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arXiv:2501.18141 · Letters in Mathematical Physics

On the mean-field antiferromagnetic gap for the half-filled 2D Hubbard model at zero temperature

Edwin Langmann A, Jonatan Lenells

Published Apr 20, 2026• 0 citations
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Abstract

Abstract We consider the antiferromagnetic gap for the half-filled two-dimensional (2D) Hubbard model (on a square lattice) at zero temperature in Hartree–Fock theory. It was conjectured by Hirsch in 1985 that this gap, $$\Delta $$ Δ , vanishes like $$\exp (-2\pi \sqrt{t/U})$$ exp ( - 2 π t / U ) in the weak-coupling limit $$U/t\downarrow 0$$ U / t ↓ 0 ( $$U>0$$ U > 0 and $$t>0$$ t > 0 are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.

DOI
10.1007/s11005-026-02085-5

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Abstract We consider the antiferromagnetic gap for the half-filled two-dimensional (2D) Hubbard model (on a square lattice) at zero temperature in Hartree–Fock theory. It was conjectured by Hirsch in 1985 that this gap, Δ\Delta Δ Δ , vanishes like exp⁡(−2πt/U)\exp (-2\pi \sqrt{t/U})exp(−2πt/U​) exp ( - 2 π t / U ) in the weak-coupling limit U/t↓0U/t\downarrow 0U/t↓0 U / t ↓ 0 ( U>0U>0U>0 U > 0 and t>0t>0t>0 t > 0 are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.
We have studied the two-dimensional Hubbard model on a square lattice with nearest-neighbor hopping. We first discuss the properties of the model within the mean-field approximation: Because of the form of the band structure, some peculiar features are found. We then discuss the simulation algorithm used and compare simulation results with exact results for 6-site chains to test the reliability of the approach. We present results for thermodynamic properties and correlation functions for lattices up to 8\ifmmode\times\else\texttimes\fi{}8 in spatial size. The system is found to be an antiferromagnetic insulator for all values of the coupling constant at zero temperature in the half-filled-band case, but the long-range order is much smaller than predicted by mean-field theory. We perform a finite-size-scaling analysis to determine the character of the transition at zero coupling. For non-half-filled-band cases, our results suggest that the system is always paramagnetic, in contradiction with Hartree-Fock predictions. The system does not show tendency to ferromagnetism nor triplet superconductivity in the parameter range studied. We also discuss some properties of the attractive Hubbard model in the half-filled-band case.
Abstract We study Hartree–Fock theory at half-filling for the 3D anisotropic Hubbard model on a cubic lattice with hopping parameter t in the x - and y -directions and a possibly different hopping parameter tzt_ztz​ t z in the z -direction; this model interpolates between the 2D and 3D Hubbard models corresponding to the limiting cases tz=0t_z=0tz​=0 t z = 0 and tz=tt_z=ttz​=t t z = t , respectively. We first derive all-order asymptotic expansions for the density of states. Using these expansions and units such that t=1t=1t=1 t = 1 , we analyze how the Néel temperature and the antiferromagnetic mean field depend on the coupling parameter, U , and on the hopping parameter tzt_ztz​ t z . We derive asymptotic formulas valid in the weak coupling regime, and we study in particular the transition from the three-dimensional to the two-dimensional model as tz→0t_z \rightarrow 0tz​→0 t z → 0 . It is found that the asymptotic formulas are qualitatively different for tz=0t_z = 0tz​=0 t z = 0 (the two-dimensional case) and tz>0t_z > 0tz​>0 t z > 0 (the case of nonzero hopping in the z -direction). Our results show that certain universality features of the three-dimensional Hubbard model are lost in the limit tz→0t_z \rightarrow 0tz​→0 t z → 0 in which the three-dimensional model reduces to the two-dimensional model.
The magnetic states occurring in the two-dimensional Hubbard model in the vicinity of half-filling of the conduction band are studied in the case of weak correlations, using Hartree-Fock theory. It is shown that at zero temperature the commensurate antiferromagnetic state is unstable against domain-wall formation for arbitrarily small deviations from half-filling. The resulting incommensurate antiferromagnet remains initially insulating. The predictions of the model for the polarization and modulation wave vector of the incommensurate antiferromagnet are discussed.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH.
Starting from the Hubbard model in the weak-coupling limit, we derive a spin-fermion model where the collective spin excitations are described by a nonlinear sigma model. This result is used to compute the fermion spectral function $A(\mathbf{k},\ensuremath{\omega})$ in the low-temperature regime where the antiferromagnetic (AF) coherence length is exponentially large (``renormalized classical'' regime). At the Fermi level, $A({\mathbf{k}}_{F},\ensuremath{\omega})$ exhibits two peaks around $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Delta}}_{0}$ (with ${\ensuremath{\Delta}}_{0}$ the mean-field gap), which are precursors of the zero-temperature AF bands, separated by a pseudogap.
Abstract We consider the extended Hubbard model on a two-dimensional square lattice at half-filling. The model is investigated using the strong coupling diagram technique. We sum infinite series of ladder diagrams allowing for full-scale charge and spin fluctuations and the actual short-range antiferromagnetic order for nonzero temperatures. In agreement with earlier results, we find the first-order phase transition in the charge subsystem occurring at v = v c ≳ U/4 with v and U the intersite and on-site Coulomb repulsion constants. The transition reveals itself in an abrupt sign change of a sharp maximum in the zero-frequency charge susceptibility at the corner of the Brillouin. States arising at the transition have alternating deviations of electron occupations from the mean value on neighboring sites. Due to fluctuations, these alternating occupation deviations have short-range order. For the considered parameters, such behavior is found for U ≲ 5t with t the hopping constant. For the insulating case U ≳ 6t, in which the transition is not observed, we find a continuous growth of the Mott gap with v. The evolution of the electron density of states with increasing v is also considered.