Fractional Quantum Ferroelectricity

• Fractional Quantum Ferroelectricity is a material class where branch-quantized polarization arises from fractional ionic displacements governed by Berry-phase and symmetry principles.
• It exhibits switchable polarization differences as fractional multiples of a polarization quantum, distinguishing it from conventional soft-mode ferroelectricity.
• Advanced methodologies, including high-throughput DFT screening and symmetry analysis, have identified numerous candidate materials spanning polar and nonpolar structures.

Fractional quantum ferroelectricity denotes a class of switchable insulating states in which the polarization change is controlled by the multivalued Berry-phase theory of polarization and by symmetry-equivalent structural configurations related through lattice translations or fractional-lattice ionic displacements. In the modern formulation, the central object is not a unique spontaneous polarization vector but a branch of the formal polarization lattice, so a crystal in a nonpolar space group can still exhibit a nonzero switchable polarization difference between equivalent states. In this sense, fractional quantum ferroelectricity is distinct both from conventional soft-mode ferroelectricity and from looser uses of “fractional” that refer to topological texture charges, partial suppression of order, or quantum-critical dipolar fluctuations rather than to polarization branch structure itself (Yang et al., 8 May 2026, Yu et al., 2024, Pang et al., 17 Apr 2025, Luo et al., 14 May 2026).

1. Definition and conceptual scope

A unified definition proposed for ferroelectricity is the existence of two or more energetically equivalent states with a nonzero polarization difference between them, reversibly switchable by an external electric field. Within that broader framework, fractional quantum ferroelectricity is the subclass in which the switchable polarization difference is a fractional multiple of the polarization quantum, while integer quantum ferroelectricity denotes the corresponding integer-multiple case (Luo et al., 14 May 2026).

The distinguishing feature of fractional quantum ferroelectricity is that the switching coordinate is not merely a small local symmetry-breaking distortion. Instead, it is tied to a branch change in the formal polarization lattice, often generated by a fractional displacement of an ion or sublattice with respect to a primitive lattice vector. Because polarization is defined modulo a polarization quantum, such switching can occur even in nonpolar point groups without contradicting crystallographic symmetry (Yang et al., 8 May 2026, Yu et al., 2024).

Class	Structural relation	Characteristic polarization change
Conventional ferroelectricity	Small symmetry-breaking distortion, usually in a polar point group	$\Delta P = 2P_s$
Fractional quantum ferroelectricity	Fractional-lattice displacement between symmetry-equivalent states	$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$
Integer quantum ferroelectricity	Quantized switching in the same modern-polarization framework	$\Delta P = nP_Q$

This definition also clarifies what fractional quantum ferroelectricity is not. It is not, in the strict sense, a claim about anyons, emergent gauge fields, or fractionally charged quasiparticles. It is also not equivalent to every quantum or unconventional ferroelectric phenomenon. Several neighboring literatures use “quantum ferroelectricity” for quantum paraelectrics, light-induced ferroelectricity, or carrier-driven suppression of dipolar order, but those phenomena are conceptually different from branch-quantized polarization switching (Gu et al., 2023, Ghosh et al., 5 Jan 2026, Shin et al., 2021).

2. Modern polarization, polarization quanta, and generalized symmetry

The modern theory of polarization treats bulk polarization as a multivalued lattice quantity,

$$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$

where $\mathbf{a}_i$ are primitive lattice vectors and $\Omega$ is the primitive-cell volume. The polarization quantum along $\mathbf{a}_i$ is therefore

$\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$

In this formulation, formal polarization is defined only modulo $\mathbf{Q}$, and experimentally relevant switching corresponds to a difference between branches rather than to an absolute dipole density (Yang et al., 8 May 2026).

This directly motivates the generalized Neumann principle,

$(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$

or equivalently $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$0, where $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$1 is a crystallographic symmetry operation and $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$2 is a polarization quantum. Conventional Neumann symmetry is recovered as the special case $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$3. The consequence is that a nonpolar or even centrosymmetric crystal can possess a nonzero formal polarization branch provided the full polarization lattice, rather than one chosen representative, is symmetry invariant (Yang et al., 8 May 2026, Pang et al., 17 Apr 2025).

The review literature makes this point with centrosymmetric KNbO$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$4: Berry-phase calculations give a formal polarization of $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$5, exactly half of the polarization quantum $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$6. This does not violate inversion symmetry because inversion maps that branch to another equivalent branch differing by one quantum (Yang et al., 8 May 2026).

A full translation of an ion with topological oxidation state $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$7 by a primitive lattice vector gives

$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$8

Fractional quantum ferroelectricity corresponds to a fractional step of that cycle. For a displacement $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$9,

$\Delta P = nP_Q$0

This is the central topological pumping relation used to classify FQFE. The same framework supports the type-I and type-II distinction: in type-I FQFE, $\Delta P = nP_Q$1 is fractional; in type-II FQFE, the displacement is still fractional but $\Delta P = nP_Q$2 is integer because of oxidation state or multiplicity (Yang et al., 8 May 2026, Yu et al., 2024).

3. Symmetry criteria, screening strategies, and broadened taxonomies

A practical symmetry strategy begins from the space group $\Delta P = nP_Q$3 of a low-symmetry state $\Delta P = nP_Q$4 and the space group $\Delta P = nP_Q$5 of the corresponding symmetrized lattice. Symmetry-related partner states are generated as

$\Delta P = nP_Q$6

and one then tests whether the transition $\Delta P = nP_Q$7 contains a fractional-lattice displacement. The displacement criterion used in high-throughput work is

$\Delta P = nP_Q$8

with $\Delta P = nP_Q$9 and $$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$0. This strategy makes the distinction between lattice symmetry and actual crystal symmetry operational, and it directly targets the structural mechanism underlying FQFE (Yu et al., 2024).

Applied to 171,527 materials, this workflow identified 202 potential FQFE candidates that are already experimentally synthesized: 12 in polar point groups and 190 in non-polar point groups. The same study also reported 2759 DFT-predicted robust FQFE materials from Materials Project, C2DB, and GNoME (Yu et al., 2024). These results are central because they recast FQFE from an isolated anomaly into a sizeable symmetry-defined materials class.

A later unified-definition screen began from 26,103 experimentally synthesized materials after band-gap, size, and chemistry filtering and found 100 conventional ferroelectrics and 68 QFEs with barriers below $$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$1. Among the QFEs, 48 were rotational-case candidates in nonpolar space groups, 12 exhibited both conventional FE and QFE behavior, and 8 were translation-related IQFE candidates (Luo et al., 14 May 2026). That work also introduced a broader category in which quantized polarization arises from arbitrary ionic displacements rather than only from simple fractional or integer displacements. This does not redefine FQFE itself, but it places FQFE within a wider quantum-ferroelectric taxonomy (Luo et al., 14 May 2026).

A related symmetry reformulation solves

$$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$2

for each point group and then checks compatibility across all group operations. In that approach, conventional ferroelectricity and FQFE appear as continuous and discrete solution sectors of the same modulo-quantum symmetry equation, rather than as incompatible symmetry classes (Pang et al., 17 Apr 2025).

4. Representative materials and disputed realizations

Monolayer $$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$3-In$$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$4Se$$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$5 is the most discussed candidate and the clearest source of controversy. It has $$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$6 symmetry, which forbids an intrinsic in-plane polarization vector under conventional point-group analysis, yet multiple experiments reported in-plane switchable polarization. In the formal-polarization framework, a continuous path connects two symmetry-equivalent structures with in-plane formal polarization changing from $$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$7 to $$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$8, so

$$\mathbf{P} \in \left\{ \mathbf{P}_0 + \sum_{i=1}^3 n_i \mathbf{Q}_i \;\middle|\; \mathbf{Q}_i = \frac{e}{\Omega} \mathbf{a}_i,\; n_i \in \mathbb{Z} \right\},$$9

The switching coordinate is a displacement of the middle-layer Se atom by $\mathbf{a}_i$0, implying $\mathbf{a}_i$1, consistent with the nominal Se oxidation state (Yang et al., 8 May 2026).

The same case remains experimentally unsettled. Reported lateral PFM evidence is complicated by crosstalk between out-of-plane and in-plane signals; angle-resolved lateral PFM showed no azimuthal dependence for $\mathbf{a}_i$2-In$\mathbf{a}_i$3Se$\mathbf{a}_i$4; the material is leaky and can host competing phases; and large-scale molecular dynamics found that a single-domain monolayer is difficult to switch because the nucleation barrier is dominated by interfacial energy. A domain-wall-mediated interpretation has therefore been advanced, in which experimentally relevant functionality arises from charged domain-wall motion rather than from bulk single-domain FQFE switching (Yang et al., 8 May 2026).

Bulk AlAgS$\mathbf{a}_i$5 is the canonical validated type-I example. Its low-symmetry phase has $\mathbf{a}_i$6, while the symmetrized lattice has $\mathbf{a}_i$7. Only the Ag ions undergo the key fractional displacement. Berry-phase calculations give

$\mathbf{a}_i$8

with polarization quantum

$\mathbf{a}_i$9

so $\Omega$0 and

$\Omega$1

CI-NEB yields a switching barrier of $\Omega$2 in the text and about $\Omega$3 in the abstract. AlAgS$\Omega$4 also has out-of-plane conventional ferroelectricity with

$\Omega$5

and the in-plane FQFE and out-of-plane FE are interlocked (Yu et al., 2024).

Monolayer HgI$\Omega$6 is the standard type-II example in a nonpolar point group. The low-symmetry phase belongs to $\Omega$7 with nonpolar point group $\Omega$8, and the symmetry-related partner involves

$\Omega$9

Berry-phase calculations give

$\mathbf{a}_i$0

with

$\mathbf{a}_i$1

so $\mathbf{a}_i$2 and $\mathbf{a}_i$3. The barrier is $\mathbf{a}_i$4 per Hg atom, phonons show no imaginary frequencies, and molecular dynamics at $\mathbf{a}_i$5 K for $\mathbf{a}_i$6 ps preserves the structure (Yu et al., 2024).

HfZnN$\mathbf{a}_i$7 illustrates a different point: FQFE can be switched through coupling to conventional polarization. In $\mathbf{a}_i$8, symmetry allows in-plane branch states $\mathbf{a}_i$9 and $\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$0, while the conventional out-of-plane component changes from $\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$1 to $\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$2, giving $\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$3, equivalent to $\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$4. The reported NEB barrier is about $\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$5 meV (Pang et al., 17 Apr 2025).

5. Sliding, multiferroic, and spin-coupled extensions

A recent extension places FQFE in bilayer altermagnets and sliding van der Waals structures. In sliding fractional quantum multiferroicity, a high-symmetry AA bilayer is transformed into two low-symmetry polar states $\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$6 and $\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$7 by half-cell translations of one layer. For bilayer Ca(CoN)$\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$8, the total displacement from $\mathbf{Q}_i = \frac{e}{\Omega}\mathbf{a}_i.$9 to $\mathbf{Q}$0 is $\mathbf{Q}$1, giving a half-translation along $\mathbf{Q}$2. With

$\mathbf{Q}$3

the two low-energy states have

$\mathbf{Q}$4

numerically $\mathbf{Q}$5 and $\mathbf{Q}$6 per unit cell for $\mathbf{Q}$7. The same system also carries out-of-plane polarization $\mathbf{Q}$8, and the preferred sliding barrier is only $\mathbf{Q}$9 along the direct diagonal path, compared with $(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$0 for the sequential path (Lu et al., 25 Dec 2025).

The multiferroic significance lies in the simultaneous control of polarization, spin splitting, Hall response, and magneto-optical response. In bilayer Ca(CoN)$(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$1, the conduction- and valence-band spin splittings are about $(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$2 and $(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$3 meV, corresponding to effective fields of $(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$4 and $(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$5 T, and the two sliding-related FQFE states are predicted to exhibit opposite anomalous Hall conductivity and Kerr response (Lu et al., 25 Dec 2025). This suggests that FQFE can be embedded into nonvolatile electrical spin control without requiring a sustained gate field.

A broader multiferroic program combines FQFE with altermagnetism. The symmetry statement is that if switched structures satisfy $(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$6 or $(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$7, then reversing the FQFE polarization reverses the altermagnetic spin splitting without rotating the Néel vector. First-principles examples include bulk MnTe, Cr$(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$8S$(\mathcal{R}-\mathcal{I})\mathbf{p}=\mathbf{Q},$9, Mn$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$00Bi$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$01NO$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$02, and two-dimensional AB$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$03 bilayers such as MnX$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$04 (X = Cl, Br, I), CoCl$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$05, CoBr$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$06, and FeI$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$07. MnTe is highlighted with a Néel temperature of about $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$08 K, electrically switchable spin splitting of about $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$09 eV, and a tunnel-junction proposal with tunneling magnetoresistance exceeding $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$10 (Dong et al., 19 Oct 2025).

These developments extend FQFE beyond a purely dielectric classification. They indicate that branch-quantized polarization can act as a symmetry-controlled order parameter in sliding ferroics, altermagnets, and electrically switchable multiferroics.

6. Distinctions from adjacent phenomena, misconceptions, and unresolved problems

Several recent ferroelectric phenomena are closely related in language but different in content. Ferroelectric skyrmions in a two-dimensional multiferroic model were reported to carry topological charges that can be integer, half-integer, and other fractional values, but the fractional quantity there is the topological charge $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$11 of a dipole texture, not a fractionally quantized bulk polarization. That work is therefore better described as fractional topological charges of ferroelectric skyrmions than as a realization of bulk fractional quantum ferroelectricity (Liu, 2021).

Carrier-doped polar metals provide a different neighboring case. There, mobile carriers induce quantum fluctuations that suppress ferroelectric order and generate polarons dressed by disrupted dipoles; the critical densities quoted are $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$12 for screening of global polarization and $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$13 for destruction of FE order. This is a mechanism for quantum reduction and spatial fragmentation of ferroelectric order, not for branch-quantized FQFE (Gu et al., 2023). Similarly, Ba$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$14CuSb$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$15O$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$16 shows $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$17 and $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$18, interpreted as spin-correlation-driven ferroelectric quantum criticality in a quantum paraelectric, but not as fractionalization of the polarization sector (Ghosh et al., 5 Jan 2026).

A second common misconception is that any Berry-phase ferroelectricity is automatically fractional. That is not the case. The quantum-geometric theory of out-of-plane stacking ferroelectricity in bilayers maps the polarization problem to an SSH/Rice–Mele-type model and gives a clear Berry-phase origin for $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$19, but the broken-sublattice-symmetry regime generally yields continuously tunable Berry phase rather than fractionally quantized branch values (Zhou et al., 2023). Likewise, interfacial ferroelectricity in non-magnetic graphene/hBN moiré superlattices shows remnant polarization, magnetic-field enhancement, and strong feedback on Shubnikov–de Haas oscillations and Hall transport, but no demonstrated fractional quantum ferroelectric phase in the strict sense (Jiang et al., 1 Jul 2025).

The main unresolved issues are therefore not definitional but physical. The review literature repeatedly emphasizes the gap between formal polarization branches and experimentally accessible switching. A topologically allowed branch change does not guarantee that the path is insulating, that nucleation barriers are manageable, or that the measured response reflects bulk switching rather than charged domain walls and interfaces. In $$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$20-In$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$21Se$$\Delta \mathbf{P}_{\rm FQFE} = \frac{N}{m}\mathbf{Q}$$22, for example, the dominant functionality may lie in charged domain-wall motion rather than in single-domain in-plane switching (Yang et al., 8 May 2026). The unified-definition program sharpens this point by making reversible switching between energetically equivalent states the operational criterion for ferroelectricity itself (Luo et al., 14 May 2026).

Fractional quantum ferroelectricity is therefore best understood as an extension of ferroelectric theory, not a violation of it. Its core content is the coexistence of symmetry-equivalent switchable states with branch-quantized polarization differences in the Berry-phase formalism, often generated by fractional-lattice ionic motion. Its present research frontier lies in determining when those topological polarization branches are physically switchable, how they couple to domain walls and interfaces, and how far they can be integrated with correlated, sliding, and spin-active quantum materials (Yang et al., 8 May 2026, Yu et al., 2024, Pang et al., 17 Apr 2025).