Spontaneously formed excitonic density wave with vortex-antivortex lattice in twisted semiconductor bilayers

Abstract

Exciton condensation, characterized by uniform phase coherence across macroscopic length scales, has enabled the discovery of a variety of excitonic states, greatly enriching our understanding of correlated many-body physics. More exotic quantum phenomena are anticipated when the phase factor develops spatial dependence. However, whether excitonic condensates with spatially modulated phase profiles can emerge spontaneously remains an open question. In this work, we uncover novel forms of excitonic density waves featuring nontrivial phase patterns in twisted semiconductor bilayers. Remarkably, we show that kinetic frustration inherent to these systems stabilizes excitonic condensates arranged into a vortex-antivortex lattice. This represents a class of correlated states previously unknown in two-dimensional semiconductors, wherein the phase degrees of freedom of exciton condensates play a defining role. Such states spontaneously break both time-reversal and inversion symmetries, leading to non-reciprocal exciton transport, an effect we term the excitonic diode effect. Furthermore, we compute and identify characteristic impurity-induced states in these unconventional condensates, providing distinct signatures for their experimental detection.

Figures (4)

• Figure 1: (a) A twisted TMD bilayer with electric gating. The band structure for the untwisted material is plot on the right for each layer. For type-II alignment, it leads to a spin-porlarized direct gap semiconductor for each valley, as shown in (b)(c). The low-energy excitons are formed by a spin-down hole in the valence band and a spin-down electron in the conduction band, leading to the spin-triplet pairing. When the conduction band is partially filled, the finite momentum excitons shown in (b) are more energetically favorable than the zero-momentum excitons depicted in (c). (d) The calculated exciton dispersion displays a moat-like band as indicated by (e). The dashed curve in (d) is the fitting plot using Eq.\ref{['eq1']}, and the inset to (d) shows the momentum space probability of the electron wavefunction, corresponding to the state marked by the red dot in (e).
• Figure 2: (a) This tight-binding model generates the single-particle dispersion as shown by the red dashed curve in Fig. 2(a), which well reproduces the dispersion of the lowest moire band. (b) The plot of the exciton wave function corresponding to the states on the lowest band in (a). (c) and (d) respectively show the calculated hopping amplitudes and interactions for the tight-binding model (Eq. \ref{['eq3']}), as a function of the twist angle $\theta$. The inset to (c) shows the ratio $t_2/t_1$, which can be lower than 0.1 for $1.8^{\circ}\lesssim\theta\lesssim2.6^{\circ}$. The inset to (d) displays the on-site repulsion, which dominates over all other interactions.
• Figure 3: The DMRG results on the ground states corresponding to Eq. \ref{['eq5']}, obtained on a finite sized cylinder of $L_x=10$ and $L_y=8$. (a)(c)(e) show the spin structure factors, i.e., $S^{\alpha\beta}(\mathbf{q})=\sum_{\mathbf{R},\mathbf{R}^{\prime}} \langle S^{\alpha}_{\mathbf{R}}S^{\beta}_{\mathbf{R}^{\prime}}\rangle e^{i\mathbf{q}\cdot(\mathbf{R}-\mathbf{R}^{\prime})}/N$ with $i,j=x,y,z$, for $(t_2/t_1, \lambda) = (0.06, 0)$, $(0.12, 0)$, and $(0.12, 1.2)$, respectively. The numerical details and parameters are included in Supplemental Materials. The peaks of $S^{ij}(\mathbf{q})$ reveal three distinct phases, i.e., the $120^{\circ}$ order, the in-plane stripe order and the stripe-z order, respectively. (b) illustrates the $120^{\circ}$ order, which prevails for small $\lambda$ and $t_2/t_1<0.1$. (d) depicts the in-plane stripe order, which stabilizes for small $\lambda$ and $t_2/t_1>0.1$. (f) displays the stripe-z order, which becomes dominant for $\lambda>1$.
• Figure 4: Demonstration of physical properties of the excitonic VAL state. (a) shows the real-space distribution of the condensate wave function $\Psi_{\mathbf{r}}$ for the VAL state. The color represents for its modulus while the arrows for its phase factor. The phase winding generates a VAL on a dual honeycomb lattice as denoted by the red dots. (b) shows the same as (a) but for the in-plane stripe EDW state. (c) The calculated exciton current density against the direction of bias field. (d) The impurity-induced YSR-like states for the excitonic VAL state, where the two generated impurity peaks are labeled by their energies, $\omega_{\alpha}$ and $\omega_{\beta}$. (e) and (f) are the color plot of $\omega_{\alpha}$ as a function of the impurity position for the excitonic VAL and the in-plane stripe EDW state, respectively. The plotted spatial region is marked by the dashed hexagon in (a) and (b).