Lattice dynamics with molecular Berry curvature: chiral optical phonons

Under the Born-Oppenheimer approximation, electrons stay at their instantaneous ground state $|\Phi_{0}(\{\bm{R}\})\rangle$ at a given time with lattice configuration $\{\bm{R}\}$. When the lattice configuration evolves, the electronic ground state evolves adiabatically and accumulates a geometrical phase, which in turn can modify the lattice dynamics. The geometrical phase manifests itself as a gauge field $\bm{A}_{l,\kappa}$ in the lattice Hamiltonian (it was originally reported by Mead and Truhlar MeadTruhlar and an alternative derivation is shown in Appendix A)

$\displaystyle H_{L}=\sum_{l,\kappa}\frac{1}{2M_{\kappa}}\left(\bm{p}_{l,\kappa}-\hbar\bm{A}_{l,\kappa}(\{\bm{R}\})\right)^{2}+V_{\text{eff}}(\{\bm{R}\})$

(1)

where $\bm{p}_{l,\kappa}=-i\hbar\bm{\nabla}_{l,\kappa}$ with $\bm{\nabla}_{l,\kappa}=\partial/\partial\bm{R}_{l,\kappa}$ is the canonical momentum of the $\kappa$-th atom at the $l$-th unit cell with a coordinate $\bm{R}_{l,\kappa}$ and a mass $M_{\kappa}$. The scalar potential $V_{\rm eff}(\{\bm{R}\})$ is contributed from the Coulomb interaction of the ions and the electrons whereas the vector potential $\bm{A}_{l,\kappa}(\{\bm{R}\})=i\langle\Phi_{0}(\{\bm{R}\})|\bm{\nabla}_{l,\kappa}\Phi_{0}(\{\bm{R}\})\rangle$ is the molecular Berry connection that describes the geometrical phase of the electronic ground state. Although the molecular Berry connection is gauge dependent, it can give rise to a gauge invariant molecular Berry curvature TQin2012 ; DXiao2010

$$\displaystyle G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{R}_{l},\bm{R}_{l^{\prime}})=2\text{Im}\Big{\langle}\frac{\partial\Phi_{0}}{\partial R_{l^{\prime},\kappa^{\prime}\beta}}\Big{|}\frac{\partial\Phi_{0}}{\partial R_{l,\kappa\alpha}}\Big{\rangle}$$

(2)

where the indices $\alpha,\beta$ represent the Cartesian components of the coordinates. One can proceed further under the assumption that every lattice point vibrates around its equilibrium position with $\{\bm{R}\}=\{\bm{R}^{0}_{l,\kappa}+\bm{u}_{l,\kappa},l=1,\dots,N;\kappa=1,\dots,r\}$ where the equilibrium position $\bm{R}^{0}_{l,\kappa}\equiv\bm{R}^{0}_{l}+\bm{d}_{\kappa}$ with $\bm{R}^{0}_{l}$ being the equilibrium position of the $l$-th unit cell, $\bm{d}_{\kappa}$ being the relative position of the $\kappa$-th ion, and $\bm{u}_{l,\kappa}$ being its displacement. At the equilibrium configuration, the Berry curvature $G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{R}_{l},\bm{R}_{l^{\prime}})$ exhibits translational symmetry that depends on $\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}}$ only.

In the following, we consider a symmetric gauge Holz1972 , which exists near the equilibrium position as shown in Appendix B, such that

$$\displaystyle A_{l,\kappa\alpha}=-\frac{1}{2}\sum_{\kappa^{\prime},\beta,l^{\prime}}G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})u_{l^{\prime},\kappa^{\prime}\beta}.$$

(3)

By taking the advantage of the translational invariance, we express the lattice Hamiltonian in momentum space

	$\displaystyle H_{L}=$	$\displaystyle\sum_{\bm{k},\kappa}\frac{1}{2M_{\kappa}}\left[\bm{p}_{\kappa}(-\bm{k})-\hbar\bm{A}_{\kappa}(-\bm{k})\right]\left[\bm{p}_{\kappa}(\bm{k})-\hbar\bm{A}_{\kappa}(\bm{k})\right]$
		$\displaystyle+V_{\text{eff}}\left(\{\bm{u}(\bm{k})\}\right)$		(4)

where the momentum-space Berry connection

$\displaystyle{A}_{\kappa\alpha}(\bm{k})\doteq\frac{1}{\sqrt{N}}\sum_{l}{A}_{l,\kappa\alpha}e^{-i\bm{k}\cdot\bm{R}^{0}_{l}}=-\frac{1}{2}\sum_{\kappa^{\prime},\beta}G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{k})u_{\kappa^{\prime}\beta}(\bm{k})$

with

	$\displaystyle\bm{u}_{\kappa}({\bm{k}})$	$\displaystyle=\frac{1}{\sqrt{N}}\sum_{l}\bm{u}_{l,\kappa}e^{-i\bm{k}\cdot\bm{R}^{0}_{l}}$		(5)
	$\displaystyle\bm{p}_{\kappa}({\bm{k}})$	$\displaystyle=\frac{1}{\sqrt{N}}\sum_{l}\bm{p}_{l,\kappa}e^{-i\bm{k}\cdot\bm{R}^{0}_{l}}$
	$\displaystyle G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{k})$	$\displaystyle=\frac{1}{N}\sum_{l}\sum_{l^{\prime}}G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})e^{-i\bm{k}\cdot(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})}.$

We further express the momentum-space molecular Berry curvature in a gauge invariant form by employing a set of many-body wavefunction $\{|\Phi_{n}\rangle\}$ with the completeness relation $\sum_{n}|\Phi_{n}\rangle\langle\Phi_{n}|=1$ and associated eigenenergy $E_{n}$. By using the identity $\langle\Phi_{n}|\mathcal{M}_{\bm{k},\kappa\alpha}|\Phi_{n^{\prime}}\rangle=\langle\frac{\partial\Phi_{n}}{\partial u_{-\bm{k},\kappa\alpha}}|\Phi_{n^{\prime}}\rangle(E_{n}-E_{n^{\prime}})$ for $n^{\prime}\neq n$, the molecular Berry curvature reads

	$\displaystyle G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{k})$	$\displaystyle=i\sum_{n\neq 0}\left[\frac{\langle\Phi_{0}|\mathcal{M}_{\bm{k},\kappa\alpha}|\Phi_{n}\rangle\langle\Phi_{n}|\mathcal{M}_{-\bm{k},\kappa^{\prime}\beta}|\Phi_{0}\rangle}{(E_{n}-E_{0})^{2}}\right]$
		$\displaystyle-\{\mathcal{M}_{\bm{k},\kappa\alpha}\leftrightarrow\mathcal{M}_{-\bm{k},\kappa^{\prime}\beta}\}$		(6)

where $E_{0}$ is the energy of the electronic ground state, $E_{n}$ is for the excited states, $\mathcal{M}_{\bm{k},\kappa\alpha}=\frac{\partial H_{e}}{\partial u_{-\bm{k},\kappa\alpha}}|_{u_{-\bm{k},\kappa\alpha}\rightarrow 0}$ represents the electron-phonon coupling with $H_{e}$ being the electronic Hamiltonian that depends on the atomic coordinates. By further taking $\{|\Phi_{n}\rangle\}$ as Slater determinant, the above formula can be expressed in terms of single-particle Bloch wavefunctions as detailed in Appendix D. This expression can readily be applied to a specific model using the first-principles approach.

We further simplify the notation by normalizing the coordinates and expressing the Hamiltonian in terms of matrices. We first define column vectors $p_{\bm{k}}=\left(\dots p_{\kappa\alpha}(\bm{k})/\sqrt{M_{\kappa}}\dots\right)^{T}$ and $u_{\bm{k}}=\left(\dots\sqrt{M_{\kappa}}u_{\kappa\alpha}(\bm{k})\dots\right)^{T}$. For the two dimensional system studied in this work, there are $2r$ elements. We also define the matrix $\tilde{G}_{\bm{k}}$ with elements $\tilde{G}_{\bm{k}}(\kappa\alpha,\kappa^{\prime}\beta)=\frac{\hbar}{2\sqrt{M_{\kappa}M_{\kappa^{\prime}}}}G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{k})$ (see Appendix E). By expressing the potential energy in a quadratic form Callaway1991 $V_{\text{eff}}\left(\{\bm{u}(\bm{k})\}\right)=\frac{1}{2}u_{\bm{k}}^{\dagger}K_{\bm{k}}u_{\bm{k}}$, the lattice Hamiltonian reads

$\displaystyle H_{L}$

$\displaystyle=\sum_{\bm{k}}\frac{1}{2}\left(\begin{matrix}u_{\bm{k}}\\ p_{\bm{k}}\end{matrix}\right)^{\dagger}\left(\begin{matrix}D_{\bm{k}}&\tilde{G}^{\dagger}_{\bm{k}}\\ \tilde{G}_{\bm{k}}&\mathbb{1}\end{matrix}\right)\left(\begin{matrix}u_{\bm{k}}\\ p_{\bm{k}}\end{matrix}\right)$

(7)

where $D_{\bm{k}}=K_{\bm{k}}+\tilde{G}^{\dagger}_{\bm{k}}\tilde{G}_{\bm{k}}$. It is noted that $D_{\bm{k}}$ here is different from that in Ref. TQin2012, . The corresponding canonical equations of motion are Kittel1987 :

$\displaystyle\left(\begin{matrix}\dot{u}_{\bm{k}}\\ \dot{p}_{\bm{k}}\end{matrix}\right)=\left(\begin{matrix}\frac{\partial H_{L}}{\partial{p}_{-\bm{k}}}\\ -\frac{\partial H_{L}}{\partial u_{-\bm{k}}}\end{matrix}\right)=\left(\begin{matrix}\tilde{G}_{\bm{k}}&\mathbb{1}\\ -D_{\bm{k}}&\tilde{G}_{\bm{k}}\end{matrix}\right)\left(\begin{matrix}u_{\bm{k}}\\ p_{\bm{k}}\end{matrix}\right).$

(8)

We then introduce the canonical transformation

	$\displaystyle u_{\bm{k}}$	$\displaystyle=\sum_{\nu}\sqrt{\frac{\hbar}{\omega_{0}}}\left(\gamma^{*}_{\nu}b^{\dagger}_{-\bm{k},\nu}+\gamma_{\nu}b_{\bm{k},\nu}\right)$		(9)
	$\displaystyle p_{\bm{k}}$	$\displaystyle=\sum_{\nu}i\sqrt{\hbar\omega_{0}}\left(\bar{\gamma}^{*}_{\nu}b^{\dagger}_{-\bm{k},\nu}-\bar{\gamma}_{\nu}b_{\bm{k},\nu}\right)$		(10)

to diagonalize the Hamiltonian as

$\displaystyle H_{L}=\sum_{\bm{k},\nu}\hbar\omega_{\bm{k},\nu}\left(b^{\dagger}_{\bm{k},\nu}b_{\bm{k},\nu}+\frac{1}{2}\right)$

(11)

where $b^{\dagger}_{-\bm{k},\nu}$ and $b_{\bm{k},\nu}$ are the creation and annihilation operators of the phonon modes. These operators are constrained by the commutation relation $[b_{\bm{k},\nu},b^{\dagger}_{\bm{k}^{\prime},\nu^{\prime}}]=\delta_{\bm{k},\bm{k}^{\prime}}\delta_{\nu,\nu^{\prime}}$ and the Heisenberg equations of motion

	$\displaystyle\dot{b}_{\bm{k},\nu}$	$\displaystyle=-i\omega_{\bm{k},\nu}b_{\bm{k},\nu}$		(12)
	$\displaystyle\dot{b}^{\dagger}_{-\bm{k},\nu}$	$\displaystyle=i\omega_{-\bm{k},\nu}b^{\dagger}_{-\bm{k},\nu}.$

Assisted by these identities, the phonon energy $\omega_{\bm{k},\nu}$ and the polarization vector $\psi_{\nu}=\left(\begin{matrix}\gamma_{\nu},\bar{\gamma}_{\nu}\end{matrix}\right)^{T}$ need to satisfy the eigenvalue equation

$\displaystyle\omega_{\bm{k},\nu}\psi_{\nu}=\left(\begin{matrix}i\tilde{G}_{\bm{k}}&\omega_{0}\\ \frac{D_{\bm{k}}}{\omega_{0}}&i\tilde{G}_{\bm{k}}\end{matrix}\right)\psi_{\nu}$

(13)

which can be obtained by substituting Eqs. (9) and (10) into Eq. (8). The eigenvalues show particle-hole symmetry like property Qin2011 , i.e., $\omega_{\nu,\bm{k}}=-\omega_{-\nu,-\bm{k}}$. Only the positive branches are physically allowed since only the wavefunctions for those branches can make the commutation relation $[b_{\bm{k},\nu},b^{\dagger}_{\bm{k}^{\prime},\nu^{\prime}}]=\delta_{\bm{k},\bm{k}^{\prime}}\delta_{\nu,\nu^{\prime}}$ valid with the normalization condition $\psi_{\nu}^{\dagger}\sigma_{x}\psi_{\nu}=1$.

One can transform the non-Hermitian problem to a Hermitian one. By multiplying Eq. (13) with $\sigma_{x}$ from left, one can find that

$\displaystyle\omega_{\bm{k},\nu}\sigma_{x}\psi_{\nu}=\Omega_{\bm{k}}\psi_{\nu}$

(14)

with the Hermitian matrix

$\displaystyle\Omega_{\bm{k}}=\left(\begin{matrix}\frac{D_{\bm{k}}}{\omega_{0}}&-i\tilde{G}_{\bm{k}}^{\dagger}\\ i\tilde{G}_{\bm{k}}&\omega_{0}\end{matrix}\right).$

Multiplying Eq. (14) with $\Omega_{\bm{k}}^{\frac{1}{2}}\sigma_{x}$ from left side and introducing a new set of eigenstates $\tilde{\psi}_{\nu}=\Omega_{\bm{k}}^{\frac{1}{2}}\psi_{\nu}$, where $\Omega_{\bm{k}}^{\frac{1}{2}}$ is also Hermitian, we come to a Hermitian eigenvalue problem as

$\displaystyle\omega_{\bm{k},\nu}\tilde{\psi}_{\nu}=\Omega_{\bm{k}}^{1/2}\sigma_{x}\Omega_{\bm{k}}^{1/2}\tilde{\psi}_{\nu}=H_{\text{eff}}\tilde{\psi}_{\nu}$

(15)

where the effective Hamiltonian ${H}_{\text{eff}}$ is Hermitian.

In this section, we present a case study on the dynamics of the honeycomb lattice. In the harmonic approximation, the atoms are considered connected by springs with longitudinal and transverse spring constants $K_{L}$ and $K_{T}$, respectively. Details of this model can be found in Appendix E.

The time-reversal symmetry is broken by the electronic property described by the Haldane model Haldane1988 with a tight-binding Hamiltonian

	$\displaystyle H_{e}=$	$\displaystyle-\sum_{\langle i,j\rangle}ta_{i}^{\dagger}b_{j}+\text{h.c.}$		(16)
		$\displaystyle-\sum_{\langle\langle i,j\rangle\rangle}t^{\prime}e^{i\phi_{ij}}a_{i}^{\dagger}a_{j}-\sum_{\langle\langle i,j\rangle\rangle}t^{\prime}e^{-i\phi_{ij}}b_{i}^{\dagger}b_{j}$
	$\displaystyle=$	$\displaystyle\sum_{\bm{q}}\left(\begin{matrix}a^{\dagger}_{\bm{q}}&b^{\dagger}_{\bm{q}}\end{matrix}\right)\mathcal{H}(\bm{q})\left(\begin{matrix}a_{\bm{q}}\\ b_{\bm{q}}\end{matrix}\right)$

where $a_{i}$ ($a_{i}^{\dagger}$) and $b_{i}$ ($b_{i}^{\dagger}$) are electron creation (annihilation) operators of $A$ and $B$ sublattices respectively in the $i$-th unit cell. The first line represents the nearest neighbor hopping with the hopping energy $t$ while the second line represents the next-nearest neighbor hopping with a flux $\phi_{ij}$ attached to it. We set $\phi_{ij}=\pm\pi/2$ for clockwise/anti-clockwise hoppings. The lattice Hamiltonian can also be expressed in momentum space with kernel $\mathcal{H}(\bm{q})$ and $\bm{q}$ running over the first Brillouin zone. The single-particle Bloch eigenstates for the conduction and valence bands are denoted as $\phi^{c,v}_{\bm{q}}$ with corresponding eigen-energies $\varepsilon^{c,v}_{\bm{q}}$. The Bloch bands are plotted in Fig. 1(a).

The phonons couple to the electronic system through the dependence of the hopping energies $t$ and $t^{\prime}$ on the lattice displacement $\{\bm{u}\}$. The nearest-neighbor hopping energy $t$ depends on the relative distance $d$ between the two atoms. When the inter-atomic distance changes by $\delta d$ due to atomic displacement, $t$ changes by $\partial_{d}t\delta d$. Here we set $t^{\prime}$ as a constant for simplicity.

In this work, we consider the electronic insulating system with the lower Bloch band being completely filled. In the non-interacting case, the many-body ground state $|\Phi_{0}\rangle$ and the excited one $|\Phi_{n}\rangle$ can be expressed as the Slater determinant of single-particle states. One thus can calculate the Berry curvature shown in Eq. 6 by using the single-particle states. We can take the Berry curvature induced by the motion of A sublattices along $x$ and $y$ directions as an example, which can be expressed as

		$\displaystyle G^{Ax}_{Ay}(\bm{k})=\frac{i}{N}\sum_{\bm{q}}\frac{\left[{\phi}^{v\dagger}_{\bm{q}}\mathcal{M}_{\bm{k},Ax}\phi^{c}_{\bm{q}+\bm{k}}\right]\left[{\phi}^{c\dagger}_{\bm{q}+\bm{k}}\mathcal{M}_{\bm{-k},Ay}\phi^{v}_{\bm{q}}\right]}{(\varepsilon^{c}_{\bm{q}+\bm{k}}-\varepsilon^{v}_{\bm{q}})^{2}}$
		$\displaystyle~{}~{}~{}-\frac{i}{N}\sum_{\bm{q}}\frac{\left[{\phi}^{v\dagger}_{\bm{q}+\bm{k}}\mathcal{M}_{\bm{-k},Ay}\phi^{c}_{\bm{q}}\right]\left[{\phi}^{c\dagger}_{\bm{q}}\mathcal{M}_{\bm{k},Ax}\phi^{v}_{\bm{q}+\bm{k}}\right]}{(\varepsilon^{c}_{\bm{q}}-\varepsilon^{v}_{\bm{q}+\bm{k}})^{2}}$		(17)

where $\mathcal{M}_{\bm{\pm k},Ax/Ay}$ represent the electron-phonon couplings as detailed in Appendix D that couple the electronic states with a momentum difference of $\bm{k}$. In Fig. 1(b), we plot the contribution from each electronic momentum $\bm{q}$ to the molecular Berry curvature $G^{Ax}_{Ay}(\bm{k})$ at $\bm{k}=0$. This corresponds to the phonon induced virtual direct inter-band transition process with the peak contribution concentrating at $K$ and $K^{\prime}$ valley. The dependence of $G^{Ax}_{Ay}(\bm{k})$ on $\bm{k}$ is plotted in Fig. 1(c) where one can find that the Berry curvature shows a peak at the phonon Brillouin zone center. By using reasonably realistic parameters, e.g., $t=3~{}$eV, $t^{\prime}=0.02~{}$eV, $\partial_{d}t=1~{}$eV/Å, we find that the peak of molecular Berry curvature corresponds to an effective magnetic field of $\sim 10^{3}~{}$Tesla. Thus, the effect of molecular Berry curvature can be large in materials with narrow electronic band gap, e.g., topological materials. In the small gap limit, we find an analytical formula for the peak value $G^{Ax}_{Ay}(\bm{k}=0)=\frac{3}{2\pi a^{2}}C(\partial_{d}t/t)^{2}$ where $C$ is the Chern number of the system as plotted in the Fig. 1(d). It is noted that the molecular Berry curvature vanishes at $K$ and $K^{\prime}$ points due to the vanishing of the matrix elements in $\partial_{x,y}\mathcal{H}$ between conduction and valence bands at different valleys in this model.

The formula above can be adopted by the first principle calculation directly, which is thus essential for exploring the phonon Hall effect in magnetic materials. The $G^{Ax}_{Ay}$ is an analogy of the effective magnetic field in the Raman spin-lattice coupling model RamanSpinLattice ; Sheng2006 ; Kagan2008 ; LZhang ; PHE_Heff_Huber_16 ; PhononModelDiode . In the Raman spin-lattice coupling model, the motion of $u_{l,Ax}$ can only be influenced by $\dot{u}_{l,Ay}$ on the same site. In contrast, the molecular Berry curvature distributes sharply around the Brillouin zone center. This distribution indicates that, by Fourier transforming back to the real space, the displacement $u_{l,Ax}$ can be influenced by the velocity $\dot{u}_{l^{\prime},Ay}$ that is far away.

Moreover, the $G$ matrix also has off-diagonal blocks with nonzero matrix element $G^{Ax}_{By}$, which reflects the correlation between the motions of the A and the B atoms. The amplitude of this term is comparable with $G^{Ax}_{Ay}$ such that $G^{Ax}_{Ay}=-G^{Ax}_{By}$ at $\bm{k}=0$. The off-diagonal block is thus important, which was completely neglected in the traditional Raman spin-lattice coupling, and can lead to a gap opening of the degenerate acoustic bandsTQin2012 ; Qin2011 . Therefore, the molecular Berry curvature is a better choice to explore the influence of electronic states on phonons in a unified way.

Once the Berry curvature is calculated for our model we can find the phonon spectrum using Eq. (15). The result of the numerical calculation of this equation is presented in FIG. 2. In this figure, we can see a gap opening between the optical branches at the $\Gamma$ point. This band gap is proportional to the molecular Berry curvature. To show this relation, we employ the perturbation method since $\tilde{G}^{\dagger}_{\bm{k}}\tilde{G}_{\bm{k}}\ll K_{\bm{k}}$. From Eq. (14), one can find that

$\displaystyle\omega^{2}_{\bm{k},\nu}\gamma_{\nu}=K_{\bm{k}}\gamma_{\nu}+2i\omega_{\bm{k},\nu}\tilde{G}_{\bm{k}}\gamma_{\nu}.$

(18)

The second term on the right hand side is treated as a perturbation. The unperturbed eigenvalues $\omega_{1,2}$ and eigenstates $\gamma^{0}_{1,2}$ for the optical branches at the $\Gamma$ point can be obtained from $\omega_{\bm{k}}^{2}\gamma^{0}_{\nu}=K_{\bm{k}}\gamma^{0}_{\nu}$. The solutions are $\omega_{1,2}=\omega_{\Gamma}$ and $\gamma^{0}_{1}=\sqrt{\frac{\omega_{0}}{4\omega_{\Gamma}}}\left(\begin{matrix}1&0&-1&0\end{matrix}\right)^{T}$ and $\gamma^{0}_{2}=\sqrt{\frac{\omega_{0}}{4\omega_{\Gamma}}}\left(\begin{matrix}0&1&0&-1\end{matrix}\right)^{T}$. In the presence of the perturbation, the general eigenstate can be expressed as a linear combination $\tilde{\gamma}=c_{1}\gamma^{0}_{1}+c_{2}\gamma^{0}_{2}$ with $c_{1}$ and $c_{2}$ being some constants to be determined. By expanding the phonon eigenvalue at the $\Gamma$ point to the first order as $\omega=\omega_{\Gamma}+\delta\omega$, one can find that

$\displaystyle\delta\omega\left(\begin{matrix}c_{1}\\ c_{2}\end{matrix}\right)=\frac{2i\omega_{\Gamma}}{\omega_{0}}\left(\begin{matrix}{\gamma^{0}_{1}}^{\dagger}\tilde{G}_{\bm{k}}\gamma^{0}_{1}&{\gamma^{0}_{1}}^{\dagger}\tilde{G}_{\bm{k}}\gamma^{0}_{2}\\ {\gamma^{0}_{2}}^{\dagger}\tilde{G}_{\bm{k}}\gamma^{0}_{1}&{\gamma^{0}_{2}}^{\dagger}\tilde{G}_{\bm{k}}\gamma^{0}_{2}\end{matrix}\right)\left(\begin{matrix}c_{1}\\ c_{2}\end{matrix}\right)$

(19)

where ${\gamma^{0}_{i}}^{\dagger}\gamma^{0}_{j}=\frac{\omega_{0}}{2\omega_{\Gamma}}\delta_{ij}$ is employed. Since $\tilde{G}_{\bm{k}}^{\dagger}=-\tilde{G}_{\bm{k}}$, the matrix on the right hand side is Hermitian. Thus the diagonal terms are zero. We find that, by setting $c_{1}=1/\sqrt{2}$ and $c_{2}=\pm i/\sqrt{2}$, the above matrix can be diagonalized with the phonon energy shifts $\delta\omega=\pm\frac{\hbar}{M}\text{Re}\left[G(\bm{k}=0)\right]$. Therefore, the optical phonons are split and the phonon polarizations become right- and left-handed polarized. The splitting of the phonon branches at the Brillouin zone center is expected to be observable in optical spectral experiments.

We would like to point out that, in the absence of the molecular Berry curvature, the dynamical matrix can be written as a real matrix at the Brillouin zone center. As a result, the phonons are always linearly polarized. In the presence of the molecular Berry curvature, however, phonons at the $\Gamma$ point become circularly polarized. This is different from the chiral phonon at the Brillouin zone corner, which has a degenerate state at the opposite momentum. Phonon_Chiral_AngularMom_Lifa_15

By using the phonon wavefunction, one can also define a phonon Berry connection and phonon Berry curvature LZhang . In Fig. 2(b), we plot the phonon Berry curvature along the high-symmetric line for the higher optical branch. Peaks appear at the points where the phonon polarization changes from circular around the $\Gamma$ point to linear away from that point. The phonon Berry curvature contributes to a nonzero Chern number $1$. The Chern number for the lower optical branch is $-1$ whereas the acoustic branches have zero Chern numbers. Associated with the spectrum splitting, the Berry curvature of optical branches can also contribute to the thermal Hall effect Strohm2005 ; Inyushkin2007 ; PHE_SpinLiquid_Exp_17 ; ChiralPhononCuprate_NP_20 .

The circular polarization of phonons also give rise to non nonzero phonon angular momentum LZhang2014 that can be expressed as

$$\displaystyle\bm{J}_{\text{ph}}=\sum_{l\alpha}\bm{u}_{l\alpha}\times\bm{\dot{u}}_{l\alpha}.$$

(20)

For a 2-dimensional system, the vertical component of the angular momentum becomes $J_{z}^{\text{ph}}=\sum_{l,\kappa}(u^{x}_{l\kappa}\dot{u}^{y}_{l\kappa}-u^{y}_{l\kappa}\dot{u}^{x}_{l\kappa})$. It can also be written in a matrix product form

$$\displaystyle J_{z}^{\text{ph}}=\sum_{\bm{k}}u^{\dagger}_{\bm{k}}L\dot{u}_{\bm{k}}=\sum_{\bm{k}}u^{\dagger}_{\bm{k}}L\left(p_{\bm{k}}+\tilde{G}_{\bm{k}}u_{\bm{k}}\right)$$

(21)

where $L$ is a real $2r\times 2r$ antisymmetric matrix for a system with $r$ atoms per unit cell. By using the second quantized expression for the canonical variables of atoms (Eqs. (9)-(10)), we calculate the angular momentum for each phonon branch (Fig. 3(a)). We find that the phonon angular momentum of both acoustic branches vanish whereas the circularly polarized optical branches are nearly quantized. This is in contrast with the phonon angular momentum obtained by using the Raman spin-lattice model which neglects the nonlocal effective magnetic field LZhang2014 ; RamanSpinLatticePhononL . In those calculations, the acoustic phonons at the Brillouin zone center split and can carry nonzero energy and nonzero angular momentum. The splitting of the acoustics bands is induced by breaking the Galilean translational symmetry due the Raman spin-lattice coupling term that meant to serve as an analogy to a uniformly charged lattice under a real magnetic field. This symmetry is respected by the molecular Berry curvature.

By summing over the angular momentum of all the phonon branches, we find a nonzero value as illustrated in Fig. 3(b). This indicates a finite zero-point lattice angular momentum in the zero temperature limit. At finite temperature, the thermal averaged phonon angular momentum is

$\displaystyle\langle J_{z}^{\text{ph}}\rangle=-\sum_{\bm{k},\nu}\gamma^{\dagger}_{\nu}L\gamma_{\nu}\left(\frac{i2\hbar\omega_{\bm{k},\nu}}{\omega_{0}}\right)\Big{(}\frac{1}{2}+f(\omega_{\bm{k},\nu})\Big{)}.$

(22)

Here we used $b_{\bm{k},\nu}b^{\dagger}_{\bm{k},\mu}=\delta_{\mu,\nu}+b^{\dagger}_{\bm{k},\mu}b_{\bm{k},\nu}$, $\langle b^{\dagger}_{\bm{k},\mu}b^{\dagger}_{\bm{k^{\prime}},\nu}\rangle=\langle b_{\bm{k},\mu}b_{\bm{k^{\prime}},\nu}\rangle=0$ and $\langle b^{\dagger}_{\bm{k},\mu}b_{\bm{k},\nu}\rangle=f(\omega_{\bm{k},\nu})\delta_{\mu,\nu}$, where $f(\omega_{\bm{k},\nu})=\frac{1}{e^{\beta\hbar\omega_{\bm{k},\nu}}-1}$ is the Bose-Einstein distribution. In low temperature regime, the phonon angular momentum has a finite value as shown in Fig. 3(c). In this limit, an analytic expression for the phonon angular momentum at the $\Gamma$ point can be found by using the perturbative approximation as:

$\displaystyle\langle J_{z}^{\text{ph}}\rangle_{\Gamma}=\frac{\hbar^{2}}{M\omega_{\Gamma}}\text{Re}\left(G(\bm{k}=0)\right)$

(23)

which is consistent with the numerical calculations of Eq. (22). As the temperature increases, the thermal averaged phonon angular momentum tends to go to zero. As $T\rightarrow\infty$, Eq. (22) approximates to

$\displaystyle\langle J_{z}^{\text{ph}}\rangle=-\sum_{\bm{k},\nu}\gamma^{\dagger}_{\nu}L\gamma_{\nu}\left(\frac{2ik_{B}T}{\omega_{0}}+\frac{i\hbar^{2}\omega^{2}_{k,\nu}}{6\omega_{0}k_{B}T}\right)$

(24)

where the first term vanishes because $\sum_{\nu}\gamma^{\dagger}_{\nu}L\gamma_{\nu}=0$ (see Appendix G) and the second term decreases as $1/T$.

The state of electrons is governed by the time-dependent Schrödinger equation. By assuming a normalized condition, it can be derived from the time-dependent variational principle with Lagrangian $\mathcal{L}_{\rm{e}}=\langle\Phi_{0}|i\hbar d_{t}-H_{\rm{e}}|\Phi_{0}\rangle$ by minimizing the action with respect to any variation of $\langle\Phi_{0}|$ in the bra space. Under the Born-Oppenheimer approximation, the electronic state lies at the instantaneous ground state of the Hamiltonian $H_{e}$ that depends on the lattice configuration $\{\bm{R}\}$. With known instantaneous ground state, one can integrate out the electronic degree of freedom to get the effective Lagrangian of lattice that reads

	$\displaystyle\mathcal{L}$	$\displaystyle=\sum_{l,\kappa}\frac{M_{\kappa}}{2}\dot{\bm{R}}_{l,\kappa}^{2}+\langle\Phi_{0}|i\hbar d_{t}-H_{e}|\Phi_{0}\rangle$		(25)
		$\displaystyle=\sum_{l,\kappa}\frac{M_{\kappa}}{2}\dot{\bm{R}}_{l,\kappa}^{2}+\langle\Phi_{0}|i\hbar d_{t}|\Phi_{0}\rangle-V_{\rm eff}(\{\bm{R}\})$
		$\displaystyle=\sum_{l,\kappa}\frac{M_{\kappa}}{2}\dot{\bm{R}}_{l,\kappa}^{2}+\hbar\bm{A}_{l,\kappa}\cdot\dot{\bm{R}}_{l,\kappa}-V_{\rm eff}(\{\bm{R}\})$

where $\bm{R}_{l,\kappa}$ labels the position of the $\kappa$-th atom in the $l$-th unit cell with mass $M_{\kappa}$. $\bm{A}_{l,\kappa}=\langle\Phi_{0}|i\nabla_{\bm{R}_{l,\kappa}}|\Phi_{0}\rangle$ is the the Berry connection. $V_{\rm eff}(\{\bm{R}\})$ is the total energy of the electrons and ions at the configuration $\{\bm{R}\}$ that forms the potential landscape of the ion. In the equilibrium configuration $\{\bm{R}^{0}\}$, $V_{\rm eff}$ takes its minimum. From the Lagrangian, one can reveal the Hamiltonian Eq. 1 by Legendre transformation, which agrees with that derived by Mead and Truhlar MeadTruhlar .

We consider a lattice where each atom vibrates around its equilibrium position with a displacement $u_{l}$ where, in this paragraph, we use shorthand notation for these indices as $\{l,\kappa\alpha\}\rightarrow l$ and $\{l^{\prime},\kappa^{\prime}\beta\}\rightarrow l^{\prime}$. In the small $\{u_{l}\}$ limit, we can expand the Berry connection $A_{l}=\langle\Phi_{0}|i\partial_{l}|\Phi_{0}\rangle$ to the linear order of $\{u_{l}\}$ as $A_{l}=A_{l}^{0}+\partial_{l^{\prime}}A_{l}u_{l^{\prime}}$ where the coefficients $\partial_{l^{\prime}}A_{l}$ are taken in the limit of ${\{u_{l}\}\rightarrow 0}$ and thus are independent of $\{u_{l}\}$. It is noted that the Berry connection $A_{l}$ is expressed in a parameter space of high dimension. It is questionable whether there exists a gauge transform such that $\tilde{A}_{l}=A_{l}-\partial_{l}\chi=-1/2\sum_{l^{\prime}}G_{l,l^{\prime}}u_{l^{\prime}}$ with gauge invariant $G_{ll^{\prime}}=\partial_{l}A_{l^{\prime}}-\partial_{l^{\prime}}A_{l}=\partial_{l}\tilde{A}_{l^{\prime}}-\partial_{l^{\prime}}\tilde{A}_{l}$. The answer is yes. One can first define $\delta A_{l}=A_{l}-\tilde{A}_{l}$. By definition, $\delta A_{l}=A_{l}^{0}+i\sum_{l^{\prime}}u_{l^{\prime}}(\frac{1}{2}\langle\partial_{l^{\prime}}\Phi_{0}|\partial_{l}\Phi_{0}\rangle+\frac{1}{2}\langle\partial_{l}\Phi_{0}|\partial_{l^{\prime}}\Phi_{0}\rangle+\langle\Phi_{0}|\partial_{l^{\prime}}\partial_{l}\Phi_{0}\rangle)$. It can be verified that $\partial_{l}\delta A_{l^{\prime}}-\partial_{l^{\prime}}\delta A_{l}=0$. According to Poincaré’s Lemma, there always exists locally a scalar function $\chi$ such that $\delta A_{l}=\partial_{l}\chi$. One can thus perform such a gauge transformation $e^{i\chi}|\Phi_{0}\rangle$ to obtain the Berry connection in the symmetric form.

We can now express the Berry connection (gauge field) in a symmetric gauge. The gauge invariant Berry curvature can be written as

	$\displaystyle G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})=\left[\frac{\partial A_{l^{\prime},\kappa^{\prime}\beta}}{\partial u_{l,\kappa\alpha}}-\frac{\partial A_{l,\kappa\alpha}}{\partial u_{l^{\prime},\kappa^{\prime}\beta}}\right]$
	$\displaystyle~{}~{}~{}~{}=i\left[\Big{\langle}\frac{\partial\Phi_{0}}{\partial u_{l,\kappa\alpha}}\Big{|}\frac{\partial\Phi_{0}}{\partial u_{l^{\prime},\kappa^{\prime}\beta}}\Big{\rangle}-\Big{\langle}\frac{\partial\Phi_{0}}{\partial u_{l^{\prime},\kappa^{\prime}\beta}}\Big{|}\frac{\partial\Phi_{0}}{\partial u_{l,\kappa\alpha}}\Big{\rangle}\right]$		(26)

Near the equilibrium position, the Berry connection in the symmetric gauge is

$\displaystyle A_{\kappa\alpha}(\bm{R}^{0}_{l})=-\frac{1}{2}\sum_{l^{\prime},\kappa^{\prime}\beta}G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})u_{\kappa^{\prime}\beta}(\bm{R}^{0}_{l^{\prime}})$

(27)

and in momentum space:

$\displaystyle A_{\kappa\alpha}(\bm{k})=-\sum_{\kappa^{\prime},\beta}\frac{1}{2}G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{k})u_{\kappa^{\prime}\beta}(\bm{k})$

(28)

In the presence of time reversal symmetry, the Berry curvature $G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})=0$ as shown below. We consider the electronic ground state that preserves time reversal invariance and is nondegenerate. Therefore, under time reversal operation $\Theta$, the ground state $|\Phi_{0}\rangle$ becomes $|\tilde{\Phi}_{e}\rangle=|\Theta\Phi_{0}\rangle=e^{i\phi}|\Phi_{0}\rangle$ with a possible phase difference. Therefore, the Berry connection obtained from $|\tilde{\Phi}_{e}\rangle$ is $\tilde{A}_{l,\kappa\alpha}=A_{l,\kappa\alpha}+\partial_{l,\kappa\alpha}\phi$. Alternatively, $\tilde{A}_{l,\kappa\alpha}=i\langle\Theta{\Phi}_{e}|\partial_{l,\kappa\alpha}|\Theta{\Phi}_{e}\rangle=i(\langle{\Phi}_{e}|\partial_{l,\kappa\alpha}|{\Phi}_{e}\rangle)^{*}$ by the definition of the time reversal operator $\Theta$ with ^∗ being the complex conjugate. Thus, $\tilde{A}_{l,\kappa\alpha}=-A_{l,\kappa\alpha}$. As a result, $A_{l,\kappa\alpha}=\partial_{l,\kappa\alpha}\phi/2$. The corresponding Berry curvature $G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})=\partial_{l,\kappa\alpha}A_{l^{\prime},\kappa^{\prime}\beta}-\partial_{l^{\prime},\kappa^{\prime}\beta}A_{l,\kappa\alpha}=0$.

Since the Berry curvature in real space is real number and $G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})=-G^{\kappa^{\prime}\beta}_{\kappa\alpha}(\bm{R}^{0}_{l^{\prime}}-\bm{R}^{0}_{l})$, it can be shown by definition that $G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{k})=-G^{\kappa^{\prime}\beta}_{\kappa\alpha}(-\bm{k})=-G^{\kappa^{\prime}\beta}_{\kappa\alpha}(\bm{k})^{*}$. Thus, $G(\bm{k})=-G(\bm{k})^{\dagger}$.

Considering the translational symmetry, one can find that when all the displacement vectors $\bm{u}_{l,\kappa}$ change by the same small amount $\delta\bm{u}$, the Berry connections do not change, i.e., $A_{l,\kappa\alpha}(\{\bm{u}\}+\delta\bm{u})=A_{l,\kappa\alpha}(\{\bm{u}\})$. Thus, $\delta\bm{u}\sum_{l^{\prime},\kappa^{\prime}\beta}\partial_{l^{\prime},\kappa^{\prime}\beta}A_{l,\kappa\alpha}=0$. Therefore, $\sum_{l^{\prime},\kappa^{\prime}\beta}G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})=2{\rm{Im}}\sum_{l^{\prime},\kappa^{\prime}\beta}\partial_{l^{\prime},\kappa^{\prime}\beta}A_{l,\kappa\alpha}=0$.

The molecular Berry curvature can be expressed in general by the many-body wavefunction $\{\Phi_{n}\}$ where $n=0$ stands for the ground state and the $n>0$ are excited states

	$\displaystyle G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{k})$	$\displaystyle=\frac{1}{N}\sum_{l}\sum_{l^{\prime}}G^{\kappa\kappa^{\prime}}_{\alpha\beta}(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})e^{-i\bm{k}\cdot(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})}$		(29)
		$\displaystyle=\frac{1}{N}\sum_{l,l^{\prime},n\neq 0}i\left[\Big{\langle}\frac{\partial\Phi_{0}}{\partial u_{l,\kappa\alpha}}\Big{|}\Phi_{n}\Big{\rangle}\Big{\langle}\Phi_{n}\Big{|}\frac{\partial\Phi_{0}}{\partial u_{l^{\prime},\kappa^{\prime}\beta}}\Big{\rangle}-(u_{l,\kappa\alpha}\leftrightarrow u_{l^{\prime},\kappa^{\prime}\beta})\right]_{u\rightarrow 0}e^{-i\bm{k}\cdot(\bm{R}^{0}_{l}-\bm{R}^{0}_{l^{\prime}})}$
		$\displaystyle=\frac{i}{N}\sum_{n\neq 0}\left[\frac{\langle\Phi_{0}|\sum_{l}\frac{\partial H_{e}}{\partial u_{l,\kappa\alpha}}e^{-i\bm{k}\cdot\bm{R}^{0}_{l}}|\Phi_{n}\rangle\langle\Phi_{n}|\sum_{l^{\prime}}\frac{\partial H_{e}}{\partial u_{l^{\prime},\kappa^{\prime}\beta}}e^{i\bm{k}\cdot\bm{R}^{0}_{l^{\prime}}}|\Phi_{0}\rangle}{(E_{n}-E_{0})^{2}}\right]$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-~{}\left[\frac{\langle\Phi_{0}|\sum_{l^{\prime}}\frac{\partial H_{e}}{\partial u_{l^{\prime},\kappa^{\prime}\beta}}e^{i\bm{k}\cdot\bm{R}^{0}_{l^{\prime}}}|\Phi_{n}\rangle\langle\Phi_{n}|\sum_{l}\frac{\partial H_{e}}{\partial u_{l,\kappa\alpha}}e^{-i\bm{k}\cdot\bm{R}^{0}_{l}}|\Phi_{0}\rangle}{(E_{n}-E_{0})^{2}}\right]$
		$\displaystyle=\frac{i}{N}\sum_{n\neq 0}\frac{\langle\Phi_{0}|\mathcal{M}_{\bm{k},\kappa\alpha}|\Phi_{n}\rangle\langle\Phi_{n}|\mathcal{M}_{-\bm{k},\kappa^{\prime}\beta}|\Phi_{0}\rangle-\langle\Phi_{0}|\mathcal{M}_{-\bm{k},\kappa^{\prime}\beta}|\Phi_{n}\rangle\langle\Phi_{n}|\mathcal{M}_{\bm{k},\kappa\alpha}|\Phi_{0}\rangle}{(E_{n}-E_{0})^{2}}$

where $\mathcal{M}_{\bm{k},\kappa\alpha}=\sum_{l}\frac{\partial H_{e}}{\partial u_{l,\kappa\alpha}}e^{-i\bm{k}\cdot\bm{R}^{0}_{l}}=\sqrt{N}\frac{\partial H_{e}}{\partial u_{-\bm{k},\kappa\alpha}}$ in the $u_{l}\rightarrow 0$ limit. In this limit, the $\mathcal{M}_{\bm{k},\kappa\alpha}$ involves only single electron scattering process.

In the non-interacting system, the ground state is a product state composed of single-particle states below the chemical potential $\mu$ with $|\Phi_{0}\rangle=\Pi_{\varepsilon_{m,\bm{q}}<\mu}c^{\dagger}_{m,\bm{q}}|0\rangle$ and $c^{\dagger}_{m,\bm{q}}$ being the creation operator of the state at the momentum $\bm{q}$ of the $m$-th band. The excited states can be expressed $|\Phi_{n^{\prime}}\rangle$ is the many body states with one occupied excited state and a hole. One can then express the Berry curvature in single particle wavefunction as

$\displaystyle G^{\kappa\alpha}_{\kappa^{\prime}\beta}(\bm{k})=\frac{i}{N}\sum_{\bm{q}}\sum_{\begin{subarray}{c}\varepsilon_{m}<\mu\\ \varepsilon_{m^{\prime}}>\mu\end{subarray}}\frac{\phi^{\dagger}_{m,\bm{q}}\mathcal{M}_{\bm{k},\kappa\alpha}\phi_{m^{\prime},\bm{q+k}}\phi_{m^{\prime},\bm{q+k}}^{\dagger}\mathcal{M}_{-\bm{k},\kappa^{\prime}\beta}\phi_{m,\bm{q}}}{(\varepsilon_{m,\bm{q}}-\varepsilon_{m^{\prime},\bm{q+k}})^{2}}-\frac{\phi^{\dagger}_{m,\bm{q+k}}\mathcal{M}_{-\bm{k},\kappa^{\prime}\beta}\phi_{m^{\prime},\bm{q}}\phi_{m^{\prime},\bm{q}}^{\dagger}\mathcal{M}_{\bm{k},\kappa\alpha}\phi_{m,\bm{q+k}}}{(\varepsilon_{m,\bm{q+k}}-\varepsilon_{m^{\prime},\bm{q}})^{2}}$

(30)