairs with full powers to dictate the direction of the commission."
The statement said the NDMD emergency meeting, where their resolution was taken, held in Oporoza, Delta and was attended by youth leaders and elders from the region.
Members of the NDMD resolved that they would not accept another sole administrator for the NDDC.
The NDMD maintained that any other change in the management of the NDDC should be the appointment of a board rather than a sole administrator, as directed by the Act establishing the commission.
"This is the least we expect from The Presidency and the Ministry of Niger Delta. Any announcement of another interim administrator other than the inauguration of a substantive board will be viewed by us as provocative and the highest level of insensitivity.
"We have stomached this outright neglect of our opinions and feelings for a long time because we believe that Dr. Efiong Akwa will be the last sole administrator of this commission.
"Any other change of management must be the inauguration of a board.
According to the body, any administration other than a substantive board will lead to a revolution of unimaginable proportion in the region, saying the entire Niger Delta was watching with keen interests and was prepared to rise to defend its common interests.
Members of the NDMD described as "utter wickedness" the plot by some few selfish individuals to want to institute and perpetuate the status quo.
"We want to believe that some selfish individuals are bent in totally damaging the image of President Muhammadu Buhari in the Niger Delta.
"Already, the way some of these individuals have handled the affairs of the NDDC has left much to be desired, " the statement said.
It said the people were quiet because they believed that the next change of leadership would be the announcement of a board.
"Don't taste the wrath of a lion by touching its tail. Any attempt to appoint another interim administrator will be likened to waking up a sleeping dog.
The group then urged Buhari to urgently constitute the board for the commission.
They also said it was unacceptable the moves in some quarters to reduce the powers of NDDC and make it subservient to the Ministry of the Niger Delta Commission.
The NDMD said it was keenly monitoring the situation, adding that the planned selfish move by these unnamed persons was not in the best interest of the region.
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"The 1970 Rule" will be the key criterion for museums contemplating the acquisition of the antiquities. Was the object known prior to 1970? Is there a recorded history?
A study of the year of acquisition for the items (now numbering well over 100) returned to Italy from Boston's Museum of Fine Arts, the Cleveland Museum of Art, the J. Paul Getty Museum (Malibu), New York's Metropolitan Museum of Art, and the Princeton University Art Museum shows that 14% of the items were acquired in the 1970s (including the Euphronios krater). (This excludes the pieces returned from Shelby White, the Royal-Athena Galleries, and the University of Virginia Art Museum in Charlottesville).
27% of the returns were acquired in the 1990s, well after the USA signed up to the 1970 UNESCO Convention in 1983. Indeed more than 85% of the acquisitions were made in the 1980s and 1990s, after the 1973 Resolution of the Archaeological Institute of America.
And it may make institutions such as the Michael C. Carlos Museum at Emory University rethink its collections management policy which hinges on the (now discredited) 1983 date.