Chirality and quasi-long-range order in finite-flux Gutzwiller states for magnetized frustrated magnets

TL;DR

The paper investigates magnetized Dirac spin liquids on the triangular lattice by constructing Gutzwiller-projected spinon wavefunctions with a uniform gauge flux $\phi$, yielding spinon Landau levels with spin splitting. Using variational Monte Carlo and a correlation-matrix framework to reconstruct quasi-local parent Hamiltonians, it identifies the $|C|=1$ Landau-level state as energetically favored at fixed magnetization, displaying dominant quasi-long-range $120^\circ$ spin correlations and a finite staggered scalar spin chirality, signaling emergent gauge flux effects. In contrast, the $|C|=2$ sector shows spin-nematic correlations with unusual long-range monopole-related order, suggesting distinct competing orders in the higher-Chern sectors. The results offer numerical diagnostics and qualitative signatures for spinon-gauge-field physics in magnetized triangular-lattice systems and provide guidance for interpreting field responses in Dirac spin-liquid candidates.

Abstract

We study Gutzwiller-projected wavefunctions for triangular-lattice U(1) Dirac spin liquids in a Zeeman field, where we allow the U(1) gauge field to develop a gauge flux, resulting in (spin-split) spinon Landau levels. We find that at a given magnetization, the optimal candidate state has a finite flux chosen such that the spinon filling lies in a $|C|=1$ Landau-level gap: it gives the lowest variational energy and the smallest energy variance within our correlation-matrix reconstruction for local Heisenberg-type models. By symmetry, we argue that the finite gauge flux results in a non-zero (staggered) scalar spin chirality, as also numerically observed, and further find that the $|C|=1$ state exhibits dominant quasi-long-ranged $120^\circ$ magnetic correlations. Studying the next-to-optimal wavefunction with a $|C|=2$ Landau-level gap, we observe unusual spin-nematic correlations. Our results may provide guidance for analyzing the magnetic-field response of DSL candidate materials and offer numerical diagnostics that can connect to the underlying theory of spinons coupled to an emergent U(1) gauge field.

Figures (13)

• Figure 1: (a) Hofstadter-butterfly spectrum of the staggered-$\pi$ triangular-lattice spinon Hamiltonian on an $18\times18$ cluster: single-particle energies $\epsilon$ vs flux density $\phi/2\pi$. (b) Corresponding Wannier diagram, where the vertical axis denotes the normalized state index (cumulative filling) $y$ and the color scale encodes the size of the energy gap between adjacent levels. For rational flux $\phi/2\pi$, $y$ of each gap obeys $y = C\,(\phi/2\pi)+s$ with integer $C$, so the slope of each straight line equals the Chern number $C$ of that gap. The circles and squares mark two $C=\pm 1$ gaps, where the spin-up and spin-down spinon Fermi levels lie at magnetizations $m=1/3$ and $2/3$, realized at $\phi = \pi/3$ and $2\pi/3$, respectively. (c,d) Variational energies per site of Eq. \ref{['eq:h-nn']} as a function of $\phi$ at magnetizations $m = 1/3$ and $2/3$, respectively. Symbols denote the linear system size ($N_x=N_y$). Vertical dashed lines mark gap-opening fluxes with the smallest nonzero $|C|$ (notably $|C|=1$). For $N_x<30$ we sample all commensurate fluxes satisfying close-shell conditions, whereas for $N_x\ge 30$ we restrict to fluxes where the spinon Fermi level lies inside a well-defined Chern gap. (e) Minimal fluctuation eigenvalue $\lambda_{\min}$ obtained when only range-$2$ couplings are included. (f) $\lambda_{\min}$ as a function of the maximum coupling range for the Fermi-pocket state and for the $|C|=1$,$2$ Landau-level states. Dotted (solid) lines correspond to $6\times6$ ($12\times12$) lattices.
• Figure 2: (a) Monopole matrix element $\langle S_i^+\rangle_{\text{mono}}$, where the color encodes the phase, shown relative to site $(0,0)$, and the marker size encodes the magnitude. (b,d,f,h): real-space transverse spin correlator $C_{ij}^{\perp}$ on $12\times 12$ lattices with reference site at $(0,0)$. The color scale is clipped to the range $[-0.15,0.15]$; the on-site value $C_{00}^{\perp}=0.5$ lies outside this range and is shown at the saturation color. (c,e,g,i): corresponding momentum-space structure factor $C^{\perp}(\mathbf{q})$ on $18\times 18$ lattices. (b) and (c) have slightly different magnetizations $m$, chosen to be the closest available values; the mismatch arises from finite-size closed-shell magnetization levels at different lattice sizes.
• Figure 3: Finite-size scaling of transverse spin order. (a) $120^\circ$ order normalized structure factor $C^{\perp}(\mathbf{q}=K)/N$ versus $(N_x\times N_y)^{-1/2}$. (b) Site-averaged monopole matrix element $\overline{|\langle S_i^+\rangle_{\text{mono}}|}$ versus $(N_x\times N_y)^{-1/4}$. Insets show the same data plotted against $(N_x\times N_y)^{-1}$. Dashed lines are linear fits using $N_x=N_y\ge 12$, shown as guides to the eye.
• Figure 4: (a) Scalar spin chirality $\chi_{ijk} \equiv \langle \vec{S}_i \cdot (\vec{S}_j \times \vec{S}_k) \rangle$ on a $6\times 6$ lattice with $|C|=1$ and $m=2/3$. (b) Plaquette-averaged magnitude $\overline{|\chi|}$ versus magnetization $m$ on a $36\times 36$ lattice. The gray curve shows the classical umbrella-state chirality for reference. (c) Ratio $\mathcal{R}$ on the same lattice.
• Figure 5: (a) Phases of the nearest-neighbour monopole matrix element $\langle S_i^+ S_j^+\rangle_{\mathrm{mono}}$, shown relative to the bond $(i,j)=((1,0),(0,0))$. (b) Finite-size scaling of the bond-averaged magnitude of the nearest-neighbour monopole matrix element, $\overline{|\langle S_i^+ S_j^+\rangle_{\mathrm{mono}}|}$, and of the corresponding four-spin correlator $\overline{|\langle S_i^+ S_j^+ S_k^- S_l^- + S_i^- S_j^- S_k^+ S_l^+\rangle|}$, which serves as a analogue of a structure factor normalized by $N$. Dashed lines are linear fits.