Multiferroic Dark Excitonic Mott Insulator in the Breathing-Kagome Lattice Material Nb₃Cl₈

Mahtab A. Khan NanoScience Technology Center, University of Central Florida, Orlando, FL USA Department of Physics, Federal Urdu University of Arts, Sciences and Technology, Islamabad, Pakistan. Naseem Ud Din Department of Physics, Florida Atlantic University, Boca Raton, FL USA Dirk Englund Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. Michael N. Leuenberger NanoScience Technology Center, Department of Physics, College of Optics and Photonics, University of Central Florida, Orlando, FL USA.

Abstract

Motivated by the recent discovery of flat bands (FBs) in breathing Kagome lattices (BKLs), we present a detailed first-principles study of the optical response of single-layer (SL) Nb₃Cl₈ using the GW-Bethe-Salpeter equation (GW-BSE) method, incorporating self-energy corrections and excitonic effects. Our findings reveal a rich spectrum of strongly bound excitons. The key results are fourfold: (i) SL Nb₃Cl₈ exhibits a dark spin-triplet Frenkel exciton ground state with binding energy substantially larger than the GW-renormalized band gap, giving rise to a negative exciton energy peak at $-0.14$ eV and indicating an excitonic Mott insulator phase potentially stable at room temperature ( $k_{B}T=0.025$ eV); (ii) the brightest exciton peak appears at 1.2 eV, in excellent agreement with experimental optical absorption spectra. (iii) We map the low-energy Frenkel exciton system onto a Hubbard model with spin-1 particles on a triangular lattice, resulting in frustrated spin configurations due to antiferromagnetic spin-spin exchange interaction. (iv) As the spin-triplet Frenkel excitons have electric dipoles that interact with each other via electric dipole-dipole interaction, we obtain antiferroelectric ordering, possibly stable at room temperature. Thus, we propose that Nb₃Cl₈ is a multiferroic dark spin-triplet excitonic Mott insulator.

Introduction

Flat bands (FBs), characterized by nearly constant energy across momentum states, lead to highly localized electronic states with large effective masses and suppressed particle hopping. These FBs foster strong electron correlations and enhanced light-matter interactions, giving rise to novel quantum phenomena. Kagome lattices (KLs), composed of corner-sharing triangles forming a hexagonal structure, can host FBs through geometry and interference-driven electron localization.

Refer to caption — Figure 1: a) Top and side view of Nb₃Cl₈, shaded grey (orange) region (rhombus) is the unit cell (Brillouin zone) of Nb₃Cl₈ consisting of 3 Nb (dark green balls) and 8 Cl (light green balls) atoms. Breathing Kagome lattice is formed by Nb atoms creating irregular hexagons (black lines) with two sets of sides of different lengths, resulting in two sets of outer equilateral triangles with different areas (shaded red and blue). SL Nb₃Cl₈ has C_3v symmetry. b) PBE (dashed colored curves, blue (red) shows spin-up (down)) E ${}_{g}^{\textrm{PBE}}=$ 0.27eV and GW (solid blue curves) E ${}_{g}^{\textrm{GW}}=$ 2.27eV band structure of SL Nb₃Cl₈, showing substantial increase in the electronic band gap. Black solid lines correspond to the continuum of states.

The breathing Kagome lattice (BKL), characterized by alternating triangles with unequal bond lengths, observed in Nb halides Nb₃X₈Sun et al. (2022); Regmi et al. (2022, 2023) (such as Nb₃Cl₈, Nb₃I₈, and Nb₃Br₈, where Nb atoms form a BKL) introduces a unique geometry that enhances electron localization. In Nb₃Cl₈, the BKL’s network of corner-sharing triangles (see Fig. 1) generates destructive interference in electronic wavefunctions, suppressing kinetic energy and producing dispersionless flat bands (FBs) in momentum space. The breathing distortion breaks inversion symmetry, isolating FBs from dispersive bands and enhancing electron correlations. This suppressed kinetic energy increases Coulomb interactions, enabling exotic quantum phenomena such as strongly bound excitons, magnetism, and potential topological phases. Remarkably, the flat bands in Nb₃Cl₈ are both distinct and experimentally accessible, in contrast to twisted bilayer semiconductors and traditional Kagome lattices (KLs), where FBs often overlap with a multitude of other bands near the Fermi level or are far away from the Fermi energy, complicating their detection. Moreover, the layered structure of Nb₃Cl₈ allows it to be thinned down to the monolayer limit, further enhancing its appeal and versatility in physical sciences, material science, and engineering.

The Exciton Insulator (EI) phase represents a fascinating quantum state of matter where the Coulomb interaction between electrons and holes drives a spontaneous formation of bound excitonic pairs, leading to a macroscopic condensation Mott (1961); Keldysh and Kopaev (1968). The concept of the excitonic insulator was formally introduced by Jérôme, Rice, and Kohn Jérôme et al. (1967), who demonstrated that such a phase can emerge in systems with either a small bandgap (narrow-gap semiconductors) or a band overlap (semimetals), where strong electron-hole interactions induce a collective instability. Rice later extended these ideas to discuss the transition between the excitonic insulator and the electron-hole liquid phases, emphasizing the role of density and screening effects in stabilizing these quantum states Rice (1970). Advances in experimental and theoretical techniques have revealed the signatures of EI phases in materials like transition metal dichalcogenides (e.g., monolayer TiSe₂) Cercellier et al. (2007); Monney et al. (2011), topological insulators Lee et al. (2019), and engineered heterostructures Fogler et al. (2014). Recent studies also highlight the interplay between excitonic condensation and other collective phenomena, such as charge density waves and spin-orbit coupling, providing a rich platform for exploring correlated electronic states Kogar et al. (2017); Varsano et al. (2020). These insights are underpinned by first-principles many-body calculations, including the GW-Bethe-Salpeter equation framework, which captures the delicate balance between Coulomb interactions and band structure effects.Hybertsen and Louie (1986); Rohlfing and Louie (1998); Benedict et al. (1998); Albrecht et al. (1998); Yang et al. (2009); Spataru et al. (2004); Deslippe et al. (2007); Yang et al. (2009); Steven G. Louie (2006); Steinhoff et al. (2014) The EI phase offers exciting prospects for applications in optoelectronics and quantum information, particularly in strongly correlated two-dimensional materials.

Several theoretical studies, including those by Gao et al.Gao et al. (2023) and Grytsiuk et al.,Grytsiuk et al. (2024a) have described Nb₃Cl₈ as an electronic Mott insulator using single- and multi-orbital Hubbard models, while neglecting excitonic effects, to explain features such as the optical absorption peak at 1.12 eV.Sun et al. (2022) These studies assume a metallic phase with half-filled bands at the Fermi level, leading to the emergence of lower and upper Hubbard bands through strong correlation effects.

In stark contrast to previous theoretical studies, our work reveals a fundamentally different perspective on SL Nb₃Cl₈ by using ab-initio Density Functional Theory (DFT) and GW-BSE calculations without any adjustable parameters to calculate its optical response, including self-energies and excitonic effects. We argue that for SL Nb₃Cl₈ it is necessary to start from the non-collinear DFT result showing exchange interaction-induced band splitting and therefore indicating a semiconductor rather than a metallic phase as the starting point for further calculations. This band splitting invalidates the assumption of a single half-filled band at the Fermi level, challenging the applicability of traditional Hubbard models to this material. Instead, we find that SL Nb₃Cl₈ hosts a large and diverse number of strongly bound excitonic states with novel k-space characteristics not previously seen experimentally or theoretically. Our calculations indicate the presence of a dark exciton ground state at a negative energy $E_{\rm EMI}=-0.14$ eV, i.e. being 0.14 eV below the GW-renormalized band gap, with a binding energy of 2.64 eV, providing compelling evidence for an excitonic insulator (EI) phase. To distinguish from exciton condensates,Zhu et al. (1995) we propose that Nb₃Cl₈ exhibits a novel EI insulator phase, namely an Excitonic Mott Insulator (EMI) phase. Furthermore, the brightest exciton peak emerges at $E_{b,\mathrm{max}}=1.2$ eV, in excellent agreement with experimental observationsSun et al. (2022). Our study provides insight into the electronic structure and flat band-induced excitonic structure in SL Nb₃Cl₈ with a breathing Kagome lattice structure, thereby offering a valuable framework for exploring the interaction between geometry and electron correlations.

I Numerical Results

Electronic Properties:

Our model system consists of a single layer (SL) of Nb₃Cl₈ with 11 atoms in the unit cell, where a layer of Nb atoms is sandwiched between layers of Cl atoms, held together through strong covalent bonds as shown in Fig. 1 (a). A breathing Kagome lattice is formed by Nb atoms creating irregular hexagons with two sets of sides of different lengths, resulting in two sets of outer equilateral triangles with different areas, as shown in Fig. 1 (a). The electronic band structure of Nb₃Cl₈ is largely determined by the symmetry properties of the high-symmetry points and lines in the first Brillouin zone of its triangular lattice. Fig. 1 also illustrates both the unit cell and the Brillouin zone of Nb₃Cl₈. The crystal structure of single-layer (SL) Nb₃Cl₈ exhibits C_3v symmetry.

Fig. 1 (b) presents the band structure results from noncollinear SOC calculations, incorporating self-energy corrections through GW (solid curves), alongside PBE results (dashed colored curves). Nearly flat bands with a very small dispersions can be seen, well separated from the continuum of bands and are therefore available for direct experimental observation. A substantial increase of 2 eV in the band gap with the GW correction compared to the PBE highlights the importance of self-energy corrections in Nb₃Cl₈. In the Supplementary Material, we provide compelling evidence from both unpolarized (UP) and spin-polarized (SP) calculations for the magnetic nature of SL Nb₃Cl₈. In UP calculations, the electronic states at the Fermi level split in SP DFT, resulting in valence and conduction bands with opposite spin orientations across the Fermi level, which forbids optical transitions from the valence to conduction band, thus leading to the formation of a dark exciton. Interestingly, nonpolarized DFT results indicate that SL Nb₃Cl₈ behaves as a metallic system,Grytsiuk et al. (2024b); Huang et al. (2023) highlighting the critical role of spin polarization in accurately capturing the material’s magnetic and optical properties. Since we obtain that the $120^{\circ}$ spin configuration has the lowest energy, we choose to perform non-collinear DFT calculations. It is interesting to note that flat bands appear in the form of triplets, i.e. a singlet accompanied by a two-fold degenerate doublet. The existence of triplets is a consequence of three-fold rotational symmetry of the crystal.Erementchouk et al. (2015); Khan et al. (2017)

Since we are interested in the FBs originating from the BKL formed by Nb atoms, we project the FBs in the band structure, onto the d-orbitals of the Nb atoms to investigate the orbital character of the FBs (SI). It can be seen that appreciable d ${}_{z^{2}}$ contribution is present in all FBs with conduction and valence bands having the largest contribution. In the analytical modeling section, we develop a single symmetric orbital tight-binding model that corroborates the numerical results, offering further validation and insight into the system’s behavior.

Optical Properties:

Fig. 2 presents the absorption spectra with oscillator strengths for SL Nb₃Cl₈, both with and without electron-hole interactions. From the oscillator strength it is evident that the first exciton eigenvalue at $E_{\rm EMI}=-0.142$ eV is indeed dark. A negative BSE eigenvalue indicates that the ground state exciton possesses an exceptionally large binding energy of 2.64 eV, greater than the material’s bandgap. This phenomenon is known as the EI phase,Sethi et al. (2021) where excitonic interactions dominate over free charge carriers. Notably, the BSE ground state corresponds to a dark exciton which couples inefficiently with light. Dark excitons can be populated when a material is cooled, as excitons in higher-energy bright states lose energy and transition to lower-energy dark exciton states. Since the Frenkel excitons are strongly localized on the Nb atoms, we propose a new EI phase for Nb₃Cl₈ called EMI phase.

Although the low-dimensional exciton insulator phase has been proposed in numerous theoretical studies,Sethi et al. (2021); Jiang et al. (2020); Liu et al. (2024) experimental realization of such systems Sethi et al. (2021); Jiang et al. (2020); Liu et al. (2024) remains a formidable challenge, demanding precise atomic-scale engineering and control; in contrast, Nb₃Cl₈ has already been successfully realized, marking a significant step forward in this field.

The brightest exciton, the exciton with the highest oscillator strength in BSE susceptibility shown in Fig. 2 (b), has a binding energy of 1.77 eV and lies at an energy of $E_{b,\mathrm{max}}=1.2$ eV, in excellent agreement with the experimental optical absorption data for SL Nb₃Cl₈.Sun et al. (2022) This alignment with experimental observations underscores the precision and reliability of GW and BSE calculations, affirming their accuracy in capturing the system’s optical properties.

Optical selection Rules:

Nb₃Cl₈ possesses hexagonal symmetry, characterized by a three-fold rotational axis and broken inversion symmetry. Much like a hydrogen atom, an exciton confined to a two-dimensional plane exhibits excitonic angular momentum (EAM) originating from the orbital motion of the electron relative to the hole. Despite inversion-symmetry breaking, we focus on the $\Gamma$ -point to bypass the need to account for the conservation of valley angular momentum (VAM), which typically arises from broken inversion asymmetry. Additionally, the three-fold rotational symmetry transforms the incoming angular momentum into a modulus of three by absorbing the excess angular momentum into the lattice. Consequently, under normally incident light, the conservation of out-of-plane angular momentum leads to the following optical selection rule:

\Delta m\hbar=\Delta l_{ex}\hbar+3N\hbar

(1)

Upon photon absorption, the change in momentum, $\Delta m\hbar$ , of the incident photon induces a corresponding change in the angular momentum of the exciton, $\Delta l_{ex}\hbar$ , and the lattice, quantified as $3N\hbar$ , where $N=0,\pm 1,\pm 2,\dots$ . Eq. (1) illustrates the selection rules governing transitions between the d ${}_{z^{2}}$ and {d ${}_{x^{2}-y^{2}}$ ,d_xy} orbitals, which require $N=\pm 3$ . In contrast, transitions from the orbitals d ${}_{z^{2}}$ to d_yz and d_xz occur when $N=0$ . Notably, out-of-plane transitions between d ${}_{z^{2}}$ orbitals are also allowed for $N=0$ . This entire description can be rigorously explained using a group-theoretical analysis of the C_3v symmetry. Within the electric dipole approximation, transitions between orbitals belonging to different irreducible representations (IRs) are summarized in Table SI in the Supplementary Information. In the electric dipole approximation the optical transitions are governed by the matrix element $\langle\psi_{i}|x_{j}|\psi_{f}\rangle$ . The initial state $\psi_{i}$ , the final state $\psi_{f}$ , and the position operator $x_{j}$ transform according to the IRs $\Gamma(\psi_{i})$ , $\Gamma(\psi_{f})$ , and $\Gamma(x_{j})$ , respectively. An electric dipole transition between two states is allowed if the direct product $\Gamma(\psi_{i})\otimes\Gamma(x_{j})\otimes\Gamma(\psi_{f})$ includes $\Gamma(I)$ in its decomposition into a direct sum. Group theoretical description corroborates the selection rules depicted in Eq. (1). Although out-of-plane transitions between the conduction and valence bands are allowed based on orbital character, these transitions are forbidden due to spin selection rules. Fig. 1 illustrates two of the allowed transitions from the valence band (VB) to the doubly degenerate conduction bands CB+1 and CB+2 at the $\Gamma$ point, which can also be seen as absorption peaks in Fig. 2.

II Results and Discussion

II.1 The Dark Exciton Ground State

Solving the two-band BSE for Nb₃Cl₈, we find that the lowest energy excitonic state is a dark exciton, with an energy of $E_{\rm EMI}=-0.14$ eV with a binding energy of 2.64 eV. This dark exciton arises from the fact that the conduction and valence bands have opposite spin, forming a spin-triplet exciton state, which effectively doubles the magnetic dipole moment on a trimer from $s=1/2$ to $S=1$ when compared with the electronic ground state obtained by DFT only. The exciton wavefunction is centered near the $\Gamma$ -point in momentum space. The delocalization in ${\bf k}$ -space (Fig. 3 (a and b)) indicates a small exciton radius in real space. To estimate its Bohr radius, we calculate the effective masses of the electron and hole in the CB and VB at the $\Gamma$ -point. We obtain $m_{e}^{*}=2.444m_{e}$ for the electron and $m_{h}^{*}=2.559m_{e}$ , resulting in an effective reduced mass of $\mu=1.25m_{e}$ . Thus, for the Bohr radius we obtain $R_{B}=1.74$ Å, (for $\epsilon_{\parallel}=1.818$ ), indicating localization on a Nb site. Note that the Bohr radius is close to the radius of the Nb atom, which is $R_{\rm Nb}=1.46$ Å. Thus, we identify this lowest-energy dark exciton being a Frenkel exciton.

Furthermore, our theoretical analysis of the tight-binding Hamiltonian for a spin-1 particle on a trimer supports these findings. Specifically, we identify the Frenkel exciton to be in the 2a₁ ground state with orbital angular momentum $m_{L}=0$ . This state arises from the symmetric combination of the $d_{z^{2}}$ orbitals of the Nb atoms in the trimer, leading to a localized Frenkel exciton with total spin $S=1$ .

II.2 Magnetic Ordering based on Spin-Triplet Frenkel Exciton

To derive an effective Hubbard Hamiltonian for the spin-1 Frenkel exciton, we note that its binding energy of $E_{b}=2.64$ eV is much larger than the intra-trimer hopping energy $t_{0}=-0.325$ eV, which is much larger than the inter-trimer hopping energy $t_{1}=0.022$ eV. Since the spin system is based on the single $d_{z^{2}}$ Nb orbital, for which $m_{l}=0$ , and, in addition, the molecular orbital 2a₁ also has $m_{L}=0$ , we infer that SOC can be neglected, in agreement with previous theoretical studies.Grytsiuk et al. (2024a)

Therefore, we obtain the effective exchange Hamiltonian

H_{\text{eff}}=J\sum_{\langle i,j\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j},

(2)

where $J$ is the exchange coupling between neighboring trimers, $\mathbf{S}_{i}$ is the spin-1 operator on trimer $i$ , and $\langle i,j\rangle$ denotes neighboring trimers. The exchange coupling $J$ can be estimated in second-order perturbation theory as

J=\frac{t_{1}^{2}}{U_{X}}\approx 0.5\,\text{meV},

(3)

where $t_{1}=0.022$ eV is the inter-trimer hopping energy, and $U_{X}\approx 1$ eV is the estimated on-site Hubbard Coulomb repulsion energy for Frenkel excitons in Nb ions.

Since a spin-1 particle on a triangular lattice is expected to behave more classically than a spin-1/2 particle, we propose to use the Weiss mean-field approximation for the spin-1 Frenkel excitons, in which the exchange interaction between neighboring spins can be represented as an effective exchange field acting on each spin. Specifically, each spin experiences a mean field due to its six neighboring spins, which effectively adds a Zeeman-like term to the Hamiltonian. We obtain a $120^{\circ}$ spin configuration for the spin-1 Frenkel excitons as the ground state, shown in Fig. 4.

Considering the exchange interaction $J$ and applying the Weiss mean-field theory, the exchange term can be approximated as

J\sum_{\langle i,j\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}\approx M_{ex}\sum_{i}S_{i},

(4)

where $M_{ex}=J\sum_{j=1}^{6}\left<S_{j}\right>=\frac{JzS}{2}=3J$ is the mean field experienced by the spin-1 particle on each site, $z=6$ is the coordination number (number of nearest neighbors), and $S=1$ is the spin of the Frenkel exciton. Compared with the spin-1/2 system on a triangular lattice, the spin-1 system is more stable because the mean exchange field is twice as large. Thus, in the mean-field approximation, the exchange interaction reduces to an effective Zeeman term, which is consistent with the spin splitting between the CB and VB obtained in the spin-DFT and noncollinear DFT calculations.

Our result is in agreement with recent experimental data suggesting antiferromagnetic coupling with a Weiss temperature of $T_{W}=-18.9$ K or $T_{W}=-13.1$ K in Nb₃Cl₈.Sun et al. (2022); Haraguchi and Yoshimura (2024) Its magnitude corresponds to an energy of about $|k_{B}T_{W}|=1$ meV, which is approximately of the order of magnitudes of the estimated values of $J$ and $D$ . The Weiss temperature can be approximated by the formula $k_{B}T_{W}=-zJS(S+1)/3$ for a spin- $S$ system.Mugiraneza and Hallas (2022) Thus, we obtain $k_{B}T_{W}=-2J=-1$ meV, in excellent agreement with experimental data.

II.3 Electric Dipole Ordering based on Spin-Triplet Frenkel Exciton

While in the absence of SOC, spin-triplet excitons are optically inactive (dark) due to spin selection rules, they exhibit an electric dipole moment due to their p-like orbital wave function. Spin triplet excitons are characterized by their total spin quantum number $S=1$ , which arises from the parallel alignment of the electron and hole spins. Since electron and hole are fermions, if the spin part of the exciton is symmetric, the orbital part must be antisymmetric, e.g. p-like.

The spin triplet state comprises three possible spin projections, corresponding to the eigenvalues of the total spin projection operator $S_{z}$ , i.e. $|S=1,m_{S}=+1\rangle=|\uparrow\Uparrow\rangle$ , $|S=1,m_{S}=0\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\Downarrow\rangle+|\downarrow\Uparrow\rangle\right)$ , and $|S=1,m_{S}=-1\rangle=|\downarrow\Downarrow\rangle$ .

The antisymmetric nature of the spatial wavefunction means the lowest-energy orbital state is $p$ -like, with an angular momentum quantum number $\ell=1$ and magnetic quantum number $m_{L}=-1,0,+1$ . Thus, the Frenkel excitons can carry an electric dipole moment due to the separation of the electron and hole. The electric dipole-dipole interaction between two electric dipoles ${\bf p}_{i}$ and ${\bf p}_{j}$ , located at positions ${\bf r}_{i}$ and ${\bf r}_{j}$ , is given by

H_{\text{dd}}=\frac{1}{4\pi\epsilon_{0}\epsilon_{r}}\left[\frac{{\bf p}_{i}\cdot{\bf p}_{j}}{r_{ij}^{3}}-3\frac{({\bf p}_{i}\cdot{\bf r}_{ij})({\bf p}_{j}\cdot{\bf r}_{ij})}{r_{ij}^{5}}\right],

(5)

where ${\bf r}_{ij}={\bf r}_{j}-{\bf r}_{i}$ is the displacement vector between the two electric dipoles and $r_{ij}=|{\bf r}_{ij}|$ is the distance between them. $\epsilon_{r}$ is the relative permittivity of the material. In the case of Nb₃Cl₈ the dielectric constant exhibits uniaxial anisotropy with the in-plane value estimated by BSE of $\epsilon_{\parallel}=1.818$ and out-of-plane value of $\epsilon_{\perp}=1.1285$ .

For a triangular lattice, the geometry of the lattice dictates the relative orientations of the dipoles. Similar to the spins, the electric dipoles are also frustrated on a triangular lattice. By comparing the electric dipole-dipole interaction energies $U_{dd,120^{\circ}}=-95$ meV for $120^{\circ}$ in-plane orientation of electric dipoles and $U_{dd,\mathrm{AFE}}=-122$ meV for out-of-plane anti-alignment of electric dipoles, we infer that the electric dipoles of the Frenkel excitons favor the out-of-plane antiferroelectric configuration. The uniaxial anisotropy energy gap can be estimated to be $\Delta_{\rm aniso}=U_{dd,120^{\circ}}-U_{dd,\mathrm{AFE}}=27$ meV, corresponding to a phase transition temperature of $T_{c,\mathrm{AFE}}=313$ K $=40^{\circ}$ Celsius.

II.4 The Brightest Exciton

The brightest exciton is a bright exciton state with the highest oscillator strength, which occurs at the energy of E ${}_{\rm brightest}=$ 1.2 eV, with a binding energy of 1.77 eV, corresponding to a spin-singlet exciton. The wavefunction amplitude for this exciton is delocalized in momentum space, with significant contributions from electron-hole pairs near the K-points, reflecting the band structure of the material. The excitonic wavefunctions for the lowest dark exciton and the brightest exciton are shown in Fig. 3.

In stark contrast to previous theoretical studies, where the optical absorption peak at 1.12 eV was attributed to the optical transition between the Hubbard bands related to the 2a₁ state,Gao et al. (2023); Grytsiuk et al. (2024a) our GW-BSE calculation shows that the peak at 1.12 eV is due to the bright exciton that is made of an electron in CB states and a hole in VB-1 and VB-2 states.

III Conclusion

The ab-initio GW-BSE calculation for Nb₃Cl₈ reveals two important features: (i) The lowest-energy exciton is dark due to the spin-triplet configuration of the electron-hole pair. This exciton lies at E ${}_{\rm dark}=-$ 0.14 eV and has a binding energy of 2.64 eV, indicating an exciton insulator state.Gao et al. (2023); Haraguchi and Yoshimura (2024) (ii) The brightest exciton lies at an energy of $E_{\rm brightest}=1.2$ eV with a binding energy of 1.77 eV, which is in an excellent agreement with experimental observations. Sun et al. (2022)

Our ab-initio GW-BSE results redefine the quantum phase of the ground state of Nb₃Cl₈ to be a dark excitonic insulator; more precisely, an excitonic insulator made of Frenkel excitons that are localized on each trimer in the BKL, giving effectively rise to the new quantum phase of excitonic Mott insulator with the Frenkel exciton spin triplets ( $S=1$ ) arranged in a $120^{\circ}$ spin configuration. This result is consistent with experimental data.Gao et al. (2023); Haraguchi and Yoshimura (2024)

Using the classical electric dipole-dipole interaction, we estimate that the electric dipoles of the Frenkel excitons to align in a out-of-plane antiferroelectric configuration with a anisotropy barrier of $\Delta_{\rm aniso}=27$ meV, suggesting stability at room temperature.

We predict that the combination of flat bands and electric dipole ordering should give rise to enhancement of second-harmonic generation (SHG), third-harmonic generation (THG), and high-harmonic generation (HHG) in Nb₃Cl₈, in particular because the antiferroelectric dipole configuration on a triangular lattice breaks the inversion symmetry. While SL Nb₃Cl₈ should exhibit nonlinear response for all orders $n$ , i.e. nonzero $\chi^{(n)}$ for all integers $n>2$ due to inversion symmetry breaking, we anticipate that for two and in general an even number of AB-stacked layers of Nb₃Cl₈ the even orders of nonlinear response should vanish, because the inversion symmetry is restored. The reason is that AB stacked (or bulk) Nb₃Cl₈ exhibits D_3d point group symmetry,Jeff et al. (2023) which contains the inversion symmetry. Note that bulk Nb₃Cl₈ is typically AB-stacked at room temperature.Sheckelton et al. (2017)

We envision applications in quantum information processing and in neuromorphic machine learning. The excitonic Mott insulating phase of Nb₃Cl₈ at room temperature, coupled with strong second-order nonlinearities ( $\chi^{(2)}$ ), offers a promising route toward robust photonic quantum information processing. Enhanced $\chi^{(2)}/\alpha$ ratios and improved $Q$ -factors, as envisioned by recent theoretical proposals,Krastanov et al. (2021) could enable deterministic two-photon gates and the generation of fault-tolerant error-correctable states, including Gottesman-Kitaev-Preskill (GKP) qubits.Eaton et al. (2019); Takase et al. (2023) $\alpha$ denotes the mode coupling loss rate.

Strong $\chi^{(2)}$ -mediated nonlinearities in Nb₃Cl₈ may also benefit photonic machine learning architectures that demand ultrafast and energy-efficient computation. Current coherent optical processors have demonstrated rapid linear algebra operations and reconfigurable interferometric networks.Carolan et al. (2015); Shen et al. (2017) As device losses decrease and Nb₃Cl₈-based materials achieve higher nonlinear coupling, on-chip nonlinear activation functions and in situ training can be implemented to realize fully integrated optical neural networks.Lin et al. (2018); Bandyopadhyay et al. (2024)

IV Acknowledgments

M. N. L. acknowledges support by the Air Force Office of Scientific Research (AFOSR) under award no. FA9550-23-1-0455 and award no. FA9550-23-1-0472. Calculations were performed at the Stokes high performance computer cluster of the University of Central Florida. The final calculations were performed on the Darwin high performance computer cluster provided by the ACCESS program of the National Science Foundation (NSF). M. N. L. and M. A. K. acknowledge support by the NSF ACCESS program under allocation no. PHY230182 and no. PHY240242.

V Computational Details

We consider a unit cell of SL Nb₃Cl₈, consisting of 11 atoms in a hexagonal structure, with edge lengths $a=b=6.81$ Å. A vacuum layer of $20$ Å is introduced in the $z$ -direction to minimize interactions between the periodic images of Nb₃Cl₈.

We first perform mean field DFT in generalized gradient approximation (GGA) as implemented in VASP,VASP (2022) on a 9 $\times$ 9 $\times$ 1 k-grid with a cut-off energy of 400 eV. For PBE Calculations a band gap of 0.27 eV is obtained Fig. 1 (b) (dashed line). The obtained results are consistent with previous DFT calculations.Sun et al. (2022)

The GW calculations were initiated using DFT-derived wavefunctions. Self-consistency was achieved by iteratively updating the GW eigen-energies, thereby removing dependence on the single-particle energies obtained from the initial DFT computations. For GW calculations we consider a 9 $\times$ 9 $\times$ 1 grid with a cut-off energy (ENCUT) of 400 eV. For response function the we set ENCUTGW $=$ 200 eV. The parameter ENCUTGW controls the basis set for the response functions in exactly the same manner as ENCUT for the wave functions. We use 324 empty bands,

Details of the convergence of conduction band, valence band and band gap with respect to the number of bands $N_{b}$ and number of iterations $N_{I}$ are shown in the Supplementary Information.

For BSE calculations, we consider 20 bands above and below the Fermi level, it is important to note that results seem to converge at 12 bands above and below the Fermi level.

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Multiferroic Dark Excitonic Mott Insulator in the Breathing-Kagome Lattice Material Nb3Cl8