Acoustic Sum Rules and Invariance Conditions

\[ \Phi_{i \alpha j \beta}(\mathbf{R}) = \Phi_{j \beta i \alpha }(-\mathbf{R}) \]

1. Acoustic Sum Rules

Below are Acoustic Sum Rules (ASR) obeyed by the force constant matrix.

Translation Invariance:

\[ \sum_{\mathbf{R}, j} \Phi_{i \alpha j \beta}(\mathbf{R}) = 0, \]

where \(i,j\) denite the atomic indices and \(\alpha,\beta,\gamma,\delta\) denore the polarization directions. \(\mathbf{R}\) is the position vector of the unitcell. The summation over \(\mathbf{R}\) is in the wigner seitz cell.

Rotation Invariance:

\[ \sum_{\mathbf{R}, j} \left( \Phi_{ i \alpha j \beta} (\mathbf{R}) \tau^{\gamma}_{\mathbf{R} j} - \Phi_{ i \alpha j \gamma} (\mathbf{R}) \tau^{\beta}_{\mathbf{R} j} \right) = 0, \]

where \(\boldsymbol{\tau}_{\mathbf{R} j} = \mathbf{R} + \boldsymbol{\tau}_j\) is the position of atom \(j\) in unitcell located at \(\mathbf{R}\).\(\boldsymbol{\tau}_j\) is the position of atom with respect to the unit cell

Huang Conditions:

\[ \sum_{\mathbf{R}, i, j} \Phi_{i \alpha j \beta} (\mathbf{R}) \tau^{\gamma}_{\mathbf{R} i j} \tau^{\delta}_{\mathbf{R} i j} = \sum_{\mathbf{R}, i, j} \Phi_{ i \gamma j \delta} (\mathbf{R}) \tau^{\alpha}_{\mathbf{R} ij} \tau^{\beta}_{\mathbf{R} ij} \]

where \(\boldsymbol{\tau}_{\mathbf{R} i j} = \boldsymbol{\tau}_{i} - \boldsymbol{\tau}_{j} - \mathbf{R}\) is the distance between two atoms.

2. Enforcing of Acoustic sum rules.

It is often the case that the force constants obtained from the standard abinito calcuations often break acoustic sum rules which leads to non-zero frequencies for the acoustic modes at the \(\Gamma\) point.

In order to enfore ASR, the first step is to explicitly write invariance conditions in form of matrix-vector product i.e \(A\Phi = 0\), where A is the contraint matrix and \(\Phi\) is the force constant vector which is obtain be deflated force-constraints matrix into a vector.

1. Translation Operator

The translation operator tensor A is defined as:

\[ A^{i\alpha\beta}_{\mathbf{R}i'\alpha'j'\beta'} = \delta_{ii'} \delta_{\alpha\alpha'} \delta_{\beta\beta'} \]

Constraint:

\[ \sum_{\mathbf{R}, i', \alpha', j', \beta'} A^{i \alpha \beta}_{\mathbf{R} i' \alpha' j' \beta'} \Phi_{\mathbf{R} i' \alpha' j' \beta'} = 0 \quad \Rightarrow \quad 9 N_A \]

2. Rotation Operator

The rotation operator tensor A is:

\[ A^{i\alpha\beta\gamma}_{\mathbf{R}i'\alpha'j'\beta'} = \delta_{ii'} \delta_{\alpha\alpha'} \left( \delta_{\beta\beta'} \tau^{\gamma}_{\mathbf{R}j'} - \delta_{\gamma\beta'} \tau^{\beta}_{\mathbf{R}j'} \right) \]

Symmetry & Constraints:

\[ A^{i\alpha\beta\gamma} = - A^{i\alpha\gamma\beta} \quad \longrightarrow \quad 3 \cdot N_A \cdot (3) = 9 N_A \]

Resulting Equation (\(A\Phi=0\)):

\[ \sum_{\mathbf{R}, j'} \left( \Phi_{\mathbf{R} i \alpha j' \beta} \tau^{\gamma}_{\mathbf{R} j'} - \Phi_{\mathbf{R} i \alpha j' \gamma} \tau^{\beta}_{\mathbf{R} j'} \right) = 0 \]

3. Huang Operator

The Huang operator tensor A is:

\[ A^{\alpha\beta\gamma\delta}_{\mathbf{R}i'\alpha'j'\beta'} = \left[ \delta_{\alpha\alpha'} \delta_{\beta\beta'} \tau^{\gamma}_{\mathbf{R}i'j'} \tau^{\delta}_{\mathbf{R}i'j'} - \tau^{\alpha}_{\mathbf{R}i'j'} \tau^{\beta}_{\mathbf{R}i'j'} \delta_{\gamma\alpha'} \delta_{\delta\beta'} \right] \]

Symmetry & Constraints:

For \((\alpha, \beta) \longleftrightarrow (\gamma, \delta)\), \(A^{\alpha\beta\gamma\delta}\) is anti-symmetric i.e,

\[ A^{\alpha\beta\gamma\delta} = - A^{\gamma\delta\alpha\beta} \]

For \((\alpha \leftrightarrow \beta)\) and \((\gamma \leftrightarrow \delta)\) exchanges, \(A^{\alpha\beta\gamma\delta}\) is symmetric.

\[ \left. \begin{matrix} (\alpha \leftrightarrow \beta) \\ (\gamma \leftrightarrow \delta) \end{matrix} \right\} \longrightarrow \underline{\underline{15}} \]

Total constraints without reducution : 81 * Due to anti-symmetry property for \((\alpha, \beta) \leftrightarrow (\gamma, \delta)\), it reduces to 36 constraints. * Due to symmetry property for \((\alpha \leftrightarrow \beta)\) and \((\gamma \leftrightarrow \delta)\) further reduces to 15 constants.