Beyond Haerter-Shastry: kinetic magnetism, multimer stripes and metallicity in the doped triangular Hubbard model

The Haerter-Shastry baseline: antiferromagnetism without magnetic interactions

Haerter and Shastry established that a single hole moving in the infinite-U triangular Hubbard model produces weak metallic antiferromagnetism of kinetic origin, with a three-sublattice 120° spin texture resembling the triangular-lattice Heisenberg antiferromagnet [7]. Their effective-spin reduction and exact diagonalization on clusters up to 27 sites yielded a finite order-parameter intercept, suggesting long-range order from hole motion alone [7]. This was a single-hole result: the mechanism was demonstrated at the level of one hole in a half-filled background, not at finite hole density [7]. A later study showed that tuning a staggered flux can switch the infinite-U ground state abruptly between Nagaoka-type ferromagnetism and Haerter-Shastry-type antiferromagnetism, with both states metallic and of kinetic origin, indicating the mechanism is not tied to a single lattice realization [2].

How far does kinetic 120° order survive hole doping?

The anchor paper uses DMRG on the infinite-U triangular Hubbard model to estimate the stability of the kinetically induced 120° antiferromagnet at finite hole density n_h [1]. The spin structure factor S(q) peaks at the K points at low n_h and the extrapolated order parameter remains non-zero, consistent with long-range antiferromagnetic order up to n_h ≈ 0.25–0.3 [1]. Beyond that window, weight at K diminishes and spreads, and the six-fold rotational symmetry is broken, marking an intermediate phase [1]. At n_h = 1/4, real-space spin correlations show a robust 'diamond-type' pattern throughout the cluster, while at n_h = 1/3 the diamond correlations persist along vertical columns but weaken between columns, forming multimer stripes [1]. The intermediate phase is thus characterized by local multi-spin correlations that organize into stripe-like patterns before melting into a featureless paramagnet at higher doping [1].

Metallicity persists across the phase diagram

The paper reports evidence of gapless charge excitations (metallicity) throughout the finite-density phase diagram [1]. The quasiparticle residue Z increases monotonically with hole density, but the DMRG data cannot establish whether Z vanishes at some small non-zero n_h; if it does not, the ground state would be a heavy Fermi liquid that may cross over to a non-Fermi liquid at small finite temperature [1]. The authors note that recent work has found numerical evidence for T-linear resistivity on the infinite-U square lattice at low hole density, and expect the essence of that phenomenon to carry over to the triangular case, though a systematic comparison is left to future work [1]. The metallic weight α, extracted from the linear fit of the charge structure factor near q_x → 0, tracks a crossover from a hole-dominated to an electron-dominated regime as n_h increases [1].

Competition and collaboration between kinetic magnetism and superexchange

At large but finite U/t, superexchange J = 4t²/U competes with the kinetic mechanism [1]. The paper identifies a crossover scale U_c/t ≈ 50 for n_h = 0.11, where spin ordering tendencies are maximized: lowering U from the infinite-U limit strengthens spin moments by suppressing non-Heisenberg terms and double occupancy, while increasing U suppresses double occupancy further, so the two trends produce a maximum in antiferromagnetic correlations at intermediate U/t [1]. This crossover is visible in nearest-neighbor and next-nearest-neighbor spin correlations, in S(K)/N, and in the inflection of local charge fluctuations [1]. The authors emphasize that on the triangular lattice, unlike the square lattice, both superexchange and kinetic frustration stabilize qualitatively the same 120° antiferromagnetic ground state, so the two effects can collaborate rather than compete [1]. They also caution that a recent proposal attributing the doping dependence of the Curie-Weiss temperature in a triangular moiré material to additive superexchange and Haerter-Shastry contributions is subtler than that additive form suggests [1].

Where the conclusions stop: infinite U, finite DMRG, and real materials

The anchor paper's conclusions rest on the infinite-U approximation and finite-size DMRG on cylinders, with the intermediate phase's precise location and extent sensitive to finite-size effects and the choice of cluster [1]. The extrapolation of the order parameter assumes a homogeneous, translationally invariant ground state and uses a fitting form that neglects higher-order corrections in inverse system size, so it is approximate over the sizes studied [1]. The crossover scale U_c/t ≈ 50 is described as a broad feature and a ballpark estimate rather than a sharp transition [1]. Extension to finite U/t, to long-range or screened Coulomb interactions common in moiré materials, and to a thermal phase diagram remains open [1]. A separate DMRG and exact-diagonalization study of the quarter-filled attractive Hubbard model illustrates how finite-size effects and interaction-driven crossovers can be diagnosed with complementary methods, a methodological caution that applies to the present infinite-U results as well [6]. The precursor work on magnonic Cooper pairs in the infinite-U triangular lattice [3], the double-exchange ferromagnetism study in a hexagonal lattice [4], and the charge-anyon superconductor proposal [5] indicate that the infinite-U triangular Hubbard model is being explored for other orders, so the kinetic magnetism phase diagram reported here is one part of a broader ongoing investigation.

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This research page is built on 7 peer-reviewed studies — published from 2005 to 2026, 5 from 2024 or later — selected as the most relevant from 7 studies that passed quality screening, drawn from 70 papers retrieved from a database of over 500 million.

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