Introduction

Ferroelectricity, which is characterized by a reversible spontaneous polarization through application of an electric field, not only is important in fundamental physics, but also finds a wide range of applications, such as piezoelectric sensors/actuators1,2,3 and non-volatile memory4,5,6. In common cases, the polarization originates from small atomic displacements, leading to the ordinary ferroelectricity, as illustrated in Fig. 1a. In such case, the direction of the polarization is consistent with the symmetry of the ferroelectric phase, e.g., the polarization has to be along the two-fold axis of a system with C2v symmetry. This is in accord with the well-known Neumann’s principle7, which states that the symmetry elements of any physical property of a crystal must include all the symmetry elements of the point group of the crystal.

Fig. 1: Concept of FQFE.
figure 1

Schematics of a FE, b QFE and c FQFE, supposing the ion with ±1 charges. The green and red balls represent movable ions and ligand ions, respectively. Top and side views of the FQFE example: d low-symmetry phase L1, e high-symmetry phase H, f low-symmetry phase L2. The black border indicates the unit cell. F includes two are fixed atoms (layers) F, and M is a movable atom (layer). The blue and red arrows depict the atomic displacements of M from H to L1 and L2, respectively. The green arrows show ∆P, the atomic displacements of M between L1 and L2. Since only M moves, the blue, red, and green arrows can also represent P1(polarization of L1), P2(polarization of L2), and ∆P (polarization difference between low symmetry phases), respectively. ∆P cannot be invariant under a point symmetry operation (C3z) of the low-symmetry phase, which leads to the FQFE. g The latticed form of P1, P2 and ∆P. The black dashed parallelogram depicts the “lattice” of polarization. The blue and red points represent P1 and P2, respectively. ∆P can be any vector between points with different color. Therefore, ∆P is non-zero and fractionally quantized, i.e. \(\frac{1}{3}{{{{{\bf{Q}}}}}}\) along the [120] direction.

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In contrast to the small atomic displacements of ordinary ferroelectricity, recent studies report novel ferroelectricity with large atomic displacements that are comparable to lattice constants8,9,10,11,12,13. For example, the Cu ions are found to successively migrate through van der Waals layers in CuInP2S69,13, implying the polarization there can be very large and exhibits different values with different switching voltage. Regarding such large displacements, one may recall the modern theory of polarization (MTP), which is developed to resolve the multi-value problem of polarization when considering periodic boundary condition in computations. In MTP, the concept of polarization quantum, \({{{{{\bf{Q}}}}}}=\frac{e}{\Omega }{{{{{\bf{a}}}}}}\), where e is the unit charge, Ω is the volume of the unit cell and a is a lattice vector, denotes the polarization caused by a unit charge traveling through the unit cell. With such understanding, the aforementioned systems actually exhibit ferroelectricity with an integer quantized polarization, dubbed as the quantum ferroelectricity (QFE, see Fig. 1b).

However, recent experimental and computational works14,15,16,17,18,19,20 report unexpected large in-plane component of polarization in monolayer α-In2Se3, which possesses C3v symmetry and allows only out-of-plane polarization accord to the Neumann’s principle. Such contradiction indicates that the understandings of ordinary ferroelectricity and QFE are insufficient and new theory of ferroelectricity is highly desired. The development from ordinary FE to QFE is reminiscent of the series of Hall, quantum Hall and, fractional quantum Hall effects. It is thus promising to expect the existence of fractional quantum ferroelectricity (FQFE, see Fig. 1c). If it exists, one may wonder (i) how to identify FQFE by symmetry analysis, (ii) which fractional numbers to choose from, and (iii) how the Neumann’s principle breaks down. Correctly answering such questions will help deepen the understanding of novel ferroelectricity.

In this work, we propose and demonstrate the existence of FQFE, through systematical group theory analysis and density functional theory (DFT) verifications. Our group theory analysis over all 230 space groups indicates that FQFE can exist in 28 point groups, of which 8 are polar while 20 are non-polar. It is also found that FQFE always contradicts with the Neumann’s principle, i.e., the direction of polarization is not limited by the symmetry of ferroelectric phase, which is in strong contrast with ordinary FE. DFT calculations are further performed and demonstrate FQFE in monolayer α-In2Se3, cubic AgBr and many other systems.

Results

Framework of FQFE

The conceptual realization of the FQFE phenomenon in both tetrahedral and hexagonal systems is depicted in Fig. 1c. To elucidate the fundamental essence of FQFE, we discuss in detail a hexagonal system as an illustrative example. Within the FQFE, the ferroelectric behavior is exhibited through two distinct phases, denoted as L1 and L2, both adopting the MoS2-type structural configuration. These phases, characterized by lower symmetry (see Fig. 1d, f), belong to space group P-6m2 with the point group D3h. Comprising both M and F ions, these phases involve the roles of M and F ions in the fractional polarization dynamics during the phase transition. Notably, M ions are mobile and contribute to the polarization switching, while the fixed F ions remain stationary. To foster comprehension of the FQFE concept, we introduce an intermediary phase, H, which presents higher symmetry (as seen in Fig. 1e) and shares the composition of M and F ions with L1 and L2.

In the context of the M-F ion composition, M1, M0 (M0 being the midpoint between M1 and M2), and M2 symbolize the positions of M ions within L1, H, and L2, respectively. The polarization attributes of L1 and L2, along with the polarization difference between them, are ascribed to P1 ~ M1-M0, P2 ~ M2-M0, and ∆P ~ M2-M1, as indicated by the blue, red, and green arrows in Fig. 1e. Note that P denotes dipole moment here, rather than conventional polarization. The MTP indicates multi-values for P1, P2, and ∆P, which are visually illustrated through a lattice representation in Fig. 1g. It becomes evident that ∆P consistently maintains a non-zero in-plane value and lacks invariance under symmetry operations, such as the three-fold rotation around the z-axis (c3z), belonging to the point group characterizing the low symmetry phases (L1 or L2). As we will find, ∆P is fractionally quantized besides the trivial integral lattice translation, which is protected by the c3z operation.

Group theory analysis of FQFE

Firstly, we use group theory to present the basic FQFE framework and extract key features of FQFE. Then, we identify feasible space group-point group pairs that can realize FQFE realization in the case of one mobile atom. This framework is then generalized to the case of multiple mobile atoms. Finally, we address methods for assessing FQFE presence within a system and determining the other ferroelectric phase (e.g., L2) for a given structure (L1).

Let us discuss the framework in the language of group theory. The space group (and corresponding point group) of the structure F, as well as that of the low symmetry phases L1 and L2, are denoted as GF (PF) and GL (PL), respectively. GF can be either a symmorphic or non-symmorphic space group. For simplicity, we first consider the symmorphic case, in which the translational parts of all space group operations are lattice translations. In this case, the high symmetry phase H can be constructed by setting M0 at the Wyckoff position with site symmetry PF. Since M consists of a single atom, symmetry operations that maintain the invariance of M0 (M1,2) and F must be symmorphic space group operations within GF. These operations collectively form the space group of H (L1,2), which inherently constitutes symmorphic space groups. For simplicity, we consider only the point groups of H (L1,2). Such point groups are the site symmetry groups of Wyckoff positions M0 (M1,2) within GF, denoted as PF (PL). M1 and M2 are attributed to different sites of the same Wyckoff position, since that the energies of L1 and L2 are degenerate. By contrast, if GF is a non-symmorphic space group, it becomes unfeasible to construct the H structure with the space group GF, since there is only one mobile atom in the H structure. With above preparation, considering the scenario where a given space group GF features M1 and M2 (representing two coordinates of the identical Wyckoff position), the FQFE can be realized if the coordinate difference (M2-M1) involves fractional components such as 1/2 and 1/3.

Exploration of potential GF-PL pairs for realizing FQFE with a single M atom follows these steps: (i) List the Wyckoff positions21 of a given space group (GF); (ii) Select a Wyckoff position with at least two distinct coordinates (M1 and M2), each exhibiting at least one fixed fractional component; (iii) Assess whether ∆P ~ M2-M1 possesses a fractional component; if so, FQFE is likely to occur, and the site symmetry group of M1,2 is PL. Taken the non-symmorphic Ccc2 space group (No.37) for an example: (i) The associated Wyckoff positions are listed in Table 1; (ii) It is found that Wyckoff positions 4b and 4c exhibit coordinates with fixed fractional components, while 4a and 8d do not; (iii) For Wyckoff position 4b, ∆P between the two sites yields (0,0,□) and thus lacks a fractional component (component □ is related to variables and is not considered in the FQFE frame); For Wyckoff position 4c, ∆Py = 3/4-1/4 = 1/2, facilitating FQFE with PL = c2z (two-fold rotation along z). In such example, ∆P pertains to the conventional cell, which is twice larger than the unit cell, due to that Ccc2 is bottom centered with fractional translational symmetry of (1/2,1/2,0). In such case, the atom M shifts from (1/4,1/4,z) to (1/4,3/4,z+1/2) and such movement is different from (1/2,1/2,0), rendering ∆P also being fractional within the unit cell.

Table 1 Wyckoff positions of space group Ccc2 (No.37)21
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Applying the above principle to all 230 space groups [14], it yields 571 GF-PL pairs, which include 191 possible GF and 28 possible PL, while do not include C1, C6v, D6h, Oh point groups, see Table S1. Interestingly, among the 28 possible PL, 8 PL are polar point groups (e.g., C3v) and 20 PL are non-polar (e.g., Td). The latter type suggests that fractionally quantized polarization can exist in previously assumed non-polar systems, which is now brought back to the playground of ferroelectricity by FQFE. From Table S1, we can see that FQFE has \({{{{{\mathbf{\Delta }}}}}}{{{{{\bf{P}}}}}}=\frac{{{\rm{n}}}}{{{\rm{m}}}}{{{{{\bf{Q}}}}}}\) (m = 2,3,4,6,8 and n is an integer) in the conventional cell, as well as being fractional within the unit cell. Notably, ∆P is non-invariant under PL in all these cases, which is due to the fact that the fractional components of M1 and M2 are fixed and determined by the symmetries in GF. Hence, such symmetry analysis indicates that FQFE always contradicts with Neumann’s law.

For FQFE systems with multiple mobile atoms, the structure can also be decomposed into F and M, only the latter now possessing multiple atoms. Each atom within M, denoted as Mi, corresponds to the single movable atom in the previous simple case, resulting in fractionally quantized polarization ∆Pi. The cumulative polarization ∆P is the sum of ∆Pi, which is also naturally being fractionalized. Notably, all Mi atoms should maintain the site symmetry of the corresponding Wyckoff positions of the space group GF. For instance, in a MXene22,23,24,25 like material of Sc2CO2 [see SM], F comprises two Sc atoms and one C atom, with GF = P-3m1, while two O atoms in M share positions with site-symmetry GL(PL)=P3m1(C3v). Each O atom contributes \(\frac{1}{3}{{{{{\bf{Q}}}}}}\) and it yields the total \({{{{{\mathbf{\Delta }}}}}}{{{{{\bf{P}}}}}}=\frac{2}{3}{{{{{\bf{Q}}}}}}\) [see SM].

Lastly, utilizing Table S1, we outline how to determine the presence of FQFE in a given low symmetry structure and how to identify the other symmetry-related low-symmetry phase(s). For the given low symmetry phase L1, one divides L1 into M and F, by assigning F with higher symmetries (GF is a supergroup of GL). Each Wyckoff position of Mi in GF should match one of those listed in Table S1 and have at least one fractional component choosing from \(\left\{\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{3}{4},\frac{1}{6},\frac{5}{6},\frac{1}{8},\frac{3}{8},\frac{5}{8},\frac{7}{8}\right\}\). If such a division is feasible, FQFE may be present. Then, apply GF symmetry operations that do not present in GL to L1 to obtain other low symmetry phase(s), L2. Note that applying different symmetry operations to L1 might lead to different L2.

FQFE in monolayer α-In2Se3

Monolayer α-In2Se3 has gained significant attention as a 2D ferroelectric material. It conforms to No.156 space group of P3m1 (point group C3v) with a three-fold rotation axis perpendicular to the monolayer, which indicates pure out-of-plane polarization according to Neumann’s principle. However, intriguingly, monolayer α-In2Se3 was predicted to exhibit not only out-of-plane polarization, but also a substantial in-plane component14, which has been subsequently confirmed through experiments17,18,19. Such observed in-plane polarization behavior in α-In2Se3 contradicts Neumann’s principle, raising questions about the novel ferroelectricity and its underlying physics.

Applying the FQFE theory, we proceed to construct the ferroelectric phase L2 from the initial state L1 in monolayer α-In2Se3. L1 comprises five atomic layers arranged as Se-In-Se-In-Se, as depicted in Fig. 2a. L2 can be constructed following these steps: (i) Dividing L1 into M and F, with the latter involving the upper two layers and the lower two layers, i.e., Se-In---In-Se. Besides other symmetries in GL, F further exhibits inversion symmetry I, leading to GF = GL + IGL = P-3m1 (No. 164). M is represented by the middle Se atom, occupying Wyckoff position 2d within GF (see Table S2), which is a case listed in Table S1. (ii) The application of inversion, a symmetry operation in GF but absent in GL, on L1 yields L2, as depicted in Fig. 2a. Since GF is a symmorphic space group, high symmetry phase H (see Fig. 2b) can be constructed by placing M, i.e. the middle layer Se atom, at Wyckoff positions 1a or 1b, each possessing site symmetry PF = D3d. The polarization switching process, illustrated in Fig. 2a, involves the middle Se atom moving from being directly above the lower In atom to being situated directly below the upper In atom. This in-plane displacement yields \(\frac{{{{{{\bf{1}}}}}}}{{{{{{\bf{3}}}}}}}{{{{{\bf{a}}}}}}{{{{{\boldsymbol{+}}}}}}\frac{{{{{{\bf{1}}}}}}}{{{{{{\bf{3}}}}}}}{{{{{\bf{b}}}}}}\) and consequently generates a non-zero in-plane fractionally quantized polarization, which contradicts the P3m1 symmetry of the system. According to such analyses, the observed in-plane polarization of monolayer α-In2Se3 actually arises from the presently proposed FQFE.

Fig. 2: Structure and ferroelectricity of monolayer α-In2Se3.
figure 2

a Top and side views of monolayer α-In2Se3, corresponding to the L1, H, and L2 phases, respectively. \(\frac{1}{3}{{{{{\bf{a}}}}}}+\frac{1}{3}{{{{{\bf{b}}}}}}\), the in-plane displacement of M (Se) in the ferroelectric phase transition is shown by the blue arrow. b NEB calculation of the energy barrier and evolution of the polarization intensity along the path similar to the one in Ref. 14. The amplitude represents the in-plane polarization magnitude along the [110] direction. The positive and negative signs indicate the polarization toward [110] and the [−1–10] direction, respectively. Here the polarizations of L1 and L2 are \({{{{{{\bf{P}}}}}}}_{1}=\frac{1}{3}{{{{{\bf{Q}}}}}}\) and \({{{{{{\bf{P}}}}}}}_{2}=-\frac{1}{3}{{{{{\bf{Q}}}}}}\), respectively. Q is the polarization quantum along the [110] direction. The polarization difference is \({{{{{\boldsymbol{\triangle }}}}}}{{{{{\bf{P}}}}}}=\frac{2}{3}{{{{{\bf{Q}}}}}}\).

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DFT calculations are further performed to verify the contradiction between the in-plane polarization of monolayer α-In2Se3 and its symmetry. The minimum energy pathway between L1 and L2 is determined using the climbing image nudged elastic band (CI-NEB) method26,27. The energy barrier is determined to be 68 meV/f.u. (see Fig. 2b), which is consistent with the value reported in Ref. 23. Our focus then shifts to the in-plane polarizations of monolayer α-In2Se3, which are assessed using the Berry phase approach28,29,30. As illustrated in Fig. 2b, the magnitude of in-plane polarizations of monolayer α-In2Se3 exhibits continuous variation along the L1-L2 pathway. Specifically, the in-plane polarizations of L1 and L2 phases along the [110] direction yield 2.37 and −2.37 eÅ per unit cell, respectively, in agreement with previous results [15]. The polarization quantum \({{{{{\bf{Q}}}}}}=\frac{e}{\Omega }{{{{{\bf{a}}}}}}\) along the [110] direction is 7.11 eÅ/unit-cell, where a represents the lattice vector along the [110] direction. Therefore, the DFT-derived in-plane polarization is \(\frac{1}{3}{{{{{\bf{Q}}}}}}\), which is consistent with the FQFE theory. The polarization calculations demonstrate that monolayer α-In2Se3 indeed behaves as a ferroelectric with in-plane polarizations, a phenomenon at odds with the symmetry of P3m1. The polarization values from DFT calculations further verify the FQFE in α-In2Se3.

FQFE in AgBr

Applying the FQFE theory, we proceed to predict the existence of FQFE within a non-polar point group. From Table S1, we choose PL = Td that exhibits the highest symmetry, and the corresponding symmophic space group GF = Fm-3m (No. 225). Illustratively, the zinc blende structure (space group F-43m) is a typical crystal structure with Td symmetry. As shown in Fig. 3a, it possesses chemical formula AB, where A locates at (0,0,0) and B at (1/4,1/4,1/4). Either A or B can represent M (or F) atoms, with GF = Fm-3m. Here, we designate A as F and B as M. The application of a GF operation that is not presented in GL, such as inversion centered at the origin, to L1 (L2) results in the formation of the other low symmetry phase L2 (L1). The construction of H involves placing M at Wyckoff positions with site symmetry PF = Oh, i.e. (1/2,1/2,1/2). Notably, H adopts a rocksalt structure with space group Fm-3m. During the transition from L1 to L2, as depicted in Fig. 3a, atom B shifts from (1/4,1/4,1/4) to (3/4,3/4,3/4). This displacement introduces a fractional shift of (1/2,1/2,1/2), rendering a fractionally quantized polarization. Notably, the lower symmetry phases L1 and L2 that exhibit polarizations actually belong to non-polar Td symmetry, showing the novelty of FQFE.

Fig. 3: Structure and ferroelectricity of bulk AgBr.
figure 3

a Schematic structure of AgBr during the L1-H-L2 phase transition. The L1,2 phase belongs to F-43m and the H phase belongs to Fm-3m. The process can be considered as Br moving along the path \(\left(\frac{1}{4},\frac{1}{4},\frac{1}{4}\right)\to \left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\to \left(\frac{3}{4},\frac{3}{4},\frac{3}{4}\right)\). b The energy barrier calculated by NEB and the evolution of the polarization in the primitive cell along the path in (a), where the primitive cells of L1, H, L2 are depicted. Here the polarizations of L1 and L2 are \({{{{{{\bf{P}}}}}}}_{1}=\frac{1}{4}{{{{{\bf{Q}}}}}}\) and \({{{{{{\bf{P}}}}}}}_{2}=-\frac{1}{4}{{{{{\bf{Q}}}}}}\), respectively. Q is the polarization quantum along the [111] direction. The polarization difference is \({{{{{\boldsymbol{\triangle }}}}}}{{{{{\bf{P}}}}}}=\frac{1}{2}{{{{{\bf{Q}}}}}}\).

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To investigate the above FQFE phenomenon further, we explore the Materials Project database31 for materials featuring both F-43m and Fm-3m phases. Among these materials, AgBr emerges as our choice. Its ground state is a conventional rocksalt structure with space group Fm-3m32,33,34. Other materials exhibiting both F-43m and Fm-3m phases, such as ZnX (X = O, S, Se) and AlX (X = P, As, Sb), are listed in the supplementary material. Then, we turn to DFT calculations to exam the ferroelectricity of AgBr. By evaluating phonon spectra and employing molecular dynamics simulation, we confirm that the low symmetry F-43m phase is dynamically stable and exhibits good thermal stability at room temperature (see Fig. S1). The lattice constants of the high and low symmetry phases measure 5.84 and 6.31 Å, respectively. The energetically degenerate ground states of the system manifest as the L1,2 phases. Notably, the L1,2 phase is 74 meV/f.u. lower in energy than the H phase.

The demonstration of ferroelectricity in AgBr hinges on showcasing the presence of a switchable spontaneous polarization. Figure 3a portrays the kinetic pathways connecting the L1 and L2 phases. Given that the high symmetry phase H exists as an energy local minimum, the L1-L2 pathway can be deconstructed into two steps: L1-H and H-L2. During the L1-H transition, the L1 phase transforms into the H phase through the displacement of the Br atom from (1/4,1/4,1/4) to (1/2,1/2,1/2) along the [111] direction, overcoming an energy barrier of 155 meV/f.u. The subsequent H-L2 transition involves the H phase transforming into another low symmetry phase (L2 phase), as the Br atom shifts from (1/2,1/2,1/2) to (3/4,3/4,3/4) along the same direction, necessitating an energy barrier of 76 meV/f.u. The L1-L2 polarization switching solely contends with the highest energy barrier, i.e., the 155 meV/f.u. barrier during the initial step. This energy barrier is similar to those observed in traditional bulk ferroelectric materials like PbTiO3 and BiFeO335, indicating that the fractionally quantized polarization of AgBr is likely to be switchable in experiments.

Figure 3b shows the evolution of polarization across the L1-L2 pathway for the AgBr primitive cell. The MTP is adopted to calculate polarization. Evidently, the system’s polarization in the primitive cell experiences continuous changes as Br atoms move from (1/4,1/4,1/4) to (3/4,3/4,3/4). The polarizations of the L1 and L2 phases along the [111] direction are determined to be 69.80 and −69.80 μC/cm2, respectively. According to MTP, the polarization quantum of AgBr system is 279.21 μC/cm2 along the [111] direction, derived from \({{{{{\bf{Q}}}}}}=\frac{e}{\Omega }{{{{{\bf{a}}}}}}\), where a is the lattice vector along the [111] direction in the primitive cell. Consequently, both L1 and L2 phases display fractionally quantized polarization. Therefore, these findings establish AgBr as possessing spontaneous and switchable polarization, qualifying it as a material with FQFE in non-polar systems. This revelation underscores the potential for numerous other materials (such as those listed in Table S4) to harbor similar FQFE, thereby presenting opportunities for exploration and advancement across various technological domains. FQFE is also suitable for organic-inorganic systems, e.g. NH4Br with similar structures to AgBr [see SM].

Discussion

According to MTP, electric polarization in a periodic system is multi-valued. However, the difference in the polarization between two ferroelectric states is solely determined by the specific switching path. Taking In2Se3 as an example, though there are three possibilities in the direction of its polarization, it will be uniquely determined when applying a specific external field. Note that given the initial and final ferroelectric states, the polarization difference can only differ by an integer quantum polarization for different switching paths. Such a feature is in line with the “double-path” ferroelectrics discussed in Ref. 36.

It is well-known that ordinary ferroelectrics are piezoelectrics, pyroelectrics, and often ferroelastics. For QFE and FQFE materials with polar groups, they display piezoelectricity, pyroelectricity, and ferroelasticity, similar to ordinary FE materials. However, QFE and FQFE materials with non-polar groups generally lack piezoelectricity and pyroelectricity but may be ferroelastic as the lattice vectors in the two low-symmetry phases may be swapped [see SM]. This suggests that other direction-dependent properties (e.g., magnetic, transport, mechanical, optical properties) of FQFE materials are possible to be switched by electric fields.

In conclusion, we propose the concept of FQFE, which leads to polarization not only in polar systems but also in non-polar ones. A distinctive attribute of FQFE lies in its fractionally quantized polarization component. Our group theory analysis suggests the potential existence of FQFE in 28 point groups. Through the application of the FQFE theory, we explain the ferroelectric behavior in the monolayer α-In2Se3. Furthermore, employing our theory and first-principles calculations, we predict the FQFE within AgBr of the non-polar Td (F-43m) phase. The discovery of FQFE significantly expands the scope of ferroelectrics, opening avenues for exploring their properties and potential applications across diverse fields.

Methods

DFT calculations

Density functional theory (DFT) calculations are performed using the Vienna ab initio simulation package (VASP)37. The employed exchange and correlation functional adopts the generalized gradient approximation (GGA) as parametrized by Perdew, Burke, and Ernzerhof (PBE)38. The adopted pseudopotentials are constructed using the projected enhanced wave (PAW) method39,40. The plane-wave energy cutoff is set to 500 eV. The structures are fully relaxed until the residual force on each atom is less than 0.01 eV/Å. The energy convergence criteria is set to 10−6 eV. The Brillouin zone is sampled using a Gamma-centered scheme with a 15 × 15 × 1 k-point mesh for In2Se3 system and a 9 × 9 × 9 k-point mesh for AgBr. For the In2Se3 monolayer, a 20 Å vacuum space is used and dipole corrections are adopted between adjacent layers to avoid interactions between neighboring periodic images. The energy barrier between different polarization states is calculated using the climbing image nudged elastic band (CI-NEB) method41. To examine the dynamical stability of AgBr, we calculate the phonon spectrum using Phonopy42 with a 2 × 2 × 2 supercell. To demonstrate the thermal stability of the FQFE systems, we perform ab initio molecular dynamics simulations for AgBr and monolayer In2Se3 systems at 300 K using 3 × 3 × 3 and 4 × 4 × 1 supercells, respectively.