Weyl semimetallic, Néel, spiral, and vortex states in the Rashba-Hubbard model

TL;DR

This work analyzes the Rashba-Hubbard model on a half-filled square lattice to understand how strong spin-orbit coupling reshapes magnetic order and topology. By combining Krylov-Schur exact diagonalization and determinant quantum Monte Carlo, the authors map the phase diagram as the hopping ratio and Hubbard interaction vary, revealing Néel, spiral, and spin-vortex magnetic phases and a Weyl semimetal regime at extreme Rashba coupling. They establish a Weyl-to-spin-vortex quantum phase transition in the pure Rashba limit, with critical behavior consistent with the Gross-Neveu universality class, supported by finite-size scaling and data collapse analyses. The study also investigates topological aspects via Berry curvature, showing Weyl nodes at weak interactions that are suppressed at larger $U$, yielding a topologically trivial spin-vortex state; overall, the results highlight the rich interplay between spin-orbit coupling, topology, and strong correlations in itinerant electrons.

Abstract

We investigate the evolution of magnetic phases in the Hubbard model under strong Rashba spin-orbit coupling on a square lattice. By using Lanczos exact diagonalization and determinant quantum Monte Carlo (DQMC) simulations, we explore the emergence of various magnetic alignments as the ratio between the regular hopping amplitude, $t$, and the Rashba hopping term, $t_R$, is varied over a broad range of Hubbard interaction strengths, $U$. In the limit $t_R \rightarrow 0$, the system exhibits Néel antiferromagnetic order, while when $t \sim t_R$, a spiral magnetic phase emerges due to the induced anisotropic Dzyaloshinskii-Moriya interaction. For $t_R > t$, we identify the onset of a spin vortex phase. At the extreme limit $t = 0$($t_R \neq 0 $), we perform finite-size scaling analysis in the Weyl semimetal regime to pinpoint the quantum critical point associated with the spin vortex phase, employing sign-free quantum Monte Carlo simulations - the extracted critical exponents are consistent with a Gross-Neveu-type quantum phase transition.

Figures (10)

• Figure 1: Fermi surface and density of states for $\theta/\pi = 0.0$ (a), $\theta/\pi = 0.2$ (b), and $\theta/\pi = 0.5$ (c), using the parameterization $t = \tilde{t} \cos{\theta}$ and $t_R = \sqrt{2} \tilde{t} \sin{\theta}$. Contour plots of the spin structure factor for the $x$ (d) and $z$ (e) components are presented in the $\theta\times U$ space for an $L=4$ lattice after averaged twisted boundary conditions. The different color schemes map different magnetic wave vectors ${\bf q}$, while their intensity indicates the corresponding structure factor, $S^\alpha({\bf q})$ -- see color bars. (f) The spectral energy gap, $\Delta^E / \tilde{t}$, between the ground and the first excited states. Panels (g), (h), and (i) depict schematic representations of the magnetic order corresponding to the black stars in panels (d) and (e), identifying the (g) Néel, (h) spiral, and (i) vortex phases. For the spiral phase (h), the complex magnetic alignment is derived from spin correlations using the relation $\mathbf{s}_\mathbf{i} \propto \langle S^z_0 {\bf S}_\mathbf{i} \rangle$Wan2022.
• Figure 2: (a) Contour plots of the spin structure factors for each spin component, calculated for $L = 12$, $\beta \tilde{t} = 24$, $U/\tilde{t} = 2$, and varying values of $\theta$. (b) Wave vector corresponding to the maximum structure factor for each spin component as a function of $\theta/\pi$; the star markers depict the values of $\theta$ used in (a). Note that the wave vector ${\bf q}$ is always commensurate, with $q$ in the spiral phases with peaks at $(q,\pi)$ or $(\pi, q)$ spanning the range $[4\pi/L, 10\pi/L]$.
• Figure 3: Spin structure factor as a function of the wave vector for different values of $U$, with fixed parameters $L = 16$ and $\beta\tilde{t} = 32$. The panels illustrate the individual contributions from the (a) $x$, (b) $y$, and (c) $z$ components. The inset shows the selected path within the first Brillouin zone. (d) Average spin structure factor, computed as described in Eq. \ref{['eq:spinsf']}.
• Figure 4: (a) Correlation ratio as a function of $U$ for various lattice sizes, with fixed parameters $\beta = 2L$ and $\theta = \pi/2$. The inset displays the extrapolation of the critical value of $U$ in the thermodynamic limit. (b) Contour plot of the cost function as a function of $\theta$ and $U$. (c) Data collapse of the correlation ratio, highlighting the universal behavior.
• Figure 5: Correlation ratio (upper panels) and double occupancy (lower panels) as functions of $U/\tilde{t}$. The colored empty markers represent DQMC data at different temperatures, while the black solid lines display results from exact diagonalization. The left panels show DQMC data for $\Delta\tau = 0.1$, and the right panels present results for a fixed $\Delta\tau = 0.05$. All data correspond to a $4 \times 4$ lattice with periodic boundary conditions and $\theta = \pi/2$.