Hedgehog lattices induced by chiral spin interactions

TL;DR

This work investigates how chiral spin interactions in a classical Heisenberg-like model on a cubic lattice stabilize a 4Q hedgehog lattice of monopole-like defects. Using Metropolis Monte Carlo, the authors show that hedgehogs form a robust NaCl-like bipartite arrangement with eight ordering-wavevectors, and they demonstrate that hedgehog density is nonlinearly controlled by the Dzyaloshinskii-Moriya strength while the four-spin chiral coupling acts as a hedgehog chemical potential. The hedgehog lattice melts at finite temperature, with first-order signatures in parts of the phase diagram and potential metastability, raising the prospect of hedgehog liquids or incompressible quantum hedgehog liquids in the quantum regime. The results connect to experiments in MnGe, MnSi1-xGex, and SrFeO3 and highlight the importance of higher-order chiral interactions in stabilizing 3D topological spin textures, offering guidance for realizing and tuning such states in real materials and their quantum analogs.

Abstract

We analyze a classical Heisenberg spin model on the simple cubic lattice which is invariant under time reversal and contains multiple chiral spin interactions. The modelled dynamics is appropriate either for local moments coupled to itinerant Weyl electrons, or localized electrons with a strong spin-orbit coupling that would produce a Weyl spectrum away from half filling. Using a Monte Carlo method, we find a robust $4Q$ bipartite lattice of hedgehogs and antihedgehogs which melts through a first order phase transition at a critical temperature in certain segments of the phase diagram. The density of hedgehogs is a non-linear function of the Dzyaloshinskii-Moriya interaction, and a linear function of the multiple-spin chiral interaction which plays the fundamental role of a ``magnetic flux'' or a hedgehog chemical potential. These findings are related to the observations of hedgehog lattices in MnGe, MnSi$_{1-x}$Ge$_x$ and SrFeO$_3$, and indirectly support the possible existence of incompressible quantum-disordered hedgehog liquids.

Figures (14)

• Figure 1: The fast Fourier transform (FFT) amplitude of the magnitude $|\boldsymbol{\chi}|$ of the spin chirality pseudovector (\ref{['SpinChirality']}) in three representative hedgehog lattices: (a) $(D,\Phi)=(1.0,0.4)$, $N=75^3$ at $T=0$, (b) $(D,\Phi)=(1,1)$, $N=42^3$ at $T=0$ and (c) at $(D,\Phi)=(1,1)$, $N=24^3$ at $T=1.5$ ($J=1$, $K=C=0$ in all cases). The peaks of the chirality FFT generally have a roughly spherical shape and sit at eight vertices of a cube in momentum space. The simulation parameters were: (a) $800$ measurements in $2\times10^8$ iterations after annealing, (b) $2000$ measurements in $8.75\times10^8$ iterations after annealing, and (c) $10^3$ measurements in $2.5\times 10^8$ iterations with $32\%$ acceptance of single-spin updates. Note that the state (c) is metastable at the given model parameters, but becomes stable at larger values of $D$ or $\Phi$.
• Figure 2: (a) A perfect $4Q$ arrangement of hedgehog (red) and antihedgehog (green) singularities in lattice cells that corresponds to the $Q=\pi/2$ state of Fig.\ref{['ChiralityPlots']}(c). (b) A sample arrangement of singularities on the dual cubic lattice, computed with (\ref{['TopCharge']}) in the equivalent state from a Monte Carlo calculation.
• Figure 3: Chirality $\chi$ peak FFT magnitude per lattice site as a function of the system size $N$, shown in the hedgehog lattice at two temperatures, $T=0$ (blue) and $T=1$ (red). The Fourier transforms of the chirality components were computed at the ordering wavevectors ${\bf Q}_{\textrm{peak}}=(\pi/3)\times(\pm1,\pm1,\pm1)$ of the ground-state hedgehog lattice with $J=D=\Phi=1$ and $K=C=0$. The Monte Carlo simulations took about $300$ measurements of the spin configuration in $1500\times N$ iterations, with $23\%$ acceptance of single-spin updates at $T=1$ (simulated annealing was performed at $T=0$).
• Figure 4: The number of hedgehogs (and antihedgehogs) in the state of Fig.\ref{['ChiralityPlots']}(c) as a function of the system size $N$, obtained by simply counting the number (\ref{['TopCharge']}) of singularities with $\mathcal{N}=+1$ in the crystal lattice unit-cells. This method slightly overestimates the expected number $N_{\textrm{h}}=N/(2\times 2^3)$ for the given $Q=\pi/2$ metastable hedgehog lattice, due to disorder with random embeddings of extra hedgehog-antihedgehog dipoles.
• Figure 5: Typical relative phases of the complex-valued FFT peaks in a single chirality pseudovector component, for the state shown in Fig.\ref{['ChiralityPlots']}(c). The two depicted patterns, observed in calculations with system sizes (a) $N=24^3$ and (b) $N=32^3$, are randomly selected by the precise manner in which the translational symmetry is broken by the hedgehog lattice. The left and right columns are obtained at $k_z=+\pi/2$ and $k_z=-\pi/2$ respectively. The circled signs denote sign changes of the phase (the same-colored circles in each row have opposite signs because the real-space chirality is real).