Itinerant magnetism in the triangular lattice Hubbard model at half-doping: Application to twisted transition-metal dichalcogenides

TL;DR

The study investigates itinerant magnetism in the triangular-lattice Hubbard model at half-doping using unrestricted Hartree-Fock, DMRG, and infinite PEPS. It reveals a sequence of large-scale spin-density-wave states whose ordering vectors evolve from $M$-point nestings toward $\Gamma$ with increasing $U$, including 3Q-I (spin-tetrahedral, insulating) and several competing higher-order textures that become metallic, culminating in ferromagnetism at strong coupling; at 1/2 hole-doping, magnetism is suppressed and absent within the explored parameter range. The results are cross-validated across methods, with DMRG and PEPS generally supporting the HF-identified spiral and ferromagnetic tendencies, albeit with finite-size and finite-$D$ Caveats. The findings have relevance for twisted TMD moiré systems, where tuning interactions and long-range Coulomb effects can realize itinerant magnetism without insulating phases, aligning with recent experiments and offering a framework to interpret complex magnetic textures in frustrated itinerant systems.

Abstract

We use unrestricted Hartree-Fock, density matrix renormalization group, and variational projected entangled pair state calculations to investigate the ground state phase diagram of the triangular lattice Hubbard model at "half doping" relative to single occupancy, i.e. at a filling of $(1\pm \frac{1}{2})$ electrons per site. The electron-doped case has a nested Fermi surface in the non-interacting limit, and hence a weak-coupling instability towards density-wave orders whose wavevectors are determined by Fermi surface nesting conditions. We find that at moderate to strong interaction strengths other spatially-modulated orders arise, with wavevectors distinct from the nesting vectors. In particular, we identify a series closely-competing itinerant long-wavelength magnetically ordered states, yielding to uniform ferromagnetic order at the largest interaction strengths. For half-hole doping and a similar range of interaction strengths, our data indicate that magnetic orders are most likely absent.

Figures (7)

• Figure 1: Ground-state phase diagram of the TLHM as a function of $U$ at 1/2 electron doping. The magnetic orders of the different phases are illustrated in Table. \ref{['tbl:HForders']}. The abbreviation $n$Q stands for spin-density waves with $n$ momentum component. The I stands for noncoplanar, and II stands for collinear. MQS:multi-Q noncoplanar stripe. FP: full spin polarization. (a) Phase diagram obtained from unrestricted Hartree-Fock simulations. (b) Phase diagram inferred from tensor network calculations. The obtained large-scale SDWs include collinear orders akin to 6Q-II as well as (coplanar) spiral order.
• Figure 2: (a) Density of states for the model at $U=0$. A Van Hove singularity is located at single-particle energy $E=2$. (b) Fermi surface (blue dashed line) in the Brillouin zone for $\overline{\langle n_i\rangle}=\frac{3}{2}$. The Fermi energy is $E_F=2$. There are three nesting vectors $\mathbf{Q}_{M_i} = -\mathbf{Q}_{M_i}$ modulo reciprocal lattice vectors.
• Figure 3: Tensor network results for 1/2 electron doping. (a,b) DMRG data. (a) Polarization $\overline {\langle S_{z}(i) \rangle}$ (polarization per site) as a function of $U$ for different geometries: four-leg zigzag cylinder with length 20, 36-site triangular flake, and 48-site hexagonal flake. The $\overline {\langle S_{z}(i) \rangle}$ is evaluated for the highest-weight ground state averaged without edge sites. (b) Static structural factor $S(\bm{k})=\sum_{m}e^{i \bm{k}\cdot (\bm{r}_m-\bm{r}_n)}[\langle \bm{S}(m) \cdot \bm{S}(n)\rangle-\langle \bm{S}(m)\rangle \cdot \langle \bm{S}(n)\rangle]$ averaged for the central sites $m$ of the cylinder geometry, $U=8$. The $\bm{k}$ can be projected to two orthogonal components $k_{\perp}$ and $k_{\parallel}$. The four possible values of $k_{\parallel}$ correspond to four colored cuts in the Brillouin zone, shown in the insets. (The red dashed lines are along the same cut through M1 and M2.) (c) PEPS variational energy data for different unit cells. The dashed lines denote HF energies for comparison, with full spin polarization at $U=7,9$. Different PEPS unit cells $m \times n$ are used, which sets the maximal possible unit cells of the obtained states. The spiral-like states obtained for $2 \times 8$ PEPS unit cell at $U=9$ have smaller actual unit cells, i.e., $1 \times 8$. The $2 \times 2$ and $3 \times 6$ states for $U=4,7$ are collinear. We perform extrapolation to the $D^{-1}=0$ limit based on $E(D)=E_0+ c/D^\alpha$ ansatz suppm.
• Figure S1: Finite triangular lattices studied by DMRG. (a) 36-site triangular flake. (b) 48-site hexagonal flake. (c) Four-leg zigzag cylinder with length 20 (periodic boundary condition imposed for the tangential direction).
• Figure S2: DMRG data example for ferromagnetization for the 48-site hexagonal flake. The plot shows the lowest energy in a given sector of the total spin $S_{\mathrm{tot}}$ at $U=10$ for various SU(2)-invariant bond dimensions. The largest energy errors are estimated to be in the order $10^{-3}$. The triangular and cylindrical geometry has one and two orders of smaller errors.