Short-scale quantum kinetic theory including spin-orbit interactions

TL;DR

This paper develops a comprehensive quantum kinetic theory for spin-1/2 particles that incorporates spin-orbit interactions and particle dispersive effects, providing new insights into plasma wave behavior and damping mechanisms.

Contribution

It introduces a gauge-invariant Wigner-based kinetic model including spin-orbit effects, with a novel term involving electric and magnetic fields, extending previous theories.

Findings

Derived dispersion relations for electrostatic and electromagnetic waves.

Compared Landau damping effects with and without spin-orbit interactions.

Highlighted the role of hidden momentum in plasma dynamics.

Abstract

We present a quantum kinetic theory for spin-$1/2$ particles, including the spin-orbit interaction, retaining particle dispersive effects to all orders in $\hbar$, based on a gauge-invariant Wigner transformation. Compared to previous works, the spin-orbit interaction leads to a new term in the kinetic equation, containing both the electric and magnetic fields. Like other models with spin-orbit interactions, our model features "hidden momentum". As an example application, we calculate the dispersion relation for linear electrostatic waves in a magnetized plasma, and electromagnetic waves in a unmagnetized plasma. In the former case, we compare the Landau damping due to spin-orbit interactions to that due to the free current. We also discuss our model in relation to previously published works.

Equations

$$\hat{H}=\frac{{\hat{\mathbf{\pi }}}^2}{2m}+q\hat{\varphi }-{\mu }_e\mathbf{\sigma }\cdot \hat{\mathbf{B}}+\frac{{\mu }_e}{2m{c}^2}\left(\hat{\mathbf{\pi }}\times \hat{\mathbf{E}}-\hat{\mathbf{E}}\times \hat{\mathbf{\pi }}\right)\cdot \mathbf{\sigma }$$

$$W(\mathbf{x},\mathbf{p},t{)}_{\alpha \beta }=\mathrm{tr}\left[\hat{W}(\mathbf{x},\mathbf{p},t){\rho }_{\alpha \beta }\right]=\int d^3\mathbf{r}⟨\mathbf{r}|\hat{W}{\hat{\rho }}_{\alpha \beta }|\mathbf{r}⟩,$$

$$\hat{W}(\mathbf{x},\mathbf{p},t)$$

$$=\int \frac{d^{3}u}{(2\pi \hslash {)}^3}\frac{d^{3}v}{(2\pi \hslash {)}^3}{e}^{-\frac{i}{\hslash }(\mathbf{u}\cdot \mathbf{p}+\mathbf{v}\cdot \mathbf{x})}\hat{T}(\mathbf{u},\mathbf{v}),$$

$$\hat{T}(\mathbf{u},\mathbf{v},t)=\exp \left[\frac{i}{\hslash }(\mathbf{u}\cdot \hat{\mathbf{\pi }}+\mathbf{v}\cdot \hat{\mathbf{x}})\right].$$

$$f(\mathbf{x},\mathbf{p},\mathbf{s})$$

$$=\frac{1}{4\pi }\sum _{\alpha ,\beta }[{\delta }_{\alpha \beta }+\mathbf{s}\cdot {\mathbf{\sigma }}_{\alpha \beta }]W_{\beta \alpha }$$

$$\hat{T}(u,v)$$

$$\displaystyle=\exp\left[-\frac{i}{2\hbar}u\cdot v\right]\exp\left[\frac{i}{\hbar}u\cdot\hat{p}\right]\exp\left[\ \frac{i}{\hbar}v\cdot\big{(}\hat{x}-qu\cdot\textstyle\int_{0}^{1}d\tau\,\hat{A}(\hat{x}+\tau u)\big{)}\right]$$

$$=\exp \left[\frac{i}{2\hslash }u\cdot v\right]\exp \left[\frac{i}{\hslash }v\cdot \hat{x}\right]\exp \left[\frac{i}{\hslash }u\cdot \hat{\pi }\right]$$

$$=\exp \left[-\frac{i}{2\hslash }u\cdot v\right]\exp \left[\frac{i}{\hslash }u\cdot \hat{\pi }\right]\exp \left[\frac{i}{\hslash }v\cdot \hat{x}\right].$$

$$\frac{d}{dt}\hat{W}=\frac{\partial }{\partial t}\hat{W}-\frac{i}{\hslash }[\hat{W},\hat{H}].$$

$$f=\mathrm{Tr}[\hat{S}\hat{W}\rho ]$$

$${\partial }_{t}f=\mathrm{Tr}\left[\left(\hat{S}{\partial }_t\hat{W}-\frac{i}{\hslash }\hat{S}[\hat{W},\hat{H}]-\frac{i}{\hslash }[\hat{S},\hat{H}]\hat{W}\right)\rho \right]$$

$${\hat{H}}_{\text{SO}}=\frac{{\mu }_e}{2m{c}^2}(\hat{\mathbf{\pi }}\times \hat{\mathbf{E}}-\hat{\mathbf{E}}\times \hat{\mathbf{\pi }})\cdot \mathbf{\sigma }.$$

$$[\hat{W},{\hat{H}}_{\text{SO}}]$$

$$[\hat{S},{\hat{H}}_{\text{SO}}]$$

$$\hat{S}[\hat{W},{\hat{H}}_{\text{SO}}]=({\sigma }_j+s_j+i{ϵ}_{ijk}{s}_i{\sigma }_k)[\hat{W},{\hat{V}}_j].$$

$$2i\epsilon_{ijk}s_{i}\hat{V}_{j}\sigma_{k}\hat{W}\\ =i\epsilon_{ijk}s_{i}\sigma_{k}\left(\big{(}\hat{V}_{j}\hat{W}+\hat{W}\hat{V}_{j}\big{)}+[\hat{V_{j}},\hat{W}]\right)$$

$${\partial }_{s_i}\hat{S}={\sigma }_i-s_i{s}_j{\sigma }_j,$$

$$({\partial }_{s_i}+s_i)\hat{S}$$

$${ϵ}_{ijk}{\partial }_{s_i}{s}_j\hat{S}$$

$$\left(\partial_{t}f\right)_{\textrm{SO}}=-\frac{i}{\hbar}\operatorname{Tr}\Big{[}\big{(}(\partial_{s_{i}}+s_{i})\hat{S}[\hat{W},\hat{V_{i}}]\\ +\epsilon_{ijk}\partial_{s_{i}}s_{j}\hat{S}[\hat{W},\hat{V_{k}}]_{+}\big{)}\rho\Big{]}.$$

$$[\hat{T},\hat{\mathbf{\pi }}\times \hat{\mathbf{E}}]$$

$$[\hat{W},{\hat{O}}^{†}]=-[\hat{W},\hat{O}{]}^{†},$$

$$\hat{\Theta }(u)|x⟩$$

$$\hat{\Theta }(u)f(\hat{x})$$

$$[\hat{T},{\hat{E}}_i]=\hat{T}(u,v)\left({\hat{E}}_i(\hat{x})-{\hat{E}}_i(\hat{x}-u)\right)$$

$${\hat{\pi }}_i\exp \left(\frac{i}{\hslash }u\cdot \hat{\pi }\right)$$

$$\exp \left(-\frac{i}{\hslash }u\cdot \hat{\pi }\right){\hat{\pi }}_i$$

$${\hat{O}}_{ij}^B$$

$${\hat{O}}_{ij}^D$$

$$\langle x^{\prime}|\hat{Q}^{B}_{ij}|x^{\prime\prime}\rangle=\mathcal{F}\left[\langle x^{\prime}|\hat{O}^{B}_{ij}|x^{\prime\prime}\rangle\right]\\ =-q\mathcal{F}\bigg{[}e^{\frac{i}{\hbar}v\cdot(x^{\prime}+u/2)}\int_{0}^{1}d\tau\,(1-\tau)[u\times B(x^{\prime}+\tau u)]_{i}\langle x^{\prime}|e^{i\Phi}e^{\frac{i}{\hbar}u\cdot\hat{p}}\left(\hat{E}_{j}(\hat{x})-\hat{E}_{j}(\hat{x}-u)\right)|x^{\prime\prime}\rangle\bigg{]}\\ =-q\mathcal{F}\left[e^{\frac{i}{\hbar}v\cdot(x^{\prime}+u/2)}\int_{0}^{1}d\tau\,(1-\tau)[u\times B(x^{\prime}+\tau u)]_{i}\langle x^{\prime}|e^{i\Phi}\left(\hat{E}_{j}(\hat{x}+u)-\hat{E}_{j}(\hat{x})\right)|x^{\prime\prime}-u\rangle\right]\\ =-q\mathcal{F}\left[e^{\frac{i}{\hbar}v\cdot(x^{\prime}+u/2)}\int_{0}^{1}d\tau\,(1-\tau)[u\times B(x^{\prime}+\tau u)]_{i}\big{(}E_{j}(x^{\prime}+u)-E_{j}(x^{\prime})\big{)}\langle x^{\prime}|e^{i\Phi}|x^{\prime\prime}-u\rangle\right].$$

$$\langle x^{\prime}|\hat{Q}^{B}_{ij}|x^{\prime\prime}\rangle=-q\mathcal{F}\int_{-1/2}^{1/2}d\tau\,\left(\frac{1}{2}-\tau\right)[u\times B(x+\tau u)]_{i}\big{(}E_{j}(x+u/2)-E_{j}(x-u/2)\big{)}\langle x^{\prime}|e^{\frac{i}{2\hbar}u\cdot v}e^{\frac{i}{\hbar}v\cdot x}e^{i\hat{\Phi}}|x^{\prime\prime}\rangle\\ =-q\mathcal{F}\left[\int_{-1/2}^{1/2}d\tau\,\left(\frac{1}{2}-\tau\right)[u\times B(x+\tau u)]_{i}\big{(}E_{j}(x+u/2)-E_{j}(x-u/2)\big{)}\langle x^{\prime}|\hat{T}(u,v)|x^{\prime\prime}\rangle\right]$$

$$E_j(x+u/2)-E_j(x-u/2)=u_k{\int }_{-1/2}^{1/2}d\tau E_{j,k}(x+\tau u),$$

Full text

I Introduction

Dense plasmas, where quantum effects are important, can, for example, be found in different types of solid-state plasmas Shukla and Eliasson (2011); Manfredi and Hurst (2015), various astrophysical environments Chabrier et al. (2002); Uzdensky and Rightley (2014), and certain forms of laser-plasma interactions Di Piazza et al. (2012); Weber et al. (2018). Such plasmas have been modelled using hydrodynamic Haas (2011) and kinetic equations Bonitz (2016), where, in the present paper, the focus is on the latter class of theories. Quantum kinetic equations can be derived, for example, from the Green’s function formalism Kadanoff and Baym (1962); Keldysh et al. (1965) or from the density matrix approach Hohenester and Pötz (1997); Ekman et al. (2017). Recently, several kinetic models for plasmas have been put forward, within the Hartree approximation Zamanian et al. (2010); Ekman et al. (2017); Andreev (2016a); Hurst et al. (2014) or the Hartree-Fock approximation Ekman et al. (2015); Andreev (2016b); Haas (2019). Depending on the scope of the theory, the density matrix can be based on the Schrödinger equation Bonitz (2016), the Pauli equation Zamanian et al. (2010), or the Dirac equation, where the latter can be applied in the weakly Asenjo et al. (2012) or fully Ekman et al. (2017) relativistic approximation.

In order to produce a quantum kinetic theory of electrons from the Dirac equation, certain restrictions need to be applied. Firstly, pair-production must be negligible, such that a Foldy-Wouthuysen transformation Foldy and Wouthuysen (1950); Silenko (2008) can be applied to separate electron states from positron states. This puts restrictions on the maximal electric field strength, which should be well below the critical field $E_{\text{cr}}=m^{2}c^{3}/|q|\hbar$, and the characteristic spatial scales of the fields, which should be much longer than the Compton length $L_{c}=\hbar/mc$. Here $m$ and $q$ are the electron mass and charge, respectively, $c$ is the speed of light in vacuum, and $\hbar$ is the reduced Planck’s constant. Given these restrictions, there are two different regimes that can be studied. Firstly, there is the fully relativistic regime, where the relativistic factor $\gamma$ fulfills $\gamma-1\sim 1$, in which case the de Broglie length $\lambda_{\text{dB}}=\hbar/p_{\mathrm{ch}}$ is of the same order as $L_{c}$. Here $p_{\mathrm{ch}}$ is the characteristic momentum of the electrons – determined by the Fermi temperature $T_{F}$ or the thermodynamic temperature $T$ whichever is larger. Since, according to the applicability conditions, this means that the spatial scales must be longer than the de Broglie length, particle dispersive effects cannot be included in the fully relativistic regime Ekman et al. (2017). However, for the second case, the weakly relativistic regime with $\gamma-1\ll{1}$, we have $\lambda_{dB}\gg L_{c}$, and as a consequence, a theory including particle dispersive effects based on the Dirac equation can be formulated. This is the main goal of the present paper. We thus generalize the work presented in Ref. Zamanian et al. (2010) based on the Pauli equation, by including several new effects, such as spin-orbit interaction, Thomas precession and the polarization currents associated with the spin. We also generalize the results of Ref. Asenjo et al. (2012), by also covering the short scale physics down to spatial scales of the order of the de Broglie length.

Our approach is as follows. We start from the Pauli Hamiltonian including the spin-orbit interaction. In Section II, we apply a combined Wigner- and Q-transform (for the spin) using the gauge-independent approach of Stratonovich Stratonovich (1956); Serimaa et al. (1986), to formulate our scalar kinetic theory.

In Section III, in order to demonstrate the usefulness of the present theory, we have calculated two examples from linearized theory: electromagnetic waves in a non-magnetized plasma and electrostatic waves propagating parallel to an external magnetic field. Classically, an external magnetic field does not affect parallel propagating electrostatic waves. By contrast, the present model predicts a condition, depending on the magnetic field, for resonant wave-particle interaction, generalizing those of previous works. We use this result to calculate the magnetic field dependence of the damping rate.

Finally, in Section IV, we compare our model with some theories from the recent research literature.

II The gauge invariant Wigner function and the spin transformation

We will consider the semi-relativistic Pauli Hamiltonian

[TABLE]

including the spin-orbit interaction. Here $\hat{\bm{\pi}}=\hat{\mathbf{p}}-q\hat{\mathbf{A}}$ is the gauge invariant momentum operator, $\hat{\phi},\hat{\mathbf{A}}$ are the scalar and vector potentials and $\mu_{e}$ is the electron magnetic moment. Note that the spin-orbit interaction is written in symmetric form so as to make the Hamiltonian Hermitian – which is necessary to have unitary time evolution.

Our goal is to formulate a scalar quantum kinetic theory based on the gauge invariant Wigner function Stratonovich (1956); Serimaa et al. (1986), that is,

[TABLE]

where $\alpha,\beta$ are the Pauli spin indices, and this trace is only over the spatial degrees of freedom. We work in the Heisenberg picture so that the operator $\hat{W}(\mathbf{x},\mathbf{p},t)$ is time-dependent, but the density matrix $\rho$ is not. The operator $\hat{W}(\mathbf{p},\mathbf{x},t)$ can be expressed as a Fourier transform

[TABLE]

where the operator $\hat{T}(\mathbf{u},\mathbf{v},t)$ is given by

[TABLE]

To obtain a scalar function from the matrix-valued $\hat{W}_{\alpha\beta}$, one can use the spin transform

[TABLE]

which is a Husimi $Q$-function for the spin Husimi (1940). This transform was used in Ref. Zamanian et al. (2010) for a theory including only the magnetic dipole interaction.

The remainder of this section presents the calculations involved in deriving an evolution equation for $f$, with the conclusion presented in Section II.4. The calculations in this section are similar to those in Ref. Serimaa et al. (1986), where the spin-independent terms in the kinetic equation are found.

Our notation is as follows. Throughout, quantities with hats are operators and quantities without hats are $c$-numbers, except the Pauli spin operator $\bm{\sigma}$ which is always an operator (a finite dimensional one, i.e., a matrix). We will also use the summation convention for repeated indices, and use commas to denote derivatives, e.g., $E_{i,j}=\partial E_{i}/\partial x_{j}$.

Various other forms of the operator $\hat{T}$ (Baker-Campbell-Haussdorff formulas) can be found Serimaa et al. (1986) and will be useful in the following:

[TABLE]

One can note that these occur in pairs related by operator orderings.

The time evolution of $\hat{W}$ is given by the Heisenberg equation of motion,

[TABLE]

Here the only effect of the partial time derivative is to give the $\partial_{t}\mathbf{A}$ contribution to the electric field in the Lorentz force. In the Heisenberg picture, the operator $\bm{\sigma}$ is time-dependent, so we must remember to include $d\bm{\sigma}/dt$ when deriving the evolution equation for $f$. Hence, if we let $\hat{S}=\frac{1}{4\pi}(1+\mathbf{s}\cdot\bm{\sigma})$, then

[TABLE]

i.e. $f$ is the expectation value of $\hat{S}\hat{W}$. Note that $\hat{S}$ and $\hat{W}$ are both Hermitian and commute, which implies that $f$ is real. The Heisenberg equation of motion then gives

[TABLE]

where in this and the previous equation, the trace is over both the spin indices and the spatial degrees of freedom.

The commutator is linear in $\hat{H}$, and the terms in the evolution equation for $\hat{W}$ corresponding to the lowest order (“non-relativistic”) part of the Hamiltonian have already been computed in Zamanian et al. (2010). Therefore we will here be concerned only with the spin-orbit contribution

[TABLE]

We will the denote the contribution from this Hamiltonian by $\left(\partial_{t}f\right)_{\textrm{SO}}$.

The spin-orbit interaction Hamiltonian has the form $\bm{\sigma}\cdot\hat{\mathbf{V}}$, and hence we find

[TABLE]

Then using $\sigma_{i}\sigma_{j}=\delta_{ij}+i\epsilon_{ijk}$, we get

[TABLE]

For the second commutator, we write $\hat{\mathbf{V}}\hat{W}$ as the sum of its symmetric and anti-symmetric parts

[TABLE]

and clearly, the second part cancels with cross product part of $\hat{S}[\hat{W},\hat{H}_{\textrm{SO}}]$ in Eq. 7.

Now we just need to use

[TABLE]

which implies

[TABLE]

and thus

[TABLE]

The remaining, laborious, task is to evaluate and take the Fourier transform of $\hat{T}(\hat{\bm{\pi}}\times\hat{\mathbf{E}})\pm(\hat{\bm{\pi}}\times\hat{\mathbf{E}})\hat{T}$, then express the result in terms of differential operators – functions of $\partial_{p}$ and $\partial_{x}$ – acting on $\hat{W}$. These operators can be taken outside the trace, so that they act on $f$, and this will give the kinetic equation. Since

[TABLE]

we need to evaluate the commutators with $\hat{\bm{\pi}}$ and $\hat{\mathbf{E}}$. This is sufficient, because for any operator $\hat{O}$

[TABLE]

since $\hat{W}$ is Hermitian, and we get the commutator with the Hermitian ordering (which appears in Eq. 4) by reading off the “imaginary part” of $[\hat{W},\hat{\bm{\pi}}\times\hat{\mathbf{E}}]$.

II.1 The commutator $[\hat{W},\hat{\mathbf{E}}]$

In Sections II.1, II.2 and II.3, we will typically omit boldface for vector quantities; the type of each quantity should be apparent from context.

Using Eq. 0b, we can write the operator $\hat{T}$ with $\hat{\Theta}(u)=\exp(\frac{i}{\hbar}u\cdot\hat{p})$ on the right. Since $\hat{p}$ is the generator of translations, the operator $\Theta(u)$ acts like

[TABLE]

and thus, using Eq. 0b to put the translation operator on the left, with the other factors commuting with $\hat{E}$,

[TABLE]

Now we need an expression for the product $\hat{\bm{\pi}}\times[\hat{T},\hat{\mathbf{E}}]$ in terms of $\hat{T}$. By putting the commutator in the form Eq. 17, we can use any expression for $\hat{T}$ as is most appropriate. The one that is most appropriate here is Eq. 0e since we can lift the result from Ref. Serimaa et al. (1986) their (4.30b), and its Hermitian conjugate:

[TABLE]

We thus have that the operator $\pi_{i}[\hat{T},E_{j}]$ can be expressed as $\pi_{i}[\hat{T},E_{j}]=\hat{O}^{B}_{ij}+\hat{O}^{D}_{ij}$ where

[TABLE]

Now let $|x^{\prime}\rangle,|x^{\prime\prime}\rangle$ be eigenstates of $\hat{x}$ (at the appropriate time). Use Eq. 0b so that the $\hat{x}$ operators are on the left in $\hat{T}$.

Let $\hat{Q}^{B}_{ij}$ be the Fourier transform in $u,v$ of $\hat{O}^{B}_{ij}$. Then, letting $e^{i\hat{\Phi}}=\exp\left[\ -\frac{iq}{\hbar}u\cdot\int_{0}^{1}d\tau\,\hat{A}(\hat{x}+\tau u)\right]$ its matrix elements are

[TABLE]

At this point we see that the Fourier transform in $v$ can be carried out and will give an expression proportional to $\delta(x^{\prime}-x+u/2)$. Hence the matrix elements of $\hat{Q}^{B}_{ij}(x,p)$ remain unchanged if we change $x^{\prime}$ to $x-u/2$ in the arguments of the fields. In the integral over $\tau$, we also change variables to $\tau-1/2$ for a somewhat simpler expression.

We can then restore the operators making up $\hat{T}$, resulting in

[TABLE]

Since the Fourier transform sends $u\mapsto i\hbar\partial_{p}$, we can now express this operator in terms of $\hat{W}$ and its derivatives. First, use that

[TABLE]

and note that this quantity is real. In the integral with $B$, $1/2$ is even and $-\tau$ is odd, so even and odd powers of $i\tau\hbar\partial_{p}$ in the expansion of $B$ survive the integration, respectively. Since we want the imaginary part, it is the term with $-\tau$ that we should keep, and in conclusion

[TABLE]

We now turn to $\hat{O}^{D}_{ij}$. Note that from the expression for $\hat{T}$ in Eq. 0e we have

[TABLE]

and hence

[TABLE]

The first term can be handled using the argument above, except that the Fourier transform in $v$ will send $v_{i}$ to $i\hbar\partial_{x_{i}}\delta(x)$. The result is that

[TABLE]

but this quantity is real, and so makes no contribution to the evolution equation. For the term with the $u$-derivative, we integrate by parts in the Fourier transform, i.e.,

[TABLE]

In the first term, $p_{i}$ can be taken outside the Fourier transform, and the remaining operator is treated like above resulting in

[TABLE]

We can write this in a somewhat simpler form as

[TABLE]

where we have introduced the notation

[TABLE]

To summarize, we obtain

[TABLE]

where

[TABLE]

The quantities $\tilde{\mathbf{E}}$ and $\Delta\tilde{\mathbf{p}}$ are the same as in Refs. Serimaa et al. (1986); Zamanian et al. (2010). Since Eq. 33 is to be contracted with the Pauli matrices, in the kinetic equation it represents the spin-orbit force.

II.2 The commutator $[\hat{W},\hat{\bm{\pi}}]$

We use Eq. 0e for $\hat{T}$, so that

[TABLE]

also using (4.25) from Ref. Serimaa et al. (1986). We can now look at the matrix elements of $\mathcal{F}\left[\hat{O}^{A}_{ij}\right]$, but it is of the form in Eq. 28 except we have only one $\hat{E}$ operator, $\hat{E}(\hat{x})$. Therefore,

[TABLE]

We see that the imaginary part of $E_{i,j}$ term cancels the imaginary part of the corresponding term in Eq. 30, since when dividing by $i$, even powers of $i\hbar\partial_{p}$ should be kept. For the second term, we can integrate by parts to show that

[TABLE]

Again keeping even powers of $i\hbar\partial_{p}$ since we are dividing by $i$,

[TABLE]

This is the “hidden momentum” velocity correction term $\propto\mathbf{E}\times\mathbf{s}$ Babson et al. (2009) (to be discussed more below), but with $\mathbf{E}=\tilde{\mathbf{E}}+\Delta\tilde{\mathbf{E}}$ where $\Delta\tilde{\mathbf{E}}$ is higher order in $\hbar$.

Following steps like those above, and using Eq. 37 again, one readily expresses the matrix elements of $\hat{O}^{C}_{ij}$ in terms of those of $\hat{W}$. The result is that

[TABLE]

which, in the kinetic equation, will give

[TABLE]

i.e., it is the correction to the magnetic force due to the relation between momentum and velocity. Here $\tilde{\mathbf{B}}$ is defined in the same way as $\tilde{\mathbf{E}}$, Eq. 32.

II.3 The spin transform

It remains to evaluate the terms in the kinetic equation arising from the spin transform, proportional to

[TABLE]

where $\operatorname{H.c.}$ denotes the Hermitian conjugate. Using Eq. 18 again we establish that

[TABLE]

where $\mathcal{D}_{k}$ is the operator

[TABLE]

Hence the product $(\hat{\bm{\pi}}\times\hat{\mathbf{E}}-\hat{\mathbf{E}}\times\hat{\bm{\pi}})_{j}\hat{T}$ is given by

[TABLE]

where

[TABLE]

Differentiating $\hat{E}_{l}(\hat{x})\hat{T}=\hat{T}\hat{E}_{l}(\hat{x}-u)$ with respect to $u_{k}$ we obtain

[TABLE]

so that

[TABLE]

Putting this into Eq. 44, all terms are similar to ones already seen, and we will provide only an outline of the remaining calculations. The operator of the type $\hat{E}_{l}(\hat{x})\frac{\partial\hat{T}}{\partial u_{k}}$ we can handle as in Eq. 29, resulting in

[TABLE]

and following the argument leading to Eq. 38, when taking the real part, we will get

[TABLE]

For the term proportional to $v_{k}$ in Eq. 44, we use that $\mathcal{F}:v_{k}\mapsto i\hbar\partial_{x_{k}}\delta(x)$ as leading up to Eq. 28. When taking the real part, the term where the derivative acts on $E$ cancels with the derivative term from Eq. 47 and what remains is

[TABLE]

which can be simplified using Eq. 31.

Following steps like those for $\hat{O}^{B}$ (Eq. 20 and following), the final term in Eq. 44, proportional to $I_{k}$, will give us a contribution of the form

[TABLE]

When taking the real part of the Fourier transform of this, there will be two terms,

[TABLE]

which is in line with previous generalisations, and

[TABLE]

which is of a new type. However, both are order $\hbar^{2}$, so their limits could not have been found in previous works that were either $\hbar$ or didn’t include the spin-orbit interaction.

Thus, the spin torque is modified to add a term proportional to $(\tilde{\mathbf{p}}+\Delta\tilde{\mathbf{p}})\times(\tilde{\mathbf{E}}+\Delta\tilde{\mathbf{E}})+\frac{q}{2}(\tilde{\mathbf{B}}\times\hbar\partial_{p})\times\cdot(\tilde{\mathbf{E}}\hbar\overset{\leftarrow}{\partial_{x}}\cdot\overset{\rightarrow}{\partial_{p}})$. The overall sign and coefficient of this term is found by matching its long-scale limit to the model in Ref. Asenjo et al. (2012)

II.4 Conclusion

The evolution equation for $f$ follows directly by taking the expectation value of the evolution equation for $\hat{W}$. It is

[TABLE]

For reference, the notation used is

[TABLE]

with $\tilde{\mathbf{B}}$, $\Delta\tilde{\mathbf{B}}$ defined in the same way as $\tilde{\mathbf{E}}$, $\Delta\tilde{\mathbf{E}}$. Functions of operators are defined in terms of their Taylor series.

There are five new terms in Eq. 53 as compared to Ref. Zamanian et al. (2010). In order, they represent: the “hidden momentum” (see below) correction to the velocity in the diffusion and magnetic force terms, respectively; the spin-orbit force; the spin torque due to spin-orbit interaction, including the Thomas precession; and the last term, which is non-linear in the fields and higher-order in $\hbar$. This term lacks an analog in Ref. Asenjo et al. (2012); the others are short-scale generalizations of terms found in Ref Asenjo et al. (2012) similar to how Ref. Serimaa et al. (1986) generalizes the Vlasov equation.

We have denoted the electron magnetic moment by $\mu_{e}$ to allow for an anomalous magnetic moment, i.e., $\mu_{e}=\frac{g}{2}\mu_{B}$ where $\mu_{B}$ is the Bohr magneton and $g=2+\frac{\alpha}{\pi}+\ldots$. That $g\neq 2$ leads to a mismatch between the cyclotron frequency and Larmor frequency, resulting in non-classical resonances Brodin et al. (2008), as will be seen below.

The system is closed with Maxwell’s equations with polarization and magnetization,

[TABLE]

The free charge $\rho_{f}$ is given by

[TABLE]

introducing $d\Omega=d^{3}p\,d^{2}s$. Finding the continuity equation for $\rho_{f}$ using Eq. 53, the free current $\mathbf{J}_{f}$ is given by

[TABLE]

and while it may appear strange that we do not have $\mathbf{p}\parallel\mathbf{v}$, this is the function on phase space in Weyl correspondence with $\hat{\mathbf{v}}=\frac{i}{\hbar}[\hat{H},\hat{\mathbf{x}}]$. This is an example of “hidden momentum”, common to systems with magnetic moments Shockley and James (1967); Coleman and Van Vleck (1968); Shockley (1968); Babson et al. (2009), and is found already in the long-scale length limit Asenjo et al. (2012), and also in Ref. Ekman et al. (2017).

Finally, the polarization and magnetization densities are given by

[TABLE]

Here, a factor $3$ appears due to the normalization of the spin transform; the magnetization is proportional to the expectation value of $\bm{\sigma}$.

It should be noted that a model very similar to Eq. 53 was recently presented by Hurst et al. Hurst et al. (2017), starting from the same Hamiltonian. Their model, however, is formulated in terms of a scalar quantity $f_{0}$ and a vector quantity $\mathbf{f}$, the mass and spin densities, respectively. Mathematically, this is writing the Hermitian matrix-valued Wigner function in the basis $\{I,\bm{\sigma}\}$, and the correspondence between models is that $f_{0}=\int f\,d^{2}s$ and $\mathbf{f}=3\int\mathbf{s}f\,d^{2}s$. Hurst et al. find the same charge and current densities as we do 111Taking into account that their convention for the sign of the charge is different from ours. including the same “hidden momentum” in the free current or velocity. They also find the new non-linear term in the spin torque.

Hurst et al. Hurst et al. (2017) derive their model using a completely different method, based on a gauge invariant expression for the Moyal bracket Müller (1999). That our results agree, up to trivial transformations, lends confidence to the results. We return to this in Section IV.

III Linear waves

To illustrate the usefulness of Eq. 53 we study linear waves in a homogeneous plasma. We consider two cases: first electrostatic waves in a magnetized plasma, then electromagnetic waves in an unmagnetized plasma. From now on, we will let $c=1$ to simplify the notation.

The momentum is expressed in cylindrical coordinates $\mathbf{p}=(p_{\perp},\varphi_{p},p_{z})$ and the spin in spherical coordinates $\mathbf{s}=\sin{\theta_{s}}\cos{\varphi_{s}}\hat{\mathbf{z}}+\sin{\theta_{s}}\sin{\varphi_{s}}\hat{\mathbf{y}}+\cos{\theta_{s}}\hat{\mathbf{z}}$. In the linearization of Eq. 53 variables are separated according to $f=f_{0}(p_{\perp},p_{z},\theta_{s})+f_{1}(x,p,s,t)$, $\mathbf{E}=\mathbf{E}_{1}$ and $\mathbf{B}=B_{0}\hat{\textbf{z}}+\mathbf{B}_{1}$, where the subscripts [math] and $1$ denote unperturbed and perturbed quantities respectively. Eq. 53 then becomes

[TABLE]

In general the background distribution can be divided into its spin-up and spin-down components Lundin and Brodin (2010)

[TABLE]

In the sums below it will be implicit that $\nu$ takes the values $\pm 1$.

III.1 Electrostatic waves

In this geometry we consider: $\mathbf{B}_{0}=B_{0}\hat{\mathbf{z}}$, $\mathbf{k}=k\hat{\mathbf{z}}$ and $\mathbf{E}=E\hat{\mathbf{z}}$. The perturbed parameters follow the plane-wave ansatz according to $f_{1}=\tilde{f}e^{i\textbf{k}\cdot\textbf{x}-\omega t}$. The left hand side of Eq. 63 can then be written

[TABLE]

where $\omega_{ce}$ is the cyclotron frequency, $\omega_{cg}=\frac{g}{2}\omega_{ce}$ and $RHS$ is the right hand side of Eq. 63.

Making an expansion of $\tilde{f}_{1}$ in eigenfunctions of the operators of the right hand side of Eq. 63 Lundin and Brodin (2010) we write

[TABLE]

Substituting Eq. 66 in Eq. 65, we can now express $f_{1}$ in terms of $f_{0}$

[TABLE]

where

[TABLE]

and $\Delta\omega_{ce}=\omega_{cg}-\omega_{ce}$. Next, noting that the magnetization current $\mathbf{J}_{m}=\nabla\times\mathbf{M}=0$, we calculate the total current $\textbf{J}=\textbf{J}_{f}+\textbf{J}_{p}$, where $\textbf{J}_{f}$ and $\textbf{J}_{p}$ are the free and polarization current respectively, in order to obtain the dispersion relation. Note that the operators contained in $\tilde{\mathbf{E}}$ and $\Delta\tilde{\mathbf{E}}$ act on $F_{0}$ as translation of the kinetic momentum, see Appendix A for more details. Introducing the notation

[TABLE]

the dispersion relation is then

[TABLE]

where

[TABLE]

Here $\chi_{fL}$ and $\chi_{pL}$ are the contributions due to the free and polarization currents, respectively. We stress that in the first factor in $\chi_{pL}$, the sum over $\pm$ and $\nu$ corresponds to four terms in total. Below we will use the indices $f$ and $p$ with the same meaning also for other quantities. We note that $\chi_{fL}$ is the same susceptibility as for spinless quantum plasmas Eliasson and Shukla (2010). However, $\chi_{pL}$ generalizes previous results Asenjo et al. (2012) to cover also the short-scale physics.

Next, in order to examine how the spin affects the dynamic of the plasma, the Landau damping will be calculated and the contributions of $\chi_{fL}$ and $\chi_{pL}$ will be compared. Starting from Eq. 71, we Taylor expand $\epsilon(k,\omega_{r}+i\gamma$) around $\omega_{r}$ Nicholson (1983), which leads to

[TABLE]

Analogously to Eq. 71, $\operatorname{Im}\epsilon$ can be divided into $\operatorname{Im}\epsilon=\operatorname{Im}(\chi_{fL})+\operatorname{Im}(\chi_{pL})$. Using the Plemelj formula, we get

[TABLE]

To see how $\operatorname{Im}\chi_{pL}$ contributes to $\gamma$ compared to $\operatorname{Im}\chi_{fL}$, we define

[TABLE]

The next step is to specify $F_{0\nu}$. For simplicity, we will consider a Maxwell-Boltzmann distribution $F_{0\nu}=C_{\nu}e^{-p^{2}/p_{\textnormal{th}}^{2}}e^{-\nu\mu_{e}B_{0}/k_{B}T}$ where $C_{\nu}$ is the normalization constant according to $\int d^{3}pd^{2}s\,F_{0\nu}=N_{0\nu}$, where $N_{0\nu}$ are the densities of spin-up ($\nu=+1$) and spin-down particles ($\nu=-1$). This condition implies the textbook result that the magnetization is proportional to $\tanh\frac{\mu_{e}B}{k_{B}T}$. We then have

[TABLE]

Finally, the real part of the dispersion relation $\omega_{r}(k)$ has to be specified explicitly in order to obtain $\Gamma$. Assuming that $\chi_{pL}$ in Eq. 71 is small compared to $\chi_{fL}$ 222This will be the case at least if $\hbar\omega_{p}/m\ll 1$. we neglect it for the purposes of computing $\omega_{r}$. This gives us

[TABLE]

Assuming $\omega\gg kv_{th}/m\pm\hbar k^{2}/2m$, we can make a Taylor expansion of the denominators up to first non-vanishing order. The real part of the dispersion relation $\omega_{r}(k)$ is then

[TABLE]

Using this expression of $\omega$ in Eq. 76, we can plot $\Gamma$, see Fig. 1. In general the spin contribution to the damping dominates (i.e., $\Gamma\to 1$) for long wavelengths, whereas the free current contribution dominates (i.e., $\Gamma\to 0$) for shorter wavelengths. The transition between these regimes is dependent on the normalized magnetic field $B_{n}=\mu_{B}B/m$ and the quantum parameter $H=\hbar\omega_{p}/mv_{\textnormal{th}}^{2}$. We have performed the computation in Fig. 1 for various other values of $v_{\textnormal{th}}$, however, this reveals that the location of the transition region, expressed in $k_{n}=kv_{\textnormal{th}}/\omega_{p}$, is almost independent of $v_{\textnormal{th}}$.

III.2 Electromagnetic waves

In this geometry we look at electromagnetic waves in unmagnetized plasmas with $\textbf{k}=k\hat{\mathbf{z}}$, $\mathbf{E}=E\hat{\mathbf{x}}$ and $\mathbf{B}=B\hat{\mathbf{y}}$. We will allow for a background distribution depending on both $p_{\perp}$ and $p_{z}$, but assume it to be spin-independent, as there is no background magnetic field. Dividing $f$, $E$ and $B$ into perturbed and unperturbed quantities, Eq. 63 becomes

[TABLE]

Using an expansion of $f_{1}$ like that in Eq. 66 $f_{1}$ is computed as,

[TABLE]

where

[TABLE]

and

[TABLE]

Next we calculate the total current $\mathbf{J}_{T}=\mathbf{J}_{f}+\mathbf{J}_{p}+\mathbf{J}_{m}$, which together with Maxwell’s equations gives us the dispersion relation

[TABLE]

where

[TABLE]

Here the indices $f,m,p$ stand for the contributions to the susceptibility due to the free, magnetization, and polarization currents, respectively.

To shed some further light on the dispersion relation Eq. 84, we consider the limit of the vanishing background pressure. We then immediately obtain

[TABLE]

The quantum contribution to Eq. 86 comes from the magnetic dipole force and the spin-orbit interaction coupling the transverse motion with the longitudinal degrees of freedom. This is illustrated by the free-particle dispersion relation in the denominator of the second term. While Eq. 86 highlights the physics in a simple manner we need to stress that the quantum modification of the classical dispersion relation Eq. 84 is of most interest for high density plasmas, in which case the Fermi pressure is important even if the Fermi pressure is small. Thus, for realistic applications, we typically must use the full expressions Eqs. 85a to 85c for the susceptibilies, rather than Eq. 86.

IV Discussion

The model derived here, Eqs. 53 to 62, is a generalization of the governing Eqs. of Asenjo et al. Asenjo et al. (2012) allowing for short macroscopic scale lengths, i.e. of the order of the de Broglie length. Alternatively, we can view the model as a generalization of Zamanian et al. Zamanian et al. (2010) into the weakly relativistic regime. A kinetic theory based on the same Hamiltonian has previously been constructed by Hurst et al. Hurst et al. (2017). Their approach leads to two coupled equations for a scalar $f_{0}$ and a vector $\mathbf{f}$, see Section II.4. While this is a notable difference, nevertheless the evolution equations of Hurst et al. Hurst et al. (2017) generalize Hurst et al. Hurst et al. (2014) in the same way as our results generalize those of Zamanian et al. Zamanian et al. (2010). However, as Hurst et al. Hurst et al. (2014) concluded, their model is equivalent to that of Zamanian et al. Zamanian et al. (2010), we can similarly conclude that the model here is equivalent to that of Hurst et al. Hurst et al. (2017), apart from the small but significant detail that we include the contribution from the anomalous magnetic moment.

To demonstrate the usefulness of our model we have calculated the linear dispersion relation for two simple cases: Electrostatic waves propagating parallel to an external magnetic field and electromagnetic waves in an un-magnetized plasma. The electrostatic dispersion relation possesses two roots. The usual Langmuir root and a root induced by the spin resonances with $\omega\simeq\Delta\omega_{c}$ Asenjo et al. (2012). Here we have focused on the Langmuir root, showing that even when the real part of the frequency is almost unaffected by the spin dynamics, the wave-particle damping can still be significantly modified.

For the electromagnetic wave mode we note that all the new terms can be expressed in terms of the longitudinal susceptibility. Physically it means that the spin-terms couples the longitudinal and the transverse degrees of freedom. However, for scale lengths much longer than the Compton length ($k\ll mc/\hbar$) this coupling is relatively weak.

A new feature of the present theory as compared to previous works is seen already for linear wave propagation, and comes from the denominators $(\omega-kp_{z}/m\pm\Delta\omega_{c}\pm\hbar k^{2}/2m)^{-1}$ seen in Eq. 72. We note the importance of not letting $\mu_{e}=\mu_{B}$ here, as otherwise we would get $\Delta\omega_{c}=0$. The physical origin of the resonances corresponding to $(\omega-kp_{z}/m\pm\Delta\omega_{c}\pm\hbar k^{2}/2m)=0$ comes from energy momentum conservation written in the form

[TABLE]

Here $(k_{1},\omega_{1})$ and $(k_{2},\omega_{2})$ are the wavenumbers and frequencies of free particles before and after the interaction with the wave field, fulfilling the free particle dispersion relation $\omega_{1,2}=\hbar k_{1,2}^{2}/2m$. The generalization from previous results (e.g., Refs. Eliasson and Shukla (2010); Brodin et al. (2017)) comes from the term $\pm\hbar\Delta\omega_{c}$ in (87), corresponding to resonant particles gaining an amount $\hbar\omega_{c}$ of perpendicular kinetic energy, at the same time as the magnetic dipole energy drops by $\hbar\omega_{cg}$, or vice versa. For our particular case of electrostatic waves propagating parallel to $\mathbf{B}_{0}$, the kinetic energy and magnetic dipole energy always change in opposite directions, as reflected by the appearance of $\Delta\omega_{c}$ in (87). However, in a more general setting, the energy relation will be of the form

[TABLE]

whereas Eq. 88 will be unaffected. Here $n$ is an integer and all sign combinations for the last three terms in (89) are possible.

Finally we point out a problem of future interest; to compute the ponderomotive force starting from Eqs. 53 to 62. Previous results Stefan et al. (2011) indicates that the spin-orbit interaction can give a substantial contribution. Hence it would be of interest to see to what extent the short-scale physics of the present model modify the results of Ref. Stefan et al. (2011).

Acknowledgements.

All authors would like to acknowledge financial support by the Swedish Research Council, grant number 2016-03806. G. Brodin, H. Al-Naseri, and J. Zamanian would also like to acknowledge support by the Knut and Alice Wallenberg Foundation through the PLIONA project.

Appendix A

In this section we show how $\tilde{\mathbf{E}}$ acts on $f_{0}(p)$ as a translation of momentum. The same applies to $\tilde{\mathbf{B}}$ since it has a similar form. We can rewrite $\tilde{\mathbf{E}}\,\partial f_{0}/\partial p_{z}$ as

[TABLE]

where $p_{q}=\hbar k/2$. However, for $\Delta\tilde{\mathbf{E}}f_{0}$, we can use integration by parts

[TABLE]

For the case of $\tilde{\mathbf{E}}\frac{\partial f_{0}}{\partial p_{\perp}}$,

[TABLE]

Transforming the coordinates $(p_{z},\tau)$ to $(u_{\pm},\tau^{\prime})$, where $u_{\pm}=p_{z}\pm\hbar k\tau$

[TABLE]