Development of an $ab initio$ method for exciton condensation and its application to ${\mathrm{TiSe}}_2$

Exciton condensation is a phenomenon that indicates the spontaneous formation of electron-hole pairs, which can lead to a phase transition from a semimetal to an excitonic insulator by opening a gap at the Fermi surface. Although the idea of an excitonic insulator has been proposed for several decades, current theoretical approaches can only provide qualitative descriptions, and a quantitative predictive tool is still lacking. To shed light on this issue, we developed an ab initio method based on finite-temperature density functional theory and many-body perturbation theory to calculate the critical behavior of exciton condensation. Utilizing our methodology on monolayer $\mathrm{TiSe}{}_2$, we identify a phase transition involving lattice distortion and nontrivial electron-hole correlation at a temperature exceeding the critical temperature of phonon softening. By breaking down the components within the gap equation, we demonstrate that exciton condensation, mediated by electron-phonon interaction, is the underlying cause of the charge-density-wave state observed in this compound. Overall, the methodology introduced in this work is general and sets the stage for searching for potential excitonic insulators in natural material systems.

Figure 1

The structure and workflow of this paper. Upper-right panel: We extend Kohn-Sham's formalism of DFT by incorporating the nonlocal (NL) charge density and potential. Using the linked-cluster expansion, we demonstrate that the NL potential contributes to an energy modification equivalent to the GW correction and derive a gap equation with mass renormalization that can describe the exciton condensation phenomenon accompanied by lattice distortion, as discussed in Secs. 2 and 3. Upper-left panel: We implement the gap equation with numerical data from standard DFT and DFPT, including the KS energy, anharmonic phonon frequency, $e$-ph coupling, and screened Coulomb interaction, computed by existing first-principles packages. Bottom panel: Combining the theory and numerical inputs, we investigate the exciton condensation in monolayer ${\mathrm{TiSe}}_2$. We perform computations by considering different corrections within the formalism step by step and present the EI/CDW phase diagram as the final result. The detailed discussion is presented in Sec. 4c.

Figure 2

Feynman diagrams based on the interacting Hamiltonian, Eq. (39b). (a) The lowest order diagrams in the linked-cluster expansion, containing products of MF density and potentials, from which we can derive the MF gap equation, Eq. (3a). [(b)–(e)] Next-to-leading-order bubble diagrams resulting from the $e$-ph coupling, where $\tilde{\mathrm{\Pi }}$ is the self-energy defined in Eq. (40b). These diagram contribute to the mass renormalization terms, ${\mathcal{Z}}_b$ and ${\mathcal{Z}}_{\mathbf{k}}^{cv}$. The bottom column presents diagrams due to the Coulomb interaction. In (f), we consider the screened Coulomb interaction $W_q$ while in (g) the bare Coulomb interaction is used. This choice of Coulomb interaction is inspired by Ref. [62]. The meaning of the symbols are summarized as follows: $\chi$ is the NL density [Eq. (34b)], ${\chi }_b$ is the phonon displacement [Eq. (24)], $\mathrm{\Delta }$ is the NL potential [Eq. (35)], ${\mathrm{\Delta }}_b$ is the phonon linear potential [Eq. (23)], $G_k$ is the standard propagator without band change and momentum transfer [Eq. (52)], $F_k$ is the anomalous propagator connecting state of momentum $\mathbf{k}$ and $\mathbf{k}+\mathbf{M}$ [Eq. (46)], and $D_q$ is the phonon propagator.

Figure 3

(a) Band structure of monolayer ${\mathrm{TiSe}}_2$ in disordered phase computed in DFT (gray) and the GW approximation (deep red) with spin-orbit coupling included. The electron pocket at the M point and the hole pocket at the $\mathrm{\Gamma }$ point shrink after the GW correction included. (b) The temperature dependence of the SCP phonon dispersion is encoded by the color map. The black line is computed by DFPT in the harmonic approximation, for which the appearance of the soft mode at the M point indicates the instability of the crystal structure against the formation of $2\times 2$ CDW at zero temperature. Inset: The normal vibration mode of the soft mode at the M point. (c) Temperature-dependent phonon frequency of the soft mode at $\mathbf{q}=\mathbf{M}$. The blue line represents the frequency obtained from DFPT and corrected by the Coulomb-screened self-energy [Eqs. (28) and (29)]. The orange line represents the frequency obtained from the SCP theory with anharmonic effect included [Eqs. (31) and (32)]. The green line represents the frequency obtained from the SCP theory along with the Coulomb screened self-energy [Eq. (42)]. Since the green line denotes the fully renormalized phonon frequency without exciton correlation, it indicates a phase transition at $T_c^{\mathrm{IP}}=190\mathrm{K}$ where lattice distorts due to phonon softening in independent particle (IP) approximation. When calculating the gap equation of exciton condensation, we adopt the fully renormalized phonon frequency to correct the $e$-ph coupling by Eq. (43).

Figure 4

Coulomb screened self-energy correction at finite temperature, Eq. (28), measured with respect to the zero-temperature value. Although the Fermi surface nesting is not significant (circle vs oval), remarkable temperature dependence can be found for phonon near $\mathbf{M}=(\pi /2,0)$ connecting the electron packet and hole pocket.

Figure 5

Temperature depending phonon displacement potential computed under different corrections. The inclusion of the phonon mass renormalization term suppresses the potential in the whole temperature range, while other corrections provide a rigid shift. Table: Summaries of critical temperature and corresponding phonon frequency of each correction type.

Figure 6

Mass renormalization coefficients near the critical temperature. (a) The exciton condensation mass renormalization coefficients can be categorized into three kinds: those coming from the $e-e$(h-h) pairing, energy modulation, and pairing involving states near the Fermi surface, respectively. The overall values decrease when temperature rises. (b) Phonon displacement mass renormalization coefficient increases along with temperature.

Figure 7

EI/CDW potential calculated from first principles. The computed critical temperature is $T_c=418\mathrm{K}$ while the experimental measure is $T_c^{\exp .}=220\mathrm{K}$. The fitted critical component $b=0.56$ is in great agreement with the square root trend in the experiment [48].

Figure 8

The phonon self-energy diagrams due to the interacting term, Eq. (39b) where $n,m$ are the band index for the electron loops and ${\omega }_m$ and ${\nu }_n$ are the Matsubara frequency of electron and phonon, respectively.