Evolution of magnetic correlation in doped Hubbard model with altermagnetic spin splitting

Abstract

The evolution of magnetic correlation in strongly correlated electron systems with altermagentic spin splitting remains largely unexplored. Here we investigate how spin splitting generated by spin-dependent next-nearest-neighbor hopping $t'$ reshapes the Fermi surface nesting and van Hove singularities in the two-dimensional square-lattice Hubbard model, leading evolution of magnetic instabilities. Using the constrained-path quantum Monte Carlo method, we find the dominant magnetic correlation as functions of the filling and $t'/t$ by computing the momentum-resolved spin structure factor. The analysis reveals a transition from antiferromagnetic $(π,π)$ order in the isotropic, half-filled system to non-collinear spiral $(π,q)$ order upon increasing the spin-dependent anisotropy or doping away from half-filling, ultimately entering a short-range correlation regime where stripe and spiral correlation coexist. These findings highlight a possible route to realizing spiral correlation in altermagnetic systems, potentially providing a platform for spintronic devices that exploit non-collinear spin textures.

Figures (14)

• Figure 1: Schematic representation of the spin-anisotropic $t'$ Hubbard model and its resulting band splitting. (a) Square lattice with on-site interaction $U$ and sign-alternating next-nearest-neighbor (NNN) hopping $\pm t'$, giving opposite effective NNN hopping for the two spin species while maintaining zero net magnetization. (b) Noninteracting band dispersion along the high-symmetry path of the Brillouin zone and the corresponding spin-resolved Fermi surface (inset) for $t' = 0$, where the spectrum is spin degenerate. (c) Noninteracting band dispersion and spin-resolved Fermi surface for $t' = 0.5$, where the term $4t'\sin k_x \sin k_y$ induces momentum-dependent spin splitting and distinct Fermi-surface contours for spin up (red) and spin down (blue).
• Figure 2: The evolution of van Hove singularities (VHS) and Fermi surfaces with varying NNN hopping $t'$. The panels correspond to (a) $t'=0.1$, (b) $t'=0.5$, and (c) $t'=0.9$. For each $t'$, solid (dashed) contours denote the Fermi surfaces of the upper (lower) band at fillings $n = 0.5$ and $n = 0.9$, while the shaded regions indicate the corresponding filled Fermi surface. Stars mark the VHS. As $t'$ increases, the VHS manifests not only at high symmetry points and the Fermi surfaces become strongly distorted, degrading $(\pi,\pi)$ nesting and thereby favoring $( \pi, q)$ or $( q, \pi)$ spiral tendencies and, at larger $t'$, stripe collinear configurations $(\pi,0)$ or $(0,\pi)$.
• Figure 3: The evolution of the electronic density of states (DOS) as a function of the NNN hopping amplitude $t'$. The panels (a)–(c) display the calculated total DOS for $t' = 0.1$, $t' = 0.5$, and $t' = 0.9$, respectively. The energy $E$ is normalized by the NN hopping $t$. Vertical dashed lines mark the energy $E=0$, typically corresponding to the Fermi level at half-filling. The prominent peaks in the DOS spectra identify the positions of VHS, which arise from saddle points in the momentum space dispersion relation. The panels (d)–(f) depict the corresponding $\nabla_{\mathbf{k}}\varepsilon_{\mathbf{k}\sigma}$ along the diagonal momentum cut. The red circles mark the critical points where the $\nabla_{\mathbf{k}}\varepsilon_{\mathbf{k}\sigma}$ vanishes, directly corresponding to the VHS locations.
• Figure 4: The ordering vector distribution of dominant $S^z(\mathbf k)$ peak locations in the $(n,t')$ plane for an $L=16$ lattice. The color scale shows the $S^z(\mathbf k)$ peak coordinate $Q$ at which $|S^{z}(\mathbf{k})|$ is maximal: $Q=1$ corresponds to Néel order at $(\pi,\pi)$, $Q=0$ to collinear stripe correlation at $(\pi,0)$ or $(0,\pi)$, and $0<Q<1$ indicates spiral correlation, e.g. $(\pi,q)$ or $(q,\pi)$. The shaded region labeled coexistence regimes where several $S^z(\mathbf k)$ peak types have comparable maximal weight , indicating strong competition among magnetism tendencies. Ordered states with different magnetic correlation are distinguished by a red line: regions to the right of the red line represent long-range order, while those to the left indicate short-range order. Overall, the diagram illustrates the evolution from Néel/stripe regimes to spiral correlation behavior as $t'$ varies.
• Figure 5: The momentum-resolved spin imbalance $\Delta n(\mathbf{k})$ and magnetic correlation at weak spin anisotropy for fixed $t'=0.10$. (a)–(c) $\Delta n(\mathbf{k})$ for fillings $n=0.945$, $0.789$, and $0.539$, respectively, overlaid with the noninteracting Fermi surfaces (dashed) and the dominant magnetic wave vector (red arrows). (d)–(f) Corresponding $S^{z}(\mathbf{k})$ for the same parameters. Near half filling, $S^{z}(\mathbf{k})$ is peaked at $(\pi,\pi)$, while upon decreasing $n$ the dominant response shifts toward ordering vector of the form $(\pi,q)$ or $(q,\pi)$, accompanied by a splitting of the $S^z(\mathbf k)$ peak structure.