Gauge flux generations of weakly magnetized Dirac spin liquid in a kagomé lattice

TL;DR

This work analyzes how a U($1$) Dirac spin liquid on the kagomé lattice responds to weak magnetic perturbations, showing both perturbative flux generation via DM interactions and Zeeman coupling and a non-perturbative spontaneous flux generation that stabilizes a massive spinon Landau level state. By combining mean-field theory with renormalized mean-field theory, it demonstrates that the DM–Zeeman mechanism induces a staggered flux that opens a spinon mass gap, while the system energetically favors a uniform spontaneous flux state forming massive Landau levels and leading to an in-plane antiferromagnetic order with a gapless gauge photon. The study also characterizes spinon continua and Berry-curvature–driven thermal Hall signals in the perturbed DSL and confirms RMFT relationships among fluxes, DM strength, and magnetization. Altogether, it reveals a coherent picture of matter–gauge coupling in a lattice U($1$) gauge theory, with implications for experimental probes and broader applicability to other matter–gauge systems.

Abstract

Inspired by the recent progress on the Dirac spin liquid and the kagomé lattice antiferromagnets, we revisit the U(1) Dirac spin liquid on the kagomé lattice and consider the response of this quantum state to the weak magnetic field by examining the matter-gauge coupling. Even though the system is in the strong Mott insulating regime, the Zeeman coupling could induce the internal U(1) gauge flux with the assistance of the Dzyaloshinskii-Moriya interaction. In addition to the perturbatively-induced non-uniform flux from the microscopic interactions, the system spontaneously generates the uniform U(1) gauge flux in a non-perturbative fashion to create the spinon Landau levels and thus gains the kinetic energy for the spinon matters. Renormalized mean-field theory is employed to validate these two flux generation mechanisms. The resulting state is argued to be an ordered antiferromagnet with the in-plane magnetic order, and the gapless Goldstone mode behaves like the gapless gauge boson and the spinons appear at higher energies. The dynamic properties of this antiferromagnet, and the implication for other matter-gauge-coupled systems are discussed.

Figures (13)

• Figure 1: Dzyaloshinskii-Moriya vectors on the kagomé lattice, with the in-plane ($D_\parallel$) and out-of-plane ($D_z$) components (the black arrow indicates the bond directionality (from site $j$ to $i$) that defines vector $\mathbf D_{ij}$.
• Figure 2: The U(1) gauge flux distribution and energy bands of $[0,\pi]$ DSL. (a) The flux pattern for $[0,\pi]$ DSL state. $t_{ij}=\eta t$, ${\eta=\pm 1}$ for bold (dashed) lines. $\bf{a_1,a_2}$ is the enlarged unit cell basis vectors. (b) Brillouin zone (grey rectangle region) for the enlarged unit cell. (c) Energy bands of $[0,\pi]$ state along $M-M'$ line for one spin sector.
• Figure 3: The U(1) gauge flux distribution and energy bands of one spin sector for the $[\phi,\pi-2\phi]$ flux pattern. (a) $[\phi,\pi-2\phi]$ flux pattern, where $t_{ij}=\eta_{ij} te^{i\zeta_{ij}\phi/3}$ and $\eta_{ij}=\pm 1$ for bold (dashed) lines. $\zeta_{ij}=1$ if the hopping from $j$ to $i$ is along the arrows and $\zeta_{ij}=-\zeta_{ji}$. (b) Gapped energy bands along the $M-M'$ line. When $\phi\neq 0$, the Dirac points near the Fermi surface will open a gap $E_g$ and all the bands become topological with a finite Chern number.
• Figure 4: Spinon energy bands and Fermi pockets with finite staggered flux $\phi$ and a finite magnetization. (a) Spinon energy bands along $M-M'$ line of the magnetized $[\phi,\pi-2\phi]$ state. Blue (red) bands are the bands of the spin-up (down) sector. (b) The nearest spin-up and spin-down bands near the Fermi surface cross each other, forming Fermi pockets and resulting in finite magnetization. A relatively large $B$ field is chosen such that the band splitting and the outcome are more visible, and this applies to the remaining figures.
• Figure 5: Spin dynamic structure factor $\mathcal{S}^{-+}$ of massive FP state ( magnetized $[\phi,\pi-2\phi]$ state ). (a) The continuum spectrum of $\mathcal{S}^{-+}$ along $X$-$\Gamma$-$M$ line. (b) $S^{-+}$ with constant energy $\omega/t=0.2$. (c) The momentum dependence of $S^{-+}( q, \omega)$ at $\omega /t =0.2$ with $B/t=0.6$. There are concentric ring signals at $\Gamma, M$ points.