---
abstract: 'We derive the quantum spin hydrodynamic equations in a ferromagnetic metal. From these equations we show the existence of a new massive spin-current mode. This mode can be observed in neutron scattering experiments and we discuss the difficulties in seeing it. At the end we discuss the existence of this mode in localized ferromagnets.'
address:
- 'Department of Physics, Boston College, Chestnut Hill, MA 02167'
- 'Department of Radiology, Brigham and Women's Hospital and Harvard Medical School, 75 Francis St., Boston MA 02115'
author:
- 'Kevin S. Bedell'
- 'Krastan B. Blagoev'
title: 'Quantum spin hydrodynamics and a new spin-current mode in ferromagnetic metals'
---

\[@twocolumnfalse

\] Propagating collective modes in condensed matter systems are often a consequence of spontaneously broken symmetry. In ferromagnetic systems, the spin rotational symmetry is spontaneously broken leading to the existence of spin waves. They were first observed in iron using neutron scattering experiments in the nineteen martinezwilliam@example.net. Theoretically spin waves were predicted@Bloch1930 [@Slater1930] in lattice models with a ferromagnetic ground state as well as in ferromagnetic Fermi liquids@Abrikosov [@and; @Dzyaloshinskii1959]. In paramagnetic metals, spin waves propagate in an external magnetic martinezwilliam@example.net. Not all propagating modes are a consequence of spontaneously broken symmetry and an example is zero sound in liquid martinezwilliam@example.net.

In this paper we will demonstrate the existence of a new collective spin mode in ferromagnetic metals. We start from a quasi-classical kinetic equation, the Landau kinetic equation, for the momentum distribution function in a ferromagnetic metal. From this we can derive the dynamic equations for the evolution of the spin density and spin current. In the low temperature limit the spin dynamics is collisionless and dominated by spin precession, and we will call this the quantum spin hydrodynamic (QSH) regime. At higher temperature this dynamics crosses over to the classical spin hydrodynamics (CSH) of Halperin and Hohenberg@Halperin [@and; @Hohenberg1969]. In the QSH regime we show that a new massive mode exists along with the usual magnon. While gaped, it exists outside of the particle-hole, Stoner, continuum and therefore it can propagate. From a macroscopic point of view, this mode is induced by the collective oscillations of the spin current and it should be observable in neutron scattering experiments.

In the long wave length limit one can derive the existence of the spin waves from the conservation law@Halperin [@and; @Hohenberg1969] for the magnetization in the CSH limit. However, the spin current is not a conserved quantity and therefore one cannot derive propagating modes related to the fluctuations of the spin current from the conservation laws. Here the Landau kinetic equation first used by Abrikosov and Dzyaloshinksii@Abrikosov [@and; @Dzyaloshinskii1959] for weak ferromagnetic metals is used to derive the full QSH theory.

We consider a three dimensional, weak itinerant ferromagnet below its Curie temperature. To avoid complications we ignore the effects related to the lattice. The phenomena which we will describe is independent of the lattice, although lattice effects can add more features to the ones that we study. We assume that the itinerant ferromagnet is well described by ferromagnetic Fermi liquid theory@Abrikosov [@and; @Dzyaloshinskii1959; @Abrikosov1986]. The kinetic equations for the magnetization and the spin current can be derived from the equation for the Fourier transform of the Wigner distribution function $n_{\vec{p}}^{\alpha \beta }(\vec{r},t)$@Silin1957 $$\frac{\partial n_{\vec{p}}^{\alpha \beta }(\vec{r},t)}{\partial t}-
\frac 1{i\hbar }F_{\vec{p}}^{\alpha \beta }[n;\epsilon ]=I_Q[n_{\vec{p}^{\prime }}],$$ where $$\begin{aligned}
F_{\vec{p}}^{\alpha \beta }[n;\epsilon ] &=&\sum_\gamma \int Dp^{\prime
}Dqd^3r^{\prime }d^3\rho e^{i\left[ (\vec{p}-\vec{p}^{\prime })\cdot 
\vec{\rho}+\vec{q}\cdot (\vec{r}-\vec{r}^{\prime })\right] }  \nonumber \\
&&\times (\epsilon _{\vec{p}^{\prime }+\hbar \vec{q}/2}^{\alpha \gamma }
(\vec{r}^{\prime }+\frac{\hbar \vec{\rho}}2,t)n_{\vec{p}^{\prime }}^{\gamma
\beta }(\vec{r}^{\prime },t)  \nonumber \\
&&-n_{\vec{p}^{\prime }}^{\alpha \gamma }(\vec{r}^{\prime },t)
\epsilon _{\vec{p}^{\prime }-\hbar \vec{q}/2}^{\gamma \beta }(\vec{r}^{\prime }-
\frac{\hbar \vec{\rho}}2,t)),\end{aligned}$$ $$n_{\vec{p}}^{\alpha \beta }(\vec{r},t)=
\int Dqe^{i\vec{q}\cdot \vec{r}}\left\langle a_{\vec{p}-\vec{q}/2,\alpha }^{\dag }
a_{\vec{p}+\vec{q}/2,\beta }\right\rangle ,$$ with $a_{\vec{p}+\vec{q}/2,\beta }$ the single fermion operator for a particle with spin $\beta $. Here $Dq=\frac{d^3q}{(2\pi \hbar )^3}$ and $\epsilon _{\vec{p}}^{\alpha \beta }(\vec{r},t)$ is the effective quasiparticle Hamiltonian. Also $I_Q$ is the quantum collision integral which contains contributions from two body collisions and from the rate of change due to the spin precession@Baym [@and; @Pethick1991].

Following Ref.@Baym [@and; @Pethick1991] we expand all single body operators in the basis set spanned by the Pauli matrices $$\epsilon _{\vec{p}}^{\alpha \beta }(\vec{r},t)=
\epsilon _{\vec{p}}\delta^{\alpha \beta }(\vec{r},t)+
\vec{h}_{\vec{p}}(\vec{r},t)\cdot \vec{\tau}^{\alpha \beta },$$ $$n_{\vec{p}}^{\alpha \beta }(\vec{r},t)=
n_{\vec{p}}\delta ^{\alpha \beta }(\vec{r},t)+
\vec{\sigma}_{\vec{p}}(\vec{r},t)\cdot \vec{\tau}^{\alpha \beta }$$ at every point $(\vec{r},t)$ of space and time with $\vec{\sigma}_{\vec{p}}$ the magnetization and $$\vec{h}_{\vec{p}}=-\gamma \frac \hbar 2\vec{{\cal {H}}}+2\int Dp^{\prime }
f_{\vec{p}\vec{p}^{\prime }}^a\vec{\sigma}_{p^{\prime }}$$ the effective magnetic field which is a sum of an external magnetic field ${\cal \vec{H}}$ and an internal magnetic field generated by the quasiparticle interactions $f^a_{\vec{p}\vec{p}^{\prime }}$. We expand the quasiparticle interactions in spherical harmonics and separate the spin symmetric ($f_l^s$) and spin antisymmetric ($f_l^a$) parts. The result is $$N(0)f_{\vec{p}\vec{p}^{\prime }}^{{\bf \tau \tau }^{\prime
}}=\sum_l(F_l^s+F_l^a\vec{\tau}\cdot \vec{\tau }^{\prime })P_l( 
\hat{p}\cdot \hat{p}^{\prime }),$$ with $$f_{\vec{p}\vec{p}^{\prime }}^{\tau \tau ^{\prime }}
\approx f_{\vec{p}\vec{p}^{\prime}}^s+
f_{\vec{p}\vec{p}^{\prime }}^a\vec{\tau}\cdot \vec{\tau}^{\prime }.$$ Here we treat the quasiparticle interaction as spin rotation invariant. This is possible in the case of small magnetizations and external magnetic fields as explained in@Blagoev [@et; @al.1999]. Here $N(0)$ is the average density of states, averaged over the two Fermi surfaces. We must note that the condition that the ground state of the metal is ferromagnetic means that $N(0)f_0^a=F_0^a<-1$.

For an inhomogeneous system the spin quantization axis is position dependent and therefore the spin density $$\vec{\sigma}(\vec{r},t)=2\int Dp\vec{\sigma}_{\vec{p}}(\vec{r},t)$$ is position dependent. From the kinetic equation one can obtain the continuity equation for the spin density which reduces to the usual spin conservation law $$\frac{\partial \vec{\sigma}(\vec{r},t)}{\partial t}+\frac \partial {\partial
x_i}j_{\vec{\sigma},i}(\vec{r},t)=
\gamma \vec{\sigma}(\vec{r},t)\times {\cal \vec{H}},$$ where the spin current is $$j_{\vec{\sigma},i}(\vec{r},t)=
2\int Dp\left[ \frac{\partial 
\epsilon _{\vec{p}}}{\partial p_i}\vec{\sigma}_{\vec{p}}(\vec{r},t)+
\frac{\partial \vec{h}_{\vec{p}}}{\partial p_i}n_{\vec{p}}(\vec{r},t)\right]$$ and a summation over repeated indices is used. The precession term in the $r.h.s.$ of Eq.(10) follows from the non-commutativity of the particle density operator and the quasiparticle Hamiltonian. Assuming that the magnetization is small and that we are close to equilibrium we can linearize the spin current which leads to $$j_{\vec{\sigma},i}(\vec{r},t)=2\int Dp v_{\vec{p}_i}\vec{\sigma}_{\vec{p}}
\left(1+\frac{F_1^a}{3}\right),$$ because in the Fermi liquid only momenta in the vicinity of the Fermi surface contributes to the magnetization. Here $\vec{v}_p=\vec{\nabla}_p\epsilon_p^0$, $n_p^0$ is the equilibrium distribution function, and $F_1^a$ is a Fermi liquid parameter.

Starting from the kinetic equation for the spin density distribution function@Baym [@and; @Pethick1991] $$\begin{aligned}
\frac{\partial \vec{\sigma}_{\vec{p}}}{\partial t}+
v_{\vec{p}_i}\frac{\partial}{\partial r_i}
\left(\vec{\sigma}_{\vec{p}}-
\frac{\partial n_p^0}{\partial \epsilon_p}\vec{h}_{\vec{p}}\right)&=& \nonumber \\
-\frac{2}{\hbar}\vec{\sigma}_{\vec{p}}\times\vec{h}_{\vec{p}}+
\left(\frac{\partial\vec{\sigma}_{\vec{p}}}{\partial t}\right)_{coll.}\end{aligned}$$ then multiplying by the quasiparticle velocity and integrating over the quasiparticle momentum the linearized equation for the spin current takes the form $$\frac \partial {\partial t}j_{\vec{\sigma},i}(\vec{r},t)+
\frac \partial {\partial x_m}\Pi _{\vec{\sigma}im}=
\left[ \frac \partial {\partial t}j_{\vec{\sigma},i}\right] _{prec.}+
\left[ \frac \partial {\partial t}j_{\vec{\sigma},i}\right] _{coll.},$$ where the spin stress tensor is $$\Pi _{\vec{\sigma}im}=2\left( 1+\frac{F_1^a}3\right) \int
Dpv_{p_i}v_{p_m}\left( \vec{\sigma}_{\vec{p}}-
\frac{\partial n_{\vec{p}}^0}{\partial \epsilon _p}\vec{h}_{\vec{p}}\right).$$

Next we apply a small, transverse (to the external field and therefore to the magnetization) magnetic field $\delta\vec{{\cal H}}=\delta {\cal H}_x\hat{e}_x+\delta {\cal H}_y\hat{e}_y$ which induces a change in the magnetization $\delta \vec{\sigma}_{\vec{p}}=
\delta \sigma _{\vec{p}x}\hat{e}_x+\delta \sigma _{\vec{p}y}\hat{e}_y$, where $\hat{e}_i$ are the unit vectors and $\delta \vec{\sigma}_{\vec{p}}=\vec{\sigma}_{\vec{p}}-\vec{\sigma}_{\vec{p}}^0$. To make the expansion in spherical harmonics more transparent we define $$\delta \sigma^{\pm }_{\vec{p}}=
-\frac{\partial n_{\vec{p}}^0}{\partial\epsilon_p^0}\nu_{\vec{p}}^{\pm}$$ and $$\sigma^0_{\vec{p}}=-\frac{\partial n_{\vec{p}}^0}{\partial\epsilon_p^0}\frac{m}{N(0)},$$ where $\delta \sigma^{\pm }_{\vec{p}}=\delta\sigma _{\vec{p}x}\pm i\delta \sigma _{\vec{p}y}$, $\sigma^0_{\vec{p}}$ is the magnitude of the magnetization in the absence of the perturbing field, and $m$ is the equilibrium magnetization. We expand in spherical harmonics the quantities $$\nu_{\vec{p}}^{\pm}=\sum_l\nu_l^{\pm}P_l(\hat{p}\cdot \hat{k})$$ and substitute the corresponding expressions in Eq.(13). Here $\hat{k}$ is the unit vector in the direction of the spin quantization axis. Now Eq.(13) involves terms grouped according to the value of the integer number $l$. Next we neglect the terms with $l>1$. Although the term $l=2$ is very important as we discuss later, the derivation which we present below is clearer and contains all the steps needed for the derivation of the dispersion including higher spherical harmonics. Then using the expansion in spherical harmonics, Eq.(18), and in the presence of an external magnetic field, Eq.(14) reduces to $$\begin{aligned}
&&\frac \partial {\partial t}\vec{j}_{\vec{\sigma},i}+
\left( 1+\frac{F_1^a}3\right) \left| 1+F_0^a\right| \frac{v_F^2}3
\frac \partial {\partial x_i}\vec{\sigma}(\vec{r},t)  \nonumber \\
&=&\gamma \vec{j}_{\vec{\sigma},i}\times {\cal H-}\frac 2\hbar 
\left( f_0^a-\frac{f_1^a}3\right) 
\vec{j}_{\vec{\sigma},i}\times \vec{\sigma}(\vec{r},t)-\left( 1+
\frac{F_1^a}3\right) \frac{\vec{j}_{\vec{\sigma},i}}{\tau _D},  \nonumber \\
&&\end{aligned}$$ Here we have used a relaxation time approximation for the collision contribution to the spin current.

Before we examine the full spin dynamics coming from Eqs.(10) and (19) it is useful to contrast this with the behavior of a paramagnetic metal. In a paramagnetic Fermi liquid the first two terms on the $r.h.s.$ of Eq.(19) are proportional to $\omega_L=2\gamma\cal H$, where $\cal H$ is the external magnetic field. At high temperature the CSH regime is realized and at low temperature the system crosses over to the precession dominated regime. The cross over scale is set by $\omega_L\tau_D\approx1$. This behavior is similar to that in paramagnetic metals@Baym [@and; @Pethick1991; @Leggett; @and; @Rice1968]. In the ferromagnetic metal the cross over scale is nearly independent of the external magnetic field. To see this we first note that at high temperature, Eq.(19) in the uniform case reduces to $$\frac \partial {\partial t}\vec{j}_{\vec{\sigma},i}=
-\left( 1+\frac{F_1^a}3\right) \frac{\vec{j}_{\vec{\sigma},i}}{\tau _D}$$ which means that $\vec{j}_{\vec{\sigma},i}$ exponentially decays and its explicit time dependence can be ignored. In this limit, which is called the classical spin hydrodynamic limit with $\frac{\partial \vec{j}}{\partial t}=0$ the equations reduce to those used by Halperin and Hohenberg@Halperin [@and; @Hohenberg1969] and it is easy to derive the dispersion, $\omega \sim q^2$ for the spin waves@Abrikosov [@and; @Dzyaloshinskii1959]. The quadratic dependence for the Goldstone mode follows from the fact that the order parameter $m$ is a conserved charge of the spin rotation algebra. As the system crosses over to low temperature it enters into the QSH regime in which the spin current changes in time due to the precession around the effective field with $\omega_1^+=2m(f_1^a/3-f_0^a)$. For zero external field the cross over from QSH to CSH occurs when $\omega_1^+\tau_D\approx1$. The range of temperatures where $\omega_1^+\tau_D>>1$ is what we have referred to as the QSH regime. In this QSH regime we can not ignore the $\frac \partial {\partial t}\vec{j}_{\vec{\sigma},i}$ term in Eq.(19). This as we will see is the source of the new spin-current mode.

There is another way to see the difference between spin waves in a ferromagnetic and a paramagnetic metal. Consider a ferromagnetic metal in an external magnetic field. The magnetization of the metal will align with the external field. If we apply a small pulse in a direction not collinear with the external magnetic field, the magnetization will start to precess around the external field. Next we reduce to zero the external magnetic field. In a paramagnetic metal the polarization will also go to zero and the spin wave will become damped. However, in a ferromagnetic metal the magnetization will continue to precess about the same axis, although no external magnetic field is present to define that axis. Now one can imagine looking at the system from a frame of reference rotating with the magnetization. In this frame of reference there will be no magnetization precession, but there will be a precession of the spin current, because the spin current is not a conserved quantity and as seen from Eqn.(19) it precesses with the frequency $\omega_1^+$.

It is instructive to look first at the modes of the system in the simpler case when $F_l^a=0$, for $l>1$ and the external field is ${\cal H}=0$. In that case the equations simplify to $$\frac{\partial \delta \vec{\sigma}(\vec{r},t)}{\partial t}+
\frac \partial {\partial x_i}j_{\vec{\sigma},i}(\vec{r},t)=0,$$ $$\begin{aligned}
\frac \partial {\partial t}\vec{j}_{\vec{\sigma},i}+
c_s^2\frac \partial {\partial x_i}\delta \vec{\sigma}
&&=-\frac 2\hbar (f_0^a-\frac{f_1^a}{3})\vec{j}_{\vec{\sigma},i}\times \left( \vec{\sigma}^0+
\delta \vec{\sigma}\right) \nonumber \\
&&-(1+\frac{F_1^a}{3})\frac{\vec{j}_{\vec{\sigma},i}}{\tau _D},\end{aligned}$$ where $c_s^2=\left| 1+F_0^a\right| (1+\frac{F_1^a}3)\frac{v_F^2}3$ is the spin wave velocity, with $v_F$ the Fermi velocity. Here we have introduced $\delta \vec{\sigma}=\vec{\sigma}(\vec{r},t)-\vec{\sigma}^0$, with $\vec{\sigma}^0=m\hat{k}$, where $m$ is the magnetization. If the system is in the QSH regime and $\delta \vec{\sigma}\ll \vec{\sigma}^0$ one can write for the spin current the following hydrodynamic equation $$\frac \partial {\partial t}\vec{j}_{\vec{\sigma},i}+
c_s^2\frac \partial {\partial x_i}\delta \vec{\sigma}=
{\cal -}\frac 2\hbar (f_0^a-\frac{f_1^a}{3})\vec{j}_{\vec{\sigma},i}\times \vec{\sigma}^0.$$ Differentiating this equation with respect to $x_i$ and using the equation for the spin density we obtain $$\frac{\partial ^2}{\partial t^2}\delta \vec{\sigma}+
c_s^2\frac{\partial ^2}{\partial x_i^2}\delta \vec{\sigma}+
\frac 2\hbar (f_0^a-\frac{f_1^a}{3})
\frac{\partial \delta \vec{\sigma}}{\partial t}\times \vec{\sigma}^0=0.$$ Below we will set $\hbar =1$. Applying again a transverse magnetic field as above Eq.(16) we can write $$\frac{\partial ^2\delta \sigma^{+}}{\partial t^2}+
c_s^2\frac{\partial^2\delta \sigma^{+}}{\partial x_i^2}-
2im(f_0^a-\frac{f_1^a}{3})\frac{\partial \delta \sigma^{+}}{\partial t}=0.$$ Assuming that the solution is of the form $$\delta \sigma^{+}=m^{+}e^{i(\vec{q}\cdot \vec{r}-\omega t)}$$ with $\vec{q}=q\hat{e}_x$ we obtain the dispersion equation $$\omega ^2+c_s^2q^2+2m(f_0^a-\frac{f_1^a}{3})\omega =0.$$ The two modes have the dispersion $$\omega _l=\left| m(f_0^a-\frac{f_1^a}{3})\right| 
\left( 1\pm \sqrt{1-\frac{c_s^2q^2}
{\left|m(f_0^a-\frac{f_1^a}{3})\right| ^2}}\right) .$$ where the $l=0(+)$ solution corresponds to the spin waves and the $l=1(-)$ solution corresponds to a new collective spin mode. For small $q$ we have the usual Goldstone mode $$\omega _0(q)=\frac{c_s^2}{\omega_1^{+}}q^2$$ and the new mode has the dispersion $$\omega _1(q)=\omega _1^{+}-\frac{c_s^2}{\omega _1^{+}}q^2.$$ Under close examination one finds that to the $q^2$ term there is a contribution from the $l=2$ spherical harmonics in Eq.(18). Taking that into account gives the full expression for this mode $$\omega _1(q)=\omega _1^{+}-\left[ \frac{c_s^2}{\omega _1^{+}}+
\frac{4N(0)v_F^2}{30m}\left(\frac{3}{F_1^a}+1\right)\right] q^2.$$ The above equation is valid for $qv_F<<m|f^a_1|$. In the limit $f_1^a\rightarrow0$ the mode merges with the Stoner continuum and is Landau damped. For $f_1^a\neq0$ the mode propagates with a dispersion that depends on the sign of $f_1^a$.

It is useful to look at the contribution of this mode to the $f$-sum rule. It is known that the spin waves do not satisfy this sum rule. However the spin waves plus the new mode exhaust the $f$-sum rule and the spectral weight is shifted from the Stoner excitations to this new mode. To see this we first introduce the dynamic structure function $$S^{\tau\tau^{\prime}}(\vec{q},\omega )=-\frac 1\pi 
\mathop{\rm Im}
\chi ^{\tau\tau^{\prime}}(\vec{q},\omega )\theta (\omega ),$$ where $\chi (\vec{q},\omega )$ is the dynamic spin susceptibility obtained from the kinetic equation by noting that $\chi=\frac{\delta\sigma}{\delta{\cal H}}$. Here $\theta(\omega )$ is the step function. From the expression for the dynamic structure function, calculated from the kinetic equation Eq.(13), we can derive the f-sum rule for the transverse dynamic spin response function, $S^{+-}(\vec{q},\omega)$. This is given by $$\int_0^\infty d\omega \omega S^{+-}(\vec{q},\omega )=
(1+\frac{F_1^a}3)\frac{nq^2}{2m^{*}},$$ where $n$ is the total density and $m^*$ is the effective mass. This has the same form as the one obtained for the spin dynamic structure function in the paramagnetic phase@Baym [@and; @Pethick1991; @Leggett1966].

This sum rule is not exhausted by the Goldstone mode alone. If we now include the new mode at zero temperature we have $$S^{+-}(\vec{q},\omega )=\alpha_{\vec q}^{+-}\delta (\omega -\omega _0(q))
+\beta_{\vec q}^{+-}\delta (\omega -\omega_1(q))$$ and this exhausts the $f$-sum rule, Eq.(33). This is significant, because the spectral weight of the Stoner excitations has been transfered to the two modes, leaving $$\int_0^\infty d\omega \omega S_{Stoner}^{+-}(\vec{q},\omega )\propto q^4.$$ This might be a plausible explanation for the difficulty of the direct observation of the Stoner excitations in a neutron scattering experiments@Mook1988 in Fe and Ni since the oscillator strength of these single excitations has been reduced.

The observation of the new mode described in this paper is difficult because it has a small spectral weight at small $q$. This follows from the $q^2$ dependence of $\beta_{\vec q}^{+-}$ term in Eq.(34). Another obstacle in observing the mode is that it originates from the oscillations of the spin current, which is not a conserved quantity and therefore it is easily damped.

In conclusion, in this paper we described a new collective mode in weak ferromagnetic metals in the quantum spin hydrodynamic regime. While our calculations are for small moment itinerant ferromagnets this mode will exist in other ferromagnetic metals in which the moment is not small. In that case the calculations are significantly more involved. This new mode should exist as well in local moment ferromagnets. Mathematically this can be seen if one relaxes the condition used by Halperin and Hohenberg@Halperin [@and; @Hohenberg1969] that the partial time derivative of the spin current is zero. This as well as a calculation of this mode at finite temperature we leave for a future publication.

We would like to thank A. Balatsky, A. Bishop, R. Cowley, P.B. Littlewood, R. McQeeney, J.L. Smith, and S. Trugman, for the fruitful discussions. P. Petkova participated at the early developments of this paper and we would like to thank her for the discussions during that period. K.S. Bedell would like to thank the Aspen Center for Theoretical Physics where some of the ideas developed in this paper were born. This work was supported by the DOE Grant DEFG0297ER45636.

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{'source': 'arXiv_out/cond-mat0003187'}