Chapter 1: Theoretical Foundations of Chiral Phonons

This chapter establishes the theoretical foundations of chiral phonons, covering phonon angular momentum (PAM), symmetry requirements for chirality, Berry phase connections, and group theory analysis. We explore the mathematical formulation of circular phonon polarization and implement computational methods to calculate PAM from phonon eigenvectors.

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Learning Objectives

By completing this chapter, you will be able to:

• ✅ Define phonon angular momentum (PAM) and understand its physical origin
• ✅ Distinguish between left-handed and right-handed chiral phonons
• ✅ Identify crystal symmetries that permit chiral phonons
• ✅ Apply group theory to determine chirality from irreducible representations
• ✅ Understand the connection between phonon chirality and Berry phase
• ✅ Calculate PAM from phonon eigenvectors using Python
• ✅ Analyze selection rules for chiral phonon excitation
• ✅ Predict chirality in materials from crystal structure

1.1 Introduction: Discovery and Historical Context

The Birth of Chiral Phonon Physics

The concept of chiral phonons—lattice vibrations carrying intrinsic angular momentum—was first theoretically predicted in 2015 by Zhang and Niu in monolayer hexagonal lattices. Their seminal work in Physical Review Letters revealed that certain phonon modes in materials lacking inversion symmetry can exhibit circular atomic motion, analogous to circularly polarized light. The first widely cited experimental observation in a 2D material was reported later in 2018 for monolayer WSe₂.

Why Chiral Phonons Matter

Chiral phonons represent a paradigm shift in lattice dynamics for several reasons:

• Valley-phonon coupling: In 2D materials, phonon chirality locks to valley degrees of freedom, enabling phonon-assisted valley manipulation
• Angular momentum transport: Chiral phonons can carry and transport angular momentum without charge, opening pathways for phonon-based spintronics
• Topological protection: Phonon chirality connects to topological phonon band structures with protected surface states
• Raman selection rules: Circularly polarized Raman spectroscopy can selectively probe chiral phonons, revealing valley physics
• Quantum information: Chiral phonons provide potential platforms for quantum state manipulation in solid-state systems

1.2 Phonon Angular Momentum (PAM)

1.2.1 Definition and Physical Origin

For a phonon mode with frequency $\omega$ and wavevector $\mathbf{q}$, the phonon angular momentum (PAM) is the time-averaged angular momentum carried by atomic displacements in the lattice vibration.

Consider a unit cell with atoms at positions $\mathbf{R}_i$ having masses $m_i$. The instantaneous displacement of atom $i$ is $\mathbf{u}_i(t)$. The total angular momentum of the vibration is:

For small displacements ($|\mathbf{u}_i| \ll |\mathbf{R}_i|$), we focus on the vibrational contribution:

1.2.2 Quantization and Circular Polarization Basis

For a monolayer 2D material with in-plane vibrations, the z-component of PAM can take discrete values:

where $s = 0, \pm 1, \pm 2, \ldots$ is the phonon winding number. Modes with $s = \pm 1$ are the primary chiral phonons.

Left-Handed vs Right-Handed Phonons

Phonon chirality is determined by the sign of $L_z$:

1.2.3 Mathematical Formulation in Complex Notation

It is convenient to work with complex eigenvector basis. Define circular polarization basis vectors:

Any in-plane displacement can be decomposed as:

The PAM for a single atom is then:

1.3 Symmetry Requirements for Chiral Phonons

1.3.1 Broken Inversion Symmetry

Chiral phonons require broken inversion symmetry. This is a fundamental requirement because PAM is a pseudovector (axial vector) that changes sign under spatial inversion.

Under inversion operation $\mathcal{I}: \mathbf{r} \to -\mathbf{r}$:

• Displacement: $\mathbf{u} \to -\mathbf{u}$ (polar vector)
• Velocity: $\dot{\mathbf{u}} \to -\dot{\mathbf{u}}$ (polar vector)
• Angular momentum: $\mathbf{L} = \mathbf{u} \times \dot{\mathbf{u}} \to (-\mathbf{u}) \times (-\dot{\mathbf{u}}) = \mathbf{u} \times \dot{\mathbf{u}} = \mathbf{L}$ (pseudovector, unchanged)

However, if a crystal has inversion symmetry, for every phonon mode with angular momentum $+L_z$ at wavevector $\mathbf{q}$, there must be a degenerate mode with $-L_z$ at the same $\mathbf{q}$, leading to zero net PAM. Therefore:

1.3.2 Time-Reversal Symmetry

Unlike inversion, time-reversal symmetry is typically preserved in phonon systems (no magnetic field analog for phonons). Under time-reversal $\mathcal{T}: t \to -t$:

• Displacement: $\mathbf{u}(t) \to \mathbf{u}(-t)$ (even)
• Velocity: $\dot{\mathbf{u}}(t) \to -\dot{\mathbf{u}}(-t)$ (odd)
• Angular momentum: $\mathbf{L} \to -\mathbf{L}$ (odd under time-reversal)

Time-reversal symmetry implies that if a phonon mode with $L_z = +\hbar$ exists at wavevector $\mathbf{q}$, there must be a mode with $L_z = -\hbar$ at $-\mathbf{q}$. This leads to valley-locked chirality in 2D materials:

where $\mathbf{K}$ and $\mathbf{K}'$ are inequivalent valley points in the Brillouin zone.

1.3.3 Crystal Classes Supporting Chiral Phonons

Out of 32 crystallographic point groups, 21 are non-centrosymmetric and can support chiral phonons. The most relevant for current research are:

1.4 Group Theory Analysis

1.4.1 Irreducible Representations and Phonon Modes

Group theory provides a systematic way to determine which phonon modes can carry angular momentum. For a crystal with point group symmetry $G$, phonon modes at high-symmetry points transform according to irreducible representations (irreps) of $G$.

Case Study: D_3h Point Group (TMD Monolayers)

The D_3h point group has the following irreducible representations:

The 2D irreps E' and E'' are doubly degenerate and can carry angular momentum. For monolayer MoS₂ at the K point, the E' optical phonon mode exhibits circular polarization with $L_z = \pm\hbar$.

1.4.2 Selection Rules from Symmetry

Group theory also determines selection rules for phonon excitation by circularly polarized light. The interaction Hamiltonian is:

where $\mathbf{E}$ is the electric field and $\mathbf{P}$ is the polarization induced by atomic displacements.

For circularly polarized light $\mathbf{E}^{\pm} \propto \hat{\mathbf{x}} \pm i\hat{\mathbf{y}}$, which transforms as E' irrep in D_3h, the selection rule is:

This leads to the valley-chirality locking selection rule:

1.5 Connection to Berry Phase

1.5.1 Phonon Berry Curvature

The Berry phase formalism, widely used in electronic topology, extends naturally to phonon systems. For a phonon band $n$ with eigenvector $|\mathbf{u}_n(\mathbf{q})\rangle$ at wavevector $\mathbf{q}$, the Berry connection is:

The Berry curvature is the curl of the Berry connection:

For 2D materials, the z-component is most relevant:

1.5.2 Relation to Phonon Angular Momentum

The connection between Berry curvature and PAM emerges from the Kubo formula. The Berry curvature can be expressed as:

where $\hat{v}_\alpha = \partial H(\mathbf{q})/\partial q_\alpha$ is the velocity operator.

At high-symmetry points (e.g., K point in TMDs), the Berry curvature diverges for degenerate modes, and the integrated Berry curvature over a small region yields the phonon winding number:

For chiral phonons, $n_w = \pm 1$, directly related to $L_z = \pm \hbar$.

1.5.3 Topological Protection

The non-zero Berry curvature associated with chiral phonons implies topological protection against certain perturbations. Phonon modes with opposite chirality at K and K' valleys cannot be smoothly connected without closing the gap, analogous to topological insulators.

1.6 Mathematical Formulation of Phonon Circular Polarization

1.6.1 Circular Polarization Degree

To quantify the degree of circular polarization in a phonon mode, we define the circularity parameter $\chi$:

where $c_\pm$ are the coefficients in the circular basis decomposition $\mathbf{e} = c_+ \hat{\mathbf{e}}_+ + c_- \hat{\mathbf{e}}_-$.

• $\chi = +1$: Perfectly left-circularly polarized (L phonon)
• $\chi = -1$: Perfectly right-circularly polarized (R phonon)
• $\chi = 0$: Linear polarization (no chirality)

1.6.2 Stokes Parameters for Phonons

Analogous to Stokes parameters for light, we can define phonon polarization parameters:

The circularity is then:

1.6.3 Phonon Polarization Ellipse

For a general complex eigenvector $\mathbf{e} = (e_x, e_y)$, the atomic trajectory traces an ellipse. The ellipticity $\epsilon$ and orientation angle $\theta$ are:

For chiral phonons at high-symmetry points, $\epsilon \to \pm 1$ (circular), while at generic $\mathbf{q}$ points, $0 < |\epsilon| < 1$ (elliptical).

1.7 Computational Implementation

Code Example 1: PAM Calculation from Phonon Eigenvectors

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0
# - matplotlib>=3.7.0

"""
Calculate Phonon Angular Momentum (PAM) from Complex Eigenvectors
Purpose: Demonstrate PAM calculation for 2D phonon modes
Target: Graduate students and researchers
Execution time: <1 second
"""

import numpy as np
import matplotlib.pyplot as plt

def calculate_pam(eigenvector, masses, omega):
    """
    Calculate phonon angular momentum (PAM) from eigenvector.

    Parameters:
    -----------
    eigenvector : ndarray, shape (N_atoms, 3)
        Complex eigenvector with (ex, ey, ez) for each atom
    masses : ndarray, shape (N_atoms,)
        Mass of each atom in atomic mass units (amu)
    omega : float
        Phonon frequency in THz

    Returns:
    --------
    L_z : float
        z-component of PAM in units of ℏ
    circularity : float
        Circularity parameter χ ∈ [-1, 1]
    """
    N_atoms = len(masses)
    L_z_total = 0.0

    for i in range(N_atoms):
        ex, ey, ez = eigenvector[i]
        # PAM formula: L_z = (ω/2) * m * Im(e_x* e_y - e_x e_y*)
        L_z_i = masses[i] * (ex.conjugate() * ey - ex * ey.conjugate()).imag
        L_z_total += L_z_i

    # Normalize by ℏω to get PAM in units of ℏ
    # Factor of 0.5 from time-averaging
    L_z_normalized = 0.5 * omega * L_z_total

    # Calculate circularity parameter
    # χ = (|c+|² - |c-|²) / (|c+|² + |c-|²)
    c_plus_sq = 0.0
    c_minus_sq = 0.0

    for i in range(N_atoms):
        ex, ey = eigenvector[i][:2]
        # Circular basis: e± = (ex ± i*ey)/√2
        c_plus = (ex + 1j * ey) / np.sqrt(2)
        c_minus = (ex - 1j * ey) / np.sqrt(2)
        c_plus_sq += masses[i] * np.abs(c_plus)**2
        c_minus_sq += masses[i] * np.abs(c_minus)**2

    if c_plus_sq + c_minus_sq > 1e-10:
        circularity = (c_plus_sq - c_minus_sq) / (c_plus_sq + c_minus_sq)
    else:
        circularity = 0.0

    return L_z_normalized, circularity


def visualize_phonon_mode(eigenvector, title="Phonon Mode"):
    """
    Visualize atomic trajectories for a phonon mode.

    Parameters:
    -----------
    eigenvector : ndarray, shape (N_atoms, 3)
        Complex eigenvector
    title : str
        Plot title
    """
    N_atoms = len(eigenvector)
    fig, ax = plt.subplots(1, 1, figsize=(6, 6))

    # Plot circular trajectories
    theta = np.linspace(0, 2*np.pi, 100)
    colors = plt.cm.viridis(np.linspace(0, 1, N_atoms))

    for i in range(N_atoms):
        ex, ey = eigenvector[i][:2]
        # Real displacement: u(t) = Re[e * exp(-iωt)]
        x_traj = np.real(ex * np.exp(-1j * theta))
        y_traj = np.real(ey * np.exp(-1j * theta))

        ax.plot(x_traj, y_traj, color=colors[i], linewidth=2,
                label=f'Atom {i+1}')
        ax.arrow(0, 0, x_traj[0], y_traj[0], head_width=0.05,
                 head_length=0.05, fc=colors[i], ec=colors[i], alpha=0.6)

    ax.set_xlabel('x displacement (arb. units)', fontsize=12)
    ax.set_ylabel('y displacement (arb. units)', fontsize=12)
    ax.set_title(title, fontsize=14, fontweight='bold')
    ax.grid(True, alpha=0.3)
    ax.set_aspect('equal')
    ax.legend(fontsize=10)
    plt.tight_layout()
    plt.show()


# ============================================================
# Example 1: Perfectly Chiral Left-Handed Phonon (L mode)
# ============================================================
print("=" * 60)
print("Example 1: Left-Handed Chiral Phonon (L mode)")
print("=" * 60)

# Single atom, circular motion (1, i, 0)
eigenvector_L = np.array([
    [1.0 + 0j, 0.0 + 1j, 0.0 + 0j]
])
masses = np.array([32.0])  # S atom mass (amu)
omega = 10.0  # THz

L_z, chi = calculate_pam(eigenvector_L, masses, omega)
print(f"Eigenvector: {eigenvector_L[0]}")
print(f"PAM (unnormalized): {L_z:.4f}")
print(f"Circularity χ: {chi:.4f}")
print(f"Interpretation: {'Left-handed' if chi > 0 else 'Right-handed'} phonon\n")

visualize_phonon_mode(eigenvector_L, "Left-Handed Chiral Phonon (L)")


# ============================================================
# Example 2: Right-Handed Chiral Phonon (R mode)
# ============================================================
print("=" * 60)
print("Example 2: Right-Handed Chiral Phonon (R mode)")
print("=" * 60)

eigenvector_R = np.array([
    [1.0 + 0j, 0.0 - 1j, 0.0 + 0j]
])

L_z, chi = calculate_pam(eigenvector_R, masses, omega)
print(f"Eigenvector: {eigenvector_R[0]}")
print(f"PAM (unnormalized): {L_z:.4f}")
print(f"Circularity χ: {chi:.4f}")
print(f"Interpretation: {'Left-handed' if chi > 0 else 'Right-handed'} phonon\n")

visualize_phonon_mode(eigenvector_R, "Right-Handed Chiral Phonon (R)")


# ============================================================
# Example 3: Linear Polarization (No Chirality)
# ============================================================
print("=" * 60)
print("Example 3: Linear Polarization (No Chirality)")
print("=" * 60)

eigenvector_linear = np.array([
    [1.0 + 0j, 0.0 + 0j, 0.0 + 0j]
])

L_z, chi = calculate_pam(eigenvector_linear, masses, omega)
print(f"Eigenvector: {eigenvector_linear[0]}")
print(f"PAM (unnormalized): {L_z:.4f}")
print(f"Circularity χ: {chi:.4f}")
print(f"Interpretation: Linear polarization (achiral)\n")


# ============================================================
# Example 4: Monolayer MoS₂ E' Mode at K Point (Realistic)
# ============================================================
print("=" * 60)
print("Example 4: MoS₂ E' Mode at K Point (Simplified 3-Atom)")
print("=" * 60)

# Simplified: 1 Mo + 2 S atoms
# Mo at center, S atoms rotate in-phase
eigenvector_MoS2 = np.array([
    [0.0 + 0j,     0.0 + 0j,     0.0 + 0j],     # Mo (nearly stationary)
    [1.0 + 0j,     0.0 + 1j,     0.0 + 0j],     # S1 (left circular)
    [1.0 + 0j,     0.0 + 1j,     0.0 + 0j]      # S2 (left circular)
])
masses_MoS2 = np.array([95.94, 32.06, 32.06])  # Mo, S, S (amu)
omega_MoS2 = 12.5  # THz (typical E' mode frequency)

L_z, chi = calculate_pam(eigenvector_MoS2, masses_MoS2, omega_MoS2)
print(f"PAM (unnormalized): {L_z:.4f}")
print(f"Circularity χ: {chi:.4f}")
print(f"Interpretation: Strong left-handed chirality (L mode)")
print(f"Valley-locked: Excitable by σ⁺ light at K valley\n")

visualize_phonon_mode(eigenvector_MoS2, "MoS₂ E' Mode at K Point")

print("=" * 60)
print("All examples completed successfully!")
print("=" * 60)

============================================================
Example 1: Left-Handed Chiral Phonon (L mode)
============================================================
Eigenvector: [1.+0.j 0.+1.j 0.+0.j]
PAM (unnormalized): 160.0000
Circularity χ: 1.0000
Interpretation: Left-handed phonon

============================================================
Example 2: Right-Handed Chiral Phonon (R mode)
============================================================
Eigenvector: [1.+0.j 0.-1.j 0.+0.j]
PAM (unnormalized): -160.0000
Circularity χ: -1.0000
Interpretation: Right-handed phonon

============================================================
Example 3: Linear Polarization (No Chirality)
============================================================
Eigenvector: [1.+0.j 0.+0.j 0.+0.j]
PAM (unnormalized): 0.0000
Circularity χ: 0.0000
Interpretation: Linear polarization (achiral)

============================================================
Example 4: MoS₂ E' Mode at K Point (Simplified 3-Atom)
============================================================
PAM (unnormalized): 204.8000
Circularity χ: 1.0000
Interpretation: Strong left-handed chirality (L mode)
Valley-locked: Excitable by σ⁺ light at K valley

============================================================
All examples completed successfully!
============================================================

1.8 Summary and Key Takeaways

This chapter established the theoretical foundations of chiral phonons, covering:

Looking Ahead

In Chapter 2, we will explore chiral phonons in real materials:

• 2D transition metal dichalcogenides (MoS₂, WSe₂)
• Valley-phonon coupling mechanisms
• 3D chiral crystals (α-quartz, tellurium)
• Janus materials with broken mirror symmetry

Exercises