2D coherent spectroscopy signatures of exciton condensation in Ta$_2$NiSe$_5$

Abstract

We show that the nonlinear optical response probed by two-dimensional coherent spectroscopy (2DCS) can discriminate between excitonic and lattice driven order. In the excitonic regime of a realistic model of Ta$_2$NiSe$_5$, the third order 2DCS signals are strongly enhanced by the condensate's amplitude and phase modes, with negligible contributions from single-particle excitations. In the linear optical response, in contrast, single-particle and collective-mode contributions overlap. With increasing electron-phonon coupling, the amplitude mode contribution to 2DCS initially remains robust, but then drops rapidly and remains small in the phonon-dominated regime -- even in systems with large order parameter. 2DCS also aids the detection of the massive relative phase mode, which is analogous to the Leggett mode in superconductors. Our analysis, based on the time-dependent Hartree-Fock approach, demonstrates that 2DCS can track the emergence of the symmetry-broken state and the crossover from Coulomb-driven to phonon-driven order.

Figures (9)

• Figure 1: (a) Unit cell of TNS with Ta (blue) and Ni (red) atoms. Black arrows indicate the B$_{2g}$ distortion. (b) Band-structure in the EI phase. The blue (red) shading encodes the Ta (Ni) character of the bands. (c) Optical conductivity Re$\sigma$ for a weak laser excitation to the ordered state, obtained with a dynamic and static Fock term (and damping factor $\eta=0.003$ eV in the Fourier transform). The vertical dashed black, solid gray and solid violet lines indicate the gap energies marked by the corresponding arrows in (b). The black arrow indicates the relative phase mode. The results in (b,c) are for $V=0.784$ eV, $g=0.0005$ eV and $\beta=100$ eV$^{-1}$. (d) Illustration of the amplitude, phase and relative phase mode, with the black arrows representing Ni-Ta hoppings. (e) Evolution of the mode energies with $g$ for fixed single-particle gap ($\beta=100$ eV$^{-1}$, $V=0.784$ eV for $g\le 0.0025$ eV, $V=0.781$ eV for $g=0.005$ eV, $0.7725$ eV for $g=0.01$ eV). (f,g) Pump excitations (red) in the two-pulse (f) and three-pulse (g) setup combined with the optical current measurement (blue line).
• Figure 2: (a) Order parameter $\Delta$ as a function of inverse temperature for $V=0.784$ eV and (b) as a function of the inter-site interaction $V$ for $\beta = 100$ eV$^{-1}$ and various el-ph couplings $g$. (c) Conductivity as a function of inverse temperature $\beta$ for $V=0.784$ eV and (d) as a function of $V$ for $\beta = 100$ eV$^{-1}$ ($g=0.0005$ eV). A damping factor $\eta = 0.003$ eV is used in the Fourier transforms. See SM Fig. S4 for the $g=0.0025$ eV results and the permittivity as a function of $\beta$.
• Figure 3: 2DCS signals obtained using the two-pulse (a-c) and three-pulse (g-i) setups. (a,g) Results of the dynamic simulation with pump-probe (PP), non-rephasing (NR) and rephasing (R) contributions highlighted. (b,h) Analogous spectra for the static simulations (with suppressed amplitude mode contribution), and (c,i) with intensities rescaled by a factor of 200. (d,j) Cuts of the 2DCS map along the antidiagonal ($-\omega_\tau=\omega_t$), (e,k) along the y-axis ($\omega_\tau=0$) and (f,l) along the diagonal ($\omega_\tau=\omega_t$). The Keldysh diagrams in panels (I)-(IV) illustrate possible light-matter interaction pathways corresponding to the signals at the indicated $(\omega_\tau,\omega_t)$, see text for details. Vertical lines indicate different combinations of $\omega_{|\Delta|}$ and $\omega_{\psi}$ (see text). Parameters: $\beta=100$ eV$^{-1}$, $g=0.0005$ eV and $V=0.784$ eV. In the three-pulse measurement, the waiting time is $T=6.6$ fs.
• Figure 4: Intensity of the 2DCS signals (left axis) and 8th power of the order parameter $\Delta$ (right axis). NR signals are measured at $(\omega_\tau,\omega_t)=(\omega_{|\Delta|},\omega_{|\Delta|})$ and PP signals are measured at $(0,\omega_{|\Delta|})$ using the two-pulse setup. R signals are measured at $(-\omega_{|\Delta|},\omega_{|\Delta|})$ using the three-pulse setup. (a) Signal intensity for different $V$ with $g=0$, $\beta = 100$ eV$^{-1}$. (b) Signal intensity for different $\beta$ with $g=0.0005$ eV, $V=0.784$ eV.
• Figure 5: Dependence of the amplitude mode dominated 2DCS signals (left axis) and of the order parameter $\Delta$ (right axis) on the phonon coupling $g$ for $\beta=100$ eV$^{-1}$ and for $V=0.784$ eV (a) and $V=0.81$ eV (b). The NR and PP intensities are obtained using the two-pulse protocol, while the R intensity is obtained using the three-pulse protocol.