Topological electronic structures of non-collinear magnetic phases in a multi-orbital Hubbard model with spin-orbit interactions

TL;DR

This work addresses how Dresselhaus and Rashba spin-orbit couplings interact with noncollinear antiferromagnetic orders in a multiorbital Hubbard system to shape topological electronic structures. The authors derive a generalized $t$-$J$ description via a canonical transformation and apply renormalized mean-field theory at low doping to determine magnetic textures and topological invariants, including a $Z^M_2$ index protected by effective time-reversal symmetry on the $k_z=0$ plane and a $Z_4$ index protected by inversion symmetry for tilted antiferromagnetism. Key findings show that Dresselhaus SOC yields collinear AF with $Z^M_2$ protection, while Rashba SOC produces tilted AF with inversion-protected $Z_4$, with bulk-edge correspondence indicating gapless surface states on symmetry-preserving surfaces and gapped states when symmetry is broken; the surface response also depends on the orientation relative to the ferromagnetic component. This framework offers a unified topological characterization for doped and canted antiferromagnets with SOC and has potential relevance for materials with strong spin-orbit coupling and Kitaev-like Mott physics.

Abstract

We explore topological electronic structure of magnetic phases in a multi-orbital Hubbard model with spin-orbit interactions. To account for more general antiferromagnetic orders that go beyond the collinear Néel order, two different spin-orbit interactions, Dresselhaus and Rashba spin-orbit interactions, are considered. By performing the canonical transformation, we derive the corresponding generalized t-J model. At half filling, employing self-consistent magnetic order calculations, we find distinctive spin arrangements under Dresselhaus or Rashba spin-orbit interactions. For the Dresselhaus spin-orbit interaction, the spin configuration exhibits collinear antiferromagnetic order. On the other hand, Rashba interaction results in spins antiferromagnetically aligning in xy-plane and a small interaction controlled by hopping parameter induces spin tilting, causing antiferromagnetic alignment in xy-plane but ferromagnetic alignment in z-direction. We categorize topological properties of these phases for low doping in the generalized t-J model.: for 3D collinear antiferromagnetic order, the system possesses a modified time-reversal symmetry, characterized by the Z2 index. In contrast, for systems with tilted antiferromagnetic orders, it is protected by inversion symmetry and characterized by the Z4 index. We further examine the bulk-edge correspondence for non-collinear magnetic phases, revealing that the surface state becomes gapless when the surface is parallel to the ferromagnetic component of tilted antiferromagnetic order; otherwise, the surface state exhibits a gap. Our findings offer a comprehensive topological characterization for doped and canted antiferromagnetic insulators with spin-orbit interactions, providing valuable insights into the interplay between spin arrangements, symmetries, and topological properties in systems governed by the multi-orbital Hubbard model.

Figures (3)

• Figure 1: Mean-field solution for magnetic order parameter $\langle S^{x,y,z}_{i,\tau}\rangle$: Temperature dependence of $\langle S^{x,y,z}_{i,a}\rangle$ for the Hamiltonian with (a) Dresselhaus interaction (b) Rashba interaction. Here $m^{x,y,z}_{i,a} = \langle S^{x,y,z}_{i,a}\rangle$, $U_{a}$=$U_{b}$=10, $v_{F}$=0.5, and $J=0.5$. For both cases, $T_{c}/t\approx3$. Magnetic order parameter versus $t$ with $T=0.05$ for the Hamiltonian with the Rashba interaction on (c) $i$ site (d) $j$ site (nearest neighbours to $i$). Here for $t<0.019$, $m^{x}_{i,a}= m^{y}_{i,a}= m^{z}_{i,a}=0.577$; while $m^{x}_{j,a} = m^{y}_{j,a}= -m^{z}_{j,a}=-0.577$. For $t \geq 0.019$, the z-components of magnetic orders vanish $m^{z}_{i,a} = m^{z}_{j,a} =0$; while the magnetic orders are antiferromagnetic in $x$-$y$ plane with $m^{x}_{i,a} =m^{y}_{j,a} =- m^{x}_{i,a} = - m^{y}_{j,a}$. Detailed configurations for magnetic orders in orbits $a$ and $b$ for (e) Hamiltonian with the Dresselhaus interaction. Here magnetic orders on the same site point to the same direction for both orbit $a$ and $b$ (red arrows) at $T=0.05$; while magnetic orders on nearest neighbor sites are anti-parallel (green arrows). On the other hand, for Hamiltonian with the Rashba interaction at $T=0.05$, (f) $z$-component of magnetic orders for $t \geq 0.019$ vanish for both $a$ and $b$ orbits at all sites, while magnetic components on nearest neighbor sites are anti-parallel in the $x$-$y$ plane (red arrows for $i$ site, green arrows for the $j$ site that is a nearest neighbour to $i$. (g) For $t <0.019$, $z$-component of magnetic orders do not vanish, and magnetic orders on nearest neighbor sites are antiferromagnetic in the $x$-$y$ plane but are ferromagnetic in the $z$ direction.
• Figure 2: (a)The topological phase diagram for renormalized mean-field theory with Dresselhaus interaction. Here we set $\bar{v}_F=1$. The antiferromagnetic order is in $\{1,1,1\}$ direction and is set to be $(M_{AF,x}, M_{AF,y}, M_{AF,z}) = (0.577,0.577,0.577)$ and (b)The topological phase diagram for Rashba interaction with antiferromagnetic order being in $\{1,1,0\}$ direction. Here we can find the mean-field solution only in the red rectangular region with $\bar{t} \geq 0.019$, and we choose $(M_{AF,x}, M_{AF,y}, M_{AF,z}) = (0.707,0.707,0)$ and $\bar{v}_F=1$ to illustrate the phase diagram. Here in (a) and (b), the topological index based on the $S$-symmetry is given by $Z^M_2$; while for the same parameter, it can be also characterized by the $Z_4$ index. Clearly, we see that $Z^M_2$ and $Z_4$ turns out to be the same, and $M_{AF,z}$ is irrelevant so that two phase diagrams are identical. Note that for a given ratio of $t/m$, topological phases that the system can have when doping increases from $0$ to finite $\delta$ are phases passing by the line with slope $t/m$ on the phase diagram. (c)The computed $S$-symmetry protected surface state in $\{1,0,0\}$ surface for Dresselhaus interaction with antiferromagnetic order being in $\{1,1,1\}$ direction in $13$ layers. Here $(M_{AF,x}, M_{AF,y}, M_{AF,z}) = (0.577,0.577,0.577)$. Similar structure of the surface state is also found for Rashba interaction with antiferromagnetic order being in $\{1,1,0\}$ direction with $(M_{AF,x}, M_{AF,y}, M_{AF,z}) = (0.707,0.707,0)$.(d) For both models computed in (c), the surface states become gapped when the effective time-reversal symmetry ($S$-symmetry) is broken in the $\{\Bar{1},\Bar{1},1\}$ surface. Here in the $\{\Bar{1},\Bar{1},1\}$ surface, the magnetic order is ferromagnetic and the computation is done in $31$ layers.
• Figure 3: (a)The $Z_{4}$ phase diagram for Rashba interaction with antiferromagnetic order being in $\{1,1,0\}$ direction and ferromagnetic field in z direction. Here we can find mean-field solution only in the red rectangular region with $\bar{t} \leq 0.019$. Furthermore, for a given ratio of $t/m$, topological phases that the system can have when doping increases from $0$ to finite $\delta$ are phases passing by the line with slope $t/m$ on the phase diagram. (b)The gapless surface state in $\{1,0,0\}$ surface and (c) the gapped surface state in $\{0,0,1\}$ surface with $\bar{m}/\bar{t}=2.3$. Here the computation is done in $13$ layers. The bulk band structure in semi-metallic phase (d)(f) and topological non-trivial phase (e). Here $\bar{t}=1$ and $\bar{v}_F=1$. (d)$\bar{m}=1.35$, $Z_4=1$. (e) $\bar{m}=2.25$, $Z_4=2$. (f)$\bar{m}=3.3$ and $Z_4=3$.