Spatially Indirect Exciton Condensation in Two-Dimensional Strongly Correlated Semimetals

The gist: The critical temperature for indirect exciton condensation can be strongly suppressed by intra-orbital Coulomb repulsion U of the d-orbitals, and competition among multiple pairing channels in a three-orbital model further reduces Tc.

Model and Methods

The study employs a two-orbital Hubbard model on a 2D triangular lattice using Cellular dynamical Mean-Field theory (CDMFT) and determinant quantum Monte Carlo (DQMC) [pg2]. This non-perturbative CDMFT approach is used to capture the quantum fluctuation of strongly interacting d-electrons beyond weak-coupling treatments [pg2]. The Hamiltonian is defined by terms representing hopping, on-site repulsion U, and inter-orbital interaction V [pg3]. The model assumes a pure electron picture where the carrier density in the electron orbital (orbital-a) is denoted as ne ≡ n, implying a hole orbital (layer-b) of nh = 1 − ⟨n̂b⟩ = ne = n in the orbital-b [pg3].

Results on Suppression by On-site Interaction U

The results demonstrate that on-site Hubbard U can strongly suppress the condensation temperature Tc, an effect that is particularly pronounced at higher electron-hole pair densities [pg2]. This behavior contrasts sharply with the case without on-site U, where Tc grows with pair density n at fixed V [pg2]. Specifically, increasing U from zero to 18 for a fixed V = 5 reduces Tc from Tc = 0.1 to Tc = 0.015, a decrease by nearly a factor of seven [pg4]. This decreasing trend is particularly rapid at smaller U and becomes more gradual for larger U [pg4]. The strong suppression of Tc by U primarily stems from a reduction in the exciton gap, rather than a decrease in the phase stiffness of the excitonic condensate [pg5].

Competition Among Pairing Channels

In the three-orbital model, competition among different pairing channels is uncovered, which also acts to suppress Tc of exciton condensation [pg2]. This competition is evident when considering electron-hole pairing with two d-orbitals and one p-orbital, where an orbital-selective electron-hole pairing state is revealed [pg6]. For instance, in the case where inter-orbital interactions between three orbitals are equal (Vab = Vbc), the critical temperature is reduced to Tc ≈ 0.015 [pg6]. This competition bears a conceptual resemblance to geometric frustration in magnetic systems, where competing magnetic couplings can prevent the establishment of long-range magnetic order [pg6].

Orbital-Selective Condensation

The analysis reveals an orbital-selective electron-hole pairing phenomenon in the three-orbital model [pg6]. Upon cooling, an excitonic order parameter between the d-p orbitals (Gac) emerges spontaneously below a critical temperature of Tc ≈ 0.032t, and grows with further decreasing T [pg6]. In contrast, the order parameter between two d-d orbitals (Gab) remains negligible at all temperatures, despite the symmetric inter-orbital attraction (Vab = Vac = Vbc) [pg6]. This orbital-selective excitonic condensation that prefers a d-p condensation over a d-d condensation likely stems from the effect of Hubbard U, which hinders inter-orbital exciton formation [pg6].

Implications for Experimental Observations

The calculated maximum transition temperature Tc ≈ 0.06t at typical parameters (U = 8t, V = 6t) corresponds to a physical temperature of approximately 70 ∼ 140K, which is consistent with the experimentally observed Tc ∼ 100K in Ta2Pd3Te5 [pg6]. This agreement suggests that the strong suppression of Tc by on-site Hubbard U may play an important role in resolving the apparent paradox between the large exciton binding energy and the low condensation temperature observed experimentally [pg6]. The findings illuminate the intricate interplay among Coulomb interactions, carrier density, and orbital degrees of freedom, providing valuable theoretical insights into the behavior of excitonic insulators in strongly correlated materials [pg6].

Future Directions

The authors propose to verify these competing mechanisms in future experiments detecting excitonic condensation temperature Tc versus carrier concentration n, where non-monotonic evolution may be found [pg6]. They also suggest that with the polarization-dependent angle-resolved photoemission spectroscopy (ARPES), future experiments should be able to resolve the orbital differences of the excitonic condensation in multi-orbital excitonic insulator materials, thereby providing an approach to verify the results obtained in this work [pg6]. The study concludes by noting that for TMC excitonic insulator candidates such as Ta2Pd3Te5 and Ta2NiSe5, the physical energy scales fall within their simulated parameter space [pg6]. The paper also mentions that the suppression of Tc by U persists even in the semiconductor limit due to orbital hybridization [pg6]. The work suggests that a theoretical approach considering only interorbital electron-hole binding may miss a key physical ingredient and severely overestimate the phase transition temperature Tc [pg6].

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Spatially Indirect Exciton Condensation in Two-Dimensional Strongly Correlated Semimetals Yao Zeng, Shi-Cong Mo, Xiang Chen, and W´ei W´u∗ Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices and School of Physics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China (Dated: June 19, 2026 Identifying materials hosting an excitonic insulator ground state has been one of the major pursuits in condensed matter physics in recent years. Promising candidates in transition metal chalcogenide compounds (TMC), including 1T-TiSe2, Ta2Pd3Te5, and Ta2NiSe5, share a crucial common characteristic: their low-energy physics is governed by electrons in d-orbitals subject to strong on-site Coulomb interactions. In this work, we investigate spatially indirect exciton condensation in two dimensional semimetals on triangular lattice. Using a combination of dynamical mean field theory and the determinant quantum Monte Carlo method, we study two- and three-orbital Hubbard models incorporating strong on-site (U) and inter-orbital interactions (V). Our results demonstrate that on-site Hubbard U can strongly suppress the condensation temperature Tc, an effect that is particularly pronounced at higher electron-hole pair densities. This behavior contrasts sharply with the case without on-site U, where Tc grows with pair density at fixed V. Moreover, we uncover competition among multiple electron-hole pairing channels in the three-orbital model, which also acts to suppress Tc of exciton condensation. An orbital-selective electron-hole pairing state is identified. These findings may help explain the large discrepancy between strong binding-energy and relative low transition temperature for indirect excitons in TMCs materials, offering important insights for understanding and engineering exciton condensation in materials with strongly correlated d- shell electrons.

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2 (b) U V Orbital/Layer a: electron-active Orbital/Layer b: hole-active (a) FIG. 1. (a) Schematic illustration of the two-orbital (twolayer) Hubbard model on 2D triangular lattice. U represents the on-site intra-orbital(intra-layer) Coulomb interaction, while V denotes the inter-orbital (inter-layer) Coulomb interaction in a unit cell. The orbital/layer a (upper) is designated as electron-active, while orbital/layer b (lower) acts as hole-active. Dotted line indicates a possible electron–hole pairing. (b) Schematic illustration of the non-interacting band structure we use which is a semimetal at U = 0, V = 0. Upper/lower curves indicate the conduction/valence bands (CB/VB). The binding of holes at the CB bottom (hollow circles) and electrons at the VB top (filled circles) leads to the formation of indirect excitons (indicated by the solid line between the two circles). Dashed line indicates the Fermi level.

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3 two dimensional (2D) semimetals using a combination of Cellular dynamical Mean-Field theory (CDMFT) and determinant quantum Monte Carlo (DQMC). The non-perturbative CDMFT approach enables us to capture the quantum fluctuation of strongly interacting d- electrons beyond weak-coupling treatments. We show that the exciton condensation temperature Tc can be significantly suppressed by on-site Coulomb repulsion U, although it in general remains non-zero for finite V. We also show that a finite U fundamentally modifies the density dependence of Tc. Specifically, Tc drops with increasing carrier concentration n, which contrasts with the U = 0 case, where Tc grows with pair density n at fixed V. Using a three-orbital model, we further uncover competition between different pairing channels in the three-orbital model, which provides an additional mechanism suppressing Tc. An orbital-selective EI state is also revealed in this model. Finally, we discuss the implications of these findings for experimental observations.

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4 Result The onset of the excitonic condensation phase can be identified by the divergence of the uniform excitonic pairing susceptibility χ, which can be defined as, χ = 1/N X i,j Z β 0 dτ ⟨Tτ pj (τ)pi(0)⟩ (2) where pi is the uniform spin-singlet electron-hole pair operator [25], pi = √1/2 P σ ⟨c †iaσcibσ⟩, and N is the number of unit cells in the system. Thus the vanishing inverse pairing susceptibility 1/χ → 0 can be used to determine the critical transition temperature Tc for excitonic condensation (i.e., Tc for EI).

1. Identify materials for exciton condensation by calculating interaction ratios, specifically noting that dimensionless interaction ratios of U/t ≈ 4 ∼ 25 and V /t ≈ 5 ∼ 10 map to physical energy scales consistent with experimentally observed critical temperatures like Tc ≈ 100K in Ta2Pd3Te5.

2. Predict the suppression mechanism of the condensation temperature based on on-site repulsion, as shown by the finding that the onset of the excitonic condensation phase can be identified by the divergence of the uniform excitonic pairing susceptibility χ, and that Tc decreases substantially as U increased from zero.

3. Develop a predictive model for carrier density dependence, specifically contrasting Tc drops with increasing carrier concentration n in the presence of on-site repulsion U versus the case without it, which is crucial for understanding experimental observations where a larger population of charge carriers boosts condensation when U=0.

4. Implement an orbital-selective pairing classifier by analyzing the three-orbital model results, specifically noting that an orbital-selective electron-hole pairing state is also revealed in this model, leading to the conclusion that the competition between multiple pairing channels in exciton insulator systems can significantly reduce Tc.

Identifying materials hosting an excitonic insulator ground state has been one of the major pursuits in condensed matter physics in recent years. Promising candidates in transition metal chalcogenide compounds (TMC), including 1T-TiSe 2, Ta 2Pd 3Te 5, and Ta 2NiSe 5, share a crucial common characteristic: their low-energy physics is governed by electrons in d- orbitals subject to strong on-site Coulomb interactions. In this work, we investigate spatially indirect exciton condensation in two-dimensional semimetals on triangular lattice. Using a combination of dynamical mean-field theory and the determinant quantum Monte Carlo method, we study two- and three-orbital Hubbard models incorporating strong on-site (U) and inter-orbital interactions (V). Our results demonstrate that on-site Hubbard U can strongly suppress the condensation temperature T c, an effect that is particularly pronounced at higher electron-hole pair densities. This behavior contrasts sharply with the case without on-site U, where T c grows with pair density at fixed V. Moreover, we uncover competition among multiple electron-hole pairing channels in the three-orbital model, which also acts to suppress T c of exciton condensation. An orbital-selective electron-hole pairing state is identified. These findings may help explain the large discrepancy between strong binding-energy and relative low transition temperature for indirect excitons in TMCs materials, offering important insights for understanding and engineering exciton condensation in materials with strongly correlated d- shell electrons.