Introduction

There are many exotic lattice models in condensed matter theory, ranging from single-particle1,2 to many-body3,4 cases. The studies of them not only give insight into the nature of topology, correlation and magnetization for quantum materials5, but also promote the development of algorithms in computational physics6. However, the realization of them is a challenging task because solid-state materials found in nature often have the hybridized band structures7, much more complex than the ideal models. In experiments, ultracold atoms8 and photonic crystals9 have been used to simulate certain lattice models, which can create artificial bands with tunable parameters, analogous to those formed by electrons in crystals. In addition to artificial systems, it is more urgent to develop suitable electronic platforms to explore these lattice models.

Metal-organic frameworks (MOFs) consist of metal atoms linked by organic ligands10, forming a highly ordered porous network. This category of materials has attracted tremendous attention in chemical society, because of its excellent performance in gas storage and catalysis11. Due to the chemical flexibility for choosing metal and ligand fragments, its band structures around Fermi level can be easily tuned through custom designed molecular building blocks12, showing an ideal electronic system to realize lattice models. Currently, Dirac and flat bands have been predicted in various two-dimensional (2D) MOFs13, forming topological insulator (TI) and Chern insulator phases14,15,16. However, the tiny spin-orbital coupling (SOC) in organic materials makes it extremely difficult to detect topological boundary states within the SOC gap17,18,19, greatly hindering the experimental progress in this field20.

Recently, topological band theory has been extended from first-order to higher-order21,22,23,24, where the nontrivial bulk topology of an m-dimensional nth-order TI is characterized by gapless states at (m-n)-dimensional boundary. Different from the conventional first-order TIs, the higher-order topology is protected by the crystalline21 and chiral25 symmetries, which doesn’t originate from SOC. This indicates that the large insulating band gap in 2D MOFs has been overlooked in previous studies, which may be compatible with the second-order TI (SOTI) phase, facilitating the detection of in-gap topological boundary states. To date, the reported 2D SOTIs are mainly limited to some artificial structures26,27,28,29 and a few inorganic solid-state materials30,31,32,33,34,35,36,37,38,39,40, but seldom in MOFs.

In this work, based on tight-binding (TB) model, we first illustrate the universal existence of SOTI in 2D MOFs with the star lattice configuration and clarify the formation mechanism of topological corner states. Then, combining first-principles calculations and scanning tunneling microscopy (STM) measurements, we identify our proposal in monolayer Ni3(HITP)2 and confirm the realization of localized topological corner states within the nontrivial bulk band gap. Our results introduce a new platform to explore the organic higher-order topology, which is achievable by current experimental technology.

Results

Topological phase diagram

The 2D MOFs, constructed by C3 symmetric cores, will form a hexagonal porous network, as shown in Fig. 1a. This is one most common structure synthesized in the experiments, having plenty of candidate materials41,42. This structure can also be seen as a star lattice if the lattice site is located at the vertex of each core, showing six sites per unit cell. The TB Hamiltonian of this lattice can be described by intra-core (inter-core) hopping t1 (t2), as labeled in Fig. 1a (see also Supplementary Fig. 1). Depending on these two parameters, its band structures are divided into four types (Type-I to IV), as shown Fig. 1b. The Type-I (II) bands, satisfying t1 > 0 (t1 < 0) and |t2 | >1.5 | t1 | , have two Kagome-bands with flat-band above (below) Dirac-band (Fig. 1c). The Type-III (IV) bands, satisfying t1 > 0 (t1 < 0) and |t2 | <1.5 | t1 | , have one Dirac-band and one four-band with four-band above (below) Dirac-band (Fig. 1d). The band gap between two groups of bands is closed down at |t2 | =1.5 | t1 | , corresponding to the band transition point between Type-I (II) and III (IV).

Fig. 1: Topological phase diagram and topological band structures of 2D MOF.
figure 1

a Schematic 2D MOF assembled with the C3 symmetric cores. The black solid-line denotes the unit-cell. The blue dashed-line denotes the termination of Edge-A or Edge-B. t1,2 are two hopping parameters of the star lattice. b Topological phase diagram vs t1,2, classified by Berry phase γ. Type-I, II, III, and IV denote four different bands. c Schematic Type-I and II bands, originating from the decoupled dimer states (t1 = 0). d Schematic Type-III and IV bands, originating from the decoupled trimer states (t2 = 0).

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To characterize the higher-order bulk topology of this lattice, its ZQ Berry phase (γ)43 is calculated as a topological index (Supplementary Fig. 2). Physically, the Type-I and II bands (dominated by t2) originate from an array of dimer states (solid line) in the decoupled limit of t1 = 0 (dashed line), as shown in Fig. 1c. The lattice sites maintain C2 symmetry, so the bulk topology in the band gap between two Kagome-bands is captured by Z2 Berry phase. In contrast, the Type-III and IV bands (dominated by t1) originate from an array of trimer states (solid line) in the decoupled limit of t2 = 0 (dashed line), as shown in Fig. 1d. The lattice sites maintain C3 symmetry, so the bulk topology in the band gap between Dirac-band and four-band is captured by Z3 Berry phase. The calculated nonzero Berry phases for different bands are listed in Fig. 1b. For Type-I and II, the Z2 Berry phase is γ = π. For Type-III and IV, the Z3 Berry phase is γ = 2π/3 and γ = 4π/3, respectively. Thus, the Berry phase can be rewritten as γ = 2π·n, where n = 1/2, 1/3, 2/3 is the filling factor of the Fermi-level for Type-I, II, III, IV bands, respectively. The quantized nontrivial Berry phase indicates that the ground state of the star lattice is adiabatically connected to the decoupled dimer or trimer states without breaking symmetry or closing the band gap43, showing a stable 2D SOTI phase. Generally, band gap closing and reopening in topological systems are accompanied by a phase transition between trivial and nontrivial phases44. Here, the phase transition between Type-I (II) and III (IV) bands is described by two nontrivial phases with different topological indexes, exhibiting a new higher-order topological phase transition without the trivial phase45.

Topological corner states in Type-I bands

Besides the bulk topological index, the other smoking gun signature of 2D SOTI is characterized by topological corner states30,31,32,33,34,35,36,37,38,39,40. As shown in Fig. 2a, the Type-I bands consist of two Kagome-bands with the Fermi-level at 1/2 filling. The two typical edge terminations are labeled Edge-A and Edge-B (Fig. 1a), corresponding to the bond broken dimer and trimer, respectively. The ribbon band structures with Edge-A termination are shown in Fig. 2b. There is one edge state in the bulk band gap, which is detached from the valence and conduction bands. This edge state is induced by the coupled bond broken dimers along the edge (inset of Fig. 2b), forming a wire with one lattice site per unit cell. The discrete energy-levels of the triangular cluster with Edge-A termination are shown in Fig. 2c, having two groups of threefold degenerate corner states in the bulk band gap (above and below the edge states). Each corner state is localized at one corner region with a spatial distribution on three broken dimers (inset of Fig. 2c). The coupling among them will create three groups of corner states, but the zero energy state is hybridized with edge states, making it invisible (Supplementary Fig. 3). Hence, the Fermi-level lies in edge states without the filling anomaly for corner states. Additionally, the ribbon band structures with Edge-B termination are also studied. Without the broken dimers along the edge, the edge state is eliminated from the bulk band gap, as shown in Fig. 2d. However, due to the broken dimer at the corner (inset of Fig. 2e), one group of spatially localized corner states is left in the discrete energy-levels of the triangular cluster, as shown in Fig. 2e. In this case, the Fermi-level lies exactly in corner states, holding a fractional charge of e/2 for each corner.

Fig. 2: Topological corner states in Type-I bands.
figure 2

a Type-I bands with the Fermi-level between two Kagome-bands, corresponding to 1/2 filling. b, d, f Ribbon band structures with Edge-A, Edge-B and Edge-B′ termination, respectively. The inset shows the shape of the edge structure. Edge-B′ is a slightly modified Edge-B with t2 = 2.1t1 in the outmost dimers, as denoted by solid lines in the inset of f. c, e, g Discrete energy-levels of triangular clusters with Edge-A, Edge-B, and Edge-B′ termination, respectively. The inset shows the spatial distribution of corner states. The circle size denotes the weighting factor of the corner states. The black, blue, and red colors in b–g denote the bulk, edge and corner states, respectively. The hopping parameter is set to t2 = 3t1.

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The appearance (absence) of the edge states with Edge-A (Edge-B) termination can be understood in an alternative way (Supplementary Fig. 4). In the decoupled limit of Edge-A termination, the edge, and bulk are constructed by monomer and dimer, respectively. In the decoupled limit of Edge-B termination, both the edge and bulk are constructed by dimer. Since the monomer and dimer have different energy levels, the edge and bulk states are distinguishable (indistinguishable) for ribbons with Edge-A (Edge-B) termination. To further support this analysis, we slightly decrease the hopping value in the outermost dimers (t2 = 2.1t1) of edge-B termination, making the edge dimer different from the bulk dimer40, labeled Edge-B′ (inset of Fig. 2f). As expected, a pair of edge states appear in the bulk band gap, detached from the valance and conduction bands, as shown in Fig. 2f. The discrete energy-levels of the triangular cluster with Edge-B′ termination are shown in Fig. 2g, where the spatially localized one group of corner states keeps the same filling anomaly as that in Fig. 2e, but located between the gapped edge states. Different from the well-known mass inversion mechanism for creating the domain-wall states36, the corner states here are created by bond broken dimers, which are insensitive to edge terminations.

Topological corner states in Type-III bands

Similar corner states are observed in Type-III bands, consisting of one Dirac-band and one four-band with the Fermi-level at 1/3 filling, as shown in Fig. 3a. There is one edge state in the bulk band gap for the ribbon with Edge-B termination (Fig. 3b), which is created by bond broken trimers along the edge. The discrete energy-levels of the triangular cluster with Edge-B termination have two groups of corner states, siting above and below the edge states, as shown in Fig. 3c. For Edge-A termination with the intact trimer, the above edge state is eliminated from the bulk band gap (Fig. 3d). Meanwhile, one group of corner states is left in the discrete energy-levels of the triangular cluster (Fig. 3e and its inset), and each corner state at the Fermi-level holds a fractional charge of 2e/3. For Edge-A′ termination with a slightly decreased hopping value in the outmost trimers (t2 = 0.35t1) (inset of Fig. 3f), a pair of edge states are detached from the bulk states, as shown in Fig. 3f. The discrete energy-levels of the triangular cluster with Edge-A′ termination retain one group of corner states at the Fermi-level (Fig. 3g and its inset). Furthermore, the corner states in Type-II (IV) bands are similar to those in Type-I (III) bands (Supplementary Figs. 5, 6), which can hold a fractional charge of e/2 (e/3), showing the universal existence of corner states in 2D MOFs with the star lattice configuration.

Fig. 3: Topological corner states in Type-III bands.
figure 3

a Type-III bands with the Fermi-level between Dirac-band and four-band, corresponding to 1/3 filling. b, d, f Ribbon band structures with Edge-B, Edge-A and Edge-A′ termination, respectively. The inset shows the shape of the edge structure. Edge-A′ is a slightly modified Edge-A with t2 = 0.35t1 in the outmost trimers, as denoted by solid lines in the inset of f. c, e, g Discrete energy-levels of triangular clusters with Edge-B, Edge-A, and Edge-A′ termination, respectively. The inset shows the spatial distribution of corner states. The circle size and color have the same meaning as those in Fig. 2. The hopping parameter is set to t2 = 0.5t1.

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Type-I bands in Ni3(HITP)2

After illustrating the higher-order band topology in 2D MOFs from the theoretical aspects, its material realization is further investigated by first-principles calculations and STM measurements. We found that the Type-I bands can be realized in monolayer Ni3(HITP)2 grown on Au(111) substrate. Through the reported on-surface coordination assembly method46,47, monolayer Ni3(HITP)2 frameworks are synthesized on Au(111) substrate from Ni atoms and 2,3,6,7,10,11-Hexaaminotriphenylene (HATP) molecules under ultrahigh vacuum conditions. The high quality of the synthesized Ni3(HITP)2 frameworks are verified by STM image (Supplementary Fig. 7). Figure 4a shows the zoomed-in STM image of a triangular-shaped Ni3(HITP)2 framework with two edges forming a corner, which is overlapped with its atomic structures. The three-fold symmetric HITP molecules appear as two triangles sitting in a honeycomb lattice, and the coordinated Ni atoms are sandwiched between every two adjacent HITP molecules (Supplementary Fig. 8). The Ni3(HITP)2 framework has a lattice constant of 2.17 ± 0.03 nm on Au(111) surface, corresponding to a \(\sqrt{57}\times \sqrt{57}\) R6.6° unit cell. To reveal the electronic structures of the Ni3(HITP)2 framework, its density functional theory (DFT) bands are calculated. As shown in Fig. 4b, the calculated bands of Ni3(HITP)2 resemble the Type-I bands with a band gap of 0.41 eV. Based on these bands, the simulated STM image shows good agreement with the experimental data (Supplementary Fig. 8). The Wannier-bands (open circle in Fig. 4b) are also fitted to investigate the effective orbital of Type-I bands, where a star lattice is formed by six Wannier orbital centers (red circle) in each unit cell (inset of Fig. 4b). These first-principles results are consistent with our TB model calculations (Fig. 2), confirming the existence of nontrivial bulk bands in Ni3(HITP)2. Moreover, the similar Type-I bands are also observed in the projected bands of Ni3(HITP)2/Au(111) (Supplementary Fig. 9). Thus, the higher-order topology in Ni3(HITP)2 will be robust to the presence of substrate (Supplementary Figs. 10, 11), making it possible to detect boundary states in its large bulk band gap. Additionally, the band degeneracies at Dirac-point and quadratic-touching-point in Ni3(HITP)2 can be lifted by the SOC, realizing the 2D first-order TI in the tiny SOC gap48. However, the higher-order topology is not originated from the SOC, so it will also be robust to the presence of tiny SOC (Supplementary Fig. 12).

Fig. 4: Topological corner state comparison between theory and experiment in Ni3(HITP)2.
figure 4

a STM topographic image of Ni3(HITP)2 triangular framework on Au(111) substrate (−0.3 V and 0.1 nA), which is overlapped with its atomic structures. b DFT bands of Ni3(HITP)2. The open circles are fitted Type-I Wannier-bands. The inset is fitted six Wannier orbital centers (red circles) in the unit cell. c Zoomed-in bands of b around the Fermi-Level. d, e Theoretical LDOS and experimental dI/dV spectra of the bulk, edge and corner states in Ni3(HITP)2 triangular framework. c, d, e are aligned by setting Fermi-level and zero bias voltage together. The shadow color region denotes the bulk band gap. f, g dI/dV maps at 0.23 V and 0.35 V, marked by two dotted lines in e showing the spatial distribution of experimental corner and edge states. The dashed yellow circle highlights the corner region. h, i LDOS maps at 0.23 eV and 0.35 eV, marked by two dotted lines in d showing the spatial distribution of theoretical corner and edge states. j, k are the same as f, g which are overlapped with the triangular cluster made of Wannier orbital centers (yellow circles). l, m dI/dV profiles for edge and corner states at 0.23 V and 0.35 V, which are along line-1 and line-2 in j, k.

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Topological corner states in Ni3(HITP)2

To directly identify the topological corner states49,50 in Ni3(HITP)2, we conduct current-imaging tunneling spectroscopy measurement on the Ni3(HITP)2 framework shown in Fig. 4a, and extract the differential conductance (dI/dV) spectra at the bulk (inner part of the framework), edge and corner sites. The representative spectra of bulk, edge and corner states are shown in Fig. 4e. By aligning the calculated Fermi-level and zero-bias voltage in experiments together, the theoretical bulk bands (Fig. 4c) and local density of states (LDOS) (Fig. 4d) are comparable with the experimental dI/dV spectrum of the bulk state, where the similar curve shape and valence flat band peak are observed. In this way, the energy window of the bulk band gap in the dI/dV spectrum is determined in the range of 0 to 0.41 V, as denoted by the shadow color region in Fig. 4c–e. Within the bulk band gap, the dI/dV spectra have two significant features in Fig. 4e. First, the intensities of the edge and corner states are generally larger than that of the bulk state, showing the character of topological boundary states in the nontrivial bulk band gap. Second, the dI/dV spectrum of the corner state has a peak (~0.23 V) in the middle of the bulk band gap, where its intensity is larger than that of the edge state, making the corner state detectable. Whereas at 0.35 V, the intensity of the edge state is larger than that of the corner state. All these features are consistent with the theoretically calculated LDOS for edge and corner states in the triangular cluster, as shown in Fig. 4d.

Furthermore, the dI/dV maps are taken at 0.23 V and 0.35 V to visualize the spatial distribution of these states, as shown in Fig. 4f, g, respectively. At 0.23 V, the electronic states are mainly localized at the boundary of the triangular framework, and the corner region (dashed yellow circle) is brighter than the edge region (Fig. 4f), showing the character of the corner state. At 0.35 V, in contrast, the edge region becomes brighter than the corner region (Fig. 4g), showing the character of the edge state. In both maps, the bulk states are darker than the edge and corner states. The simulated LDOS maps at 0.23 eV (Fig. 4h) and 0.35 eV (Fig. 4i) show good agreement with the experimental maps. Moreover, the Wannier orbital centers (yellow circles) in the triangular cluster also match well with the real-space positions of the corner and edge states in dI/dV maps at 0.23 V (Fig. 4j) and 0.35 V (Fig. 4k), illustrating the bond-broken dimer character for all of them. To quantitatively check the spatial localization of corner and edge states, the dI/dV profiles are also taken at the edge (line-1) and corner (line-2), as marked by the white solid lines in Fig. 4j, k. Figure 4l, m show that both corner and edge states have a peak spectrum along line-1 and line-2 with the full-width at half maximum of ~6 Å, which are much localized states in real-space. Therefore, the localized topological corner state within the nontrivial bulk band gap is directly confirmed in Ni3(HITP)2, demonstrating the first experimental evidence of topological boundary states in 2D MOFs.

Discussion

In summary, the universal higher-order topology is proposed theoretically in large gapped 2D MOFs, and its unique topological corner state is identified experimentally in 2D Ni3(HITP)2. Due to intrinsic merits of low cost, easy fabrication and mechanical flexibility for organic materials, the organic materials have been always matched with their inorganic counterparts to enable various cutting-edge research frontiers, such as organic superconductor, organic light emitting diode, organic solar cell, and organic field effect transistor. The discovery of topological corner state in organic frameworks provides a new platform to explore the long-term dreamed organic topological states in experiments, greatly extending the category of conventional topological materials. The higher-order organic topological states can also be incorporated with magnetism, correlation and superconductivity to further investigate exotic quantum phenomena for application in both quantum computation and quantum simulation. We believe our work will inspire immediate research interest for studying higher-order topology in organic materials.

Methods

Tight-binding calculations

The tight-binding (TB) Hamiltonian of star lattice is written as

$$H=-{t}_{1}\mathop{\sum }\limits_{\langle i,j\rangle }^{{{{{{\rm{intra}}}}}}-{{{{{\rm{core}}}}}}}{c}_{i}^{{{\dagger}} }{c}_{j}-{t}_{2}\mathop{\sum }\limits_{\langle i,j\rangle }^{{{{{{\rm{inter}}}}}}-{{{{{\rm{core}}}}}}}{c}_{i}^{{{\dagger}} }{c}_{j}$$
(1)

where \({c}_{i}^{{{\dagger}} }\) (\({c}_{i}\)) is the creation (annihilation) operator at lattice site i = 1 to 6 (Supplementary Fig. 1). t1,2 is intra-core (inter-core) hopping parameter. In reciprocal space, the TB Hamiltonian is written as

$$H=\left(\begin{array}{cccccc}0 & -{t}_{2}{e}^{i{{{{{\bf{k}}}}}}\cdot {{{{{{\bf{a}}}}}}}_{2}} & -{t}_{1} & 0 & -{t}_{1} & 0\\ -{t}_{2}{e}^{-i{{{{{\bf{k}}}}}}\cdot {{{{{{\bf{a}}}}}}}_{2}} & 0 & 0 & -{t}_{1} & 0 & -{t}_{1}\\ -{t}_{1} & 0 & 0 & -{t}_{2}{e}^{-i{{{{{\bf{k}}}}}}\cdot {{{{{{\bf{a}}}}}}}_{1}} & -{t}_{1} & 0\\ 0 & -{t}_{1} & -{t}_{2}{e}^{i{{{{{\bf{k}}}}}}\cdot {{{{{{\bf{a}}}}}}}_{1}} & 0 & 0 & -{t}_{1}\\ -{t}_{1} & 0 & -{t}_{1} & 0 & 0 & -{t}_{2}\\ 0 & -{t}_{1} & 0 & -{t}_{1} & -{t}_{2} & 0\end{array}\right)$$
(2)

where a1,2 is lattice vector (Supplementary Fig. 1). The six eigenvalues are obtained as

$${E}_{1,2}=-\frac{1}{2}\left({t}_{1}\pm \sqrt{9{t}_{1}^{2}+4{t}_{2}^{2}+4{t}_{1}{t}_{2} \, f({{{{{\bf{k}}}}}})}\right)\\ {E}_{3,4}=-\frac{1}{2}\left({t}_{1}\pm \sqrt{9{t}_{1}^{2}+4{t}_{2}^{2}-4{t}_{1}{t}_{2}\, f({{{{{\bf{k}}}}}})}\right)\\ {E}_{5,6}={t}_{1}\pm {t}_{2}$$
(3)

where \(f({{{{{\bf{k}}}}}})=\sqrt{3+2\,\cos ({{{{{\bf{k}}}}}}\cdot {{{{{{\bf{a}}}}}}}_{1})+2\,\cos ({{{{{\bf{k}}}}}}\cdot {{{{{{\bf{a}}}}}}}_{2})+2\,\cos ({{{{{\bf{k}}}}}}\cdot {{{{{{\bf{a}}}}}}}_{1}+{{{{{\bf{k}}}}}}\cdot {{{{{{\bf{a}}}}}}}_{2})}\). Clearly, the band structures of star lattice are closely related to honeycomb and Kagome lattices. When \(|{t}_{2}| > 1.5|{t}_{1}|\), its band structures exhibit a Kagome lattice feature, including two Kagome-bands. When \(|{t}_{2}| < 1.5|{t}_{1}|\), its band structures exhibit a honeycomb lattice feature, including one Dirac-band and one four-band. When \(|{t}_{2}|=1.5|{t}_{1}|\), the band gap between two sets of bands is closed down.

The Berry phase of star lattice is calculated through the local twist method reported in previous works43. For Z2 Berry phase, the local twist is introduced in one dimer between lattice site 1 and 2 (Supplementary Fig. 2a). The TB Hamiltonian includes bare and twisted two parts:

$$H(\theta )=-{t}_{1}\mathop{\sum }\limits_{\begin{array}{c}\langle i,j\rangle \ne (1,2)\end{array}}^{{{{{{\rm{intra}}}}}}-{{{{{\rm{core}}}}}}}{c}_{i}^{{{\dagger}} }{c}_{j}-{t}_{2}\mathop{\sum }\limits_{\begin{array}{c}\langle i,j\rangle \ne (1,2)\end{array}}^{{{{{{\rm{inter}}}}}}-{{{{{\rm{core}}}}}}}{c}_{i}^{{{\dagger}} }{c}_{j}-{t}_{2}{e}^{i\theta }{c}_{1}^{+}{c}_{2}+{{{{{\rm{h}}}}}}{{{{{\rm{.c}}}}}}.$$
(4)

where θ1∈[0,2π]. Through exact diagonalization, the many-body ground state of \(H(\theta )\) is obtained as \(|{\varPhi }_{0}(\theta )\rangle\). Then, Z2 Berry phase is defined as a contour integral of Berry connection (Supplementary Fig. 2b):

$$\gamma=i{\int }_{0}^{2\pi }d\theta \cdot \langle {\varPhi }_{0}(\theta )|{\nabla }_{\theta }{\varPhi }_{0}(\theta )\rangle \,({{{{\mathrm{mod}}}}}\,2\pi )$$
(5)

Enforced by C2 symmetry of TB Hamiltonian, Z2 Berry phase is quantized to nπ with n = 0, 1. For Z3 Berry phase, the local twist is introduced in one trimer among lattice site 1, 2 and 3 (Supplementary Fig. 2c). The TB Hamiltonian also includes bare and twisted two parts:

$$H({\theta }_{1},\, {\theta }_{2})=-{t}_{1}\mathop{\sum }\limits_{\begin{array}{c}\langle i,j\rangle \ne (1,2,3)\end{array}}^{{{{{{\rm{intra}}}}}}-{{{{{\rm{core}}}}}}}{c}_{i}^{{{\dagger}} }{c}_{j}-{t}_{2}\mathop{\sum }\limits_{\begin{array}{c}\langle i,j\rangle \ne (1,2,3)\end{array}}^{{{{{{\rm{inter}}}}}}-{{{{{\rm{core}}}}}}}{c}_{i}^{{{\dagger}} }{c}_{j}-{t}_{1}{e}^{i{\theta }_{2}}{c}_{1}^{+}{c}_{2}\\ -{t}_{1}{e}^{-i({\theta }_{1}+{\theta }_{2})}{c}_{2}^{+}{c}_{3}-{t}_{1}{e}^{i{\theta }_{1}}{c}_{3}^{+}{c}_{1}+{{{{{\rm{h}}}}}}{{{{{\rm{.c}}}}}}$$
(6)

where θ1,2∈[0,2π]. Through exact diagonalization, the many-body ground state of \(H({\theta }_{1},\, {\theta }_{2})\) is obtained as \(|{\varPhi }_{0}({\theta }_{1},\, {\theta }_{2})\rangle\). Then, Z3 Berry phase is defined as a contour integral of Berry connection along path I1,2,3 (Supplementary Fig. 2d):

$${\gamma }_{1,2,3}=i{\int }_{{I}_{1,2,3}}d\mathop{\theta }\limits^{\rightharpoonup }\cdot \langle {\varPhi }_{0}({\theta }_{1},\, {\theta }_{2})|{\nabla }_{\mathop{\theta }\limits^{\rightharpoonup }}{\varPhi }_{0}({\theta }_{1},\, {\theta }_{2})\rangle \,({{{{\mathrm{mod}}}}}\,2\pi )$$
(7)

The C3 symmetry of TB Hamiltonian leads to \({\gamma }_{1}={\gamma }_{2}={\gamma }_{3}\), while the cancellation of path I1,2,3 leads to \({\gamma }_{1}+{\gamma }_{2}+{\gamma }_{3}=0\), so Z3 Berry phase is quantized to \(\frac{2n\pi }{3}\) with n = 0, 1, 2.

To consider the effect of substrate induced small symmetry breaking perturbation, the random hopping (δt) and onsite energy (δε) Hamiltonian is written as

$${H}_{random}=\mathop{\sum}\limits_{i}\delta {\varepsilon }_{i}{c}_{i}^{{{\dagger}} }{c}_{i}+\mathop{\sum }\limits_{\langle i,j\rangle }^{{{{{{\rm{intra}}}}}}-{{{{{\rm{core}}}}}}}\delta {t}_{ij}{c}_{i}^{{{\dagger}} }{c}_{j}+\mathop{\sum }\limits_{\langle i,j\rangle }^{{{{{{\rm{inter}}}}}}-{{{{{\rm{core}}}}}}}\delta {t}_{ij}{c}_{i}^{{{\dagger}} }{c}_{j}$$
(8)

To consider the effect of intrinsic SOC in star lattice51, the SOC Hamiltonian is written as

$${H}_{soc}={{{{{\rm{i}}}}}}\lambda \mathop{\sum}\limits_{\langle \langle i,j\rangle \rangle }{e}_{ij}({c}_{i\uparrow }^{{{\dagger}} }{c}_{j\uparrow }-{c}_{i\downarrow }^{{{\dagger}} }{c}_{j\downarrow })$$
(9)

where λ is the intensity of intrinsic SOC, and eij = +1(−1) denotes the right (left) turning electrons between the next-nearest-neighbor sites. Including the above small symmetry breaking perturbation and intrinsic SOC for type-I bands, the corresponding topological corner states (Supplementary Figs. 10, 12) and Berry phase (Supplementary Fig. 11) are almost the same as those shown in Figs. 1 and 2. Therefore, our proposed higher-order topology in the large band gap between two groups of Kagome-bands is very robust by considering the effect of substrate and SOC, making it detectable in the experiment.

First-principles calculations

The first-principles calculations are carried out in framework of generalized gradient approximation with both hybrid B3LYP52 and PBE53 functional using Vienna Ab initio simulation package (VASP)54. B3LYP functional is used for free-standing monolayer Ni3(HITP)2, and PBE functional is used for Ni3(HITP)2/Au(111). All calculations are performed with a plane-wave cutoff of 500 eV on 5 × 5 × 1 Monkhorst-Pack k-point mesh.

The vacuum layer of 15 Å thick is used to ensure the decoupling between neighboring slabs. The substrate supercell is simulated by three layers of \(\sqrt{57}\times \sqrt{57}\)R6.6° Au(111) structure. The DFT-D3 method55 is used for van der Waals corrections. During structural relaxation, all atoms are relaxed until forces smaller than 0.01 eV/Å, where bottom-layer of Au atoms are fixed. The Wannier-bands and orbitals are fitted through Wannier90 package56. Based on Tersoff-Hamann approximation, the STM image is simulated for the occupied states within 0.9 eV below the Fermi level.

To simulate experimental dI/dV maps, the theoretical local density of states (LDOS) for a triangular cluster are calculated by using fitted Wannier Hamiltonian. A small onsite energy εonsite is added to lattice site at cluster boundary, namely, εonsite = 0.05 eV for trimer sites at corner, εonsite = 0.08 eV for edge sites nearest neighboring to trimer sites at corner, and εonsite = 0.16 eV for the other edge sites. Since Wannier Hamiltonian is obtained by fitting the bulk bands, the addition of these extra onsite energies can better describe boundary effects in the experiments. The existence of topological corner state is robust to local perturbations. Then, LDOS is calculated as \({{{{{\rm{LDOS}}}}}}(E,\, i)=\mathop{\sum}\limits_{n}{|{\varphi }_{n}(i)|}^{2}\delta (E-{E}_{n})\), where i denotes lattice site, \({E}_{n}\) and \({\varphi }_{n}\) denote the nth eigenvalue and eigenstate, and delta-function is approximated by a Lorentzian broadening of 0.2 eV. We consider s-orbital (\({\varphi }_{s}(r)={R}_{10}(r){Y}_{00}(\theta,\, \phi )\propto {e}^{-r/a}\) and a = 5 Å) as the basis for each lattice site, then, the theoretical LDOS maps at 0.23 eV and 0.35 eV are simulated by taking its value from the plane at 0.5 Å above triangular cluster.

Growth of Ni3(HITP)2 on Au(111)

The atomically flat Au(111) substrate is cleaned by cycles of Ar+ sputtering followed by annealing at 700 K for 30 min. The 2,3,6,7,10,11-Hexaaminotriphenylene (HATP) molecules are synthesized by previously reported method57. The HATP molecules are degassed in vacuum chamber at 500 K for removing HCl and then evaporated from molecular beam evaporator at approximately 573 K, while the neutral Ni atoms (99.99 + %, Goodfellow Cambridge Ltd.) are directly evaporated from electron beam evaporator. The high-quality monolayer Ni3(HITP)2 frameworks are assembled by sequentially co-depositing HATP molecules and Ni atoms on Au(111) substrate held at room temperature, and annealing the sample at 520 K for 30 min.

Scanning tunneling microscope measurements

In ultra-high vacuum, the scanning tunneling microscope (STM) measurements are performed at 4.9 K with a base pressure better than 1 × 10−10 mbar. The Pt/Ir tips are used in STM experiments. The bias voltage is applied to sample with respect to the tip. The differential conductance (dI/dV) signals are acquired using a lock-in amplifier with a sinusoidal modulation of 1517 Hz at 5 mV.