Magnons in multiorbital Hubbard models, from Lieb to kagome

TL;DR

This paper studies magnetic orders and excitations in a half-filled Hubbard model that interpolates between Lieb and kagome lattices. It combines self-consistent Hartree–Fock with real-time two-particle responses from the Bethe–Salpeter equation in the random-phase approximation to map the $U$–$t'$ phase diagram and resolve magnon spectra. The authors identify Goldstone magnons in symmetry-broken phases and gapped Higgs magnon branches, with Higgs modes exhibiting site- and spin-resolved character, and they reveal how band geometry controls magnetic order and fluctuations. The work provides experimentally relevant signatures for magnetic excitations in multi-orbital lattices and highlights the interplay between lattice geometry, order parameters, and collective modes.

Abstract

We investigate the magnetic orders and excitations in a half-filled Hubbard model that continuously interpolates between the Lieb and kagome lattices. Using self-consistent Hartree-Fock approximation combined with real-time two-particle response functions from the Bethe-Salpeter equation in the random phase approximation, we map the $U-t'$ phase diagram of the Lieb-kagome lattices, identifying the typical magnetic states and the corresponding magnetic excitation spectra. In addition to gapless Goldstone magnons, the ferrimagnetic and antiferromagnetic symmetry-broken phases also exhibit gapped Higgs magnon bands, which originate from amplitude fluctuations in the order parameter characterizing spontaneous symmetry breaking.

Figures (22)

• Figure 1: Illustrations of three types of magnetic excitations in the Lieb lattice: (a) the Goldstone mode, corresponding to a gapless magnon mode; (b) the Higgs mode, representing a gapped (amplitude) magnon from fluctuation of the order parameter; and (c) the Stoner mode (pair), describing single-particle spin-flip excitations.
• Figure 2: Schematic of (a) the Lieb and (b) kagome lattices. Sites of sublattices A, B, and C are marked by blue circles, red triangles, and orange triangles, respectively, and the shaded yellow region highlights the three sites (A, B and C) in a single unit cell. Solid and dashed lines refer to the nearest-neighbour (NN) and the next-nearest-neighbour (NNN) hopping amplitude $t$ and $t'$, respectively. By varying $t'$, the system is continuously tuned from the ideal Lieb ($t'=0$) to the ideal kagome ($t'=t$) lattice. (c) The first Brillouin zone and high-symmetry points corresponding to the Lieb lattice. (d) Non-interacting band structures of the Lieb lattice as functions of $t'$, plotted along the high-symmetry path in panel (c).
• Figure 3: Phase diagram in the HF approximation of the Lieb-kagome lattice at the inverse temperature $\beta=10$. The heatmap presents the magnetization as a function of $U$ and $t^\prime$. There is a first-order phase transition from the paramagnetic state to a magnetically ordered state, either ferrimagnetic or antiferromagnetic (AFM), with increasing interaction strength, and a continuous transition between the ferrimagnetic and antiferromagnetic state as a function of $t'$.
• Figure 4: Band structures and transverse spin susceptibilities $\chi^{+-}$ and $\chi^{-+}$ of three typical magnetic phases in the Lieb-kagome lattice with $U=4.0$. Top row: The paramagnetic state with $t'=1.0$, the kagome lattice limit; Middle row: The ferrimagnetic state with $t'=0.0$, the Lieb lattice limit; Bottom row: The altermagnetic state with $t'=0.5$.
• Figure 5: Average magnetization per unit cell $\langle S^z\rangle$ as a function of $t^\prime$ with different values of $U$.