Fractional Quantum Ferroelectricity in Multiferroics

• Fractional quantum ferroelectricity is defined by quantized, switchable polarization from fractional atomic displacements, enabling intrinsic magnetoelectric coupling.
• The minimal tight-binding model shows that electric-field-induced polarization switching reverses momentum-dependent spin splitting without reorienting the magnetic order.
• Material candidates like MnTe exhibit room-temperature operation with high tunneling magnetoresistance, making them promising for low-power spintronic devices.

Fractional Quantum Multiferroics (FQMF) constitute a recently established class of materials that achieve strong, intrinsic magnetoelectric coupling at room temperature by integrating fractional quantum ferroelectricity (FQFE) with altermagnetism (AM) (Dong et al., 19 Oct 2025). Unlike conventional multiferroics, where the mechanisms underlying ferroelectricity and magnetism often compete and suppress mutual effects, FQMFs overcome this limitation through symmetry-enforced coupling between large, switchable polarization changes and momentum-dependent spin splitting. The direct linkage between an electric polarization quantum (set by large fractional lattice displacements) and a reversal of the electronic spin texture enables robust, electrically driven spin control without the need to physically reorient antiferromagnetic vectors.

1. Symmetry-Driven Coupling of Ferroelectricity and Altermagnetism

FQMFs originate from the synergy of FQFE and altermagnetism in nonpolar point group crystals. FQFE allows for switchable, quantized polarization changes via fractional atomic displacements between symmetry-related lattice sites that are not integer multiples of the primitive translation vector. When this symmetry-permitted lattice shift is realized in a system that also supports altermagnetism—magnetic ordering where the electronic band structure exhibits momentum-dependent spin splitting even in the absence of spin–orbit coupling—a unique symmetry locking arises. Specifically, symmetry operations such as parity–time reversal ($\mathcal{PT}$) and a combined time-reversal with a fractional lattice translation ($\mathcal{T}\tau$) enforce that switching the FQFE polarization reverses the AM spin structure: $$\tau\mathcal{PT}: \ \epsilon_n(\mathbf{k}, \mathbf{s}) = \epsilon_n(\mathbf{k}, -\mathbf{s}), \quad \tau\mathcal{T}: \ \epsilon_n(\mathbf{k}, \mathbf{s}) = \epsilon_n(-\mathbf{k}, -\mathbf{s}) = \epsilon_n(\mathbf{k}, -\mathbf{s})$$ where $\tau$ denotes a fractional lattice translation, $\mathcal{P}$ is inversion, $\mathcal{T}$ is time-reversal, $\epsilon_n$ is the band energy, and $\mathbf{s}$ is the spin. Hence, an electric-field-induced FQFE switching directly inverts the spin splitting of the band structure.

2. Minimal Theoretical Model of FQMF

The essential physics of FQMF is captured by a minimal tight-binding model on a two-dimensional lattice: $$H = -\sum_{\langle ij \rangle, s} t_{ij}(\mathbf{r}_A) c_{i,s}^\dagger c_{j,s} - J \sum_{i,s,s'} M_i c_{i,s}^\dagger \sigma^z_{ss'} c_{i,s'} - \mu \sum_{i,s} c_{i,s}^\dagger c_{i,s}$$ where:

• $t_{ij}(\mathbf{r}_A)$ is the hopping amplitude dependent on the fractional displacement $\mathbf{r}_A$ of an atom (realizing FQFE),
• $J$ describes the exchange interaction mediating AM through the local magnetic configuration $M_i$,
• $\mu$ is the chemical potential.

Switching between ferroelectric states (L$_1 \rightarrow$ L$_2$)—characterized by half-quantum translation of atom A—simultaneously reverses the sign of the polarization and inverts the spin-resolved band structure. This coupling directly couples electric polarization changes to spin polarization reversals, without reorienting the magnetic order parameter.

3. Material Realizations and Design Principles

First-principles calculations reveal a wide class of FQMF candidates in both bulk and two-dimensional systems:

• Bulk materials: MnTe (space group P6$_3$/mmc), Cr$_2$S$_3$, Mn$_4$Bi$_3$NO$_{15}$.
• 2D AB$_2$ bilayers: MnX$_2$ (X = Cl, Br, I), CoCl$_2$, CoBr$_2$, FeI$_2$.

These materials feature nonpolar crystal symmetries conducive to FQFE and maintain robust magnetic order supporting AM. For instance, in MnTe, Te atoms are displaced by a symmetry-prescribed fraction of the lattice vector to induce quantum polarization switching coupled to spin texture inversion. The design strategy involves selecting systems where fractional displacements and non-centrosymmetric magnetic order coexist.

Material	Structure/Symmetry	Key Multiferroic Features
MnTe	Bulk, P6$_3$/mmc	FQFE; AM; large switchable spin-splitting
MnX$_2$ (X=Cl, Br, I)	2D AB$_2$ bilayer	FQFE; AM; robust in-plane polarization
Cr$_2$S$_3$	Bulk	FQFE/AM coupling

4. Functional Metrics: Magnetoelectric Coupling and Device-Relevant Quantities

Performance characteristics of FQMFs, exemplified by MnTe, include:

• Néel temperature $\sim 300$ K: Indicates robust antiferromagnetic order persisting to room temperature.
• Electrically switchable spin splitting $\sim 0.8$ eV: Substantially higher than in conventional multiferroics and typical altermagnets; critical for efficient spintronic operation.
• Insulating band gap $\sim 0.9$ eV: Preserves ferroelectricity while enabling selective carrier doping for tunnel junction devices.

The direct, symmetry-enforced reversal of spin splitting upon electrical switching is a hallmark of FQMFs, indicating strong, internal magnetoelectric coupling not reliant on spin–orbit effects or the physical rotation of the magnetic order parameter.

5. Device Proposals: Field-Controlled FQMF Tunnel Junctions

A practical application discussed is the electric-field-controlled FQMF tunnel junction (FQMFTJ), specifically constructed from MnTe:

• The junction comprises two MnTe slabs with well-defined ferroelectric (L$_1$, L$_2$) states.
• In the parallel state (both slabs in L$_1$), the spin-resolved Fermi surfaces are aligned, maximizing tunneling conductance.
• Switching the free layer (by electric field) to L$_2$ reverses its spin splitting relative to the fixed L$_1$ layer, creating momentum-space mismatch and suppressing electronic transmission.
• Tunneling Magnetoresistance (TMR) exceeds 300% near the Fermi level, calculated as

$$\mathrm{TMR} = \frac{T_P - T_{AP}}{T_{AP}} \times 100\%$$

where $T_P$ and $T_{AP}$ are tunneling probabilities for parallel and antiparallel configurations.

The high TMR is supported by both intrinsic spin structure mismatch and atomic mismatch induced by FQFE-controlled displacements.

6. Technological Implications and Outlook

FQMFs unlock voltage-controlled spintronic device architectures combining nonvolatile ferroelectric switching and robust spin polarization manipulation. Room-temperature operation—enabled by high Néel temperature and large, symmetry-protected spin splitting—positions FQMFs as viable candidates for low-power memory, reconfigurable logic, and other nanoelectronic applications. The nonpolar symmetry classes accessible by FQMF design strategies extend the landscape for functional materials beyond that of conventional multiferroics, suggesting that advances in symmetry engineering and high-throughput first-principles searches will continue expanding the repertoire of high-performance, room-temperature magnetoelectric platforms.

A plausible implication is that FQMFs, by allowing the electrical control of spin splitting independent of magnetic moment reorientation, could substantially reduce device switching energies and improve stability against external magnetic perturbations.

7. Summary Table of FQMF Attributes

Characteristic	FQMF Example (MnTe)	Conventional Multiferroic
Symmetry class	Nonpolar (P6$_3$/mmc)	Usually polar
Magnetoelectric coupling	Intrinsic, symmetry-enforced	Weaker, competing mechanisms
Spin splitting (switchable)	$\sim 0.8$ eV	$\sim$0.01–0.1 eV
Operating temperature	Room temperature	Mostly below room temperature
Tunneling magnetoresistance	$>$300%	$<$100% typical
Polarization control mechanism	Fractional quantum (FQFE)	Small atomic shifts

FQMFs represent a conceptually and technologically distinct route to achieving strong, room-temperature magnetoelectric coupling through the fusion of fractional quantum ferroelectricity and altermagnetism (Dong et al., 19 Oct 2025). This approach opens new directions for voltage-controlled spintronics and next-generation multifunctional electronic devices.