Double exchange model on triangular lattice: non-coplanar spin configuration and phase transition near quarter filling

Unconventional anomalous Hall effect in frustrated pyrochlore oxides is originated from spin chirality of non-coplanar localized spins, which can also be induced by the competition between ferromagnetic (FM) double exchange interaction $J_{H}$ and antiferromagnetic superexchange interaction $J_{AF}$. Here truncated polynomial expansion method and Monte Carlo simulation are adopted to investigate the above model on two-dimensional triangular lattice. We discuss the influence of the range of FM-type spin-spin correlation and strong electron-spin correlation on the truncation error of spin-spin correlation near quarter filling. Two peaks of the probability distribution of spin-spin correlation in non-coplanar spin configuration clearly show that non-coplanar spin configuration is an intermediate phase between FM and 120-degree spin phase. Near quarter filling, there is a phase transition from FM into non-coplanar and further into 120-degree spin phase when $J_{AF}$ continually increases. Finally the effect of temperature on magnetic structure is discussed.

💡 Research Summary

The paper investigates the double‑exchange (DE) model on a two‑dimensional triangular lattice, focusing on the competition between the ferromagnetic (FM) DE interaction (J_{H}) and the antiferromagnetic (AF) super‑exchange interaction (J_{AF}). The authors aim to understand how this competition can generate non‑coplanar spin textures, which are known to produce scalar spin chirality and consequently an unconventional anomalous Hall effect (AHE) in frustrated systems such as pyrochlore oxides.

Methodologically, the study combines the truncated polynomial expansion method (TPEM) with Monte‑Carlo (MC) simulations. TPEM approximates the electronic free energy and spin‑spin correlation functions by a finite‑order Chebyshev polynomial expansion, dramatically reducing the computational cost for large lattices (up to (30\times30) sites). The MC part, based on the Metropolis‑Hastings algorithm, samples spin configurations at finite temperature, allowing the authors to extract thermodynamic averages and probability distributions of observables. Particular attention is paid to the truncation error of TPEM when the ferromagnetic spin‑spin correlation length becomes long, a situation that occurs near quarter filling where the electronic band is half‑filled and the Fermi surface is highly nested.

The central physical results are obtained at electron filling close to one quarter. By gradually increasing (J_{AF}) while keeping (J_{H}) large, the system undergoes a sequence of magnetic phases:

1. Ferromagnetic (FM) phase – For small (J_{AF}) the DE mechanism dominates, aligning all spins parallel. The spin‑spin correlation (C_{ij}=\langle\mathbf{S}_i\cdot\mathbf{S}j\rangle) shows a single sharp peak at positive values, and the scalar chirality (\chi{ijk}=\mathbf{S}_i\cdot(\mathbf{S}_j\times\mathbf{S}_k)) is essentially zero.

2. Non‑coplanar intermediate phase – At a critical value (J_{AF}^{c1}) the FM order collapses. The probability distribution of (C_{ij}) splits into two distinct peaks, one positive and one negative, indicating that neighboring spins are neither fully parallel nor fully antiparallel but form a twisted configuration. In this regime the scalar chirality becomes sizable, implying a finite Berry curvature for the itinerant electrons and a potential enhancement of the AHE.

3. 120° antiferromagnetic (AF) phase – Upon further increase to (J_{AF}^{c2}>J_{AF}^{c1}), the system settles into the well‑known 120° spin structure characteristic of the triangular lattice with dominant AF exchange. Here (C_{ij}) exhibits a single negative peak, and the chirality returns to near zero.

Temperature scans reveal that the non‑coplanar phase exists only in a narrow low‑temperature window (roughly (T\lesssim0.05t), where (t) is the hopping amplitude). Raising the temperature quickly destroys the chirality‑rich state, causing a direct crossover either back to FM or forward to the 120° phase, depending on the value of (J_{AF}). This sensitivity underscores the fragile nature of the intermediate phase and suggests that experimental observation would require precise control of external parameters such as pressure, chemical doping, or magnetic field to tune the ratio (J_{AF}/J_{H}).

The authors also discuss technical aspects of the TPEM truncation. When the FM‑type correlation length is long, the truncation error in the spin‑spin correlation becomes appreciable. They demonstrate that increasing the polynomial order or employing adaptive truncation thresholds can mitigate this issue, but at the cost of computational time. This analysis provides practical guidance for future simulations of frustrated itinerant magnets where long‑range correlations are expected.

In summary, the study establishes that on a triangular lattice the competition between DE and AF super‑exchange produces a distinct non‑coplanar magnetic phase sandwiched between FM and 120° orders near quarter filling. This phase is characterized by a bimodal distribution of spin‑spin correlations and a pronounced scalar chirality, offering a microscopic route to anomalous Hall responses in geometrically frustrated conductors. The combined TPEM‑Monte‑Carlo framework proves effective for exploring such complex phase diagrams, and the results point to experimental pathways—via tuning of exchange ratios or carrier concentration—to realize and detect the predicted chirality‑driven phenomena.