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arXiv:2508.12421

Antiferromagnetic Long-Range Order in a Lattice Fermion Model

Yukimi Goto, Tohru Koma

Published Aug 17, 2025• 0 citations
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Abstract

We study a lattice fermion model with antiferromagnetic interactions on the three-dimensional cubic lattice. The hopping term of the Hamiltonian has a Weyl-type dispersion. We prove that the model has reflection positivity. Moreover, by relying on the property, we prove the existence of the antiferromagnetic long-range order at low temperatures in a strong coupling regime.

DOI
10.48550/arxiv.2508.12421

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We study a lattice fermion model with antiferromagnetic interactions on the three-dimensional cubic lattice. The hopping term of the Hamiltonian has a Weyl-type dispersion. We prove that the model has reflection positivity. Moreover, by relying on the property, we prove the existence of the antiferromagnetic long-range order at low temperatures in a strong coupling regime.
The method of Kennedy, Lieb, and Shastry is applied to the antiferromagnetic XXZ\mathrm{XXZ}XXZ model to examine the existence of long-range order (LRO). On the 3D simple-cubic lattice LRO is proved to exist for any $S\ensuremath{\geqq}\frac{1}{2}$ and $\ensuremath{\Delta}\ensuremath{\geqq}0$ at low temperature. On the 2D square lattice LRO exists in the ground state for any $\ensuremath{\Delta}\ensuremath{\leqq}0$ if $S\ensuremath{\geqq}1$ and for $0\ensuremath{\leqq}\ensuremath{\Delta}<0.13$ and $\ensuremath{\Delta}=0$ if S=12S=\frac{1}{2}S=21​. The existence of LRO in the ground state of the S=12XYS=\frac{1}{2} \mathrm{XY}S=21​XY model ($\ensuremath{\Delta}=0$), therefore, has been confirmed, in agreement with the conjecture by Nishimori et al. The ground state of the S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg model in two dimensions still remains to be clarified.
The existence of long-range order is proved under certain conditions for the antiferromagnetic XYZ model on the simple cubic or the square lattice. In particular, the spin-1/2 XXZ model on the square lattice is shown to have ground-state long-range order if the exchange anisotropy Δ satisfies 0≦ Δ <0.20 or Δ >1.72, which improves the result of Kubo and Kishi. The existence of long-range order of the z -component of the spin operator is proved for the XXZ model with XY -like anisotropy (0≦ Δ ≦1) under certain conditions. A similar result is shown to hold for the long-range order in the x -direction for the Ising-like model ( Δ ≧1). The XXZ model on the two-dimensional hexagonal lattice is proved to have finite ground-state long-range order for any value of Δ (≧0) if S ≧1 and for Δ >2.55 when S =1/2.
The existence of long range order (LRO) is examined in the antiferromagnetic Heisenberg model on the square or the simple cubic lattice with the next nearest neighbor interactions which are taken to be λ times the nearest neighbor ones (λ≧0). It is rigorously shown that Néel LRO exists at a low temperature on the simple cubic lattice and in the ground state on the square lattice for λ smaller than a critical value λ c . On the square lattice λ c increases with S starting from 0.072 for S =1 and approaches 0.5 in the limit S →∞. On the simple cubic lattice λ c =0.002 for S =1/2 and approaches 0.25 in the limit S →∞.
We study Fermi gases in a three-dimensional cubic optical lattice with five fermions per site, i.e., the sss band is completely filled and the ppp band with threefold degeneracy is half filled. We show that for repulsive interaction between fermions, the system will exhibit spin-3/2 antiferromagnetic order at low temperature. This conclusion is obtained both in strong interaction regime by strong coupling expansion and in weak interaction regime by the Hartree--Fock mean-field theory with analysis of the Fermi surface nesting. We also show that in the strongly correlated regime the N\'eel temperature for ppp band antiferromagnetism is 2 to 3 orders of magnitudes higher than that of sss band, which is much more promising to be attained in cold atom experiments.
We give a rigorous proof of the existence of long-range antiferromagnetic order in the ground state of several two-dimensional spin-1/2 Heisenberg systems.Three types of systems are considered.The first type consists of an even number, N , of coupled square lattices with antiferromagnetic nearest-neighbor Heisenberg interactions.In this case, we prove the existence of long-range order (LRO) in the ground state for an interplane to inplane coupling ratio r in the range 0.16 ≤ r ≤ 2.1 for N ≥ 4. Further, we prove that the antiferromagnetic bilayer with ferromagnetic next-nearest-neighbor (nnn) interplane couplings possesses LRO in the ground state for 0.21 ≤ r < 1/4, where r is the absolute value of the ratio of the ferromagnetic and antiferromagnetic couplings.The final example consists of two antiferromagnetic spin-1/2 square lattices that are coupled via an antiferromagnetic nnn interplane coupling r.For r = 1, this system is effectively a spin-1 square lattice and has an ordered ground state.We show that LRO exists in the region 0.85 ≤ r ≤ 1. * ) Note that E r q in the notation of KLS corresponds to 2 + r -E q in our notation.
We look at the general SU(2) invariant spin-1 Heisenberg model. This family includes the well-known Heisenberg ferromagnet and antiferromagnet as well as the interesting nematic (biquadratic) and the largely mysterious staggered-nematic interaction. Long range order is proved using the method of reflection positivity and infrared bounds on a purely nematic interaction. This is achieved through the use of a type of matrix representation of the interaction making clear several identities that would not otherwise be noticed. Using the reflection positivity of the antiferromagnetic interaction one can then show that the result is maintained if we also include an antiferromagnetic interaction that is sufficiently small.