Coupling of acoustic phonon to a spin-orbit entangled pseudospin

Consider a long wavelength acoustic phonon, with a spatial and time dependent displacement vector $\vec{\xi}(\vec{x},t)$. For simplicity, we shall consider a cubic crystal, and remark on modifications for other symmetries later. As is well-known, we can decompose this into three components: $\vec{\nabla}\cdot\vec{\xi}$, $\frac{1}{2}\vec{\nabla}\times\vec{\xi}$ and the tensor $\frac{1}{2}\left(\frac{\partial\xi_{l}}{\partial x_{j}}+\frac{\partial\xi_{j}}{\partial x_{l}}\right)-\frac{1}{3}\delta_{jl}\vec{\nabla}\cdot\vec{\xi}$, corresponding to an isotropic expansion (compression), rotation, and anisotropic deformation respectively [36]. Under a low energy excitations of the crystal [27], the electronic state $|\Psi\rangle$ of our ion under consideration should still be within the manifold described by eq (1) though in a frame specified by the local environment. Hence at an instantaneous time $t$, we should have (up to small terms describing the excitations to higher energy levels)

$$|\Psi(t)\rangle=\alpha^{\prime}_{\Uparrow}(t)|\Uparrow^{\prime}(t)\rangle+\alpha^{\prime}_{\Downarrow}(t)|\Downarrow^{\prime}(t)\rangle$$

(5)

where $|\Uparrow^{\prime}(t)\rangle$ ( $|\Downarrow^{\prime}(t)\rangle$) are states given by eq (1) except with $x,y,z$, $|\uparrow\rangle,|\downarrow\rangle$ replaced by $x^{\prime},y^{\prime},z^{\prime}$, $|\uparrow^{\prime}\rangle,|\downarrow^{\prime}\rangle$ rotated from the former by $\vec{\Theta}(t)\equiv\frac{1}{2}\vec{\nabla}\times\vec{\xi}(t)$. (The isotropic compression and anisotropic deformation would not affect what we would be discussing below [37] and shall be ignored from now on). Suppose that our ion is under an external field $\vec{B}$, and let $\vec{B}^{\prime}$ be the value of this field in the above mentioned rotating frame. The Schrödinger equation of motion for $|\Psi\rangle$, employing eq. (5) and noting the time dependence of the basis function $|\Uparrow^{\prime}(t)\rangle,|\Downarrow^{\prime}(t)\rangle$, implies

$$i\frac{\partial}{\partial t}\left(\begin{array}[]{c}\alpha^{\prime}_{\Uparrow}\\ \alpha^{\prime}_{\Downarrow}\end{array}\right)=-g\mu_{B}\vec{B}^{\prime}(t)\cdot\frac{\vec{\tau}}{2}\left(\begin{array}[]{c}\alpha^{\prime}_{\Uparrow}\\ \alpha^{\prime}_{\Downarrow}\end{array}\right)+\left(\begin{array}[]{cc}-i\langle\Uparrow^{\prime}|\frac{\partial}{\partial t}|\Uparrow^{\prime}\rangle&-i\langle\Uparrow^{\prime}|\frac{\partial}{\partial t}|\Downarrow^{\prime}\rangle\\ -i\langle\Downarrow^{\prime}|\frac{\partial}{\partial t}|\Uparrow^{\prime}\rangle&-i\langle\Downarrow^{\prime}|\frac{\partial}{\partial t}|\Downarrow^{\prime}\rangle\end{array}\right)\left(\begin{array}[]{c}\alpha^{\prime}_{\Uparrow}\\ \alpha^{\prime}_{\Downarrow}\end{array}\right)$$

(6)

Here $\vec{\tau}$, which rigorously should have been denoted as $\vec{\tau}^{\prime}$, are Pauli matrices in the $\Uparrow^{\prime}$, $\Downarrow^{\prime}$ subspace, but we shall not make this distinction in notations for simplicity. Since $|\Uparrow^{\prime}(t)\rangle=e^{-i\vec{\Theta}\cdot(\vec{L}+\vec{s})}|\Uparrow\rangle$ $\approx(1-i\vec{\Theta}\cdot(\vec{L}+\vec{s}))|\Uparrow\rangle$, the time derivatives can be evaluated as , e.g., $-i\langle\Uparrow^{\prime}|\frac{\partial}{\partial t}|\Uparrow^{\prime}\rangle$ $=-(\frac{\partial\vec{\Theta}}{\partial t})\cdot[\langle\Uparrow^{\prime}|(\vec{L}+\vec{s})|\Uparrow^{\prime}\rangle]$. Using eq (4) (and ignoring a terms $\propto\vec{\Theta}\times\frac{\partial\Theta}{\partial t}$ which arises due to the difference between the primed and unprimed $\Uparrow$ and $\Downarrow$ space), we obtain

$$i\frac{\partial}{\partial t}\left(\begin{array}[]{c}\alpha^{\prime}_{\Uparrow}\\ \alpha^{\prime}_{\Downarrow}\end{array}\right)=\left[-g\mu_{B}\vec{B}^{\prime}(t)\cdot\frac{\vec{\tau}}{2}+\frac{5}{6}\frac{\partial\vec{\Theta}}{\partial t}\cdot\vec{\tau}\right]\left(\begin{array}[]{c}\alpha^{\prime}_{\Uparrow}\\ \alpha^{\prime}_{\Downarrow}\end{array}\right)$$

(7)

It would be more convenient to have an equation of motion involving directly $\vec{B}$ instead. We observe that $\vec{B}^{\prime}=\vec{B}-\vec{\Theta}\times\vec{B}$ and hence $\vec{B}^{\prime}\cdot\vec{\tau}=$ $e^{i\frac{\vec{\Theta}}{2}\cdot\tau}\vec{B}\cdot\vec{\tau}e^{-i\frac{\vec{\Theta}}{2}\cdot\tau}$. Introducing

$$\left(\begin{array}[]{c}\tilde{\alpha}_{\Uparrow}\\ \tilde{\alpha}_{\Downarrow}\end{array}\right)=e^{-i\frac{\vec{\Theta}}{2}\cdot\tau}\left(\begin{array}[]{c}\alpha^{\prime}_{\Uparrow}\\ \alpha^{\prime}_{\Downarrow}\end{array}\right)$$

(8)

we obtain finally

$$i\frac{\partial}{\partial t}\left(\begin{array}[]{c}\tilde{\alpha}_{\Uparrow}\\ \tilde{\alpha}_{\Downarrow}\end{array}\right)=\left[-\frac{g\mu_{B}}{2}\vec{B}+\frac{4}{3}\frac{\partial\vec{\Theta}}{\partial t}\right]\cdot\vec{\tau}\left(\begin{array}[]{c}\tilde{\alpha}_{\Uparrow}\\ \tilde{\alpha}_{\Downarrow}\end{array}\right)$$

(9)

where we have again dropped a term involving second powers in $\Theta$ . $\frac{4}{3}$ arises from $\frac{1}{2}-(-\frac{5}{6})$ thus is due to the difference between the rotational matrix for ordinary spin-1/2 and our pseudospin (eq. (4)). The direction of the pseudospin, defined as the expectation value of $\vec{\tau}$ with the “spin” wavefunction $(\tilde{\alpha}_{\Uparrow},\tilde{\alpha}_{\Downarrow})$, is given by

$$\frac{\partial}{\partial t}\hat{\tau}=\hat{\tau}\times\left[\vec{\omega}_{0}+r\frac{\partial\vec{\Theta}}{\partial t}\right]=\hat{\tau}\times\left[\vec{\omega}_{0}+\frac{r}{2}(\nabla\times\frac{\partial\vec{\xi}}{\partial t})\right]$$

(10)

with $\vec{\omega}_{0}=g\mu_{B}\vec{B}$ and $r=-\frac{8}{3}$. The former is the standard precession due to the external field and the second extra term is due to the rotational properties of our basis functions derived above.

We construct now the Lagrangian for the coupled phonon and pseudospin system. To simplify the writing, when no confusion arises, we shall often just write “spin” for the pseudospin.

First, the acoustic phonon alone can be described by the Lagrangian density

$$L_{0,ph}=\frac{1}{2}\rho_{M}\left(\frac{\partial\xi_{j}}{\partial t}\right)^{2}-U_{elastic}$$

(11)

where $U_{elastic}=\frac{1}{2}\left[\lambda_{1}(\frac{\partial\xi_{j}}{\partial x_{l}}\frac{\partial\xi_{j}}{\partial x_{l}})+\lambda_{2}\frac{\partial\xi_{j}}{\partial x_{j}}\frac{\partial\xi_{l}}{\partial x_{l}}\right]$ is the elastic energy density. Here $\rho_{M}$ is the mass density (dimension mass times inverse volume) and sums over repeated indices are implicit. We have also ignored a term $\lambda_{3}(\frac{\partial\xi_{j}}{\partial x_{j}}\frac{\partial\xi_{j}}{\partial x_{j}})$ which is allowed in cubic symmetry for simplicity. Its effects will be discussed later. Under this simplification, for a system without coupling to spin, sound velocities are independent of direction of propagation $\hat{q}$, with longitudinal and transverse sound velocities given by $v_{L}=[(\lambda_{1}+\lambda_{2})/\rho_{M}]^{1/2}$ and $v_{T}=[\lambda_{1}/\rho_{M}]^{1/2}$ respectively.

For the spin, first we recall that, for a spin $S$ under a magnetic field along $\hat{z}$, the Lagrangian can be written as [38] $L_{s}=g\mu_{B}SB\cos\theta+S\cos\theta\frac{\partial\phi}{\partial t}$ where $\theta$ and $\phi$ are the angles for the spin direction in spherical coordinates, the first term being from the Zeeman energy and the second a Berry phase term. To produce the equation of motion (10), we need only to replace $g\mu_{B}SB\cos\theta$ by $\frac{\vec{\tau}}{2}\cdot\left[g\mu_{B}\vec{B}+\frac{r}{2}(\nabla\times\frac{\partial\vec{\xi}}{\partial t})\right]$ (now specializing to pseudospin $1/2$). The last term allows us to identify the pseudospin - phonon coupling.

The Lagrangian $L=L_{ph}+L_{s}+L_{ph-s}$ is a sum of the phonon term (11), the spin term and the phonon-spin coupling term. We then have, for a net effective spin density $\rho_{s}$ per unit volume,

$$L_{s}=\rho_{s}\frac{1}{2}\left[g\mu_{B}\vec{B}\cdot\hat{\tau}+\rm cos\theta\frac{\partial\phi}{\partial t}\right]$$

(12)

$$L_{ph-s}=\frac{r\rho_{s}}{4}\left[\hat{\tau}\cdot(\nabla\times\frac{\partial\vec{\xi}}{\partial t})\right]$$

(13)

with $\hat{\tau}$ the net (pseudo-)spin direction. The phonon-pseudospin coupling is dictated by the factor $r$ derived in the last subsection. As is evident from our derivation above, this coupling arises from the Berry phase due to the rotating frame of reference for the pseudospin in the presence of the transverse acoustic phonon. We remind the readers here that this coupling thus has an entirely different origin from the magneto-elastic coupling discussed by, e.g., [22] for magnetic materials, which describes the change in magnetic energies in the presence of stress.

The equation of motion for $\hat{\tau}$ was already obtained in (10), which reads, after Fourier transform and linearizing about the equilibrium where $\hat{\tau}=\hat{z}$,

$$-i\omega\hat{\tau}(\omega,\vec{q})=\omega_{0}(\hat{\tau}\times\hat{z})+\frac{r\omega}{2}\left[\hat{z}\times(\vec{q}\times\vec{\xi})\right]$$

(14)

where $\vec{q}$ is the wavevector and $\omega$ the angular frequency.

The equation of motion for the displacement is

$$\rho_{M}\omega^{2}\xi_{j}-\frac{r\omega}{4}\rho_{s}(\vec{q}\times\hat{\tau})_{j}=\lambda_{1}q^{2}\xi_{j}+\lambda_{2}q_{l}(q_{j}\xi_{l})$$

(15)

We now study the consequences of eq. (14) and (15). Equation (14) implies that $\hat{\tau}_{z}$ is just a constant. The components orthogonal to the field direction ($j=x,y$) obeys

$$\tau_{j}=\frac{r\omega/2}{\omega_{0}^{2}-\omega^{2}}\left[\omega_{0}(\vec{q}\times\vec{\xi})_{j}-i\omega(\hat{z}\times(\vec{q}\times\vec{\xi}))_{j}\right]$$

(16)

Putting this into eq. (15) gives us the equation of motion entirely expressed in terms of $\xi_{j}$:

$$0=\rho_{M}\omega^{2}\xi_{j}-\left[\lambda_{1}q^{2}\xi_{j}+\lambda_{2}q_{l}(q_{j}\xi_{l})\right]-\frac{r^{2}\rho_{s}}{8}\frac{\omega_{0}\omega^{2}}{\omega_{0}^{2}-\omega^{2}}\left[-q_{z}^{2}\xi_{j}+q_{z}q_{j}\xi_{z}+(q_{z}(q_{l}\xi_{l})-q^{2}\xi_{z})\delta_{jz}\right]-i\frac{r^{2}\rho_{s}}{8}\frac{\omega^{3}}{\omega_{0}^{2}-\omega^{2}}q_{z}(\vec{q}\times\vec{\xi})_{j}$$

(17)

Coupling of the pseudospin to the phonon results in the last two new terms. Here the factor $\delta_{jz}=1$ if $j=z$ and vanishes otherwise. We note the factor $q_{z}$ in the last term, which is generated from the last term in eq. (16). This factor reflects the fact that the time dependent parts of $\tau$ only have $x$ and $y$ components.

We now analyze eq. (17) in two different limits.

For small fields, $\omega_{0}$ is much smaller than the phonon frequencies, eq. (17) approximately reads

$$0=\rho_{M}\omega^{2}\xi_{j}-\left[\lambda_{1}q^{2}\xi_{j}+\lambda_{2}q_{l}(q_{j}\xi_{l})\right]+i\frac{r^{2}\rho_{s}}{8}\omega q_{z}(\vec{q}\times\vec{\xi})_{j}$$

(18)

Longitudinal sound, with $\xi$ parrallel to $\vec{q}$, is not affected. Physically, there is no rotation of the environment surrounding the pseudospin in this case. The two polarizations of the transverse sound are coupled via the spins, turning them into circular polarized ones. Writing $\vec{\xi}=\xi_{\theta}\hat{\theta}+\xi_{\phi}\hat{\phi}$, we get

$$\left(\begin{array}[]{cc}\omega^{2}-q^{2}v_{T}^{2}&-i\frac{\rho_{s}r^{2}}{8\rho_{M}}\omega q^{2}\cos\theta_{q}\\ +i\frac{\rho_{s}r^{2}}{8\rho_{M}}\omega q^{2}\cos\theta_{q}&\omega^{2}-q^{2}v_{T}^{2}\end{array}\right)\left(\begin{array}[]{c}\xi_{\theta}\\ \xi_{\phi}\end{array}\right)=0$$

(19)

Here $\theta_{q}$ is the angle between $\hat{q}$ and $\hat{z}$. To lowest order in the phonon-pseudospin coupling, the frequencies are given by

$$\omega_{\pm}=qv_{T}\left[1\pm Z\rm cos\theta_{q}\right]$$

(20)

for the modes with right ( $(\xi_{\theta},\xi_{\phi})\propto(1,i)$) and left ( $(\xi_{\theta},\xi_{\phi})\propto(1,-i)$) circular polarization, with $Z$ a $q$-dependent dimensionless parameter

$$Z\equiv\frac{\rho_{s}r^{2}q}{16\rho_{M}v_{T}}\ .$$

(21)

Thus the fractional splitting increases with $q$, reflecting that a shorter wavelength implies a larger rotation motion of the lattice $\vec{q}\times\vec{\xi}$ and hence a stronger coupling to our pseudospin. This is different from a naïve picture of hybridization between the phonon modes with the Larmor precession of the spins, where the induced splitting would decrease with increasing frequencies away from $\omega_{0}$. From eq. (20), we see that for $q_{z}>0$, the lower (higher) frequency mode is left (right)-circularly polarized. The reverse is the case if $q_{z}<0$. See Fig 1a.

For very small $q$, the phonon frequency $\sim qv_{T}$ is much smaller than $\omega_{0}$. In this case, the effective equation of motion for the phonon cooridinate can be written as

$$0=\rho_{M}\omega^{2}\xi_{j}-\left[\lambda_{1}q^{2}\xi_{j}+\lambda_{2}q_{l}(q_{j}\xi_{l})\right]-\frac{r^{2}\rho_{s}\omega^{2}}{8\omega_{0}}\left[-q_{z}^{2}\xi_{j}+q_{z}q_{j}\xi_{z}+(q_{z}(q_{l}\xi_{l})-q^{2}\xi_{z})\delta_{jz}\right]-i\frac{r^{2}\rho_{s}}{8}\frac{\omega^{3}}{\omega_{0}^{2}}q_{z}(\vec{q}\times\vec{\xi})_{j}$$

(22)

Note the sign differences between the last terms of eqs. (18) and (22) in two different frequency regimes, similar to the case of, e.g., driven harmonic oscillator for above versus below resonance. Formally the last term is one higher order in $\omega_{0}^{-1}$ than the second last, but we shall explain shortly why we keep this term. Longitudinal sound is again unaffected. The eigenvector has $\vec{\xi}$ parallel to $\vec{q}$, as can be checked by multiplying eq. (22) by $q_{j}$ and the sum over $j$ (there is no contribution from either the last or second last terms). The transverse sounds obey

$$\left(\begin{array}[]{cc}\omega^{2}-q^{2}v_{T}^{2}+\frac{\rho_{s}r^{2}}{8\rho_{M}\omega_{0}}q^{2}\omega^{2}&i\frac{\rho_{s}r^{2}}{8\rho_{M}\omega_{0}^{2}}\omega^{3}q^{2}\cos\theta_{q}\\ -i\frac{\rho_{s}r^{2}}{8\rho_{M}\omega_{0}^{2}}\omega^{3}q^{2}\cos\theta_{q}&\omega^{2}-q^{2}v_{T}^{2}+\frac{\rho_{s}r^{2}}{8\rho_{M}\omega_{0}}q^{2}\omega^{2}\cos^{2}\theta_{q}\end{array}\right)\left(\begin{array}[]{c}\xi_{\theta}\\ \xi_{\phi}\end{array}\right)=0$$

(23)

For $\theta_{q}$ not too close to $0$ or $\pi$, we can ignore the off-diagonal terms in this matrix equation as they are second order in $\omega_{0}^{-1}$. We obtain two non-degenerate modes with frequencies $\omega=qv_{T}(1+X)^{-1/2}$ (for $\vec{\xi}$ along $\hat{\theta}$) and $\omega=qv_{T}/(1+X\cos^{2}\theta_{q})^{-1/2}$ (for $\vec{\xi}$ along $\hat{\phi}$). Here $X\equiv\frac{\rho_{s}r^{2}q^{2}}{8\rho_{M}\omega_{0}}$ is a q-dependent dimensionless parameter. Thus the mode with $\vec{\xi}$ along $\hat{\theta}$ has a lower frequency than the one with $\hat{\phi}$ due to the coupling to the pseudospin. For $\theta_{q}=0$ or $\pi$, these two modes are degenerate up to $\omega_{0}^{-1}$. The off-diagonal term then turns these transverse modes to circularly polarized. For $\theta_{q}=0$, the modes with $(\xi_{\theta},\xi_{\phi})\propto(1,\pm i)$ have frequencies roughly given by $\omega\approx qv_{T}(1+X)^{-1/2}[1\mp X_{2}]$, with the dimensionless parameter $X_{2}\equiv\frac{\rho_{s}r^{2}q^{2}}{16\rho_{M}\omega_{0}}\frac{qv_{T}}{\omega_{0}}$. Note that both $X$ and $X_{2}$ are increasing functions of $q$. Similar to the case in subsection III.1, the sign in front of $X_{2}$ in this expression for $\omega$ needs to be reversed for $\theta_{q}=\pi$. Note that $X_{2}\ll X$ since we are now considering $qv_{T}\ll\omega_{0}$ and also that the circular polarization for the higher frequency mode is opposite to the anti-adiabatic case for a given $\hat{q}$. For general $\theta$, the modes are elliptically polarized. See Fig 1b.

The Lagrangian density that reproduces the equation of motion (18) can easily found to be

$$L=L_{0,ph}+\frac{r^{2}\rho_{s}}{16}\epsilon_{jkl}\left(\frac{\partial^{2}\xi_{j}}{\partial z\partial x_{k}}\right)\left(\frac{\partial\xi_{l}}{\partial t}\right)$$

(24)

The last term, in the form of an effective Lorentz force, might have been expected from phenomenological grounds. An initial guess might be a term proportional to $\hat{z}\cdot(\vec{\xi}\times\frac{\vec{\partial}\xi}{\partial t})$: this term does arise in the case of optical phonons [31, 32, 41], but here this is not acceptable since the appearance of $\vec{\xi}$ violates translational invariance. Instead, in eq. (24), a second order spatial derivative appears instead, similar to what has been discussed in [13, 40], though in our case the precise form, as derived in Sec II, is different here.

The conjugate momentum $\Pi_{j}$ is given by

$$\Pi_{j}\equiv\frac{\partial L}{\partial\dot{\xi}_{j}}=\rho_{M}\left(\frac{\partial\xi_{j}}{\partial t}\right)-\frac{r^{2}\rho_{s}}{16}\epsilon_{jkl}\left(\frac{\partial^{2}\xi_{l}}{\partial z\partial x_{k}}\right)$$

(25)

with the equation of motion (18) just the same as $\frac{\partial\Pi_{j}}{\partial t}=\frac{\partial L}{\partial\xi_{j}}$. After Fourier transforming the spatial coordinates, these two equations can be written in matrix form

$$\frac{\partial}{\partial t}\left(\begin{array}[]{cc}\rho_{M}\hat{1}&0\\ \rho_{M}\Omega&\hat{1}\end{array}\right)\left(\begin{array}[]{c}\xi\\ \Pi\end{array}\right)=\left(\begin{array}[]{cc}-\rho_{M}\Omega&\hat{1}\\ -\mathcal{Q}&0\end{array}\right)\left(\begin{array}[]{c}\xi\\ \Pi\end{array}\right)$$

(26)

where $\Omega$, $\mathcal{Q}$, $\hat{1}$ are $3\times 3$ matrices: $\Omega\equiv Z(qv_{T})\cos\theta_{q}\hat{\Omega}$ with $Z$ defined in eq (21), $\hat{\Omega}_{jk}\equiv-\epsilon_{jkl}\hat{q}_{l}$, $\mathcal{Q}_{jk}\equiv\lambda_{1}q^{2}\delta_{jk}+\lambda_{2}q_{j}q_{k}$, and $\hat{1}_{jk}=\delta_{jk}$.

Eq (26) can be rewritten as

$$\frac{\partial}{\partial t}\left(\begin{array}[]{c}\xi\\ \Pi\end{array}\right)=-i\mathcal{S}\left(\begin{array}[]{c}\xi\\ \Pi\end{array}\right)$$

(27)

with $\xi$, $\Pi$ column matrices consisting of elements $\xi_{x,y,z}$ and $\Pi_{x,y,z}$, and $\mathcal{S}$ a $6\times 6$ matrix given by

$$\mathcal{S}=\left(\begin{array}[]{cc}-i\Omega&i/\rho_{M}\\ -i\mathcal{Q}&-i\Omega\end{array}\right)$$

(28)

where, rigorously speaking, the lower left element should have been $-i\mathcal{Q}+i\rho_{M}\Omega^{2}$, and we have taken the simpler form since $\Omega^{2}$ is second order in $1/\omega_{0}$ and hence higher order than the other terms we kept.

Following [40], we search for the row vectors $(\vec{u},\vec{v})$ which satisfy, for positive frequencies $\omega$,

$$\omega(\vec{u},\vec{v})=(\vec{u},\vec{v})\mathcal{S}$$

(29)

Once $(\vec{u},\vec{v})$’s are found, the Berry curvatures $\vec{\Omega}_{B}$ can then be evaluated via the formulas collected in Appendix B. For the longitudinal mode, $(\vec{u},\vec{v})=(u_{q}\hat{q},v_{q}\hat{q})$. The transverse modes can be more easily written in terms of $u_{\theta,\phi}$ and $v_{\theta,\phi}$ defined via $\vec{u}=u_{\theta}\hat{\theta}+u_{\phi}\hat{\phi}$ and similarly for $\vec{v}$. They obey (observe that $\hat{\theta}\hat{\Omega}=-\hat{\phi}$ and $\hat{\phi}\hat{\Omega}=\hat{\theta}$)

$$\omega\left(\begin{array}[]{c}u_{\theta}\\ u_{\phi}\\ v_{\theta}\\ v_{\phi}\end{array}\right)=\left(\begin{array}[]{cccc}&-iZqv_{T}\cos\theta_{q}&-i\lambda_{1}q^{2}&\\ +iZqv_{T}\cos\theta_{q}&&&-i\lambda_{1}q^{2}\\ i\rho_{M}&&&-iZqv_{T}\cos\theta_{q}\\ &i\rho_{M}&iZqv_{T}\cos\theta_{q}&\end{array}\right)\left(\begin{array}[]{c}u_{\theta}\\ u_{\phi}\\ v_{\theta}\\ v_{\phi}\end{array}\right)$$

(30)

The right (left) circular polarized mode has eigenvector (normalized according to eq. (51))

$$\left(\frac{(\rho_{M}qv_{T})^{1/2}}{2},\pm\frac{i(\rho_{M}qv_{T})^{1/2}}{2},\frac{i}{2(\rho_{M}qv_{T})^{1/2}},\mp\frac{1}{2(\rho_{M}qv_{T})^{1/2}}\right),$$

(31)

frequencies $\omega=qv_{T}(1\pm Z\cos\theta_{q})$ (c.f. eq ( 20)) and curvature $\vec{\Omega}_{B}=\pm\hat{q}/q^{2}$.

In this regime, eq. (22) indicates that the equation for the frequency is cubic. This creates complications if we want to treat the problem in the same way as in the last subsection. However, since we are treating the pseudospin-phonon coupling as small, we can simplify the problem by noting the fact that since the last term in eq. (22) is thus already small, we can replace $\omega^{2}$ there by the “ unperturbed” transverse sound frequency $(qv_{T})^{2}$ (transverse since the last term affects only the transverse modes). Thus we now consider the effective equation of motion

$$0=\rho_{M}\omega^{2}\xi_{j}-\left[\lambda_{1}q^{2}\xi_{j}+\lambda_{2}q_{l}(q_{j}\xi_{l})\right]-\frac{r^{2}\rho_{s}\omega^{2}}{8\omega_{0}}\left[-q_{z}^{2}\xi_{j}+q_{z}q_{j}\xi_{z}+(q_{z}(q_{l}\xi_{l})-q^{2}\xi_{z})\delta_{jz}\right]-i\frac{r^{2}\rho_{s}}{8}\frac{\omega(qv_{T})^{2}}{\omega_{0}^{2}}q_{z}(\vec{q}\times\vec{\xi})_{j}$$

(32)

This equation reproduces the sound velocites discussed near the end of Sec. III.2 and we can check that the displacement eigenvectors found below are proportional to those found there.

The Lagrangian density that reproduces this equation of motion can easily found to be

$$L=L_{0,ph}+\frac{r^{2}\rho_{s}}{8\omega_{0}}\left[\frac{1}{2}\left(\frac{\partial^{2}\xi_{l}}{\partial z\partial t}\right)^{2}-\left(\frac{\partial^{2}\xi_{z}}{\partial z\partial t}\right)\left(\frac{\partial^{2}\xi_{l}}{\partial x_{l}\partial t}\right)+\frac{1}{2}\left(\frac{\partial^{2}\xi_{z}}{\partial x_{l}\partial t}\right)^{2}\right]+\frac{r^{2}\rho_{s}v_{T}^{2}}{16\omega_{0}^{2}}\nabla^{2}\vec{\xi}\cdot\vec{\nabla}\times\left(\frac{\partial^{2}\vec{\xi}}{\partial z\partial t}\right)$$

(33)

Carrying out the same procedure as in the last subsection, we obtain

$$\frac{\partial}{\partial t}\left(\begin{array}[]{cc}\rho_{M}(1+X\hat{\Lambda})&0\\ -\rho_{M}\tilde{\Omega}&1\end{array}\right)\left(\begin{array}[]{c}\xi\\ \Pi\end{array}\right)=\left(\begin{array}[]{cc}\rho_{M}\tilde{\Omega}&1\\ -\mathcal{Q}&0\end{array}\right)\left(\begin{array}[]{c}\xi\\ \Pi\end{array}\right)$$

(34)

where $\tilde{\Omega}\equiv X_{2}qv_{T}\cos\theta_{q}\hat{\Omega}$ (dimension frequency) with $X,X_{2}$ defined in III.2 and $\hat{\Omega}_{jk}$ $\mathcal{Q}_{jk}$ already defined in subsection IV.1,

$$\hat{\Lambda}\equiv\left(\begin{array}[]{ccc}\hat{q}_{z}^{2}&0&-\hat{q}_{x}\hat{q}_{z}\\ 0&\hat{q}_{z}^{2}&-\hat{q}_{y}\hat{q}_{z}\\ -\hat{q}_{z}\hat{q}_{x}&-\hat{q}_{z}\hat{q}_{y}&q_{x}^{2}+q_{y}^{2}\end{array}\right)$$

(35)

We have again the equation (27) with now

$$\mathcal{S}=\left(\begin{array}[]{cc}i[1+X\hat{\Lambda}]^{-1}\tilde{\Omega}&i/\rho_{M}[1+X\hat{\Lambda}]^{-1}\\ -i\mathcal{Q}+i\rho_{M}\tilde{\Omega}[1+X\hat{\Lambda}]^{-1}\tilde{\Omega}&i\tilde{\Omega}[1+X\hat{\Lambda}]^{-1}\end{array}\right)$$

(36)

which, in accordance with our approximations, the second term in the lower left element can be dropped.

We can solve for the eigenvectors $(\vec{u},\vec{v})$ as before. It is useful to note the vector relations $\hat{q}\hat{\Lambda}=0$, $\hat{\theta}\hat{\Lambda}=\hat{\theta}$ and $\hat{\phi}\hat{\Lambda}=\cos^{2}\theta_{q}\hat{\phi}$. Once more, for longitudinal modes, $(\vec{u},\vec{v})=(u_{q}\hat{q},v_{q}\hat{q})$ is unaffected by the pseudospin. If $\theta_{q}$ is not too close to $0$ or $\pi$, in the first approximation we can ignore the effects of $\tilde{\Omega}$. The modes are thus linearly polarized with either $\vec{u}$, $\vec{v}$ entirely along $\hat{\theta}$ or $\hat{\phi}$ with frequencies already given in subsection III.2. The normalized eigenvectors are, respectively,

$$(u_{\theta},v_{\theta})_{0}=\left(\frac{(\rho_{M}qv_{T})^{1/2}(1+X)^{1/4}}{\sqrt{2}},\frac{i}{\sqrt{2}(\rho_{M}qv_{T})^{1/2}(1+X)^{1/4}}\right)$$

(37)

and

$$(u_{\phi},v_{\phi})_{0}=\left(\frac{(\rho_{M}qv_{T})^{1/2}(1+X\cos^{2}\theta_{q})^{1/4}}{\sqrt{2}},\frac{i}{\sqrt{2}(\rho_{M}qv_{T})^{1/2}(1+X\cos^{2}\theta_{q})^{1/4}}\right)$$

(38)

for the lower and higher frequency mode. Here the subscript $0$ reminds us that we have ignored $\tilde{\Omega}$. The effect of finite $\tilde{\Omega}$ can be included by perturbation theory, using eqs. (37) and (38) as the “unperturbed” solutions. For the lower frequency mode, the wavevector can be written as $(\vec{u},\vec{v})=(u_{\theta,0}\hat{\theta},v_{\theta,0}\hat{\theta})+\beta(u_{\phi,0}\hat{\phi},v_{\phi,0}\hat{\phi})$ where $\beta$ is a small coefficient. We find that $\beta$ is imaginary with

$${\rm Im}\beta=\frac{X_{2}}{2X}\frac{\cos\theta_{q}}{\sin^{2}\theta_{q}}(1+X)^{1/4}(1+X\cos^{2}\theta_{q})^{1/4}\left[(1+X\cos^{2}\theta_{q})^{1/2}+(1+X\cos^{2}\theta_{q})^{1/2}\right]$$

(39)

hence ${\rm Im}\beta$ has the same sign as $\cos\theta_{q}$. For $q_{z}>0$, the lower frequency mode is right elliptically polarized (vice versa for $q_{z}<0$). Similarly, the higher frequency mode (the $\phi$ mode before perturbation) becomes left elliptically polarized, with the degree of ellipticity characterized by the same coefficient ${\rm Im}\beta$.

For $\theta_{q}=0$, the modes are circularly polarized, with normalized eigenvectors

$$(u_{\theta},u_{\phi},v_{\theta},v_{\phi})=\left(\frac{(\rho_{M}qv_{T})^{1/2}(1+X)^{1/4}}{2},\mp\frac{i(\rho_{M}qv_{T})^{1/2}(1+X)^{1/4}}{2},\frac{i}{2(\rho_{M}qv_{T})^{1/2}(1+X)^{1/4}},\frac{\pm 1}{2(\rho_{M}qv_{T})^{1/2}(1+X)^{1/4}}\right)$$

(40)

for the higher (left-circularly polarized) and lower (right-circularly polarized) frequency modes, respectively. The opposite signs are to be taken if $\theta_{q}=\pi$.

Eq. (39) together with (37) and (38) allow us to obtain the Berry curvature. $\vec{\Omega}_{B}$ has no $\phi$ component. For $\theta_{q}$ not too close to $0$ or $\pi$, for the lower frequency mode,

$$\vec{\Omega}_{B}\cdot\hat{\theta}=\frac{2v_{T}}{q\omega_{0}}\frac{\cos^{2}\theta_{q}}{\sin^{3}\theta_{q}}\ ,$$

(41)

$$\vec{\Omega}_{B}\cdot\hat{q}=\frac{4v_{T}}{q\omega_{0}}\frac{\cos\theta_{q}}{\sin^{4}\theta_{q}}\ .$$

(42)

Here we have only kept the lowest order finite terms and have used $\frac{1}{q^{2}}\frac{X_{2}}{X}=\frac{v_{T}}{2q\omega_{0}}$. For the higher frequency mode, there is an extra negative sign for these formulas.

For $\theta_{q}=0$, we obtain $\vec{\Omega}_{B}=\mp 1/q^{2}$ for the two modes in eq. (40) [42].