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arXiv:2604.24848

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 Côté, Nicola Lanatà, Olivier Gingras

Published Apr 27, 2026• 0 citations

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.

DOI: 10.48550/arxiv.2604.24848

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We study the ground-state magnetism of the half-filled Hubbard model on the anisotropic triangular lattice, where two out of three bonds have hopping

t

and the third one has ${t}^{\ensuremath{'}}$ in a unit triangle. Working in a spin-rotating frame and using the density matrix renormalization group method as an impurity solver, we provide a proper description of incommensurate magnetizations at zero temperature in the framework of the dynamical mean-field theory (DMFT). It is shown that the incommensurate spiral magnetic order for ${t}^{\ensuremath{'}}/t\ensuremath{\gtrsim}0.7$ survives the dynamical fluctuations of itinerant electrons in the Hubbard interaction range from the strong-coupling (localized-spin) limit down to the insulator-to-metal transition. We also find that when the anisotropy parameter ${t}^{\ensuremath{'}}/t$ increases from the N\'eel-to-spiral transition, the magnitude of the magnetic moment exhibits a maximum at the isotropic triangular lattice point ${t}^{\ensuremath{'}}/t=1$ and then rapidly decreases in the range of larger ${t}^{\ensuremath{'}}/t$. This work gives a solid foundation for further extension of the study including nonlocal correlation effects neglected at the standard DMFT level.

We use a spin-rotational invariant Gutzwiller energy functional to compute random-phase-approximation-like (RPA) fluctuations on top of the Gutzwiller approximation (GA). The method can be viewed as an extension of the previously developed

\mathrm{GA}+\mathrm{RPA}

approach for the charge sector [G. Seibold and J. Lorenzana, Phys. Rev. Lett. 86, 2605 (2001)] with respect to the inclusion of the magnetic excitations. Unlike the charge case, no assumptions about the time evolution of the double occupancy are needed in this case. Interestingly, in a spin-rotational invariant system, we find the correct degeneracy between triplet excitations, showing the consistency of both computations. Since no restrictions are imposed on the symmetry of the underlying saddle-point solution, our approach is suitable for the evaluation of the magnetic susceptibility and dynamical structure factor in strongly correlated inhomogeneous systems. We present a detailed study of the quality of our approach by comparing with exact diagonalization results and show its much higher accuracy compared to the conventional $\mathrm{Hartree}\ensuremath{-}\mathrm{Fock}+\mathrm{RPA}$ theory. In infinite dimensions, where the GA becomes exact for the Gutzwiller variational energy, we evaluate ferromagnetic and antiferromagnetic instabilities from the transverse magnetic susceptibility. The resulting phase diagram is in complete agreement with previous variational computations.

We calculate the magnetic phase diagram of the half-filled $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U$ Hubbard model as a function of ${t}^{\ensuremath{'}}$ and

U

, within the Gutzwiller approximation plus random-phase approximation. As

U

increases, the system first crosses over to one of a wide variety of incommensurate phases, whose origin is clarified in terms of double nesting. We evaluate the stability regime of the incommensurate phases by allowing for symmetry breaking with regard to the formation of spin spirals, and find a crossover to commensurate phases as

\mathrm{GA}+\mathrm{RPA}

increases and a full gap opens. The results are compared with a variety of other recent calculations, and in general good agreement is found. For parameters appropriate to the cuprates, double occupancy should be only mildly suppressed in the absence of magnetic order, inconsistent with a strong-coupling scenario.

Using a cluster extension of the dynamical mean-field theory (CDMFT) we map out the magnetic phase diagram of the anisotropic square lattice Hubbard model with nearest-neighbor intrachain

t

and interchain ${t}_{\ensuremath{\perp}}$ hopping amplitudes at half filling. A fixed value of the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}=\ensuremath{-}{t}_{\ensuremath{\perp}}/2$ removes the nesting property of the Fermi surface and stabilizes a paramagnetic metal phase in the weak-coupling regime. In the isotropic and moderately anisotropic regions, a growing spin entropy in the metal phase is quenched out at a critical interaction strength by the onset of long-range antiferromagnetic (AF) order of preformed local moments. It gives rise to a first-order metal-insulator transition consistent with the Mott-Heisenberg picture. In contrast, a strongly anisotropic regime ${t}_{\ensuremath{\perp}}/t\ensuremath{\lesssim}0.3$ displays a quantum critical behavior related to the continuous transition between an AF metal phase and the AF insulator. Hence, within the present framework of CDMFT, the opening of the charge gap is magnetically driven as advocated in the Slater picture. We also discuss how the lattice-anisotropy-induced evolution of the electronic structure on a metallic side of the phase diagram is tied to the emergence of quantum criticality.

We investigate the Hubbard model on the triangular lattice at half filling using the dynamical cluster approximation (DCA) and dual fermion (DF) methods in combination with continuous-time quantum Monte Carlo (CT QMC) and semiclassical approximation (SCA) methods. We study the one-particle properties and nearest-neighbor spin correlations using the DCA method. We calculate the spectral functions using the CT QMC and SCA methods. The spectral function in the SCA and obtained by analytic continuation of the Pad\'e approximation in CT QMC are in good agreement. We determine the metal-insulator transition and the hysteresis associated with a first-order transition in the double-occupancy and nearest-neighbor spin-correlation functions as functions of temperature. As a further check, we employ the DF method and discuss the advantages and limitation of the dynamical mean-field theory, DCA, and recently developed DF methods by comparing the Green's functions. We find an enhancement of antiferromagnetic correlations and provide evidence for magnetically ordered phases by calculating the spin susceptibility.

In this Rapid Communication, we present a comprehensive study of the Hubbard model on the isotropic triangular lattice by using the recently developed ladder dual-fermion approach. This method is a nonlocal extension of the dynamical mean-field theory and is free of finite-size effect. In addition to confirming the much-discussed phase diagram at half-filling, our work provides insights into both hole- and electron-doped regimes and, in particular, the finite-temperature phase diagrams. We find the triangular system to be short-range correlated with an associated magnetic phase diagram, which is asymmetric with respect to hole and electron doping. In contrast to the unfrustrated lattice, it can adiabatically be cooled by increasing the interactions. Strikingly, at the electron-doped side, the entropy displays a maximum. This latter example, as well as other results of our work, may provide insights for a variety of correlated triangular materials, such as the water-intercalated sodium-doped cobaltates.