Fractional Quantum Multiferroics

• Fractional Quantum Multiferroics are materials that exhibit coupled ferroelectric and magnetic orders with fractionalized excitations and nonlocal dynamics.
• Theoretical frameworks leverage the fractional Schrödinger equation and generalized symmetry principles to predict fractionally quantized polarization and magnetization states.
• Practical implications include energy-efficient multistate memory devices, low-barrier switching mechanisms, and topologically robust quantum systems.

Fractional Quantum Multiferroics (FQMF) refer to materials and quantum states in which the interplay between ferroelectric and magnetic orders exhibits signatures of strong quantum fluctuations, anomalous scaling, nonlocal dynamics, and, crucially, fractionalization of order parameters, excitations, or topological invariants. This field generalizes classical multiferroicity by leveraging concepts from fractional quantum mechanics, quantum criticality, symmetry-based fractionalization of polarization, and emergent phenomena in frustrated and disordered systems. FQMFs open new directions for understanding, realizing, and controlling exotic cross-couplings among electric, magnetic, and structural degrees of freedom at the quantum level.

1. Conceptual Framework: Fractionalization in Quantum Multiferroics

Fractionalization in FQMF encompasses several intertwined phenomena:

• Fractional quantum mechanics: The use of fractional derivatives (e.g., the quantum Riesz derivative) generalizes canonical quantum models, capturing nonlocality and emergent fractal trajectories. This results in unconventional spectra and wavefunctions, key for systems with disorder, frustration, or Lévy-type dynamics (Laskin, 2010, Kirichenko et al., 2018).
• Fractional quantum ferroelectricity (FQFE): Polarization states arising from atomic displacements that are rational fractions of a lattice vector generate fractionally quantized polarization, distinct from both classical (continuous) and conventional quantum (integer-quantized) ferroelectrics (Yu et al., 29 Apr 2024, Pang et al., 17 Apr 2025, Yu et al., 30 Aug 2025). This can be systematically described using generalized symmetry and group-theoretical frameworks.
• Fractionalization in spin and charge sectors: In quantum magnets and correlated systems, entanglement and strong frustration yield excitations with fractional (e.g., Sᶻ = 1/3) quantum numbers and partial magnetizations, and lead to composite objects such as magneto-electric-elastic quasiparticles (Maruyama et al., 2023, Cabra et al., 2021).
• Fractional topological charges: In coupled ferroelectric and magnetic textures, the topological invariants (e.g., of FE skyrmions) can take fractional or otherwise nonintegral values, further evidencing quantum-induced fractionalization (Liu, 2021).

FQMFs thus represent a regime where the multiferroic order parameters or excitations exhibit noninteger, emergent, or modulated structure as a result of complex quantum mechanical constraints, strong fluctuations, or symmetry principles.

2. Theoretical Foundations: Fractional Quantum Mechanics and Symmetry Generalizations

The theoretical machinery underlying FQMF includes:

Fractional Schrödinger Equation and Path Integrals

The general Hamiltonian

$H_\alpha(p, r) = D_\alpha |p|^\alpha + V(r)$

leads to the fractional Schrödinger equation:

$$i\hbar \frac{\partial \psi}{\partial t} = D_{\alpha}(-\hbar^2 \Delta)^{\alpha/2} \psi + V(x) \psi$$

with the Riesz fractional derivative introducing nonlocal, Lévy-flight-type dynamics into quantum evolution (Laskin, 2010).

The corresponding path integral generalizes Feynman's construction, integrating over fractal, jump-like trajectories instead of Brownian paths. This framework provides tools for predicting the effects of disorder, long-range correlations, and anomalous diffusion in multiferroic crystals (Laskin, 2010, Kirichenko et al., 2018).

Symmetry-Driven Fractionalization: Generalized Neumann’s Principle

The traditional Neumann's principle restricts polarizations to directions invariant under crystal symmetries. The generalized formulation accepts that, due to the multivalued nature of polarization (modulo a quantum), for any crystal operation ℛ:

$\mathcal{R} \mathbf{p} = \mathbf{p} + \mathbf{Q}$

where $\mathbf{Q}$ is a polarization quantum. This equation unifies both conventional and fractional quantum ferroelectricity; in nonpolar symmetries, only discrete, often fractional, polarization solutions are permitted (Pang et al., 17 Apr 2025, Yu et al., 29 Apr 2024).

The allowed fractional displacement between nonequivalent phases for an atom $A$ along direction $a$ is

$\Delta d_{A,a} = \frac{M}{N}$

yielding a polarization quantum

$\Delta P = \frac{Q}{N}$

for type-I FQFE, or $\Delta P = Q$ for type-II FQFE (where the cumulative effect restores integer quantization) (Yu et al., 29 Apr 2024).

Entanglement and Coupled Order Parameters

Fractional multiferroic states are often characterized by highly entangled quantum ground states. For example, rigorous mappings exist between entangled spin-½ antiferromagnetic ground states and partially magnetized spin-$S$ ferromagnetic systems where the magnetization is reduced to $M=1-1/(2S)$, reflecting their embedded quantum correlations (Maruyama et al., 2023).

3. Mechanisms and Material Realizations

Frustration, Disorder, and Strong Correlations

• Spin and lattice frustration: Competing exchange interactions, low-dimensional geometries, and weak anisotropies can stabilize plateau states (e.g., $M=1/3$) in Heisenberg chains. The elementary magnetic excitation breaks into solitons with fractional quantum numbers, each coupled to local polarization flips and elastic distortions, forming composite "magneto-electric-elastic" (MEE) quasiparticles (Cabra et al., 2021).
• Charge fluctuations: In charge-ordered multiferroics like LuFe₂O₄, quantum charge fluctuations tied to spin arrangements modulate both the magnitude and dynamics of polarization, introducing an effective "fractionalization" of the conventional ferroelectric response via tunable quantum pathways (Lee et al., 2012).

Fractional Topological Textures

In 2D multiferroics, magnetoelectric coupling between magnetic skyrmions and electric dipoles results in the formation of ferroelectric skyrmion lattices (SLs). These FE skyrmions can exhibit topological charges quantized not just as integers, but also as half-integers and other fractional values, modulated by electric fields and magnetic textures (Liu, 2021).

Symmetry-Driven FQFE in Real Materials

• Bulk and 2D materials: High-throughput symmetry-based screening identifies numerous experimentally realized compounds where atomic displacements lead to type-I and type-II FQFE; for instance, bulk AlAgS₂ (fractional in-plane polarization, ultra-low switching barriers) and monolayer HgI₂ (large, integer-polarization from valence-multiplied fractional shifts) (Yu et al., 29 Apr 2024).
• Switching and coupling: Demonstrations in materials like HfZnN₂ illustrate experimentally accessible switching pathways between inequivalent fractional polarization states, mediated by symmetry lowering and entanglement with conventional out-of-plane polarization (Pang et al., 17 Apr 2025).

4. Quantum Criticality and Emergent Phenomena

Quantum critical points (QCPs) where ferroelectric or magnetic transitions are suppressed to zero temperature due to fluctuations are fertile ground for FQMF behavior (Rowley et al., 2015, Narayan et al., 2017).

• Scaling laws: Near uniaxial ferroelectric QCPs (e.g., in hexaferrites), the inverse dielectric susceptibility $\chi_E^{-1} \propto T^3$ emerges, sharply contrasting with $T^2$ for pseudocubic perovskites. Magnetic and electric orders may both approach quantum criticality, allowing for strong cross-fluctuation and emergent coupled order (Narayan et al., 2017).
• Dynamic multiferroicity: At FE QCPs, fluctuations in electric dipoles generate dynamic magnetic responses due to entangled order parameters, even in nominally nonmagnetic systems (e.g., paraelectric SrTiO₃), resulting in experimentally detectable induced magnetic susceptibilities (Dunnett et al., 2018).
• Fractionalized order parameter: Quantum transitions may display exponents and critical behaviors indicative of a joint, fractionalized magnetoelectric order parameter, suggesting new universality classes (Schrettle et al., 2012).

5. Applications, Control, and Future Research Directions

FQMF systems promise:

• Tunable multistate memory: The possibility of more than binary (multi-fractional) switching states via low-barrier fractional polarization, opening nonvolatile, multi-level memory architectures (Yu et al., 29 Apr 2024, Pang et al., 17 Apr 2025).
• Energy-efficient actuation: Lower switching barriers and emergent phenomena could enable logic, memory, or sensor devices with reduced energy dissipation.
• Topologically robust functionality: Skyrmionic and fractional domain structures may support protected data pathways or quantum information schemes (Liu, 2021).

Emergent avenues include:

• Systematic discovery: Group-theoretical and high-throughput DFT-based approaches allow identification of large numbers of FQFE and candidate FQMF materials, even among previously overlooked nonpolar space groups (Yu et al., 29 Apr 2024, Pang et al., 17 Apr 2025).
• Exploring coupled quantum criticality: Investigation of crossover scaling regimes and dynamic susceptibilities in materials with near-coincident FE and magnetic QCPs (Narayan et al., 2017).
• Engineered heterostructures: Layered or interface systems where one layer’s fractional polarization field couples to another’s magnetic state, controlled by strain, chemical tuning, or fields.
• Theory and modeling: Further development of fractional path integrals, Berry-phase polarization with quantization modulo $Q$, and composite effective field theories for coupled fractional orders (Laskin, 2010, Pang et al., 17 Apr 2025).

6. Representative Models and Key Equations

Model/Expression	Context	Reference
$$i\hbar\,\partial_t\psi = D_\alpha(-\hbar^2\Delta)^{\alpha/2}\psi$$	Fractional Schrödinger Eq.	(Laskin, 2010)
$\Delta d_{A,a} = d'_{L_2,A,a} - d'_{L_1,A,a} = M/N$	Fractional atomic displacement	(Yu et al., 29 Apr 2024)
$\Delta P = Q/N$ (type-I) or $Q$ (type-II)	Fractional/Integer quant. pol.	(Yu et al., 29 Apr 2024)
$M = 1 - 1/(2S)$	Fractional magnetization	(Maruyama et al., 2023)
$\chi_E \propto 1/T^3$	Dielectric critical scaling	(Rowley et al., 2015)
$$m = \lambda\,\mathbf{p} \times \partial_t \mathbf{p}$$	Dynamic multiferroicity	(Dunnett et al., 2018)
$Q_E$ = integer, half-integer, or fractional $\times$ base unit	FE skyrmion topological charge	(Liu, 2021)
$$F = F_{\text{elec}}(P+\mathcal{AP}) + F_{\text{mag}}(M) + F_{\text{coupling}}(P,M)$$	Coupled free energy	(Yu et al., 30 Aug 2025)

7. Summary

Fractional Quantum Multiferroics represent a rapidly developing class of materials and physical models where canonical ferroic concepts are extended and enriched by the inclusion of fractional quantum mechanics, symmetry-enabled fractionalization, strong correlations, frustration, and quantum criticality. The field draws on deep group-theoretical insights, advanced numerical approaches, and experimental advances in control and measurement, yielding not only previously inaccessible functionalities (such as multi-level switching, low-barrier actuation, and topological robustness) but also a conceptual unification of “fractionalized” order in coupled electric, magnetic, and structural systems. Cutting-edge theoretical frameworks such as the generalized Neumann’s principle (Pang et al., 17 Apr 2025), symmetry-protected FQFE classification (Yu et al., 29 Apr 2024), and dynamical criticality models (Narayan et al., 2017, Dunnett et al., 2018) jointly underpin the pursuit of new quantum device architectures and fundamental understanding in complex correlated materials.