Topological Phonon EMPs in Quantum Materials

• Topological phonon EMPs are quasiparticle excitations in crystalline solids whose internal spin, orbital, or pseudospin angular momentum is rigidly locked to momentum, yielding unique topological properties.
• Effective models such as the Dirac and spin-3/2 Hamiltonians quantify their momentum-dependent winding and topological invariance, underpinning controlled manipulation in device applications.
• Engineered stacking, interface-induced chirality, and external fields enable targeted switching of EMP winding patterns, fostering innovations in spintronics, orbitronics, and quantum information.

Topological phonon emergent particles (EMPs) are quasiparticle excitations in crystalline solids where chirality—manifested as orbital, spin, or generalized pseudospin angular momentum—is rigidly locked to the momentum of the underlying phonon or electronic Bloch wave. These EMPs arise in materials and interfaces with broken inversion or mirror symmetry, topological band structures, or engineered metasurfaces supporting exotic evanescent or guided modes. The momentum-dependent winding of a local spin, orbital, or pseudospin leads to nontrivial topological properties, selective transport, and unique coupling to external fields, enabling robust functionalities for spintronics, orbitronics, photonics, and quantum information.

1. Fundamental Principles of Chirality-Momentum Locking

The central principle underlying topological phonon EMPs is the locking of an internal degree of freedom—spin, orbital, or pseudospin—to the crystal momentum of an excitation. Formally, in an eigenstate with wavevector $\mathbf{k}$, the expectation value $\langle \hat{O} \rangle$ (where $\hat{O}$ can be spin $\mathbf{S}$, orbital angular momentum $\mathbf{L}$, or a pseudospin operant in the orbital sector) satisfies a one-to-one correspondence with $\hat{\mathbf{k}}$, typically as $\langle \hat{O} \rangle \propto \hat{\mathbf{k}}$ (parallel locking) or $$\langle \hat{O} \rangle \propto (\hat{\mathbf{k}} \times \hat{n})$$ (orthogonal locking; $\hat{n}$ is surface normal or polar axis).

In two-dimensional chiral borophene, momentum-locked "orbital pseudospin" arises from the $p_x/p_y$ orbital manifold with basis states $$|\pm\rangle = (|p_x\rangle \pm i|p_y\rangle)/\sqrt{2}$$, carrying angular momentum $m = \pm 1$ about $z$ (Lima et al., 2019). The low-energy Dirac Hamiltonian $h_k = \hbar v_D(q_x \sigma_y - q_y \sigma_x)$ near the $K/K'$ points realizes a winding of the orbital pseudospin expectation $\langle \sigma_x \rangle, \langle \sigma_y \rangle$ about the Fermi surface, which is topologically quantized and chirality-dependent.

In spin-orbit-coupled surfaces, such as Bi$_2$Se$_3$, the orbital angular momentum $L$ of the surface states is locked perpendicular to $k$ and anti-aligned to the spin, forming chiral OAM textures that coexist with standard spin-momentum locking (Park et al., 2011).

2. Effective Models and Topological Characterization

EMP phenomena are rigorously captured by effective low-energy Hamiltonians featuring momentum-dependent internal degrees of freedom:

• Orbital Pseudospin Dirac Model: As in chiral borophene, the $\{|+\rangle,|-\rangle\}$ basis supports Pauli operators $\sigma_{x,y,z}$, yielding a Dirac band at $K/K'$ with eigenstates $$\langle \sigma_x \rangle_\lambda(\mathbf{q}) = -\lambda q_y/|\mathbf{q}|$$, $$\langle \sigma_y \rangle_\lambda(\mathbf{q}) = \lambda q_x/|\mathbf{q}|$$, generating a topological winding of $2\pi$ about each Dirac point, analogous to helical spin winding in TI surfaces (Lima et al., 2019).
• Spin-3/2 Multifold Fermions: In chiral topological semimetals, e.g., PtGa, multifold fermions with Hamiltonian $H_{3/2}(\mathbf{k}) = v\sum_i k_i J_i$ ($J_i$: spin-3/2 matrices) entail a purely parallel spin-momentum locking $S(\mathbf{k}) \propto \mathbf{k}/|\mathbf{k}|$, as directly confirmed by spin-resolved ARPES on Fermi arc surface states (Krieger et al., 2022).
• Weyl Fermion CDW Couplings: In CoSi, unidirectional charge density waves couple opposite-chirality Weyl nodes via a mass term $H_{\rm CDW} = \Delta(\mathbf{r})\tau_x$, which anticommutes with chirality operator $\tau_z$ and gaps the nodes. The CDW wavevector $Q_{\rm cdw}$ is locked to the bulk enantiomer by crystal symmetry (Li et al., 2021).
• Surface Wave Hamiltonians: Photonic or acoustic surface waves exhibit momentum-locked transverse spin densities $S_\perp$, generically described by spin-orbit interaction analogues, where $S(\mathbf{k})$ is locked to $\mathbf{k}$ and surface normal $\mathbf{n}$ via $$Z = \mathbf{n}\cdot(\boldsymbol{\sigma} \times \hat{\mathbf{k}})$$ (Yu et al., 2022).

3. Layer, Stacking, and Field-Controlled Switching

Stacking crystals or 2D materials with identical or opposite chirality enables engineered control of EMP winding patterns and Dirac cone textures:

• Bilayer Chirality Tuning: In borophene, bilayers with the same chirality (C$_1$–C$_1$) preserve Dirac orbital textures with energy splitting controlled by interlayer coupling and gate bias. Bilayers with opposite chirality (C$_1$–C$_2$) allow hybridization-induced gap opening, with restoration of Dirac cones via an applied external field through layer pseudospin $\tau_{x,z}$ (Lima et al., 2019).
• Interface-Induced Chirality: In TaSe$_2$/Bi$_2$Se$_3$ van der Waals interfaces, spontaneous symmetry lowering from $D_{3h}$ to $C_{3v}$ via atomic shifts induces Rashba-type spin-momentum locking absent in the parent bulk phases. Charge transfer shifts the Dirac point and allows coexistence of two oppositely chiral Fermi-surface contours (Polyakov et al., 2020).

4. EMPs in Photonic, Acoustic, and Spintronic Systems

Topological phonon EMPs generalize beyond electronic systems:

• Spin-Momentum-Locked Optical Forces: Evanescent waves with transverse spin generate lateral optical forces on chiral particles near surfaces, whose direction is locked to the particle’s handedness. The lateral force arises from the spin density $S_\perp$ and its coupling to the chiral polarizability $\alpha_{em}$, with right-hand rule enforcement in dielectric environments and left-hand rule in dispersive metals (Kalhor et al., 2015, Wang et al., 2013, Shi et al., 2022).
• Generalized Spin-Orbit Interactions: In magnonic, photonic, and phononic excitations, chirality acts as a generalized SOI; the transverse spin density $$S_\perp = \epsilon_0/(4\omega)\,\operatorname{Im}(E^*\times E)$$ (or its variants) mediates unidirectional pumping, diode, and skin effects, as well as topological transport. The chirality index $Z$ quantifies the locking among momentum, spin, and interface normal (Yu et al., 2022).

5. Surface Chirality, Emergent Topological Phases, and Orbitronics

Chirality-momentum locking fosters robust surface states and emergent topological phases:

• Surface Fermi-Arc OAM Texture: In structurally chiral semimetals like CoSi, bulk multifold chiral fermions possess OAM textures that project onto surface Fermi arcs, yielding helicoid winding of $$\langle \mathbf{L}_\parallel \rangle(\mathbf{k}_\parallel) \propto \mu \hat{\mathbf{k}}_\parallel$$, quantized by the bulk Chern number. This rigid surface chirality momentum locking enables orbitronic current generation and diode effects (Hagiwara et al., 2024).
• CDW-Driven Axion Phases: Unidirectional CDW order locked to crystal chirality in CoSi gaps out Weyl points, realizes intra-unit-cell $\pi$ phase shifts, and supports axion insulator states with modulated $\theta$ angle; mirror symmetry relates enantiomers and flips the CDW wavevector accordingly (Li et al., 2021).
• Non-Hermitian Exchange and CISS: Total mirror-symmetry breaking in molecular or crystalline systems imposes a non-Hermitian spin-momentum coupling $-i\alpha\,\boldsymbol{\sigma}\cdot \mathbf{p}$, which produces perfect spin-momentum locking of interface states, skin effect accumulation, and giant spin selectivity in transport (CISS)—universally tied to chirality (Theiler et al., 9 May 2025).

6. Device Concepts and Functional Applications

EMP phenomena underpin highly controllable platforms for novel devices:

• Orbital-Momentum Transistors: Gate-tunable stacking and chirality enable fast, dissipationless switches of orbital currents, with inverse Edelstein effect scales $10^6$ m/s in borophene exceeding conventional Rashba systems (Lima et al., 2019).
• Spin-Orbit Photonic Routing: Chiral metasurfaces with strong in-plane $x$–$y$ coupling and broken rotational symmetry support surface waves with two transverse spin components, both locked to momentum. Selection of spin sign directs information, yields polarization-resolved beam splitting, and enables valley-selective quantum emitter coupling (Kandil et al., 2021).
• Gain-Momentum-Locked Photonics: Chiral-gain media selectively amplify surface plasmons of a specified spin and propagation direction, locking unidirectional gain to transverse spin; only one orbital-angular-momentum eigenmode is amplified, facilitating unidirectional edge lasing and OAM beam generation (Serra et al., 2024).
• Spin Diodes, Skyrmion/Meron Lattices, and Quantum Magno-Photonics: EMP-mediated chiral excitations underpin broad-band diodes, topologically protected spin lattices, non-reciprocal cavity networks, and quantum channels for one-way magnon or photon transmission (Yu et al., 2022, Shi et al., 2022).

7. Analogies, Future Perspectives, and Unifying Frameworks

EMP phenomena in topological phonon systems tie together electronic, photonic, acoustic, and magnetic analogues:

• Unified SOI Paradigm: The wave-theoretical correspondence $S_\perp \sim \operatorname{Im}(\Psi^* \times \Psi)$, spin-momentum locking index $Z$, and non-Hermitian exchange unify CISS, quantum Hall, quantum spin Hall, valleytronics, and magnonic pumping regimes (Theiler et al., 9 May 2025, Yu et al., 2022).
• Robustness and Tunability: EMP effects derive their stability from symmetry protection (chirality), topological invariants (winding or Chern numbers), and are directly observable in ARPES, STM, optical force or transport experiments. Stackable, gate-tunable, or interface-engineered systems allow flexible design of emergent particle behaviors absent in the parent bulk phases (Lima et al., 2019, Polyakov et al., 2020).
• Quantum and Nonlinear Approaches: Chiral magnon-photon coupling, axion domain wall engineering, non-Hermitian skin-driven detection, and strongly nonlinear EMP regimes offer promising targets for next-generation quantum materials and information logic (Li et al., 2021, Yu et al., 2022).

In total, topological phonon emergent particles represent a class of excitations in quantum materials and metamaterials characterized by a rigid, symmetry-protected locking between internal chirality and momentum. Their presence enriches the landscape of functional materials, enables transfer across electronic, optical, and vibrational domains, and underpins a spectrum of applications in spintronics, orbitronics, cavity quantum electrodynamics, and topological information processing.