Dynamical Spin Splitting in Quantum Systems

• Dynamical spin splitting is defined as the time-dependent modulation of spin-resolved energy states driven by mechanisms such as spin–orbit coupling, Zeeman effects, and dynamic symmetry breaking.
• It is modeled using effective Hamiltonians that incorporate Rashba, Zeeman, and Floquet-driven terms, revealing momentum- and field-dependent energy gaps observable via spectroscopic probes.
• Its tunability via external fields, strain, and layer engineering enables applications in spintronics, including gate-tunable spin currents and quantum device control.

Dynamical spin splitting refers to the energy separation between spin-resolved electronic states or collective excitations that emerges or is modulated through temporal, spatial, or symmetry-breaking perturbations—often in the presence of spin–orbit interactions, external fields, or engineered heterogeneity. Unlike conventional static spin splitting, which is directly attributed to broken time-reversal or inversion symmetries (e.g., Zeeman or Rashba effects), dynamical spin splitting can manifest as a time-dependent, momentum-dependent, or field-induced modification of the spectrum and is observable through a variety of transport, thermodynamic, and spectroscopic probes. Its microscopic origins, tunability, and symmetry constraints make it a central object of paper in spintronics, quantum transport, low-dimensional magnetism, and materials with complex topology.

1. Fundamental Mechanisms for Dynamical Spin Splitting

Dynamical spin splitting arises from an interplay of several microscopic mechanisms:

• Spin–Orbit Coupling (SOC): The coupling of spin degrees of freedom to orbital motion in systems lacking inversion symmetry produces momentum-dependent splitting, as in the Rashba and Dresselhaus effects. The Rashba SOC term $\alpha(\vec{\sigma} \times \vec{k})\cdot\hat{z}$ leads to an energy separation that can be modulated by external fields or engineered symmetry, resulting in dynamical control (1105.2394, Singh et al., 2017, Absor et al., 2017, Guo et al., 2022).
• Zeeman Effect: The interaction $-\vec{\mu}_s \cdot \vec{B}$ between electronic magnetic moments and an external (or exchange) magnetic field produces a field-dependent splitting; in systems with in-plane field tilt or anisotropic g-factors, this can become nontrivial and field-orientation dependent, as seen in quantum Hall and quantum dot systems (1105.2394, Grigoryan et al., 2013).
• Dynamic Symmetry Breaking: In nonmagnetic systems, thermal or structural fluctuations—such as dynamically fluctuating atomic positions in soft perovskites—can transiently break inversion symmetry, generating Rashba- or Dresselhaus-like spin splitting at finite temperatures even in centrosymmetric crystals (McKechnie et al., 2017, Monserrat et al., 2017). In such cases, the splitting fluctuates both in magnitude and direction, yielding a time-averaged, observable effect.
• Interfacial and Layer Asymmetry: In Janus or hybrid heterostructures, built-in electric fields create a potential difference across different layers or atomic motifs. For example, in 2D Janus antiferromagnets, an intrinsic out-of-plane field leads to a layer-resolved energy offset for opposing spin layers and thus spin splitting even without net magnetization—termed electric-potential-difference antiferromagnetism (EPD-AFM) (Guo, 2023).
• Floquet–Driven and Nonequilibrium Fields: Time-periodic driving by optical fields (Floquet engineering) in antiferromagnets can dynamically break symmetries and produce effective time-reversal breaking, resulting in robust spin splitting and pure spin currents even in systems lacking SOC (Li et al., 30 Jul 2025).
• Relativistic Magnetoelectric Correction: Derived from the Dirac formalism, this relativistic correction is of the form $$H_\mathrm{me} = \mu_B\eta_0 (\mathcal{E} - \mathcal{V}(\vec{r}))\vec{\sigma}\cdot\vec{m}$$, allowing for spin splitting governed by local electric multipoles and motif connectivity. The mechanism is distinct from, but unifies, Zeeman splitting and SOC contributions, and is universal across all magnetic and nonmagnetic point groups (Acosta, 28 May 2025).

These mechanisms can combine or compete within the same material platform, giving rise to a rich variety of dynamical spin splitting behaviors.

2. Quantitative Modeling: Representative Hamiltonians and Spectroscopic Signatures

Dynamical spin splitting is typically modeled by effective Hamiltonians incorporating relevant interactions. Key examples include:

Mechanism (Context)	Effective Hamiltonian Term(s)	Key Parameters
Rashba SOC (2DEG, TMDs)	$$H_\mathrm{R} = \alpha_\mathrm{R} (\vec{k} \times \vec{\sigma})\cdot\hat{z}$$	$\alpha_\mathrm{R}$, $\vec{k}$
Zeeman	$H_\mathrm{Z} = -\mu_B \vec{\sigma}\cdot\vec{B}$	$g$, $\vec{B}$
Floquet engineering	$$H_{\rm eff}^F \sim v_F(\tau^x \eta^z q_y + \tau^y q_x) + s\lambda \tau^z - \Delta \tau^z \eta^z$$	drive amplitude, $\Delta$
Relativistic ME	$$H_\mathrm{me} = \mu_B\eta_0(\mathcal{E} - \mathcal{V}(\vec{r}))\vec{\sigma}\cdot\vec{m}$$	Multipoles $\mathcal{V}(\vec{r})$

Where relevant, the eigenenergy splitting is given (for Rashba, BiSb monolayer (Singh et al., 2017)) by

$\alpha_R = \frac{2E_R}{k_0}$

with $E_R$ the Rashba energy and $k_0$ the momentum offset. For phonon-assisted splitting (Monserrat et al., 2017), the effective low-energy dispersion is

$$\epsilon_{\pm}(k_z) = \frac{\hbar^2 k_z^2}{2m} \pm \lambda_{\rm SOC} k_z \sqrt{u_1^2 + u_2^2},$$

where $u_{1,2}$ are phonon amplitudes.

Signatures of dynamical spin splitting include:

• Oscillations or persistent splittings in the chemical potential and specific heat under varying magnetic field (2DEG) (1105.2394).
• Splitting and avoided crossing of magnetic resonance branches (electron spin resonance, ESR) in spin chains with DM interaction (Wang et al., 2022).
• Nondegenerate energy structure in Bell state manifolds for driven quantum dots (Grigoryan et al., 2013).
• Pure spin currents and non-equilibrium spin accumulations in optically driven antiferromagnets (Li et al., 30 Jul 2025).
• Direct detection of momentum offsets and energy splittings in ARPES and inverse-ARPES (perovskites, CsPbCl₃) (Monserrat et al., 2017).

3. Role of Symmetry, Multipole Expansions, and Material Classes

The appearance, magnitude, and momentum dependence of dynamical spin splitting are tightly constrained by crystal symmetry, local site symmetry, and motif connectivity:

• Local Multipole Expansion: The electric potential $\mathcal{V}_n(\vec{r})$ at site $n$ is expanded in multipoles (monopole $Q_n$, dipole $d_n$, quadrupole $Q_{nij}$, etc.), each with selection rules controlled by site and lattice symmetry (Acosta, 28 May 2025):

$$\mathcal{V}_n(\vec{r}) = \lambda_0 Q_n + \lambda_1 \sum_i d_{ni} r_i + \lambda_2 \sum_{ij} Q_{nij} r_i r_j + ...$$

The form of spin splitting ($k$-independent, linear in $k$, quadratic in $k$) is dictated by which multipole is symmetry-allowed and the motif connectivity.

• Point Group and Motif Connectivity: In altermagnets and certain antiferromagnets, spin splitting arises due to rotations or improper rotations connecting symmetry-inequivalent motifs. For example, an off-diagonal quadrupole $Q_{xy}$ in MnF₂ under $C_{2h}$ symmetry produces an anisotropic $k_xk_y$ splitting (Acosta, 28 May 2025). For Janus AFMs (Guo, 2023), the lack of mirror symmetry allows a layer-dependent out-of-plane dipole, resulting in EPD-AFM splitting.
• Role of Dynamic Local Distortions: In perovskites, dynamic symmetry breaking causes instantaneous local environments to lack inversion, activating Rashba–Dresselhaus SOC even though the average structure is centrosymmetric (McKechnie et al., 2017, Monserrat et al., 2017).
• Stacking and Layer Control: In multilayer or van der Waals systems, the stacking sequence (ABA, ABC) and the distribution of Rashba SOC across layers control the net spin splitting. For instance, Rashba SOC induced with opposite sign on outer layers of ABC-stacked trilayer graphene cancels, but is nonzero in ABA stacking (Cheng et al., 2023).

4. Tunability via External Fields, Strain, and Nonequilibrium Drives

Dynamical spin splitting is tunable across several axes:

• Magnetic Field Orientation: Tilting the applied magnetic field modulates Zeeman components and amplifies spin splitting in weak fields due to the interplay with Rashba SOC (1105.2394).
• Electric Field & Strain: In Janus structures and 2D semiconductors, external perpendicular electric fields can reversibly switch Rashba splitting on and off by changing the field direction. Piezoelectric in-plane or out-of-plane strain can also modulate the built-in field and thus the splitting (Guo et al., 2022, Guo, 2023).
• Optical Driving: Periodic optical fields (in the Floquet regime) dynamically generate and control spin splitting and spin currents; the intensity, phase, and polarization of the field set the effective splitting scale (Li et al., 30 Jul 2025).
• Layer Engineering: In multilayer graphene, selectively functionalizing or gating specific layers enables stacking-sensitive control of Rashba effects (Cheng et al., 2023).
• Charge Transfer and Interface Engineering: In graphene/LCO hybrids, the strength of interfacial hybridization and charge transfer—modulated by gate voltage—allows tuning of spin exchange splitting from 155.9 to 306.5 meV (Shin et al., 13 Mar 2024).

5. Experimental Probes and Representative Platforms

Dynamical spin splitting can be observed and characterized in a range of systems:

System/Material	Key Effect/Observation	Methodology
2DEGs	Persistent, amplified spin splitting via tilt	Thermodynamics ($\mu$, $C_V$) (1105.2394)
Janus AFMs	Layer-resolved EPD-AFM splitting, strain-tuning	First-principles, piezotronics (Guo, 2023)
Perovskites	Phonon/thermal fluctuation-induced splitting	DFT, QSGW, ARPES (McKechnie et al., 2017, Monserrat et al., 2017)
Chiral 1D InSeI	Collinear spin-momentum locking, strain response	DFT, spin-resolved bands (Zhao et al., 2023)
TMDs, BiSb monolayer	Rashba, Zeeman, warping tuned by strain/polarity	DFT, spin textures (Singh et al., 2017, Absor et al., 2017)
Multilayer graphene	Rashba SOC stacked order sensitivity	Tight-binding, DFT (Cheng et al., 2023)
Quantum dots	Rotating field-induced dynamical Ising coupling	SW transformation, Bell state energies (Grigoryan et al., 2013)
AFM spin chains, DM	Mode splitting, ESR doublets, finite momentum shift	Hydrodynamics, numerical MPS (Wang et al., 2022)
Graphene/LCO hybrid	Gate tunable exchange splitting	QHE, SdH oscillations (Shin et al., 13 Mar 2024)
Optically driven AFM	Floquet spin splitting, pure spin currents	Floquet theory, spin transport (Li et al., 30 Jul 2025)

Evidence includes split quantum oscillation frequencies, nondegenerate Landau level fillings, avoided crossings in resonance, and gate-switchable spintronic device characteristics.

6. Applications and Implications in Spintronics and Quantum Devices

Dynamical spin splitting is essential for enabling control over spin currents, spin filtering, and qubit interactions in quantum information systems:

• Spin-FETs and Logic: Large, electrically or mechanically tunable Rashba splitting (e.g., BiSb monolayer) allows for gate-controlled spin-precession and spin filtering in field-effect transistors (Singh et al., 2017).
• QSHI Spin Switches: In Janus monolayer RbKNaBi, the Rashba effect can be switched on or off by electric field polarity, enabling high-speed spintronic switches (Guo et al., 2022).
• Optically Controlled Spin Currents: Floquet engineering allows nonequilibrium pure spin current generation in AFMs without SOC, amenable to ultrafast optospintronics (Li et al., 30 Jul 2025).
• Qubit Entanglement and Control: In quantum dot arrays, dynamical Ising spin–spin coupling facilitates manipulation of Bell-state manifolds for quantum computing (Grigoryan et al., 2013).
• Spintronic Memory and Logic: The robust, high-magnitude, and gate-tunable exchange splitting in graphene/LCO heterostructures offers the basis for electrically controlled spin logic and memory elements (Shin et al., 13 Mar 2024).
• Materials Diagnostics: The anisotropy of spin dynamics in AFMs with d-wave nonrelativistic spin splitting offers unambiguous experimental fingerprints for the identification and functional exploitation of altermagnets (Denisov et al., 28 Oct 2024).

7. Outlook and Theoretical Extensions

Recent works have established the importance of symmetry, motif connectivity, and multipolar electric field expansions in determining the precise form and tunability of dynamical spin splitting (Acosta, 28 May 2025). This unified formalism not only consolidates known spin splitting phenomena (Zeeman, Rashba) but predicts new forms (e.g., quadratic, $k$-independent, or higher-order in $k$, governed by local multipolar symmetries) that can be rationally engineered through atomic-level control of structure and environment. Extensions to higher-order multipoles, strongly correlated systems, and non-equilibrium (optical, Floquet) regimes are anticipated to reveal further opportunities for controlling and exploiting spin degrees of freedom in quantum materials and devices.