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arXiv:2503.08362

Altermagnetism and beyond in the $t$-$t^\prime$-$δ$ Fermi-Hubbard model

Saisai He, Jize Zhao, Hong‐Gang Luo, Shijie Hu

Published Mar 11, 2025• 0 citations
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Abstract

In this work, we revisit the phase diagram of the $t$-$t^\prime$-$δ$ 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 $δ= 0.5$, presenting a rich ground-state phase diagram in the $U$-$t^\prime$ 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-$t'$ region. Our work offers a firm step forward in understanding the complex behaviors of correlated electrons in the $t$-$t^\prime$-$δ$ Hubbard model over a large parameter space.

DOI
10.48550/arxiv.2503.08362

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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.
Two-dimensional density-matrix renormalization group method is employed to examine the ground-state phase diagram of the Hubbard model on the triangular lattice at half-filling. The calculation reveals two discontinuities in the double occupancy with increasing the repulsive Hubbard interaction UUU at ${U}_{\mathrm{c}1}\ensuremath{\sim}7.8t$ and ${U}_{\mathrm{c}2}\ensuremath{\sim}9.9t$ (ttt being the hopping integral), indicating that there are three phases separated by first-order transitions. The absence of any singularity in physical quantities for $0\ensuremath{\le}U<{U}_{\mathrm{c}1}$ implies a metallic phase in this regime. For U>Uc2U>{U}_{\mathrm{c}2}U>Uc2​, the local spin density induced by an applied pinning magnetic field exhibits a three sublattice feature, which is compatible with the ${120}^{\ensuremath{\circ}}$ N\'eel-ordered state realized in the limit of $U\ensuremath{\rightarrow}\ensuremath{\infty}$. For Uc1<U<Uc2{U}_{\mathrm{c}1}<U<{U}_{\mathrm{c}2}Uc1​<U<Uc2​, a response to the applied pinning magnetic field is comparable to that in the metallic phase with a relatively large spin correlation length, but showing neither valence bond nor chiral magnetic order, which therefore resembles gapless spin liquid. However, the spin structure factor for the intermediate phase exhibits the maximum at the K\mathrm{K}K and ${\mathrm{K}}^{\ensuremath{'}}$ points in the momentum space, which is not compatible to spin liquid with a large spinon Fermi surface. The calculation also finds that the pairing correlation function monotonically decreases with increasing UUU and thus the superconductivity is unlikely in the intermediate phase.
Employing a combination of functional renormalization group calculations and projective determinantal quantum Monte Carlo simulations, we examine the Hubbard model on the square lattice bilayer at half filling. From this combined analysis, we obtain a comprehensive account of the ground-state phase diagram with respect to the extent of the system's metallic and (antiferromagnetically ordered) Mott-insulating as well as band-insulating regions. By means of an unbiased functional renormalization group approach, we exhibit the antiferromagnetic Mott-insulating state as the relevant instability of the free metallic state, induced by any weak finite on-site repulsion. Upon performing a careful analysis of the quantum Monte Carlo data, we resolve the difficulty of identifying this antiferromagnetic ground state for finite interlayer hopping in the weak-coupling regime, where nonmonotonic finite-size corrections are shown to relate to the two-sheeted Fermi surface structure of the metallic phase. On the other hand, quantum Monte Carlo simulations are well suited to identify the transition between the Mott-insulating phase and the band insulator in the intermediate-to-strong coupling regime. Here, we compare our numerical findings to indications for the transition region obtained from the functional renormalization group procedure.
Motivated by the recent discovery of metallic kagome lattice materials, AV3Sb5{\mathrm{AV}}_{3}{\mathrm{Sb}}_{5}AV3​Sb5​ (A = K, Rb, Cs), we investigate the ground state of the half-filled kagome lattice Hubbard model by employing the density-matrix renormalization group method. We identify a metal-insulator transition around $U\ensuremath{\sim}{U}_{c1}$ and four distinct phases as a function of U/tU/tU/t on narrower cylinders, including a metallic phase at U<Uc1U<{U}_{c1}U<Uc1​, two insulating intermediate phases (a translational symmetry breaking phase at Uc1<U<Uc2{U}_{c1}<U<{U}_{c2}Uc1​<U<Uc2​ and a quantum spin liquid phase at Uc2<U<Uc3{U}_{c2}<U<{U}_{c3}Uc2​<U<Uc3​), and the kagome antiferromagnetic phase at U>Uc3U>{U}_{c3}U>Uc3​. We confirm that the translational symmetry breaking phase is robust for wider cylinders, while the quantum spin liquid phase is smoothly connected to the kagome antiferromagnetic phase with increasing the system width. Moreover, our numerical observations indicate a continuous metal-insulator transition at Uc1{U}_{c1}Uc1​ whose nature is consistent with Slater's transition scenario. The magnetic phase transition between two insulating intermediate phases at Uc2{U}_{c2}Uc2​ is first order. Our findings may provide insights into exotic kagome lattice materials.
An unusual metallic phase is argued to develop in the one-dimensional ionic Hubbard model, at half-filling and zero magnetization, at intermediate electron-electron repulsion UUU when second-neighbors hopping is allowed and tuned close to a topological Lifshitz transition (connected with a change of the Fermi surface in the noninteracting system). The metallic state lies between a band insulator phase at low repulsion and a correlated (Mott-like) insulator phase at high repulsion. In approaching the latter, the model supports short-range antiferromagnetic order and spontaneous dimerization of both bond charge and nearest-neighbor antiferromagnetic correlations. A combination of mean-field and effective-field theory (bosonization) provides an analytical understanding of the physical processes underlying the argued phase transitions. The ground and low-energy excited states of finite-length chains are explored by density-matrix renormalization-group (DMRG) calculations, providing numerical evidence for the intermediate gapless phase. Such finite systems are attainable by cold atoms in optical lattices for a wide range of the parameter UUU.
The cluster slave-spin method is employed to investigate systematically the ground state properties of the Hubbard model on a square lattice with doping $\ensuremath{\delta}$ and coupling strength UUU being its parameters. At half-filling, a relation between the staggered magnetization MMM and the antiferromagnetic (AFM) gap ${\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}$ is established in the small UUU limit to compare with that from the Hartree-Fock theory, and a first-order metal-insulator Mott transition in the paramagnetic state is substantiated, which is characterized by discontinuities and hystereses at UMott=10t{U}_{\text{Mott}}=10tUMott​=10t. The interaction Uc{U}_{c}Uc​ for the crossover in the AFM state, separating the weak- and strong-coupling regimes, is found to remain almost unchanged with large dopings, and smaller than UMott{U}_{\text{Mott}}UMott​ at half-filling because of long range AFM correlations. Finally, an overall phase diagram in the $U\ensuremath{-}\ensuremath{\delta}$ plane is presented, which is composed of four regimes: the AFM insulator at half-filling, the AFM metal with the compressibility $\ensuremath{\kappa}>0$ or $\ensuremath{\kappa}<0$, and the paramagnetic metal, as well as three phase transitions: (i) From the AFM metal to the paramagnetic metal, (ii) between the AFM metal phases with positive and negative $\ensuremath{\kappa}$, and (iii) separating the AFM insulating phase at $\ensuremath{\delta}=0$ from the AFM metal phase for $\ensuremath{\delta}>0$.
We examine the phase diagram of the half-filled one-dimensional extended Hubbard model using quantum information entropies within the density-matrix renormalization group. It is well known that there is a charge-density-wave phase at large nearest-neighbor and a small on-site Coulomb repulsion and a spin-density wave at small nearest-neighbor and large on-site Coulomb repulsion. At intermediate Coulomb interaction strength, we find an additional narrow region of a bond-order phase between these two phases. The phase transition line for the transition out of the charge-density-wave phase changes from first order at strong coupling to second order in a parameter regime where all three phases are present. We present evidence that the additional phase-transition line between the spin-density-wave and bond-order phases is infinite order. While these results are in agreement with recent numerical work, our study provides an independent unbiased means of determining the phase boundaries by using quantum information analysis, yields values for the location of some of the phase boundaries that differ from those previously found, and provides insight into the limitations of numerical methods in determining phase boundaries, especially those of infinite-order transitions.