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arXiv:2506.16503

Mixed phases in a Fermi--Hubbard model describing altermagnetism

Edwin Langmann A, Jonatan Lenells

Published Jun 19, 2025• 0 citations
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Abstract

We study an extension of the 2D Fermi--Hubbard model, which was recently introduced in [Das et al., Phys. Rev. Lett. 132, 263402 (2024)] and shown to describe altermagnetism that can be studied in cold atom systems. Using an updated Hartree--Fock method that can detect instabilities towards phase separation, we show that the model is in a mixed phase in large parts of the parameter regime at half-filling. We argue that the occurrence of a mixed phase is an indication of exotic physics which, in this model, occurs in parameter regimes accessible in cold atom experiments.

DOI
10.48550/arxiv.2506.16503

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We study an extension of the 2D Fermi--Hubbard model, which was recently introduced in [Das et al., Phys. Rev. Lett. 132, 263402 (2024)] and shown to describe altermagnetism that can be studied in cold atom systems. Using an updated Hartree--Fock method that can detect instabilities towards phase separation, we show that the model is in a mixed phase in large parts of the parameter regime at half-filling. We argue that the occurrence of a mixed phase is an indication of exotic physics which, in this model, occurs in parameter regimes accessible in cold atom experiments.
We show that the standard textbook description of (restricted) Hartree-Fock theory for (Fermi) Hubbard-like models is in need of an update, and we present such an update allowing us to correct basic and established results in the condensed matter physics literature that are qualitatively wrong. Our update amounts to adding a test that reliably checks the thermodynamic stability of solutions of Hartree-Fock equations. This stability test makes it possible to detect, by simple means and with certainty, regions in phase space where the model exhibits mixed phases where two conventional phases coexist and translation invariance is broken in complicated ways; in such a mixed phase, unconventional physics is to be expected. Our results show that mixed phases are ubiquitous in Hubbard-like models in arbitrary dimensions.
Altermagnetism represents a type of collinear magnetism, that is in some aspects distinct from ferromagnetism and from conventional antiferromagnetism. In contrast to the latter, sublattices of opposite spin are related by spatial rotations and not only by translations and inversions. As a result, altermagnets have spin-split bands leading to unique experimental signatures. Here, we show theoretically how a d-wave altermagnetic phase can be realized with ultracold fermionic atoms in optical lattices. We propose an altermagnetic Hubbard model with anisotropic next-nearest neighbor hopping and obtain the Hartree-Fock phase diagram. The altermagnetic phase separates in a metallic and an insulating phase and is robust over a large parameter regime. We show that one of the defining characteristics of altermagnetism, the anisotropic spin transport, can be probed with trap-expansion experiments.
The applicability of the Hartree-Fock and random phase approximations to models of strongly correlated electrons is discussed. The 2D Hubbard model is analyzed. An antiferromagnetic phase (at half filling) and Fermi liquid behavior (at low electron densities) are found. In addition, the system shows an unconventional regime near half filling. It is characterized by the formation of spin and charge textures. Local magnetic moments and spin waves are also present, and the quasiparticle pole at the Fermi energy vanishes. The charge excitation spectrum is gapless. The overall features resemble those of the 1D Luttinger liquid.
Altermagnetism and beyond in the ttt-t′t^\primet′-δδδ Fermi-Hubbard model
In this work, we revisit the phase diagram of the ttt-t′t^\primet′-δδδ Fermi-Hubbard model on the square lattice to gain a more comprehensive understanding of this correlated model at half filling. This model has recently become a prominent topic of research because it hosts altermagnetic phases. Using mean-field analysis, we identify four metallic phases and two insulating phases with nontrivial magnetic orders at an intermediate value of δ=0.5δ= 0.5δ=0.5, presenting a rich ground-state phase diagram in the UUU-t′t^\primet′ plane. We also highlight the distinct features of the Fermi surface topology for each metallic phase. To go beyond the mean-field theory, we employ the density-matrix renormalization group method to simulate the ground state numerically. The phase boundaries are determined from the discontinuities and peaks in the entanglement entropy and magnetizations. In addition to the phases identified in the mean-field theory, we find a valence-bond solid state in a narrow intermediate-t′t't′ region. Our work offers a firm step forward in understanding the complex behaviors of correlated electrons in the ttt-t′t^\primet′-δδδ Hubbard model over a large parameter space.
The repulsive Fermi-Hubbard model on a square lattice has a rich phase diagram near half-filling (n=1n=1n=1): at n=1n=1n=1 the ground state is an antiferromagnetic insulator, at $0.6<n\ensuremath{\lesssim}0.8$ the ground state is a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superfluid (at least for moderately strong interactions, $U\ensuremath{\lesssim}4$), and the region $1\ensuremath{-}n\ensuremath{\ll}1$ is likely subject to phase separation. Much less is known about the nature of strong magnetic fluctuations at finite temperature and how they change with the doping level. Recent experiments on ultracold atoms have now reached this interesting fluctuation regime. In this work we employ the skeleton diagrammatic method to quantify the characteristic temperature scale TM(n){T}_{M}(n)TM​(n) for the onset of magnetic fluctuations with a large correlation length.
A strong-coupling regime of the Hubbard model around half filling is investigated in the Hartree-Fock random-phase approximation. At half filling, besides a normal solution that gives rise to antiferromagnetic instability at ${T=T}_{\mathrm{N}}^{\mathrm{RPA}}\ensuremath{\sim}U,$ we find a solution representing the paramagnetic phase of the Mott insulator, for which we obtain the N\'eel temperature ${T}_{\mathrm{N}}\ensuremath{\sim}J$ in three-dimensional lattices. The solution is interpreted as the ferromagnetic band insulator depolarized by spin wave, the bound state of a particle-hole pair. Similarly, an anomalous metallic phase is derived from the Nagaoka ferromagnetic state. We evaluate TN{T}_{\mathrm{N}}TN​ as a function of doping concentration x to find that it vanishes at $x\ensuremath{\simeq}0.05$ for U/6t=4U/6t=4U/6t=4 in a simple cubic lattice.