Correlation-enhanced spin-orbit coupling and quantum anomalous Hall insulator with large band gap and stable ferromagnetism in monolayer $\mathrm{Fe_{2}Br_{2}}$

The quantum anomalous Hall effect (QAHE) is characterized by the nonzero Chern numbers with the quantized Hall conductanceh0 , which can be used to design low-power-consumption spintronic devices due to the existence of dissipationless chiral edge states under zero magnetic field. The QAH state also provides a fertile playground to explore topological magnetoelectric effects, Majorana fermions and quantum computationh1 ; h2 ; h3 . However, the experimental observation of QAHE is rare, and its ultralow working temperature limits exploring these emergent quantum physics and promising applications. The QAHE is firstly observed in Cr-doped $\mathrm{(Bi,Sb)_{2}Te_{3}}$ thin films at a very low temperature of 30 mKh4 . In quick succession, the QAHE is realized in V-doped and Cr-and-V co-doped $\mathrm{(Bi,Sb)_{2}Te_{3}}$ thin film at about 25 and 300 mKh5 ; h6 . Recently, the intrinsic QAHE has been observed in the van der Waals (vdW) layered material $\mathrm{MnBi_{2}Te_{4}}$h7 ; h8 , and a high-Chern-number QAHE has also been obtained in a 10-layer $\mathrm{MnBi_{2}Te_{4}}$ device at about 13 Kh9 . So, seeking high-temperature QAHI with large gap is a compelling problem of condensed matter physics and materials science.

However, searching for high-temperature QAHI with large nontrivial band gap is challenged. The ferromagnetism prefers metallic systems composed of light 3$d$ elements, while topological insulator (TI) favors heavy elements for achieving strong SOC effects. Recently, a robust QAHI $\mathrm{Fe_{2}I_{2}}$ monolayer with centrosymmetry is predicted with a large nontrivial band gap of 301 meV and a high Curie temperature of about 400 Kfe . In $\mathrm{Li_{2}Fe_{2}X_{2}}$ (X=S, Se and Te) monolayers (Lithium-decorated iron-based superconductor monolayer materials), the high-temperature large-gap QAHIs are also achievedfe1 . To realize the combination of piezoelectricity with topological properties, monolayer $\mathrm{Fe_{2}IX}$ (X=Cl and Br), $\mathrm{FeI_{1-x}Br_{x}}$ (x=0.25 and 0.75) and $\mathrm{Li_{2}Fe_{2}SSe}$ are predictedfe2 ; fe3 ; fe4 , namely two-dimensional (2D) piezoelectric quantum anomalous Hall insulator (PQAHI), which provides possibility to use piezotronic effect to control QAHE. All these 2D materials share the same Fe layer structure and Fe-dominated low-energy states, and they all are room-temperature large-gap high-Chern-number QAHIs. The SOC is generally considered to be appreciable only in heavy elements, and the SOC effect should be very small in light Fe element. However, these 2D materials are predicted to large-gap QAHIs. In ref.fe1 , a large QAH gap is attributed to the enlarged effective SOC strength of $d$ orbitals by bonding with heavy elements, and then the SOC effects are significantly enhanced near Dirac cones.

To clarify this question, we take monolayer $\mathrm{Fe_{2}Br_{2}}$ as a concrete example to study the correlation effects on effective SOC strength. Why do we choose monolayer $\mathrm{Fe_{2}Br_{2}}$? Firstly, monolayer $\mathrm{Fe_{2}Br_{2}}$ has not been studied in detail, including its stability and electronic structures. Secondly, for $\mathrm{Fe_{2}Br_{2}}$, the Fe elements bond with relatively light elements Br compared to I elements in $\mathrm{Fe_{2}I_{2}}$, and it has a simpler structure than $\mathrm{Li_{2}Fe_{2}S_{2}}$. In this work, it is found that the correlation-enhanced SOC effect of Fe atoms induces large nontrivial band gap in monolayer $\mathrm{Fe_{2}Br_{2}}$, and the effective enlarged SOC strength of Fe-3$d$ orbitals is not due to bonding with heavy elements Br. Calculated results show out-of-plane magnetic anisotropy is necessary to produce QAH state. So, the magnetic anisotropy direction must be determined to achieve QAHE in these 2D materials. Taking classic $U$=2.5 eVfe ; fe5 as an example, $\mathrm{Fe_{2}Br_{2}}$ is proved to be dynamically, mechanically and thermally stable. The high Chern number (C=2) is firmly confirmed by Berry curvatures and chiral edge states. In $\mathrm{Fe_{2}Br_{2}}$, the QAH state, FM ordering and out-of-plane magnetic anisotropy are robust against biaxial strain. It is found that reduced correlation and compressive strain make for high Curie temperature. Our works provide basis for understanding large nontrivial band gap in monolayer $\mathrm{Fe_{2}XY}$ (X/Y=Cl, Br and I) and $\mathrm{Li_{2}Fe_{2}XY}$ (X/Y=S, Se and Te).

The rest of the paper is organized as follows. In the next section, we shall give our computational details and methods. In the next few sections, we shall present electronic correlation effects on electronic structures along with the case at $U$=2.5 eV, strain influence on topological properties, and Curie temperatures of monolayer $\mathrm{Fe_{2}Br_{2}}$. Finally, we shall give our discussion and conclusion.

Within density functional theory (DFT)1 , the first-principles calculations are performed by using the Vienna ab initio simulation package (VASP)pv1 ; pv2 ; pv3 with plane-wave basis set. The projector augmented wave (PAW) method is adopted in conjugation with a GGA of Perdew, Burke and Ernzerhof (PBE-GGA) as exchange-correlation functionalpbe . A plane-wave cutoff of 500 eV is used to perform geometric optimization and electronic properties calculations of monolayer $\mathrm{Fe_{2}Br_{2}}$. The energy convergence criterion is set for $10^{-8}$ eV, and the residual force is less than 0.0001 $\mathrm{eV.{\AA}^{-1}}$. The Brillouin zone (BZ) integration is carried out with 18$\times$18$\times$1 k-point sampling. To avoid interactions between two neighboring images, the vacuum region along the $z$ direction is set to be larger than 16 $\mathrm{{\AA}}$. The electronic correlation of Fe atoms is considered within the GGA+$U$ scheme by the rotationally invariant approach proposed by Dudarev et alu . The SOC is included self-consistently in the calculations to investigate magnetic anisotropy and topological properties.

The elastic stiffness tensor $C_{ij}$ is calculated by using strain-stress relationship (SSR) with GGA, and the 2D elastic coefficients $C^{2D}_{ij}$ have been renormalized by $C^{2D}_{ij}$=$L_{z}$$C^{3D}_{ij}$ with $L_{z}$ being the length of unit cell along $z$ direction. For phonon spectrum calculation, the Phonopy codepv5 is used within finite displacement method. The interatomic force constants (IFCs) with a 5$\times$5$\times$1 supercell are calculated to attain phonon dispersions. For ab initio molecular dynamics (AIMD) simulation, the calculation is carried out with a 4$\times$4$\times$1 supercell for more than 8000 fs with a time step of 1 fs by using canonical ensemble. WannierTools code is used to perform surface state and Berry curvature calculations, based on the tight-binding Hamiltonians constructed from maximally localized Wannier functions by Wannier90 codew1 ; w2 . The Curie temperature is estimated by Monte Carlo (MC) simulation with a 40$\times$40 supercell and $10^{7}$ loops, as implemented in Mcsolver codemc .

As shown in Figure 1, the unit cell of $\mathrm{Fe_{2}Br_{2}}$ contains four atoms with two co-planar Fe atoms being sandwiched between two layers of Br atoms. The $\mathrm{Fe_{2}Br_{2}}$ possesses $P4/nmm$ space group (No. 129) with centrosymmetry. The key space-group symmetry operations of $\mathrm{Fe_{2}Br_{2}}$ include space inversion $P$, $C_{4}$ rotation, $M_{x}$ ($M_{y}$) mirror and glide mirror $G_{z}=\{M_{z}|\frac{1}{2},\frac{1}{2},0\}$, which is different from monolayer $\mathrm{Fe_{2}IBr}$ with missing $P$ and glide mirror $G_{z}$fe2 . So, monolayer $\mathrm{Fe_{2}IBr}$ possesses piezoelectricity, but $\mathrm{Fe_{2}Br_{2}}$ is not piezoelectric. it is noteworthy that magnetic orientation can influence these symmetries within SOC.

In the presence of the electron correlation, the SOC effect of 3$d$ systems with special orbital symmetry and electron occupation is found to be more prominenth10 ; h11 ; h12 ; h13 . Here, we optimize lattice constants $a$ with varied $U$ (0-4 eV), and then determine magnetic ground state. The energy differences between antiferromagnetic (AFM) and FM ordering vs $U$ are plotted in Figure 2. With increasing $U$, the ground state of $\mathrm{Fe_{2}Br_{2}}$ changes from FM ordering to AFM one, and the critical $U$ value is about 3.75 eV. It is found that increasing $U$ weakens FM interaction, giving rise to important influence on Curie temperature of $\mathrm{Fe_{2}Br_{2}}$. The $a$ as a function of $U$ with FM ground state is shown in Figure 2, and $a$ increases with increasing $U$. Similar result can be found in FeClF monolayerh12 . The different magnetic orientation can affect the symmetry of 2D systems, and then produce important influence on their valley and topological propertiesh10 ; h12 . The magnetic anisotropy can be described by magnetic anisotropy energy (MAE), which plays a very important role to determine thermal stability of magnetic ordering. The MAE can be calculated by $E_{MAE}$ = $E_{x}$-$E_{z}$, where $E_{x/z}$ is the energy per Fe atom when the magnetization is along the $x/z$ direction. In Figure 2, we plot MAE as a function of $U$, which favors an out-of-plane FM state in considered $U$ range. The out-of-plane magnetic anisotropy plays a crucial role to produce QAH state, as we shall see in a while. It is found that increasing $U$ makes for out-of-plane one. This is different from one of FeClF monolayerh12 , where increasing $U$ changes its magnetic anisotropy from out-of-plane to in-plane.

Next, we prove that electron correlation can dramatically enhance the SOC effect in $\mathrm{Fe_{2}Br_{2}}$ monolayer. The energy band structures with varied $U$ are calculated by using GGA and GGA+SOC. The GGA+SOC gaps as a function of $U$ are shown in Figure 3, and both GGA and GGA+SOC energy bands at representative $U$ values are plotted in Figure 4. Ignoring electron correlation by setting $U$=0.00 eV, the SOC induce a very small splitting along $\Gamma$-X line near the Fermi level. However, a metallic state is produced, because both conduction and valence bands slightly cross the Fermi level. Once the correlation effect is included self-consistently, the SOC-induced gap is enhanced dramatically, reaching 305 meV ($U$=3.5 eV). To further confirm correlation-enhanced SOC, we artificially increase the strength of SOC, and realize a large energy gap in $\mathrm{Fe_{2}Br_{2}}$ without electron correlation ($U$=0.00 eV). The SOC strength is improved to 400%/800% of the normal one, and the corresponding energy bands are plotted in Figure 5. It is clearly seen that a gap of 47/146 meV is induced with enhanced SOC strength to 400%/800%. The gap with enhanced SOC strength to 800% at $U$=0.00 eV is close to one (166 meV) with the normal SOC strength at $U$=2.5 eV. These show that correlation indeed can trigger enhanced SOC effect, and then produce large-gap QAHI.

Within GGA, in small $U$ range, distorted Dirac cones can be observed with the valence and conduction energy bands of the minority-spin touching the Fermi level. With increasing $U$, there are Dirac cones with linear band dispersion, for example $U$=2.50 eV. For Dirac states, the presence of SOC triggers gap opening at the touching points, which means that $\mathrm{Fe_{2}Br_{2}}$ should be a QAHI. By checking the chiral edge modes (see Figure 6), we therefore confirm that monolayer $\mathrm{Fe_{2}Br_{2}}$ is a potential QAHI in considered $U$ range. It is clearly seen that two chiral edge states does exist, which connect the conduction bands and valence bands. This means that the Chern number of $\mathrm{Fe_{2}Br_{2}}$ is equal to two ($C$=2). The $U$ dependence of electronic structures in $\mathrm{Fe_{2}Br_{2}}$ is different from those in monolayer $\mathrm{VSi_{2}P_{4}}$, $\mathrm{FeCl_{2}}$ and FeClFh10 ; h11 ; h12 , and they undergo a rich topological phase transition with QAH phase only existing in certain $U$ range.

We adopt typical $U$=2.5 eVfe ; fe5 as a concrete example to detailedly investigate the physical properties of $\mathrm{Fe_{2}Br_{2}}$. To confirm the stability of $\mathrm{Fe_{2}Br_{2}}$, phonon dispersion, AIMD simulation and elastic constants $C_{ij}$ are calculated by using GGA. As shown in FIG.1 of electronic supplementary information (ESI), the phonon spectra of $\mathrm{Fe_{2}Br_{2}}$ shows no imaginary frequency, indicating its dynamical stability. By AIMD simulation, the temperature and total energy fluctuations of $\mathrm{Fe_{2}Br_{2}}$ as a function of simulation time along with final structures after 8 ps are plotted in FIG.2 of ESI at 300 K. Neither structure reconstruction nor bond breaking with small temperature and total energy fluctuations confirm its thermodynamical stability at room temperature. We use elastic constants to prove mechanical stability of monolayer $\mathrm{Fe_{2}Br_{2}}$. Using Voigt notation, the 2D elastic tensor with space group $P4/nmm$ can be expressed as:

The three independent elastic constants $C_{11}$, $C_{12}$ and $C_{66}$ are 48.69 $\mathrm{Nm^{-1}}$, 22.78 $\mathrm{Nm^{-1}}$ and 29.01 $\mathrm{Nm^{-1}}$, which satisfy the Born criteria of mechanical stability: $C_{11}>0$, $C_{66}>0$, $C_{11}-C_{12}>0$, confirming its mechanical stability.

Due to $P4/nmm$ symmetry, $\mathrm{Fe_{2}Br_{2}}$ is mechanically anisotropic. The direction-dependent in-plane Young’s moduli $C_{2D}(\theta)$ and Poisson’s ratios $\nu_{2D}(\theta)$ can be attained from the calculated $C_{ij}$ by using these expressionsela ; ela1 :

in which $m=sin(\theta)$, $n=cos(\theta)$ and $B=(C_{11}C_{22}-C_{12}^{2})/C_{66}$. The $\theta$ is the angle of the direction with the $x$ direction as $0^{\circ}$ and $y$ direction as $90^{\circ}$. The direction-dependent Young’s moduli $C_{2D}(\theta)$ and Poisson’s ratios $\nu_{2D}(\theta)$ are plotted in FIG.3 of ESI. Due to cubic symmetry, we only consider the angle range from $0^{\circ}$ to $45^{\circ}$. For $\mathrm{Fe_{2}Br_{2}}$, the softest/hardest direction is along the (100)/(110) direction with Young s moduli of 38.03 $\mathrm{Nm^{-1}}$/64.04 $\mathrm{Nm^{-1}}$. The maximum value of $C_{2D}$ is less than that of graphene (340 $\mathrm{Nm^{-1}}$)gra , indicating its extraordinary flexibilities. This provides possibility to use strain to tune physical properties of $\mathrm{Fe_{2}Br_{2}}$. The minima/maxima of $\nu_{2D}(\theta)$ is 0.104/0.468 along the (110)/(100) direction.

Without SOC, $\mathrm{Fe_{2}Br_{2}}$ is a 2D half Dirac semimetal state with a large-gap insulator for spin up and a gapless Dirac semimetal for spin down. The band crossings are protected by $M_{y}$ ($M_{x}$), forbidding $d_{z^{2}}$ and $d_{xy}$ to hybridize (see FIG.4 of ESI). In contrast to typical Dirac cones in graphene, there are four Dirac cones in 2D BZ due to $C_{4}$ symmetry. Within SOC, the Dirac gap of of 166 meV can be produced. These similar results can be found in $\mathrm{Fe_{2}I_{2}}$, $\mathrm{Fe_{2}IX}$ (X=Cl and Br), $\mathrm{FeI_{1-x}Br_{x}}$ (x=0.25 and 0.75), $\mathrm{Li_{2}Fe_{2}X_{2}}$ (X=S, Se and Te) and $\mathrm{Li_{2}Fe_{2}SSe}$fe ; fe1 ; fe2 ; fe3 ; fe4 . The magnetic anisotropy direction is very important to determine the topological properties of some 2D systems. For example, for monolayer $\mathrm{VSi_{2}P_{4}}$, $\mathrm{FeCl_{2}}$ and FeClFh10 ; h11 ; h12 , the QAH states can exist at proper $U$ range with out-of-plane magnetic anisotropy, and no special QAH states appear with in-plane case. The energy band structures with out-of-plane and in-plane magnetizations by using GGA+SOC are plotted in Figure 7. By varying the magnetization orientation from $z$ to $x$, the band gap decreases down to very small value (Strictly, the gap should be zero.). The topological edge states with out-of-plane and in-plane magnetizations are calculated, as shown in Figure 8. For in-plane case, no non-trivial chiral edge modes appear within the bulk gap of $\mathrm{Fe_{2}Br_{2}}$, meaning its Chern number $C$=0.

To corroborate this finding, the Chern number is recalculated by integrating the Berry curvature ($\Omega_{z}(k)$) of the occupied bands:

in which $\mu_{n,k}$ is the lattice periodic part of the Bloch wave functions. The distributions of Berry curvature in 2D BZ with out-of-plane and in-plane magnetization are plotted in Figure 8. For out-of-plane magnetization, the hot spots in the Berry curvature are around four gapped Dirac cone, which have the same signs. A quantized Berry phase of $\pi$ for each gapped Dirac cone can be attained, and the total Berry phase of 4$\pi$ due to four Dirac cones means a high Chern number $C$=2. For in-plane magnetization, there are four main hot spots in the Berry curvature along four $\Gamma$-M lines, and two of then have the opposite signs with the other two, giving rise to disappeared Chern number.

For $\mathrm{Fe_{2}Br_{2}}$, a large QAH gap is not because the effective SOC strength of $d$ orbitals is enlarged by bonding with heavy elements Br, and is because the electron correlation improves the SOC effect. To illustrate this, the energy band structures with only considering SOC of Fe/Br with out-of-plane magnetization are plotted in Figure 7. For only considering SOC of Fe, the gap is 140 meV, which is very close to 166 meV with full SOC. However, the gap of 35 meV is attained with only considering SOC of Br, which is about one-fifth of one with full SOC. So, the large gap is mainly from the SOC effects of Fe combined with electronic correlation.

To explain the key role of SOC of Fe, the atom- and Fe-3$d$-orbital-resolved band structures of $\mathrm{Fe_{2}Br_{2}}$ are plotted in FIG.4 of ESI by using GGA+SOC. It is found that the Fe-$d$ derived states contribute mainly to the gapped Dirac states near the Fermi level, which leads to importance of SOC of Fe to induce QAH gap. Without SOC, the two crossed spin-down bands (see Figure 4) near the Fermi level are mainly contributed by $d_{z^{2}}$ and $d_{xy}$ orbitals of Fe, and their band order is inverted between $\Gamma$ and X. When considering SOC, a Dirac gap is produced. Topological edge states of $\mathrm{Fe_{2}Br_{2}}$ monolayer with only considering SOC of Fe/Br are plotted in FIG.5 of ESI. Two chiral gapless edge modes appear within the bulk gap for only considering SOC of both Fe and Br, indicating QAH properties.

Strain can modify the distance between atoms, and then can tune kinetic and interaction energies of electrons, which can induce topological transitionv6 ; v7 ; v8 . The biaxial strain effects on QAH robustness of $\mathrm{Fe_{2}Br_{2}}$ are investigated, and $a/a_{0}$ (0.96-1.04) is used to simulate the biaxial strain with $a$/$a_{0}$ being the strained/unstrained lattice constants. The $a/a_{0}$$<$1/$>$1 means compressive/tensile strain. Based on energy difference between AFM and FM (see Figure 9), the ground state of $\mathrm{Fe_{2}Br_{2}}$ is FM ordering in considered strain range. It is found that compressive strain can enhance FM interaction, and tensile strain can weaken one, which will produce important effects on Curie temperature of $\mathrm{Fe_{2}Br_{2}}$. Based on previous research results, the magnetic orientation is very key to induce QAH state, and the MAE as a function of $a/a_{0}$ is plotted in Figure 9. In considered strain range, the out-of-plane magnetic anisotropy is robust against strain, which confirms possible existence of QAH state.

The GGA+SOC energy band structures of $\mathrm{Fe_{2}IBr}$ vs $a/a_{0}$ are plotted in FIG.6 of ESI, which show that they are all FM semiconductors. The gaps as a function of strain are shown in Figure 10, which shows a nonmonotonic behavior with $a/a_{0}$ from 0.96 to 1.04. With increasing $a/a_{0}$, the gap firstly increases, and then decreases, which is due to the change of conduction band bottom from one point along $\Gamma$-X to $\Gamma$ point. The topological edge states of strained monolayer $\mathrm{Fe_{2}Br_{2}}$ are calculated, and two chiral topologically protected gapless edge states emerge at representative 0.94 and 1.06 strains (see FIG.7 of ESI), giving Chern number C=2. These prove that the QAH state of monolayer $\mathrm{Fe_{2}Br_{2}}$ is robust against strain. Unlike $\mathrm{Fe_{2}Br_{2}}$, strain can induce a series of phase transitions from ferrovalley (FV) insulator to half-valley metal (HVM) to QAHI in these 2D materialsv6 ; v7 ; v8 , for example $\mathrm{VSi_{2}N_{4}}$.

Based on the previous discussion, both electronic correlation and strain can effectively tune FM interaction of $\mathrm{Fe_{2}Br_{2}}$, which means that they can affect observably its Curie temperature ($T_{C}$). The MC simulations are performed to estimate $T_{C}$ of monolayer $\mathrm{Fe_{2}Br_{2}}$ based on spin Heisenberg model, whose Hamiltonian can be expressed as:

where $S_{i}$/$S_{j}$, $S_{i}^{z}$, $J$ and $A$ are the spin vector of each Fe atom, spin component parallel to the $z$ direction, nearest neighbor exchange parameter and MAE.

Based on the energy difference between AFM and FM, the normalized magnetic coupling parameters ($|S|$ = 1) can be attained as $J$=($E_{AFM}$-$E_{FM}$)/8. At representative correlation strength ($U$=1.5 eV and 2.5 eV) and strain ($a/a_{0}$=0.96 and 1.04), the calculated $J$ values are 110.2 meV, 62.8 meV, 87.7 meV and 37.8 meV, respectively. We plot the normalized magnetic moment and auto-correlation as a function of temperature in Figure 11. The estimated $T_{C}$ is about 830/471 K for $U$$=$1.5/2.5 eV, and 663/287 K for $a/a_{0}$$=$0.96/1.04. So, both electronic correlation and strain have important influence on $T_{C}$ of $\mathrm{Fe_{2}Br_{2}}$. It is found that the reduced correlation strength and compressive strain can improve $T_{C}$. At typical $U$=2.5 eV, the predicted $T_{C}$ is significantly higher than that of previously reported many 2D FM semiconductors (20-160 K)m7-6 ; tc1 ; tc2 . These indicate that $\mathrm{Fe_{2}Br_{2}}$ should be a room-temperature QAHI.

Electronic correlations have significant effects on physical properties of materials with localized $d$ electrons, especially for low-dimensional systems. For example monolayer $\mathrm{VSi_{2}P_{4}}$, $\mathrm{FeCl_{2}}$ and FeClFh10 ; h11 ; h12 , different correlation strengths can drive these systems into various types of interesting ground states, such as FV, HVM and QAH states. The correlation strength can also tune magnetic anisotropy of these 2D materials. For monolayer $\mathrm{VSi_{2}P_{4}}$, several transitions in the magnetic anisotropy can be observed with varied $U$h10 . The out-of-plane FM ordering allows a nonvanishing Chern number of these 2D systemsh10 ; h12 , which is different from in-plane FM situation. Here, the correlation can be used to induce enhanced SOC effect of Fe atoms in $\mathrm{Fe_{2}Br_{2}}$, which can produce large-gap QAHI with out-of-plane magnetic anisotropy. In fact, amplifying the SOC effect in light elements can be achieved by the interplay between particular crystal symmetry and electron correlation, which contains certain partially occupied orbital multipletsh13 . In a word, our works emphasize the importance of electronic correlation and magnetic anisotropy to determine the electronic state of $\mathrm{Fe_{2}Br_{2}}$.

In summary, the intriguing large-gap of QAHI $\mathrm{Fe_{2}Br_{2}}$ is explained by the reliable first-principle calculations, which is due to correlation-enhanced SOC effects of Fe element. The $\mathrm{Fe_{2}Br_{2}}$ exhibits excellent dynamic, mechanical and thermal stabilities, and can realize room-temperature high-Chern-number ($C$=2) QAHE. At representative $U$=2.5 eVfe ; fe5 , the band gap of 166 meV and $T_{C}$ of about 471 K are obtained, respectively. The emergence of QAH states in monolayer $\mathrm{Fe_{2}Br_{2}}$ is robust against strain. The $T_{C}$ can be enhanced to 663 K at 0.96 compressive strain, and the band gap remains at about 92 meV. At 1.04 tensile strain, the $T_{C}$ of 287 is still close to room temperature, and the gap is about 152 meV. Our works provide a comprehensive understanding of correlation effects on large nontrivial band gap in monolayer $\mathrm{Fe_{2}Br_{2}}$, which can be readily extended to other monolayer $\mathrm{Fe_{2}XY}$ (X/Y=Cl, Br and I). These results are also helpful to deepen our understanding of large-gap QAHI in light-element materials.