Exciton condensation due to electron-phonon interaction

We show that coupling to vibrational degrees of freedom can drive a semimetal excitonic-insulator quantum phase transition in a one-dimensional two-band $f$-$c$-electron system at zero temperature. The insulating state typifies an excitonic condensate accompanied by a finite lattice distortion. Using the projector-based renormalization method we analyze the ground-state and spectral properties of the interacting electron-phonon model at half filling. In particular we calculate the momentum dependence of the excitonic order-parameter function and determine the finite critical interaction strength for the metal-insulator transition to appear. The electron spectral function reveals the strong hybridization of $f$- and $c$-electron states and the opening of a single-particle excitation gap. The phonon spectral function indicates that the phonon mode involved in the transition softens (hardens) in the adiabatic (nonadiabatic and extreme antiadiabatic) phonon frequency regime.

Figure 1

(a) Semimetallic $f$-$c$-electron band structure used in this work. An $f$ valence-band hole and a $c$ conduction-band electron may form an “excitonic” bound state owing to their interaction with the lattice degrees of freedom, with a Brillouin-zone boundary phonon involved for momentum-conservation reasons. Note that the schematic band structure shown mimics the situation in the EI material ${\mathrm{TmSe}}_{0.45}{\mathrm{Te}}_{0.55}$, where the quasilocalized 4$f^{13}$ state has its maximum at the $\Gamma$ point, the 5$d$ strongly dispersive state has its minimum at the X point, and exciton formation is accompanied by a $\Gamma$-X phonon.[19] Also in 1$T$-${\mathrm{TiSe}}_2$ the valence-band top and the conduction-band minimum are located at different points in the Brillouin zone.[21] (b) For the case in which, above a critical electron-phonon coupling strength $f$-$c$-electron coherence is achieved at sufficiently low temperatures, even a new symmetry-broken ground state may appear, the so-called excitonic insulator, which may be accompanied by a finite lattice distortion and a modulation of the charge density.[7]

Figure 2

Magnitude of the EI order-parameter function $d_k=\langle c_{k+Q}^{†}{f}_k^{}\rangle$ with $Q=\pi$ (cf. color bar) depending on momentum $k$ (on the $x$ axis) and the electron-phonon coupling strength $g$ (on the $y$ axis) in the adiabatic (${\omega }_0=0.5$, left-hand panel) and nonadiabatic (${\omega }_0=2.5$, right-hand panel) phonon frequency regime.

Figure 3

EI order parameter $\Delta$ (black filled symbols, left axis of ordinate) and lattice displacement $x_Q$ (red open symbols, right axis of ordinate) as functions of the electron-phonon interaction $g$ in the adiabatic (${\omega }_0=0.5$, circles) and nonadiabatic (${\omega }_0=2.5$, squares) cases. Black (red) dashed (dot-dashed) lines without symbols give the corresponding mean-field results for $\Delta$ ($x_Q$) at ${\omega }_0=0.5$ (${\omega }_0=2.5$).

Figure 4

PRM ground-state phase diagram of the two-band $f$-$c$-electron-phonon model in the $g$-${\omega }_0$ plane for the half-filled band case. Inset: Asymptotic behavior of the (squared) critical coupling strength $g_c^2$ at very large phonon frequencies (${\omega }_0\to \infty$) (cf. Appendix ). Dashed lines give the corresponding mean-field results.

Figure 5

Intensity plots of the $c$-electron (left-hand panels) and $f$-electron (right-hand panels) single-particle spectral functions $A_k^{c,f}\left(\omega \right)$ in the adiabatic regime with ${\omega }_0=0.5$. The electron-phonon coupling $g$ increases as indicated from top to bottom panels. In the lowermost two panels the corresponding mean-field results are included, without dissolving the spectral intensity however (see white dashed lines).

Figure 6

Intensity plots of the $c$-electron (left-hand panels) and $f$-electron (right-hand panels) single-particle spectral functions $A_k^{c,f}\left(\omega \right)$ for $g=0.5$ (top panels) and $g=0.8$ (bottom panels) in the nonadiabatic regime with ${\omega }_0=2.5$.

Figure 7

Intensity plots of the phonon spectral function $C_q\left(\omega \right)$ for different $g$ at ${\omega }_0=0.5$ (left-hand panels) and ${\omega }_0=2.5$ (right-hand panels). The straight white dashed line in the lower panels marks the dispersionless mean-field result.

Figure 8

Intensity plots of the coherent (left-hand panels) and the incoherent (right-hand panels) parts of the $c$- and $f$-electron single-particle spectral functions and of the phonon spectral function. Note the different color coding of the coherent and incoherent parts of $A_k^{c,f}\left(\omega \right)$. We stress that also $A_k^c\left(\omega \right)$ has a finite incoherent part for $\omega >0$ (as magnification would show), which only is noticeable in a small range above $\omega =0$ however, because—among other factors—the renormalized $f$ bandwidth is small. Results are given for ${\omega }_0=0.5$ and $g=0.6$.