---
abstract: 'We formulate a new scheme to study the combined magnetization and lattice dynamics in magnets, so-called magnetoelastics. The coupling between magnetization and lattice are considered through an expansion of electron-phonon coupling, while the magnetization is coupled to electrons through *sd*-like interaction. We show the that the time-scale of the magnetization dynamics due to coupling between magnetic degrees of freedom and electronic degrees of freedom can be transferred to lattice degrees of freedom and therefore can give rise to lattice dynamics on the same time-scale. This opens a new route to understand and treat ultrafast lattice dynamics induced by magnetization dynamics. We also show that all the parameters necessary to simulate this *coupled lattice and magnetization dynamics* can be obtained from first-principles.'
author:
- 'S. Bhattacharjee'
- 'J. Fransson'
title: Atomistic Coupling between Magnetization and Lattice Dynamics from First Principles
---

Magnetic phenomena at very fast time-scale is already a hot topic of both theoretical [@zhang2000; @munzenberg2010] as well as experimental research [@beaurepaire1996; @ultra; @johnson2011]. Magnetization dynamics at very fast scale is a primary requirement for efficient data storage since the writing the data in magnetic way involves the process of switching or magnetization reversal. Such memory devices such as magnetic random access memory (MRAM) is proposed to be a universal memory device [@akerman2005]. Similarly there are proposal as well as demonstrations of ferroelectric random access memory devices (FeRAM) in which electric polarization due to mainly ionic displacements is considered as a variable for the construction of a bit.

With the advances in the synthesis as well as theoretical understanding in multiferroic materials one goal of current research is aiming towards a possible memory device where advantages of FeRAM and MRAM will be combined [@scott; @bibes]. In the magnetic and ferroelectric case, bits are constructed from two states i.e, ($\bfM$, $-\bfM$) in magnetic case and ($\bfP$, $-\bfP$) in ferroelectric case ($\bfM$ and $\bfP$ being the magnetization and electric polarization, respectively). In contrast, multiferroics materials offer multiple states such as ($+\bfM$, $+\bfP$), ($+\bfM$, $-\bfP$), ($-\bfM$, $+\bfP$), and ($-\bfM$, $-\bfP$) can be considered. One can, moreover, write data electrically and read magnetically, thereby reducing the energy consumption.

It has been very difficult to realize dynamics in such systems since polarization dynamics in many materials is much slower compared to the magnetization dynamics. The time evolution of electric polarization is related to lattice dynamics which can be very slow (about nano seconds) compared to the very fast magnetization dynamics (in pico or even femto second scale). Recently, however, Walker *et. al* have demonstrated experimentally lattice dynamics in femto-second scale in the multiferroic material TbMnO$_3$ [@FemLatt], showing magnetic field induced ionic displacements. Here one should note that ferro-electricity in TbMnO$_3$ otherwise is not due to ionic displacement as in a proper ferroelectric like BaTiO$_3$. Coupling between lattice and magnetization has also been reported to have strong effects for iron based superconductors e.g. SmFeAsO$_{1-x}$F$_x$ and Ba$_{1-x}$K$_x$FeAs$_2$ [@liu2009], LaFeAsOF [@yildirim2009; @yundurain2011], and BaFe$_2$As$_2$ [@yildirim2009; @kim2012], and in spin-ladder systems e.g. BaFe$_2$Se$_3$ [@caron2011]. It is important to notice that those experiments open a path-way to manipulate the polarization dynamics through the magnetization dynamics in such materials on a very short time-scale.

We here develop a theoretical formalism which can be used to study ultrafast lattice dynamics in magnetic materials where magnetization is coupled to the ionic displacements. There are very few attempts in the literature where magnetization and lattice dynamics are studied as a coupled dynamical system. Ma *et. al* [@Ma] had developed a model for simulating molecular dynamics for magnetic material where a co-ordinate dependent exchange interaction between the spins determines the coupling between the spin and lattice degrees of freedom. The co-ordinate dependent exchange function is obtained by minimizing the expectation value of energy of a Stoner Hamiltonian with respect to magnitude of the magnetic moments. One should remember here that such minimization strongly neglects the spatial variation of charge density in the material and such approximation is valid only for very good metal where local charge neutrality is a natural situation. Recently Wang *et. al* [@bella] also have developed a scheme based on molecular dynamics, where ionic dynamics was obtained from Newton’s equation of motion while magnetization dynamics was obtained from stochastic Landau-Lifshitz-Gilbert (LLG) formalism [@ll; @gil]. The authors had studied complex electric and magnetic susceptibilities of multiferroic BiFeO$_3$ using such method.

The LLG formalism is a restricted approach to study the magnetization dynamics in the sense that the coupling between magnetic degrees of freedom with environmental ones are not considered in a proper way. The means of dissipation considered is only a single parameter: the Gilbert damping parameter. In recent developments [@inertia; @bnf] very fast dynamics have be demonstrated by coupling the magnetization with surroundings. Aiming at describing *magnetism dependent lattice dynamics*, we here show that one can obtain an effective coupling between the magnetization and lattice dynamics, by inclusion of charge-lattice coupling. We derive coupled equations for magnetization and lattice dynamics which can be implemented in existing first-principles codes, and all the parameters needed for the simulation can be obtained from the first principles. Thus, the purpose of this letter is therefore two-fold: (i) design a scheme to perform combined magnetization and lattice dynamics which goes beyonds methods based on Newtonian mechanics combined with classical LLG-equation, (ii) address ultrafast lattice dynamics observed recently in experiments. For the simulations in the microcanonical level (closed system), our method provides a direct way to study the energy transfer between different subsystems, such as magnetization, phonon and electrons. Also, our method can be used to study the recently observed multiglass behavior (both spin and dipole glass) in the some doped perovskites such as Mn doped SrTiO$_3$ [@kleeman2008].

Our model for the magnetic interactions begins from the assumption that the magnetization $\bfM(\bfr)$ interacts with the surrounding spin density $\bfs_s(\bfr')$ through the potential $v(\bfr)=-\int J(\bfr,\bfr')\bfsigma\cdot\bfs_s(\bfr')d\bfr'$. We, thus, write the interaction Hamiltonian $\Hamil_M=(1/2)\int\psi^\dagger(\bfr)v(\bfr)\psi(\bfr)d\bfr=-\int J(\bfr,\bfr')\bfM(\bfr)\cdot\bfs_s(\bfr')d\bfr d\bfr'$, where $\psi(\bfr)=(\psi_\up(\bfr)\ \psi_\down(\bfr))^T$ is a spinor such that we define $\bfM(\bfr)\equiv\psi^\dagger(\bfr)\bfsigma\psi(\bfr)/2$, whereas $J(\bfr,\bfr')=J(\bfr',\bfr)$.

The charge $n(\bfr)=\bfs_0(\bfr)\equiv\psi^\dagger(\bfr)\psi(\bfr)$ is subject to the potential $\phi(\bfr)=\int\phi(\bfr,\bfQ(\bfr'))d\bfr'$ due to electron-ion interactions, where $\bfQ(\bfr)$ is the ionic displacement from its equilibrium position. Here, we do not assign any specific nature of the displacement. However, in the perspective of multiferroic materials, for instance, one can think of $\bfQ(\bfr)$ representing a polar mode or a mode which corresponds to rotation of oxygen octahedra. For small displacements, we employ the expansion $\phi(\bfr,\bfQ(\bfr'))\approx\phi_0(\bfr)+\bfQ(\bfr')\cdot\nabla_{\bfr'}\phi_0(\bfr)$, where $\phi_0(\bfr)=\phi(\bfr,0)$, which gives the interaction between the charge and lattice vibrations $\Hamil_\text{ep}=\int\bfQ(\bfr')\cdot\nabla_{\bfr'}\phi_0(\bfr)n(\bfr)d\bfr d\bfr'$.

Given the general non-equilibrium conditions in the system, e.g. temporal fluctuations and currents, we define the action $$\begin{aligned}
\calS=&
    \oint(\Hamil_\text{M}+\Hamil_\text{MP})dt
    +\calS_\text{Z}
    +\calS_\text{WZWN}
    +\calS_\text{latt}\end{aligned}$$ on the Keldysh contour [@path1; @path2; @fransson2010]. Here, $\calS_\text{Z}=-\gamma\oint\int\bfB_\text{ext}(\bfr,t)\cdot\bfM(\bfr,t)d\bfr dt$ represents the Zeeman coupling to the external magnetic field $\bfB_\text{ext}$, whereas $\calS_{WZWN}=\int\oint\int_0^1\bfM(\bfr,t;\tau)\cdot[\partial_\tau\bfM(\bfr,t;\tau)\times\partial_t\bfM(\bfr,t;\tau)]d\tau dt|\bfM(\bfr)|^2d\bfr$ describes the Berry phase accumulated by the spin. The free lattice is represented by $\calS_\text{latt}=\oint\int\{[i\bfQ(\bfr,t)\cdot\dt\bfQ(\bfr,t)-\bfp_\bfr^2/2M_\ion]\delta(\bfr-\bfr')-\bfQ(\bfr,t)\cdot\bfV_{\bfr\bfr'}(t)\cdot\bfQ(\bfr',t)\}d\bfr d\bfr'dt$, with the ionic mass $M_\ion$ and the dyad $\bfV_{\bfr\bfr'}=\nabla_{\bfr}(\nabla_{\bfr'}V_0)$, and where $V_0$ is the ionic potential at equilibrium (vanishing displacements).

Our model for the coupled magnetization and lattice dynamics is obtained as a second order cumulant expansion of the partition function $\calZ[\bfM(\bfr,t),\bfQ(\bfr,t)]=\tr{\rm T_C}e^{i\calS}$, tracing out the electronic degrees of freedom. Thus, the effective action $\calS_\text{MP}$, which describes the magnetization and lattice dynamics, is given by (using $x=(\bfr,t)$) $$\begin{aligned}
\lefteqn{\calS_{MP}}
\nonumber\\=&
    -\frac{1}{2}\int
    \Bigl(
            \bfQ(x)\cdot\calD_{00}(x,x')\cdot\bfQ(x')               +\bfM(x)\cdot\calD_{s0}(x,x')\cdot\bfQ(x')  \nonumber\\&
            +\bfQ(x)\cdot\calD_{0s}(x,x')\cdot\bfM(x')              +\bfM(x)\cdot\calD_{ss}(x,x')\cdot\bfM(x')      \Bigr)
    dxdx', \end{aligned}$$ where we have defined $$\begin{aligned}
\calD_{pq}(x,x')=& 
    \int
        \Xi_p(\bfr,\bfrho)
        \bfK_{pq}(\bfrho,\bfrho';t,t)
        \Xi_q(\bfrho',\bfr')
    d\bfrho d\bfrho',
\nonumber\\&\end{aligned}$$ with the notation $\Xi_0(\bfr,\bfrho)=\nabla_\bfr\phi_0(\bfrho)$, $\Xi_s(\bfr,\bfrho)=-J(\bfr,\bfrho)$, and $\bfK_{pq}(\bfrho,\bfrho';t,t')=\eqgr{\bfs_p(\bfrho,t)}{\bfs_q(\bfrho,t)}$, $p,q=0,s$.

The effective action contains phonon-phonon, spin-phonon, and spin-spin interactions, represented by the first, second and third, and fourth terms, respectively. The phonon-phonon interaction account for anharmonic effects in the material and the interactions is mediated by electronic density-density correlations ($\calD_{00}$). The spin-phonon part is mediated by correlations between the spin and electron density ($\calD_{s0}$, $\calD_{0s}$). For collinear magnetic background, this interaction reduces to the correlations between the electronic density and the spin-polarization acting solely in the $z$-direction. For chiral magnetic structures, e.g. spin-orbit coupled systems, however, the interaction has finite components in all spatial directions. It is, hence, clear that the magnetic structure of the electronic background may have a significant influence on the mechanical subsystem. The electron mediated exchange interaction $\calD_{ss}$ was discussed in [@bnf].

It is worth mentioning that the magnetomechanical, or magnetoelastic, coupling not only depends on the background electronic structure, but also to large extent on the coupling $J$ between the local spins and the electronic structure, on the one hand, and the potential landscape $\phi(\bfr)$ and its gradient $\nabla\phi$ experienced by the charge, on the other hand. It is, thus, easily understood that materials with slowly varying potential $\phi$, such that $\nabla\phi\approx0$, are expected to have weak or negligible magnetomechanical coupling. This would be pertinent to materials with no or uniform displacement distribution, e.g. ferroelectric materials. This should be contrasted with magnetic materials in which a spatially fluctuating ionic potential leads to a non-negligible interaction parameter $\nabla\phi$ which, in turn, implies a significant magnetomechanical coupling, pertaining to e.g. TbMnO$_3$ [@FemLatt] and BaFe$_2$As$_2$ [@kim2012].

The equations of motion for the magnetization $\bfM$ and displacement $\bfQ$ are found by variation of the total action $\calS_T=\calS_{latt}+\calS_{MP}+\calS_{WZNW}+\calS_{ext}$ with respect to fast fluctuations, see e.g. Ref. [@bnf] for details. Requiring $\partial_t|\bfM(x)|^2=0$, we obtain

\[eq-eqm\] $$\begin{aligned}
\dot\bfM(x) 
    =& 
    \bfM(x)\times
    \biggl(
        -\gamma\bfB_\text{ext}(x) 
\nonumber\\& 
        +\int 
        \Bigl( 
            \calD_{s0}^r(x,x')\cdot\bfQ(x') 
            +\calD_{ss}^r(x,x')\cdot\bfM(x') 
        \Bigr)
        dx'
    \biggr), 
\\ 
M_\ion\ddot\bfQ(x)=& 
    \frac{1}{2} 
    \int
        \Bigl(
            \bfV_{\bfr\bfr'}\cdot\bfQ(\bfr',t)+\bfQ(\bfr',t)\cdot\bfV_{\bfr'\bfr} 
        \Bigr) 
    d\bfr'
\nonumber\\&
    +\frac{1}{2}
    \int 
    \biggl( 
        \calD_{0s}^r(x,x')\cdot\bfM(x')
        +\calD_{00}^r(x,x')\cdot\bfQ(x') 
    \biggr) 
    dx'. \end{aligned}$$

The equations for $\bfM$ and $\bfQ$ provide a general framework for coupled treatment of magnetization and lattice dynamics. It is legible that time evolution of both displacement and magnetization depend on surroundings in terms of space and time. Thus, Eq. (\[eq-eqm\]) emphasize that both $\bfQ$ and $\bfM$, in general, depend non-locally on both the magnetization and ionic displacements for the entire structure. This is in sharp contrast to previous exisiting methods which normally are based on molecular dynamics where $\bfQ$ and $\bfM$ are derived from effective total energy in the adiabatic approximation, i.e. instantaneous arrangements of magnetization and positions of ions. In this context it is interesting to observe that the time evolution of a local mode [@hewson2002], is non-locally influenced by the magnetization at different points in space and time. It can, moreover, be seen that the ionic dynamics can be controlled by external fields, e.g $\bfB_\text{ext}(\bfr,t)$, something that was experimentally demonstrated in the Ref. \[\].

The non-locality prescribed in the equations of motion for the magnetization and displacement, Eq. (\[eq-eqm\]), provides a firm foundation for a coupled dynamics of the two variables. For a deeper insight into the process that dominate the dynamics of the two variables, we here outline an approximation scheme which is essentially equivalent to the Landau-Lifshitz-Gilbert paradigm of magnetization dynamics. Thus, considering the magnetization and lattice dynamics to be slow compared to the electronic processes, we expand in time $\bfM(\bfr',t')\approx\bfM(\bfr',t)-\tau\dot\bfM(\bfr',t)+\tau^2\ddot\bfM(\bfr',t)/2$ and $\bfQ(\bfr',t')\approx\bfQ(\bfr,t)-\tau\bfQ(\bfr',t)+\tau^2\ddot\bfQ(\bfr',t)/2$, where $\tau=t-t'$. We define the internal magnetic field $\bfB_\text{int}(x)=-\int\calD_{ss}(x,x')\cdot\bfM(\bfr',t)dx'/\gamma-\int\calD_{s0}(x,x')\cdot\bfQ(\bfr',t)\dx'/\gamma$ such that the total magnetic field $\bfB(x)=\bfB_\text{ext}(x)+\bfB_\text{int}(x)$, and the internal force field $\bfF(x)=\int(\bfV_{\bfr\bfr'}\cdot\bfQ(\bfr',t)+\bfQ(\bfr',t)\cdot\bfV_{\bfr'\bfr})d\bfr'/2+\int[\calD^r_{0s}(x,x')\cdot\bfM(\bfr',t)+\calD^r_{00}(x,x')\cdot\bfQ(\bfr',t)]dx'/2$. Then, we can write the resulting equations of motion for $\bfM$ and $\bfQ$ as

\[eq-llg\] $$\begin{aligned}
\dot\bfM(x)=&
    \bfM(x)\times
    \biggl(
        -\gamma\bfB(x)
        +
        \int
            [
                \hat\bfG_{ss}(x,\bfr')\cdot\dot\bfM(\bfr',t)
                +\hat\bfI_{ss}(x,\bfr')\cdot\ddot\bfM(\bfr',t)
            +\hat\bfG_{s0}(x,\bfr')\cdot\dot\bfQ(\bfr',t)
            +\hat\bfI_{s0}(x,\bfr')\cdot\ddot\bfQ(\bfr',t)
            ]
        d\bfr'
    \biggr)
\\
M_\ion\ddot\bfQ(x)=&
    \bfF(x)
    +
    \frac{1}{2}
    \int
        [
            \hat\bfG_{00}(x,\bfr')\cdot\dot\bfQ(\bfr',t)
            +\hat\bfI_{00}(x,\bfr')\cdot\ddot\bfQ(\bfr',t)
            +\hat\bfG_{0s}(x,\bfr')\cdot\dot\bfM(\bfr',t)
            +\hat\bfI_{0s}(x,\bfr')\cdot\ddot\bfM(\bfr',t)
        ]
    d\bfr'
    ,\end{aligned}$$

In Eq. (\[eq-llg\]) we have introduced the intrinsic magnetic (mechanical) damping and moment of inertia tensors $\hat\bfG_{ss(00)}(x,\bfr')=-\int\tau\calD^r_{ss(00)}(x,x')dt'$ and $\hat\bfI_{ss(00)}(x,\bfr')=\int\tau^2\calD^r_{ss(00)}(x,x')dx/2$, respectively, which appear also in any uncoupled description of the magnetization and lattice dynamics. In addition to those intrinsic interactions, Eq. (\[eq-llg\]) also contain corresponding magneto-mechanical damping and moment of inertia tensors $\hat\bfG_{pq}(x,\bfr')=-\int\tau\calD^r_{pq}(x,x')dx'$ and $\hat\bfI_{pq}(x,\bfr')=\int\tau^2\calD^r_{pq}(x,x')dx/2$, thus, showing that the magnetic (mechanical) subsystem has a strong influence on the mechanical (magnetic) one. It, moreover, suggests that important time-scales for the two subsystems are governed by analogous electronic processes, since the propagators $\bfK_{pq}(x,x')$ all have similar dependencies on the electronic structure. We can, therefore, infer that the dynamical processes of the magnetic and lattice degrees of freedom occur on the same time-scales, from which we can also understand that dynamical variations in the lattice (magnetic) subsystem transfer to the magnetic (lattice) subsystem without significant delay. We expect that this conclusion will be of particular importance for understanding the magneto-mechanical dynamics that has been observed in e.g. multiferroic and superconducting materials.

Next we describe how the parameters in Eq. (\[eq-llg\]) can be calculated from a first principles point of view. The conditions defined by the DFT system, the time locality of $\calD^r_{pq}$ enables the Fourier transform $\calD^r(\bfr,\bfr';\dote{})=\int\calD(x,x')e^{i\dote{}\tau}dx'$. Thus, we can write the damping and inertial tensors in terms $\lim_{\dote{}\rightarrow0}i\partial_{\dote{}}\calD^r_{pq}(\bfr,\bfr';\dote{})=\int\tau\calD^r_{pq}(x,x')dt'$ and $\lim_{\dote{}\rightarrow0}\partial^2_{\dote{}}\int\tau^2\calD^r_{pq}(x,x')dt'$, where $$\begin{aligned}
\calD_{pq}^r(\bfr,\bfr';\dote{})=&
    4{\rm sp}
    \int
        \Xi_p(\bfr,\bfrho)
        \bfsigma^p
        \im\bfG^r_{\bfrho'\bfrho}(\omega)
        \bfsigma^q
        \im\bfG^r_{\bfrho\bfrho'}(\omega')
\nonumber\\&\times
        \Xi_q(\bfrho',\bfr')
        \frac{f(\omega')-f(\omega)}{\dote{}-\omega+\omega'+i\delta}
    \frac{d\omega}{2\pi}
    \frac{d\omega'}{2\pi}
    d\bfrho
    d\bfrho',
\label{eq-genX}\end{aligned}$$ where $\bfG^r_{\bfr\bfr'}(\omega)$ is the retarded Green function for the background electronic structure represented as a $2\times2$ matrix in spin space over which the trace ${\rm sp}$ is taken. Here also $p,q=0,s$ using $\bfsigma^0=\sigma^0$ ($2\times2$ identity) and $\bfsigma^s=\bfsigma$. We notice that this expression presents a generalized form of exchange interaction in both magnetic and lattice subsystems as well as between the two of them. Using Kramer’s-Krönig relations in the limit $\dote{}\rightarrow0$ we can write this generalized exchange interaction as $$\begin{aligned}
\calD^r_{pq}(\bfr,\bfr')=&
    \frac{1}{\pi}{\rm sp}\, 
    \im
    \int
        f(\omega)
        \Xi_p(\bfr,\bfrho)
        \bfsigma^p
        \bfG^r_{\bfrho'\bfrho}(\omega)
\nonumber\\&\times
        \bfsigma^q
        \bfG^r_{\bfrho\bfrho'}(\omega')
        \Xi_q(\bfrho',\bfr')
    d\omega
    d\bfrho
    d\bfrho'
\label{eq-genX0}\end{aligned}$$ in accordance with Ref. \[\] for the purely magnetic case.

The generalized damping tensor can be written as $$\begin{aligned}
\hat\bfG_{pq}(\bfr,\bfr')=&
    \frac{1}{\pi}
    {\rm sp}
    \int
        f'(\omega)
        \Xi_p(\bfr,\bfrho)
        \bfsigma^p
        \im\bfG^r_{\bfrho'\bfrho}(\omega)
\nonumber\\&\times
        \bfsigma^q
        \im\bfG^r_{\bfrho\bfrho'}(\omega)
        \Xi_q(\bfrho',\bfr')
    d\omega
    d\bfrho
    d\bfrho'
    .
\label{eq-genG}\end{aligned}$$ In this form, it is easy to see that the main influence from the electronic structure on both the magnetic and mechanical damping, as well as the magneto-mechanical damping is generated by the properties at the Fermi surface. In the same fashion we can write the generalized moment of inertia tensor according to $$\begin{aligned}
\hat\bfI_{pq}(\bfr,\bfr')=&
    {\rm sp}
    \int
        f(\omega)
        \Xi_p(\bfr,\bfrho)
        \Bigl[
            \bfsigma^p
            \im\bfG^r_{\bfrho'\bfrho}(\omega)
\nonumber\\&\times
            \bfsigma^q
            \partial^2_\omega\re\bfG^r_{\bfrho\bfrho'}(\omega)
            +
            \bfsigma^q
            \im\bfG^r_{\bfrho\bfrho'}(\omega)
\nonumber\\&\times
            \bfsigma^p
            \partial^2_\omega\re\bfG^r_{\bfrho'\bfrho}(\omega)
        \Bigr]
        \Xi_q(\bfrho',\bfr')
    \frac{d\omega}{2\pi}
    d\bfrho
    d\bfrho'
    ,
\label{eq-genI}\end{aligned}$$ showing that the moment of inertia depends on the occupied portion of the electronic structure. The derivatives of the GFs illustrate that the moment of inertia is very sensitive to sharp variations in the electronic structure. We, therefore, expect that the moment of inertia may have significant effect in narrow band magnetic materials, e.g. strongly correlated electron systems. Moreover, in a similar fashion as the magnetic order may strongly influence the generalized exchange it may also have a strong influence on the generalized moment of inertia. The coupling between the magnetic and mechanical subsystems indicates that the magnetic order also has a significant influence on the mechanical systems. In an analogous way we, furthermore, expect the lattice ordering to have a signifiant effect both on the mechanical as well as on the magnetic dynamics.

Despite the apparent similarities of Eqs. (\[eq-genX\]) | (\[eq-genI\]) with the corresponding equations provided in [@bnf], the purpose of presenting them here is to show that not only the magnetic interactions, particularly the exchange and damping, can be given in a form of Kubo formula, but that this also holds for the mechanical and magneto-mechanical interactions. This is very appealing from the perspective of the fluctuation-dissipation theorem as it demonstrates that the coupling between the magnetic (mechanical) fluctuations and mechanical (magnetic) dissipation, can be treated on the same footing as the pure magnetic and mechanical correspondences. This very clearly illustrates the important role of the charge density mediated interactions, both for the magnetic and mechanical systems, as well as the interactions between them.

The above two coupled equations, Eq. (\[eq-eqm\]) or Eq. (\[eq-llg\]), provide a tool for simulating the dynamics of a closed system composed of electron, magnetization, and phonon subsystems and various time-resolved quantities such as spin-spin or electric dipole-dipole autocorrelation functions can obtained easily. Also, one very interesting feature of our formulation is that, this in principle should work for the cases where spatial distribution of charge density is quite non-uniform. We have explicitly considered the charge densities in terms of the correlation function $\bfK$ in our formulation. Further extension of this work would be to develop a scheme for controlling the temperature of the system during the course simulation, which is beyond the scope of present study.

J.F. thanks J. T. Haraldsen for fruitful discussions, and the Swedish Research Council, the Wenner-Gren foundation, and Los Alamos National Laboratory for support.

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