Excitonic and magnetic phases in doped WTe$_2$ monolayers: a Hartree-Fock approach

TL;DR

This work employs zero-temperature Hartree-Fock calculations on a folded two-band k·p model with long-range Coulomb interactions to map the competition among broken-symmetry phases in doped 1T'-WTe2 monolayers. It reveals a rich neutrality phase diagram featuring a spin density wave, a quantum spin Hall insulator, a spin spiral, and an intermediate orbital–magnetic OM3 state, with a later transition to a trivial insulator at strong coupling. Upon electron doping, an easy-plane ferromagnet FM2 emerges via a Stoner-like instability and competes with the spin-spiral phase, suggesting a link between magnetic fluctuations and observed superconductivity. The results provide a framework for interpreting transport and spectroscopic data and guide future experiments probing magnetic and topological order near the QSHI gap.

Abstract

Transport and local spectroscopy measurements have revealed that monolayers of tungsten ditelluride ($1T'$-WTe$_2$) display a quantum spin Hall effect and an excitonic gap at neutrality, besides becoming superconducting at low electron concentrations. With the aim of studying the competition among different broken-symmetry phases upon electron doping, we have performed extensive Hartree-Fock calculations as a function of electron density and Coulomb interaction strength. At charge neutrality, we reproduce the emergence of a spin density wave and a spin spiral state surrounding a quantum spin Hall insulator at intermediate interaction strengths. For stronger interactions, the spin spiral is disrupted by a state breaking both inversion and time-reversal symmetries (but not their product) before the system becomes a trivial band insulator. With electron doping the quantum spin Hall insulator evolves into an easy-plane ferromagnet due to a Stoner-like instability of the conduction band. This phase competes energetically with the spin spiral state. We discuss how our results may help to interpret past and future measurements.

Figures (8)

• Figure 1: Hartree-Fock phase diagram of WTe$_2$ monolayers at $T=0$. The sequence of solutions at neutrality as a function of interaction strength (horizontal axis) is: a semimetal, an insulator with a spin density wave (SDW), a quantum spin Hall insulator (QSHI), an insulator with a spin spiral (SS), an insulator with orbital and magnetic order (OM3), and a trivial band insulator (BI). Upon doping the QSHI evolves into easy-plane ferromagnet (FM2), which competes in energy with the SS phase. The phase nomenclature along with the order parameters and the (broken) symmetries are summarized in Tab. \ref{['tab:order_parameters']}.
• Figure 2: Non-interacting electron bands and folding scheme. (a) Non-interacting bands obtained from the $k\cdot p$ model in Eq. \ref{['eq:kp_model']}. The model parameters are: $a=-3$ eV Å$^2$, $b=18$ eV Å$^4$, $m=0.03$ eV$^{-1}$ Å$^{-2}$, $v_x=0.5$ eV Å, $v_y=3$ eV Å and $\delta=-0.45$ eV. Black dashed lines represent the bands in the absence of SOC terms. Colors show the orbital composition of the bands. (b) Fermi contours at neutrality. The lattice constant is $\textrm{a}=3.471$ ($6.221$ Å along the $y$ axis). Note the slight color change in the cusp of the electron pockets indicating a small contribution of the $p_y$-orbital (same color scale as in the band structure). The larger rectangle represents the momentum cutoffs in our calculations. Dashed lines represent zone boundaries of the superlattice.
• Figure 3: Spin density waves (SDWs) vs. spin spirals (SSs). Schematic representation of a singlet SDW (top panel, in orange) polarized along the U$_s$(1) quantization axis within the plane perpendicular to $\boldsymbol{q}_c$, a doublet SDW (middle panel, in purple), and a SS (bottom panel, in blue) rotating within the plane perpendicular to the quantization axis.
• Figure 4: Lowest-energy solutions preserving translational symmetry. The FM2 and OM3 phases compete directly with the QSH and SS insulators.
• Figure 5: Orbital-resolved electron bands of (a) normal state, (b) FM2, and (c) SS solutions for $\epsilon^{-1}=0.2$ and $n_e=16.75\cdot10^{12}$cm$^{-2}$. Color indicates the orbital composition. In panel (c) the color opacity reflects the spectral weight of the single-electron states of the SS solution after band unfolding.