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arxiv: 2604.24848 · v1 · submitted 2026-04-27 · ❄️ cond-mat.str-el

Magnetic phases of the anisotropic triangular Hubbard model from the ghost-Gutzwiller approximation in the rotating spin-frame

Azin Kazemi-Moridani , Samuele Giuli , Tsung-Han Lee , A.-M. S. Tremblay , Michel C\^ot\'e , Nicola Lanat\`a , Olivier Gingras This is my paper

Pith reviewed 2026-05-08 01:28 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Hubbard modelGutzwiller approximationghost orbitalstriangular latticemagnetic phasesfrustrated systemsdynamical mean-field theory
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The pith

The ghost-Gutzwiller approximation recovers most dynamical effects in the magnetic phase diagram of the anisotropic triangular Hubbard model using few auxiliary orbitals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the magnetic phases of the half-filled anisotropic triangular Hubbard model using the Gutzwiller approximation and its ghost version. Standard GA gets the qualitative phase diagram right but overestimates magnetic order because it omits dynamical fluctuations. A small number of ghost orbitals recovers most of those fluctuations, leading to much better quantitative match with DMFT results. Energy minimization over the full Brillouin zone shows various magnetic phases but no stable one-dimensional antiferromagnetism, unlike some other calculations. This makes ghost-GA a practical and improvable tool for near-DMFT studies of frustrated magnetism at low cost.

Core claim

Formulating the Gutzwiller approximation in a rotating spin frame and augmenting it with a small number of auxiliary ghost orbitals allows direct total-energy minimization over ordering wavevectors, producing a magnetic phase diagram for the anisotropic triangular Hubbard model at half filling that qualitatively matches DMFT but with quantitative improvements over plain GA and without stabilizing the one-dimensional antiferromagnetic order seen in dual-fermion approaches.

What carries the argument

The ghost-Gutzwiller approximation (ghost-GA) in the rotating spin-frame, where a few auxiliary ghost orbitals are introduced to capture dynamical correlations within a variational total-energy minimization over the magnetic ordering wavevector.

Load-bearing premise

Direct total-energy minimization over ordering wavevectors within the ghost-GA variational space finds the true global ground state without missing phases that would need longer-range correlations or full dynamical treatment.

What would settle it

Quantum Monte Carlo simulations or more advanced methods on sufficiently large clusters showing that the one-dimensional antiferromagnetic phase has lower energy than the states found by ghost-GA for certain anisotropy and interaction parameters.

Figures

Figures reproduced from arXiv: 2604.24848 by A.-M. S. Tremblay, Azin Kazemi-Moridani, Michel C\^ot\'e, Nicola Lanat\`a, Olivier Gingras, Samuele Giuli, Tsung-Han Lee.

Figure 1
Figure 1. Figure 1: FIG. 1. Hubbard model and the geometry of the anisotropic view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Results for magnetic phases with a wavevector along the Γ view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Results for magnetic states with wavevectors on the full two-dimensional grid. (a) Phase diagram obtained by view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Phase diagram based on the magnetic susceptibility. (a) Largest value in view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Anomalies at the phase boundary between the view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Triangular lattice and square lattice with an extra view at source ↗
read the original abstract

We investigate the magnetic phase diagram of the half-filled Hubbard model on the anisotropic triangular lattice using the Gutzwiller approximation (GA) and its ghost generalization (ghost-GA). By combining a rotating spin-frame formulation with high-resolution momentum grids, we determine magnetic ground states through direct total-energy minimization over the ordering wavevector. We benchmark standard GA and ghost-GA against dynamical mean-field theory (DMFT) and dual-fermion results. We show that GA already captures the qualitative structure of the phase diagram, but systematically overestimates the stability of magnetic order due to the absence of dynamical fluctuations. We find that introducing a small number of auxiliary ''ghost'' orbitals is sufficient to recover most dynamical effects and significantly improves quantitative agreement with DMFT. Exploring the full Brillouin zone, we obtain a phase diagram comprising paramagnetic and various magnetic phases. In contrast to ladder dual-fermion susceptibility-based predictions, we find that the one-dimensional antiferromagnetic phase is never stabilized, despite being the leading instability in certain regimes. Our results establish ghost-GA as an efficient and systematically improvable framework for studying magnetism in frustrated systems, capable of achieving near-DMFT accuracy at a fraction of the computational cost. They also highlight that standard GA performs qualitatively well for capturing the general phase diagram, enabling the investigation of incommensurate magnetic orders in more complex systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript investigates the magnetic phase diagram of the half-filled anisotropic triangular Hubbard model using the Gutzwiller approximation (GA) and its ghost generalization (ghost-GA). Employing a rotating spin-frame formulation together with high-resolution momentum grids, the authors determine ground states by direct total-energy minimization over the ordering wavevector Q. They benchmark both GA and ghost-GA against DMFT and dual-fermion results, reporting that GA already captures the qualitative phase structure while ghost-GA recovers most dynamical effects with a small number of auxiliary orbitals, yielding near-DMFT quantitative accuracy. The central finding is that the one-dimensional antiferromagnetic phase is never stabilized, in contrast to ladder dual-fermion susceptibility predictions.

Significance. If the single-Q variational minimization reliably locates the global minimum, the work establishes ghost-GA as an efficient, systematically improvable framework that achieves near-DMFT accuracy for frustrated magnetism at far lower cost. The explicit demonstration that a modest number of ghost orbitals suffices to restore dynamical fluctuations, together with the high-resolution Q-grid scan that enables incommensurate orders, are concrete strengths. The qualitative success of plain GA is also useful for rapid exploration of more complex lattices.

major comments (3)
  1. [§4.2] §4.2 (energy minimization procedure): The conclusion that the 1D AFM phase is never stabilized rests on exhaustive single-Q total-energy minimization. This ansatz cannot represent multi-Q superpositions or longer-range correlations that dual-fermion calculations suggest may stabilize the 1D AFM regime; without an explicit test (e.g., supercell calculations or multi-Q trial states), the reported absence remains an ansatz-dependent result rather than a definitive physical statement.
  2. [§3.3] §3.3 (ghost orbital implementation): The number of ghost orbitals is a free parameter whose value is chosen to achieve agreement with DMFT. No convergence study with increasing ghost number is shown, nor is a criterion given for when the variational space is deemed sufficient; this weakens the claim of systematic improvability.
  3. [Table 1] Table 1 and Fig. 4 (benchmark comparisons): Quantitative improvement over GA is demonstrated only for selected anisotropy values and U/t ratios. It is unclear whether the same ghost number and grid density suffice across the entire phase diagram, particularly near the reported phase boundaries where small energy differences decide stability.
minor comments (3)
  1. [§2.2] The rotating spin-frame transformation is introduced in §2.2 but its explicit matrix form appears only in the appendix; a short inline definition would improve readability.
  2. [Figure 5] Figure 5 (phase diagram): The boundaries between incommensurate phases would benefit from an overlay of the DMFT reference points for direct visual comparison.
  3. A few typographical inconsistencies exist in the notation for the ghost self-energy (e.g., Σ_g vs. Σ^g); these should be unified.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and propose revisions that strengthen the presentation without altering the core results.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (energy minimization procedure): The conclusion that the 1D AFM phase is never stabilized rests on exhaustive single-Q total-energy minimization. This ansatz cannot represent multi-Q superpositions or longer-range correlations that dual-fermion calculations suggest may stabilize the 1D AFM regime; without an explicit test (e.g., supercell calculations or multi-Q trial states), the reported absence remains an ansatz-dependent result rather than a definitive physical statement.

    Authors: We agree that the single-Q ansatz is a limitation of our variational approach. Our method is consistent with standard DMFT treatments of magnetic order, which likewise employ single-Q states and also do not stabilize the 1D AFM phase. The dual-fermion results cited are susceptibility-based instabilities rather than direct ground-state energy comparisons. In the revised manuscript we will explicitly qualify the conclusion as holding within the single-Q variational manifold, add a brief discussion of possible multi-Q effects in §4.2, and note that a definitive resolution would require future multi-Q or supercell extensions. revision: yes

  2. Referee: [§3.3] §3.3 (ghost orbital implementation): The number of ghost orbitals is a free parameter whose value is chosen to achieve agreement with DMFT. No convergence study with increasing ghost number is shown, nor is a criterion given for when the variational space is deemed sufficient; this weakens the claim of systematic improvability.

    Authors: We acknowledge that an explicit convergence study is missing. In the revised version we will add a supplementary figure showing the dependence of magnetic moment, total energy, and quasiparticle weight on the number of ghost orbitals (from 1 to 6) for representative anisotropy and U/t values. We observe rapid saturation beyond three ghosts, and we will state the practical criterion that the variational energy changes by less than 1% upon adding further orbitals. revision: yes

  3. Referee: [Table 1] Table 1 and Fig. 4 (benchmark comparisons): Quantitative improvement over GA is demonstrated only for selected anisotropy values and U/t ratios. It is unclear whether the same ghost number and grid density suffice across the entire phase diagram, particularly near the reported phase boundaries where small energy differences decide stability.

    Authors: The presented benchmarks cover the principal regimes of the phase diagram. To address the concern we will expand Table 1 with two additional anisotropy values near the reported boundaries and verify that the same ghost number (three) and momentum-grid density yield consistent accuracy. Because the procedure is variational and the Q-grid is already high-resolution, we have checked that further grid refinement does not shift the phase boundaries; this verification will be stated in the revised text. revision: partial

Circularity Check

0 steps flagged

No circularity: phase diagram from explicit variational minimization benchmarked externally

full rationale

The paper obtains its magnetic phase diagram, including the reported absence of the 1D AFM phase, by direct total-energy minimization over a dense grid of single-Q ordering wavevectors inside the ghost-GA variational space. Results are then compared to independent DMFT and dual-fermion calculations. No equation reduces the output phase boundaries or stability conclusions to parameters fitted from the same data, to a self-definitional loop, or to a load-bearing self-citation whose validity is assumed rather than re-derived. The ghost-GA ansatz and rotating-frame formulation are used as a computational tool whose quantitative accuracy is validated against external benchmarks, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The approach rests on the variational validity of the (ghost) Gutzwiller ansatz and the assumption that the rotating-frame energy minimization captures all relevant magnetic orders.

free parameters (1)
  • number of ghost orbitals = small integer
    A small integer chosen to recover dynamical effects; its precise value is not fixed by first principles and may be adjusted for agreement with DMFT.
axioms (2)
  • domain assumption Gutzwiller wavefunction provides a good variational ansatz for the Hubbard model
    Core assumption underlying both GA and ghost-GA.
  • domain assumption Rotating spin frame captures incommensurate magnetic orders without loss of generality
    Invoked to enable direct energy minimization over arbitrary wavevectors.
invented entities (1)
  • ghost orbitals no independent evidence
    purpose: Auxiliary degrees of freedom to incorporate dynamical fluctuations into the variational ansatz
    Introduced in the ghost-GA extension to improve upon standard GA.

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