⇦Chinese Physics Letters, 2022, Vol. 39, No. 7, Article code 077101 Competition of Quantum Anomalous Hall States and Charge Density Wave in a Correlated Topological Model Xin Gao (高鑫)1, Jian Sun (孙健)1, Xiangang Wan (万贤纲)2,3, and Gang Li (李刚)1,4* Affiliations 1School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China 2National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China 3Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China 4ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 201210, China Received 12 April 2022; accepted manuscript online 31 May 2022; published online 22 June 2022 *Corresponding author. Email: ligang@shanghaitech.edu.cn Citation Text: Gao X, Sun J, Wan X G et al. 2022 Chin. Phys. Lett. 39 077101    Abstract We investigate the topological phase transition driven by non-local electronic correlations in a realistic quantum anomalous Hall model consisting of $d_{xy}$–$d_{x^{2}-y^{2}}$ orbitals. Three topologically distinct phases defined in the non-interacting limit evolve to different charge density wave phases under correlations. Two conspicuous conclusions were obtained: The topological phase transition does not involve gap-closing and the dynamical fluctuations significantly suppress the charge order favored by the next nearest neighbor interaction. Our study sheds light on the stability of topological phase under electronic correlations, and we demonstrate a positive role played by dynamical fluctuations that is distinct to all previous studies on correlated topological states.
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 DOI:10.1088/0256-307X/39/7/077101 © 2022 Chinese Physics Society Article Text Since the discovery of the quantum Hall effect and the understanding of quantized conductivity in terms of Thouless–Kohomoto–Nightingale–Nijs (TKNN) number,[1–3] the study of unconventional topological systems has become one of the central topics in modern condensed matter physics. Haldane pioneered in the fundamental understanding of time-reversal (TR) symmetry breaking and quantum anomalous Hall (QAH) states.[4] The artificial complex hopping in the Haldane model can be understood as a higher-order perturbation down-folded from the spin-orbit coupling (SOC) term in a multiorbital model. Thus, the most economical realization of the Haldane model in a realistic material system would be a two-orbital model with SOC on a honeycomb lattice. The most straightforward choice fulfilling such a condition is the spin-polarized orbitals, in which $l_{z}s_{z}$ provides the onsite SOC between the two orbitals. Such a mechanism has been realized in the TR symmetric monolayer bismuthene on the SiC(0001) substrate and experimentally confirmed as a large-gap quantum spin Hall system.[5,6] The same mechanism also applies to the correlated $d$-orbitals.[7–12] The expectation value of $l_{z}s_{z}$ between $d_{xy}$ and $d_{x^{2}-y^{2}}$ is intrinsically nonzero, as such a four-band correlated model on honeycomb lattice can be defined. Taking $(|\phi^{\rm A}_{d_{xy}}\rangle, |\phi^{\rm A}_{d_{x^2-y^2}}\rangle, |\phi^{\rm B}_{d_{xy}}\rangle, |\phi^{\rm B}_{d_{x^2-y^2}}\rangle)$ as the basis, we have the simplest tight-binding model for a QAH state on correlated $d$-orbitals. $$\begin{alignat}{1} H_{0}(k) = \begin{pmatrix} 0 & 2i\lambda & h_{\alpha\alpha}(k) & h_{\alpha\beta}(k) \cr -2i\lambda & 0 & h_{\beta\alpha}(k) & h_{\beta\beta}(k) \cr h_{\alpha\alpha}^{*}(k) & h_{\alpha\beta}^{*}(k) & 0 & 2i\lambda \cr h_{\beta\alpha}^{*}(k) & h_{\beta\beta}^{*}(k) & -2i\lambda & 0 \end{pmatrix},~~~~~~ \tag {1} \end{alignat} $$ where $\lambda$ is the SOC strength and factor 2 stems from the two overlaps of complex spherical harmonics $Y_{2}^{\pm2}$; $h_{\alpha\alpha}$, $h_{\alpha\beta}$, and $h_{\beta\beta}$ are the intra- and inter-orbital hoppings with more details explained in the Supplementary Information. Equation (1) is a spinless model describing a system fully spin-polarized with only one spin component controlling the low-energy excitations. This four-band QAH model contains three topologically different phases with Chern number $C=-1, 0, 2$. The phase transition between any two phases occurs when the valence and conduction bands cross to close the bandgap. Due to the lack of the other spin, only the local SOC contributes. Furthermore, as $d_{xy}$ and $d_{x^2-y^2}$ favor a nonzero local SOC $\langle d_{xy}|l_{z}s_{z}|\phi_{d^{x^2-y^2}}\rangle$ that is usually the largest one among all the SOC terms, this model features a large-gap QAH state that may be utilized in high-temperature spintronic device applications. Furthermore, this model corresponds precisely to the low-energy excitations of a large class of iron halogenides, represented by FeBr$_{3}$.[9] Correlated Tight-Binding Model of QAH. The reduced dimensionality and the correlated $d$-orbital imply that the QAH phases of Eq. (1) are subject to strong electronic correlations. The stability of the topological phase against correlations has been studied in the context of the Kane–Mele–Hubbard model[13–33] or the spinful Bernevig–Hughes–Zhang model,[34–39] where the non-interacting limit of the model is a topologically trivial semimetal. Distinct from these studies, our multi-orbital model already displays three topologically distinct phases in the non-interacting limit. Thus, there is no need for our model to require complex hopping, and the electronic correlations directly interact with the topological phases. To study the robustness of the QAH states against electronic correlations, here we consider the nonlocal Coulomb repulsions. $$ H_{\rm int} =V_{1}\sum_{\langle i,j\rangle}n_{i}n_{j} + V_{2}\sum_{\langle\langle i,j\rangle\rangle}n_{i}n_{j}.~~ \tag {2} $$ $V_{1}$ is the density-density interaction between the nearest-neighbor sites, which couples electrons at different sublattices. For simplicity, we assume the same interaction strength for both inter-orbital and intra-orbital correlations, and the electron occupancy is half-filling. Here $n_{i}=\sum_{\alpha}n_{i\alpha}$ is the total occupancy at site $i$. $V_{2}$ is same as $V_{1}$, but for the next nearest-neighbor interaction which couples the same sublattice. In order to compare with the spinless Haldane–Hubbard model on the same basis, we have neglected the inter-orbital local interactions and focus only on the nonlocal interactions favoring charge density wave (CDW) formation. The inclusion of the local inter-orbital repulsion favors the charge redistribution between the two orbitals, leading to a possible orbital order that we will not consider in our work. Without dedicated calculations, let us first try to intuitively understand what $V_{1}$ and $V_{2}$ do on the model. For any positive value of $V_{1}$, electrons experience a repulsion when both nearest-neighbor sites are occupied, e.g., $\langle n_{_{\scriptstyle \rm A}}\rangle=\langle n_{_{\scriptstyle \rm B}}\rangle = 0.5$. Electrons tend to distribute unevenly within each dimmer to minimize the potential energy associated with $V_{1}$. As $V_{1}$ would increase the potential energy, at half-filling, the most economical way to distribute electrons is to occupy both orbitals on one sublattice and let the other sublattice empty, e.g., $\langle n_{_{\scriptstyle \rm A}}\rangle=1, \langle n_{_{\scriptstyle \rm B}}\rangle=0$. The tendency of one sublattice is fully occupied while the other is empty leads to a periodic modulation of the electron density. As a result, a CDW with the same periodicity of the primitive unit cell will develop as shown in Fig. 1(b). As for $V_{2}$, the situation becomes slightly different. We will obtain another CDW with a larger periodicity $\sqrt{3}\times\sqrt{3}$. $V_{2}$ connects the same sublattice forming two decoupled triangular lattices. Triangular lattice is geometrically frustrated that cannot accommodate the same CDW in Fig. 1(b). The smallest CDW pattern compatible with a triangular lattice will have to allow three different electron occupancies in each triangle with classic values $\langle n_{1}^{\rm A/B}\rangle=2$, $\langle n^{\rm A/B}_{2}\rangle=1$, $\langle n^{\rm A/B}_{3}\rangle=0$, see Fig. 1(c). The three different circles with identical colors belong to the same sublattice representing three different electron densities forming a $\sqrt{3}\times\sqrt{3}$ CDW. With only $V_{2}$, the $\sqrt{3}\times\sqrt{3}$ CDW on one sublattice is independent of that on the other sublattice. However, with two independent triangular sublattices, another degenerate configuration emerges, i.e. $\langle n_{1}^{\rm A}\rangle=2$, $\langle n_{2}^{\rm A}\rangle=2$, $\langle n_{3}^{\rm A}\rangle=0$, $\langle n_{1}^{\rm B}\rangle=2$, $\langle n_{2}^{\rm B}\rangle=0$, $\langle n_{3}^{\rm B}\rangle=0$. By using the occupation number $n_{_{\scriptstyle {\rm A}_1}}n_{_{\scriptstyle {\rm A}_2}}n_{_{\scriptstyle {\rm A}_3}}n_{_{\scriptstyle {\rm B}_1}}n_{_{\scriptstyle {\rm B}_2}}n_{_{\scriptstyle {\rm B}_3}}$ to label different CDW patterns, in the following section, we will simply denote the three different CDW patterns by $V_{1}$ and $V_{2}$ as $222000$, $220200$, and $210210$, respectively.
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Fig. 1. Tight-binding model and different CDW patterns. (a) The $d_{xy}$–$d_{x^{2}-y^{2}}$ model on honeycomb lattice and the corresponding BZ. (b) Sketch of the CDW mode favored by $V_{1}$ where electrons on the two triangular sublattices distribute unevenly. Blue and red colors correspond to the two sublattices, and the size of circles denotes the charge density. (c) The CDW mode favored by $V_{2}$ has a larger periodicity as compared to that in (b), as indicated by the light purple area. The classical configuration of this $\sqrt{3}\times\sqrt{3}$ CDW can be either 210210 or 220200, see the main text for more details.
When $V_{1}$ and $V_{2}$ are simultaneously present, the two CDWs are, unfortunately, not compatible. Residing $210210$ on honeycomb lattice will unavoidably have two neighboring sites with high electron densities. Thus, $\sqrt{3}\times\sqrt{3}$ CDW-$210210$ favored by $V_{2}$ competes with the $1\times1$ CDW-$222000$ favored by $V_{1}$ on honeycomb lattice. They both compete with the nontrivial topology of the four-band QAH model of Eq. (1). It is highly interesting to know the stability of the QAH phase and the way nonlocal charge correlations destroy it. Results—Static Mean-Field Solution. To understand this problem, we first apply the simple mean-field theory by decoupling the change density of the two sublattices: $$\begin{alignat}{1} &\hat{n}_{{\rm A}_i}\hat{n}_{{\rm B}_j} \approx \langle n_{_{\scriptstyle {\rm A}_i}}\rangle\hat{n}_{{\rm B}_j} + \hat{n}_{{\rm A}_i}\langle n_{_{\scriptstyle {\rm B}_j}}\rangle - \langle n_{_{\scriptstyle {\rm A}_i}}\rangle\langle n_{_{\scriptstyle {\rm B}_j}}\rangle,~~~~~ \tag {3a}\\ &\hat{n}_{{\rm A}_i}\hat{n}_{{\rm A}_j}\approx \langle n_{_{\scriptstyle {\rm A}_i}}\rangle\hat{n}_{{\rm A}_j} + \hat{n}_{{\rm A}_i}\langle n_{_{\scriptstyle {\rm A}_j}}\rangle - \langle n_{_{\scriptstyle {\rm A}_{i}}}\rangle\langle n_{_{\scriptstyle {\rm A}_j}}\rangle.~~~~~ \tag {3b} \end{alignat} $$ One then arrives at an effective single-particle problem defined by a $12\times 12$ Hamiltonian with $\langle n_{_{\scriptstyle {\rm A}_i}}\rangle$ and $\langle n_{_{\scriptstyle {\rm B}_j}}\rangle$ as parameters. After diagonalizing the Hamiltonian and recalculating the charge density from the single-particle eigenstates, one closes the self-consistency loop. At convergence, the different solutions of $n_{_{\scriptstyle {\rm A}_i}}$ and $n_{_{\scriptstyle {\rm B}_j}}$ will identify the ground state. In this work, we have tempts with four different configurations as initial trivial states, i.e., 111111, 210210, 222000, 220200, and stabilize them through the self-consistent mean-field calculations. We then compare their free energies to determine the ground state whenever we can stabilize multiple charge configurations. We present the mean-field phase diagram in Fig. 2. Figures 2(a)–2(c) show the topological phase diagram of the $d_{xy}$–$d_{x^{2}-y^{2}}$ model under different strengths of $V_{1}$ and $V_{2}$. When both $V_{1}$ and $V_{2}$ are absent, the tight-binding model Eq. (2) shows three distinct phases with Chern numbers $C=-1$, $C=0$, and $C=2$. For most values of $a/\lambda$ and $b/\lambda$, the model Eq. (2) is in certain topological nontrivial phase. Only when $a \approx -b$, the model Eq. (1) becomes trivial. As shown in Fig. 2(b), by including interactions, CDW phase emerges in the center of the phase diagram, i.e., $a=b=0$, and gradually expands to larger values of $a/\lambda$ and $b/\lambda$, which is a natural consequence of the competition between kinetic and potential energy. Through the direct calculation of Chern number, we further confirm that, once CDW appears, the Chern number becomes zero. This new $C=0$ phase, shown by yellow in Figs. 2(a)–2(c), is distinct from the original $C=0$ phase colored in blue along the off-diagonal of the phase diagram. The latter is an ordinary band insulator, while the new one is a CDW insulator with charge pattern 222000. The emergence of the CDW-222000 in Fig. 2 breaks all three band insulating phases, redistributing charge from equal density, i.e., 111111 pattern, to 222000. As indicated by the comparison of Figs. 2(b) and 2(c), finite $V_{2}$ significantly shrinks the CDW-222000 size, confirming that $V_{2}$ competes with $V_{1}$. Consequently, $V_{2}$ stabilizes topological nontrivial phase and allow them to persist to larger values of $V_{1}$. Individually, both $V_{1}$ and $V_{2}$ destabilize the topological phases. However, when they coexist, their competition stabilizes the topological phases because they tend to destroy the CDW pattern favored by each other. Figures 2(d) and 2(e) display the $V_{1}$–$V_{2}$ phase diagram with three characteristic model parameter sets of $a/\lambda$, $b/\lambda$. We choose $a/\lambda = 3.5$, $b/\lambda = -3.5$ for $C=0$ phase, $a/\lambda = 1.0$, $b/\lambda = -6.0$ for $C=-1$ phase, and $a/\lambda = 3.2$, $b/\lambda = 2.6$ for $C=2$ phase. Both interacting phase diagrams in Figs. 2(d) and 2(e) contain two types of CDW states, i.e., CDW-222000 phase for larger $V_{1}$ and smaller $V_{2}$, and CDW-210210 state for larger $V_{2}$. In between, no charge modulation is found, where the model keeps its topological nature characterized by the tight-binding model, but with a modified dispersion. It is clear that the presence of next nearest-neighbor interaction $V_{2}$ suppresses the sublattice CDW-222000. Critical value of $V_{1}$ becomes larger with the increase of $V_{2}$. However, CDW-222000 is more robust than other types of CDW when $V_{1}$ is sufficiently large. CDW-222000 always wins the competition with other types of CDW favored by $V_{2}$. In our calculations, we did not find pure CDW-220200 phase and it is degenerate with CDW-210210 at larger $V_{1}$. However, for smaller $V_{1}$, CDW-210210 is always more stable than 220200.
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Fig. 2. Mean-field phase diagram of the $d_{xy}$–$d_{x^2-y^2}$ model under different interactions, i.e., phase diagrams as functions of $a/\lambda$ and $b/\lambda$ at (a) $V_{1}=V_{2}=0$, (b) at $V_{1}=3.0$, $V_{2}=0.0$, (c) at $V_{1}=3.0$, $V_{2}=0.5$. [(d), (e)] The phase diagrams as functions of $V_{1}/\lambda$ and $V_{2}/\lambda$ for the fixed tight-binding model parameters $a/\lambda$ and $b/\lambda$. $C=0$ and $C=-1$ phases have very similar dependences on $V_{1}/\lambda$ and $V_{2}/\lambda$, and they are shown in (d). (e) Phase diagram for $C=2$ phase.
Dynamical Fluctuations. After understanding the static mean-field phase diagram, we further study the dynamical fluctuation effect on the QAH phase. Despite of the much richer phase diagram predicted by static mean-field calculations, most of the mean-filed predictions beyond the density decoupling are known to be unstable.[16–21,23–31,40,41] In particular, topological Mott phase emerging from the complex hopping is completely suppressed by dynamical fluctuations. Thus, beyond the static mean-field study shown before, we want to know if the topological phases of our model survive under a more sophisticated consideration of correlation effect. We would emphasize that out study is distinct to the previous studies,[16–20,23–31] which started with a topological trivial state and expected to trigger topological phase by electronic correlations. The nontrivial topology builds in our tight-binding model. No complex decoupling, which is usually unstable under dynamical fluctuations, is crucially needed in our construction. Thus, we expect a more robust topological phase in this realistic model. To this end, we use exact diagonalization (ED) to solve Eqs. (1) and (2) in three different strategies. We firstly use ED as an impurity solver in a dynamical mean-field theory (DMFT) calculation with only local charge fluctuations. Secondly, going beyond DMFT to partially account for nonlocal charge fluctuations, we further solve a cellular-DMFT (C-DMFT) equation with six-site unit cell. In the third strategy, a six-site cluster with periodic boundary condition is solved directly by ED. The three numerical calculations, i.e., DMFT, C-DMFT, and ED, are complementary in nature so that we can understand the influence of non-local charge fluctuations and boundary effect from their comparison. Comparing C-DMFT to DMFT, one gets insight of the non-local effect. Comparing C-DMFT to ED with the same cluster size, one can understand the boundary effect. See the Supplementary Information for more details. Figure 3 displays the corresponding results at $V_{2}=0$. For a better comparison, we also show the static mean-field results in Fig. 3(a). Figures 3(b)–3(d) contain dynamical fluctuations beyond the static mean-field solution. As shown in Fig. 3(b), dynamical fluctuations included in the DMFT significantly enlarge the critical values $V_{\rm 1c}$ by suppressing the CDW-222000 phase. At the same time, the difference among $V_{\rm 1c}$ for the three phases also becomes more pronounced as compared to Fig. 3(a). The DMFT impurity contains only two sites and they are coupled by $V_{1}$. Embedding such a small size system into lattice artificially overemphasizes the Coulomb repulsion between neighboring sites. As clearly indicated by Fig. 3(c), the nonlocal spatial fluctuations contained in a six-site C-DMFT calculations render a smaller $V_{\rm 1c}$ than those in DMFT, but larger than those obtained in static mean-field calculations. C-DMFT is a better approximation than both DMFT and static mean-field that it contains both dynamical and some spatial fluctuations, thus, $V_{\rm 1c}$ obtained in C-DMFT is supposed to be a more reliable value. ED, on the other hand, suffers a finite cluster issue, thus boundary effect in ED is worse than C-DMFT. From Fig. 3, we conclude that dynamical fluctuations push phase boundary of CDW-222000 to a larger value of $V_{1}$, further stabilizing the topological phase.
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Fig. 3. CDW order parameters versus nearest-neighbor Coulomb repulsion ${V_{1}}$, i.e., results from (a) mean-field solution, (b) DMFT with 2-site impurity, (c) C-DMFT with 6-site cluster calculations, and (d) exact diagonalization. The inset shows the free energy evolution around the transition boundary, indicating their first-order nature. The same colors as the main plot in (d) are used.
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Fig. 4. ${V_{1}}$–$V_{2}$ phase diagram. (a)–(c) The $C=0$, $-1$, and 2 phases boundaries to CDW for three different $V_{2}/\lambda$. Increasing $V_{2}/\lambda$ clearly suppresses the CDW-222000 state (indicated as CDW), pushing $V_{\rm 1c}$ to larger values. (d) The $V_{1}$–$V_{2}$ phase diagram from C-DMFT.
As shown in Fig. 4, including $V_{2}$ the topological phases are further stabilized. In Figs. 4(a)–4(c), we show the critical $V_{\rm 1c}$ of the three different phases for different values of $V_{2}$. Increasing $V_{2}$ clearly shifts the CDW phase boundary to the right, which is consistent with the mean-field prediction discussed in Figs. 2(d) and 2(e), However, the CDW-210210 and CDW-220200 phases contained in the mean-field phase diagrams in Figs. 2(d) and 2(e) completely disappear upon the inclusion of the dynamical fluctuations, in sharp contrast to CDW-222000 phase. The latter is only slightly suppressed by $V_{2}$. We confirmed that, at least up to $V_{2}/\lambda =20$, CDW-210210 and CDW-220200 phases cannot be stabilized, which is further supported by the ED calculation on the isolated six-site cluster. In ED, there is no indication of the CDW-210210 and CDW-220200 for any value of $V_{2}$. Furthermore, the CDW-222000 phase is also completely suppressed down to $V_{2}=0$ line. Any finite $V_{2}$ will immediately kill the CDW-222000 phase, which is likely due to the boundary condition and the finite-size effect in ED. Nevertheless, both ED and C-DMFT with six-site cluster calculations consistently show the breakdown of CDW (210210, 220200) favored by the next nearest-neighbor repulsion $V_{2}$. Only the CDW-222000 emerging from the nearest-neighbor interaction $V_{1}$ survives, whose phase space, however, significantly shrinks as compared to the mean-field prediction. Discussions and Conclusions. Two conspicuous consequences of the interaction effect deserve further explication. First, from the inspection of the local density of state, we find that the phase transition from QAH to CDW does not experience a gap-closing process even in the static mean-field level. In the inset of Fig. 3(d) we show the free energy from exact diagonalization. The free energy in the non-CDW (CDW) phase at a smaller (larger) $V_{1}$ regime intersects at the transition boundary with a clear slope change, indicating a first-order phase transition. This is consistent with some previous works.[38,42–44] The QAH and the CDW phases differ in the sublattice/translational symmetry. Thus, there is no need for such a topological phase transition to close the gap. In this sense, it is similar to a recently discovered topological phase transition in the twisted transition metal dichalcogenide (TMD),[45] where a correlated insulating phase becomes a QAH insulator without closing the gap. Phenomenologically, the two phase transitions are both between a topologically trivial insulator and a QAH state. Yet, the nature of the correlated insulating phase in twisted TMD is not known and was anticipated as a non-magnetic Mott insulator or a 120-degree antiferromagnetic insulator, while the QAH phase is possibly valley polarized. Our work thus sheds new light and provides additional insights to this timely topic. According to our calculations, a CDW insulator may also be a good candidate to the correlated insulating phase observed in twisted TMD. We note that, due to the presence of three topologically distinct phases in the model, we are able to examine their responses to charge modulation on equal footing. We find that both the band-insulator with $C=0$ and the Chern insulator with $C=-1, 2$ transform to the CDW insulating states by the first-order transition. In this sense, there is no difference between topologically trivial and nontrivial phases under the electronic correlations. The CDW phase appears when the self-energy of A, B sublattices starts to deviate. Otherwise, the correlation effect mainly rigidly shifts the dispersion. Second, the breakdown of the CDW-210210 + CDW-220200 is astonishing. Their destruction is distinct to that of the topological Mott phase studied in the Kane–Mele–Hubbard model,[31] where the latter requires an unusual decomposition of $V_{2}$. However, CDW-210210 + CDW-220200 is a density modulation and is believed to be much more stable. In our calculations, the loss of longer-range CDW is unlikely attributed to the finite-cluster size effect, as smaller systems tend to overestimate non-local order. In summary, we have studied the quantum phase transition in a correlated topological model. This model is a two-orbital extension of the Haldane model, and the $C=-1$ phase was realized in the monolayer of FeBr$_{3}$ family. All three phases, i.e., $C=0$, $C=-1$, and $C=2$, were examined against the nonlocal Coulomb interaction. We found that the QAH phase is stable till the intermediate value of interactions. The destruction of the QAH phase and the emergence of the CDW phase are simultaneous. With the dynamical fluctuations, we find the CDW solution from the static mean-field becomes less stable at a smaller interaction regime, resulting in an enhanced QAH phase. The QAH phase expands in the parameter space due to the destruction of the CDW phase by the dynamical fluctuations. At the same time, the CDW favored by the next nearest-neighbor interaction $V_{2}$ becomes unstable under dynamical fluctuations. No emergent topological phase triggered by electronic correlations is found. Our calculations consistently show that the non-local electronic repulsion induces a phase transition to the topological trivial CDW phase. The transition does not accompany gap-closing, unlike the topological insulator to band insulator transition. Our work sheds light on the interacting topological phases. We point out that the density-density interaction and the normal self-energy disfavor the topological phase. Thus, the search for an interaction-triggered topological phase requires new types of interactions and anomalous self-energy to come into play. Acknowledgement. This work was supported by the National Natural Science Foundation of China (Grant No. 11874263), the National Key R&D Program of China (Grant No. 2017YFE0131300), and Shanghai Technology Innovation Action Plan (2020-Integrated Circuit Technology Support Program 20DZ1100605, 2021-Fundamental Research Area 21JC1404700). Part of the calculations was performed at the HPC Platform of ShanghaiTech University Library and Information Services and at the School of Physical Science and Technology, ShanghaiTech University. 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