Ab initio Approach to Collective Excitations in Excitonic Insulators

Abstract

An ab initio approach is presented for studying the collective excitations in excitonic insulators, charge/spin density waves and superconductors. We derive the Bethe-Salpeter-Equation for the particle-hole excitations in the quasiparticle representation, from which the collective excited states are solved and the corresponding order parameter fluctuations are computed. This method is demonstrated numerically for the excitonic insulating phases of the biased WSe2-MoSe2 bilayer. It reveals the gapless phase-mode, the subgap Bardasis-Schrieffer modes and the above-gap scattering states. Our work paves the way for quantitative predictions of excited state phenomena from first-principles calculations in electronic systems with spontaneous symmetry breaking.

Figures (10)

• Figure 1: (a) Atomic structure of the biased WSe$_2$-MoSe$_2$ bilayer. (b) Band structure of the normal state (left) and the EI state in the BEC regime (right) where $E_g=E_b/2$. (c) Plot of the order parameter $|\Delta_{vc_1k}|$ ($\,\mathrm{meV}$) in momentum space. (d) Plot of the $U$-matrix elements: $|U_{vvk}|^2$,$|U_{vc_2k}|^2$ and $|U_{vc_1k}|^2$ from left to right.
• Figure 2: (a) Optical conductivity along the armchair (AC) direction of the device in Fig. \ref{['fig1']}(a) in the BEC regime where $E_g=E_b/2$. Red and blue curves are calculated by BSE and IPA respectively with a Lorentzian broadening of $2 \,\mathrm{meV}$. (b) The excitation spectrum labeled by corresponding quantum numbers calculated from the BSE in Eq. \ref{['bse']}. Red and blue labels are for $c_1$- and $c_2$-types, respectively. (c) Same as (b) but calculated from TDA. The arrows between (b) and (c) connect the same labels.
• Figure 3: Plot of $\sum_{vc}|A_{vck}|^2$ and the real and imaginary parts of $\delta \Delta_{vc_1k}$ (meV) for a few $c_1$-type collective excitations on the momentum plane corresponding to those in Fig. \ref{['fig2']}.
• Figure 4: (a) The BCS EI regime of the device in Fig. \ref{['fig1']}(a) where the gap is tuned to $E_g\approx-30\,\mathrm{meV}$. Plotted are the band structures of the normal state (left) and the EI state without (middle) and with (right) updated dielectric screening. Note that the $c_2$-band remains unoccupied in this parameter range. (b) Optical conductivity along the AC direction. Red and blue curves are calculated from BSE and IPA respectively with a Lorentzian broadening of $2 \,\mathrm{meV}$.
• Figure 5: (a) Feynman diagrams for the $e$-$h$ propagator which yields the BSE. The two incoming legs may be viewed as the creation of a boson by $a^\dagger_S$, so that this diagram is actually the self energy of the bosonic propagator. Black solid lines are electronic propagators, which are corrected by the mean field if the system is in the broken symmetry phase. We use the blue wavy line to represent the interband interaction $V_{vck,v'c'k'}$ that includes both the direct and exchange interactions. The blue dashed line represents the same interaction, but explicitly decomposed in the $e$-$h$ and anti-$e$-$h$ channels in the diagonalized band basis. (b) Representation of collective modes in terms of bosonic propagators. The first line is the bosonic propagator expressed in terms of the original interband interaction $V_{vck,v'c'k'}$. The collective modes are contained in the vertex PhysRevB.26.4883BaSh. The second line are the equivalent diagrams after the Hubbard-Stratonovich transformation, where blue dashed lines are the propagators of the bosonic field that mediates the interaction $V_{vck,v'c'k'}$Sun.2020_SCSZY2020.