Altermagnetic phase transition in a Lieb metal

Abstract

We analyze the phase transition between a symmetric metallic parent state and itinerant altermagnetic order. The underlying mechanism we reveal in our microscopic model of electrons on a Lieb lattice does not involve orbital ordering, but derives from sublattice interference.

Figures (10)

• Figure 1: (a) Real space lattice structure with of the Lieb lattice. The purple square marks the unit cell with the $A$ site on the trivial Wyckoff position ($1a$), and the $B$ and $C$ sites on the $2c$ Wyckoff positions. Nearest- and next-nearest neighbor hoppings $t$ and $t'$ are indicated. The gray square corresponds to the unit cell of a lattice without the $A$ site, i.e., a simple square lattice. (b) Resulting band structure along the high-symmetry path indicated in panel (c) for $t^\prime = t/2$ and $\mu_A=0$. We overlay the sublattice content by color (purple: $A$, gray: $B$/$C$). The right panel displays the sublattice resolved density of states, with the $B$/$C$ polarized van-Hove singularity at the band touching point. (c) Brillouin zones (BZ) of the Lieb lattice structure (purple) and the intercalated square lattice (gray) formed by dashed lines connecting the $2c$ Wyckoff positions. We additionally plot the Fermi surface and its sublattice polarization (yellow: $B$, green: $C$), which is pure along the BZ boundaries and mixed at the van-Hove points $M$.
• Figure 2: $d$-wave altermagnetic state on the Lieb lattice. (a) Real space magnetization pattern in the $B_1$ irreducible representation of $C_{4v}$. The magnetic order parameter $\Delta$ is nonzero only on the $B$ and $C$ sites. (b) Quasiparticle bandstructure in the altermagnetic phase for $t^{\prime}=t/2$, $\mu_A=0$, and $\Delta_{\mathrm{M}} =0.2\,t$ with spin polarization indicated by purple/green ($\downarrow$/$\uparrow$) lines. The inset shows the spin polarized Fermi surface, where the nontrivial transformation behavior under $C_4$ rotations and diagonal mirrors $\mathcal{M}_{xy}$, $\mathcal{M}_{x\bar{y}}$ becomes apparent.
• Figure 3: (a) Size of the altermagnetic order parameter ($\Delta_M$) as a function of temperature ($T$) for $t'=t/2$ and $U=3t$, obtained from self-consistent mean-field simulations, showing a clear mean-field phase transition at $T_c/t \approx 0.23$. (b) Value of the magnetic order parameter in the limit of $T=0$ for $U$ across a large parameter regime. The dashed line indicates $U=3t$ as used in panel (a).
• Figure S1: Diagrammatic representation of static four-point FRG. We group the three channels by color: particle-particle channel ($P$, purple), direct particle-hole channel ($D$, yellow), and crossed particle-hole channel ($C$, green). Spin is conserved along the short edge of each vertex.
• Figure S2: Critical cutoff scale $\Lambda_c$ of (a) the proposed AM state for different values of $U_A = U_{B,C}$ at $t'=0.5$ and (b) competing phases for different values of $t'$ at $U_i=1$ depicted in \ref{['Fig:states']}. (c) Joint phase diagram the $U, t^\prime$ parameter space. We see that the AM state is stable in a broad parameter regime and extends to relatively large values of interaction.