Altermagnetic phases and phase transitions in Lieb-$5$ Hubbard model

Abstract

The emergence of altermagnetism, the collinear magnetic phase characterized by momentum-dependent spin-split bands but zero net magnetization, has fundamentally reshaped the classification of magnetic order. We propose an altermagnetic (AM) order in a repulsive Hubbard model on the Lieb-$5$ lattice. Considering only nearest-neighbor hoppings within the lattice, we show a phase transition from the nonmagnetic to a unique AM isolated band metal phase (AMIM), allowing clear identification of spin-split states. Additionally, the AM metallic phase (AMM) is also shown to appear as an intermediate phase during the transition from the normal metal to the AMIM in the presence of the diagonal hopping within each unit cell of the Lieb-$5$ lattice. The manifestation of distinct AM phases and the phase transitions, driven by Hubbard interaction and hopping integrals, have been explored in terms of spin-resolved band structure, spectral function, and the behavior of the AM order parameter. The stability of these AM phases against the spin-orbit coupling and temperature is also established.

Figures (13)

• Figure 1: (a) Lieb-$5$ lattice containing three sub-lattices invariant under a spin flip followed by a $C_4$ rotation around the center. (b) Its variant with diagonal hopping.
• Figure 2: Spin-resolved band structure of Lieb-$5$ lattice in the AM phase (a) without ($t_d=0$) and (b,c) with diagonal hopping ($t_d=0.5$). The Hubbard interaction term is considered as (a,b) $U = 5$ and (c) $U = 3$. The other parameters are: $t_{1} = 1$, $t_{2} = 0.5$, $T = 0$. (d) Schematic diagram of the first Brillouin zone (BZ) with high symmetry points and path ($k_x,k_y \in [-\pi, \pi]$).
• Figure 3: (a,b) Density plot of $\delta m$ with Hubbard $U$ and (a-b) intercell hopping integral $t_2$ for (a) $t_d=0$, (b) $t_d=0.5$ and (c-d) diagonal hopping integral for (c) $t_2 =0.25$, (d) $t_2 = 0.5$. Other parameter values are the same as in Fig. \ref{['fig:EK']}.
• Figure 4: Spin-resolved spectral function $\Delta A(\omega)$ = $A_{\uparrow}(\omega) - A_{\downarrow}(\omega)$ in $k_x, k_y$ space for Lieb-$5$ lattice with (a) $U =3$ and (b) $U = 5$ considering $t_d = 0.5$, $\omega = 0$ and $\eta = 0.01$. The other parameters are the same as in Fig. \ref{['fig:EK']}.
• Figure 5: Density profile of $\delta m$ over $U$ and SOC ($\lambda_{\text{soc}}$) for (a) $t_d = 0$ and (b) $t_d = 0.5$. The other parameters are the same as in Fig. \ref{['fig:EK']}.