Preformed excitons, orbital selectivity, and charge density wave order in $1T\text{−}{\mathrm{TiSe}}_2$

Traditional routes to charge density wave (CDW) in transition-metal dichalcogenides, relying on Fermi surface nesting or Jahn-Teller instabilities, have recently been brought into question. While this calls for exploration of alternative views, a paucity of theoretical guidance sustains lively controversy on the origin of, and interplay between, CDW and superconductive orders in transition-metal dichalcogenides. Here, we explore a preformed excitonic liquid route, heavily supplemented by modern correlated electronic-structure calculations, to an exci-tonic CDW order in $1T\text{−}{\mathrm{TiSe}}_2$. We show that orbital-selective dynamical localization arising from preformed excitonic liquid correlations is somewhat reminiscent of states proposed for $d$ and $f$ band quantum criticality at the border of magnetism. Excellent quantitative explication of a wide range of spectral and transport responses in both normal and CDW phases provides strong support for our scenario, and suggests that soft excitonic liquid fluctuations mediate superconductivity in a broad class of transition-metal dichalcogenides on the border of CDW. This brings the transition-metal dichalcogenides closer to the bad actors (where the metallic state is clearly not a Fermi liquid) in $d$ and $f$ band systems, where anomalously soft fluctuations of electronic origin are believed to mediate unconventional superconductivity on the border of magnetism.

Figure 1

(a) Tight-binding band structure, (b) Fermi surface map, and (c) density of states plot for the Ti $d$ $t_{2g}$ and Se $p$ bands. The multiorbital DMFT involves the two bands, Ti $d$ (black) and Se $p$ (yellow), that straddle the Fermi level.

Figure 2

Spectral function from DMFT calculations at different temperatures (20–300 K) without phonon coupling ($g=0$).

Figure 3

(a) Temperature evolution of spectral function of $a$ (Ti $d$: left panel) and $b$ (Se $p$: right panel) bands; (b) the imaginary part of the self-energies at $T=0$.

Figure 4

(a) Temperature dependence of the imaginary part of the self-energy close to the Fermi level. (b) Real part of the self-energy for Ti $d$ and Se $p$ bands at $T=50$ K ($<T_{\text{CDW}}$) and $T=200\mathrm{K}(>T_{\text{CDW}})$.

Figure 5

Theoretical ARPES map along $K\text{−}M\text{−}K$ direction of the Brilluoin zone at (a) 20 K, (b) 150 K, and (c) 270 K. (d) EDC at $M$ point at the temperatures shown.

Figure 6

Theoretical ARPES spectra at $M$ point at different temperatures. As temperature increases across $T_{\text{CDW}}$, the peak in the DMFT spectral function crosses the renormalized Fermi energy ($E_F$).

Figure 7

DMFT Fermi surface map at (a) 20 K, (b) 150 K, and (c) 270 K, showing good agreement with the temperature evolution of the Fermi surface in ARPES data of Rossnagel et al. [31].

Figure 8

DMFT results for optical conductivity at various temperatures across $T_{\text{CDW}}$.

Figure 9

Temperature-dependent dc resistivity within DMFT. Except for the low-$T$ part, LCAO+DMFT results agree very well with experimental results (represented by points, after Li et al. [16]), including a broad maximum, rather than a sharp nonanalytic change, across $T_{\text{CDW}}$ (see text).

Figure 10

(a) The interorbital excitonic order parameter [$\mathrm{Im}{G}_{ab}$, normalized by $\Delta \left(10\right)$ K] and its derivative (inset), and (b) the CDW order parameter $n{}_a\text{−}n{}_b(\propto {\Delta }_{\text{CDW}}$) and its derivative (inset) as a function of $T$.

Figure 11

Phonon spectral function at different temperatures. The inset shows change in peak position of the $A_{1g}$ phonon mode with temperature, in good accord with Raman data above $T_{\text{CDW}}$ (see text).