Emergent gauge flux and spin ordering in magnetized triangular spin liquids: applications to Hofstadter-Hubbard model

TL;DR

The work analyzes emergent gauge flux and spin ordering in magnetized triangular spin liquids by deriving an effective strong-coupling spin model from the Hofstadter-Hubbard framework and exploring orbital-flux-induced chiral spin liquids (CSLs) versus Zeeman-driven spinon pockets. Using fermionic partons, the authors show that orbital flux generates a staggered flux [θ,π−θ] that drives a DSL→CSL transition, while finite magnetization can induce a uniform internal flux that reorganizes spinons into Landau levels (LLs) with conical spin order; the LL state breaks U(1) spin-rotations and yields a gapless gauge photon with monopole-induced XY order. Thermal Hall signatures distinguish DSL, CSL, and LL states: CSL exhibits a quantized κ_{xy}/T at low T, LL cancels κ_{xy} due to opposite spin Chern numbers, and DSL remains zero due to symmetry; Monte Carlo studies of Gutzwiller-projected wavefunctions corroborate LL’s energetic stability and reveal 120° spin ordering. Overall, the paper provides a framework to detect emergent gauge flux and field-driven ordering in triangular spin liquids, with implications for moiré Hofstadter-Hubbard systems and related materials.

Abstract

Motivated by the recent progress in the moiré superlattice systems and spin-1/2 triangular lattice antiferromagnets, we revisit the triangular-lattice spin liquids and study their magnetic responses. While the magnetic responses on the ordered phases can be mundane, the orbital magnetic flux and the Zeeman coupling have synergetic effects on the internal gauge flux generations in the relevant spin liquid phases. The former was known to induce an internal U(1) gauge flux indirectly through the charge fluctuations and ring exchange, and thus could lead to the formation of a chiral spin liquid. The latter could spontaneously generate a uniform field-dependent internal gauge flux, driving a conically-ordered state. The competition and interplay between these two field effects are discussed through a generic spin-1/2 $J_1$-$J_2$-$J_χ$ model and with the experimental consequences. Our results could find applications in the moiré superlattice systems with the Hofstadter-Hubbard model as well as the triangular lattice antiferromagnets.

Figures (8)

• Figure 1: (a) Schematic illustration of the triangular lattice $J_1$-$J_2$-$J_\chi$ model under a magnetic field $B$. (b) The $[\theta,\pi-\theta]$ state with staggered $\theta$ flux. The phase $e^{i\phi_{ij}}$ takes ${\phi_{ij}=\mp\theta}$ when the spinon hops parallel (antiparallel) to the blue arrows, and $\phi_{ij}=0$ when the spinon hops along black arrows. For the red lines, $\phi_{ij}$ further acquires a phase of $\pi$.
• Figure 2: Spinon dispersions of the $[\theta,\pi-\theta]$ state along high-symmetry lines in the Brillouin zone. (a) DSL with ${\theta=0}$ and (b) CSL with ${\theta=\pi/6}$ at zero field. The dispersions for DSL (c) and CSL (d) with ${B=2}$. A finite staggered flux $\theta$ gaps out the Dirac cone in (a), and the resulting bands acquire non-zero Chern numbers. The finite $B$ splits the spin-$\uparrow$ (blue dashed line) and spin-$\downarrow$ (red solid line) bands, leading to Fermi-pocket states with finite magnetizations in (c) and (d).
• Figure 3: (a) Schematic evolution from a Zeeman-split Fermi-pocket (FP) state to a Landau-level (LL) state at finite magnetization. Blue and red parabolic bands (discrete levels) represent spin-$\uparrow$ and spin-$\downarrow$ spinons, respectively. The dashed horizontal line refers to the chemical potential. (b) Mean-field $J_\chi$-$B$ phase diagram with fixed $J_1=1$, $J_2=0.1$. At finite $J_\chi$, the DSL (yellow star) evolves into CSL at zero field, which remains stable up to a critical field $B_c$ (red dots), beyond which it transitions into the LL state. The parameter $\theta$ for different $J_\chi$ at the red dots are consistent with variational Monte Carlo results from Ref. hu2016VariationalMonte.
• Figure 4: Projected Monte Carlo results for the FP and LL states with ${\theta=0.1\pi}$, ${J_1=1}$, ${J_2=0.1}$, and ${J_\chi=0.3}$ on a ${18 \times 18}$ lattice. (a) Energy density $\mathcal{E}$ as a function of magnetization density $m_z$. (b) Spin structure factor $S^{XY}(\bm{q})$ for projected LL state at ${m_z\approx0.05}$, where prominent peaks appear at the $K$ points of the Brillouin zone.
• Figure 5: Temperature dependence of the spinon thermal Hall conductivity $\kappa_{xy}/T$ in units of $\pi k_B^2/6\hbar$. (a) Results for the CSL at ${B=0}$ and ${\phi_u=0}$ for different staggered flux $\theta$. As ${T \to 0}$, $\kappa_{xy}/T$ approaches the quantized value $\pm 2$, and its magnitude decreases continuously toward zero with increasing temperature. (b) Results for the magnetized LL state at uniform flux ${\phi_u = 2\pi/76}$ and field ${B = 1}$. The $\kappa_{xy}/T$ vanishes in the zero-temperature limit due to the cancellation of the Chern numbers from spin-$\uparrow$ and spin-$\downarrow$ sectors.