Quantifying the Phase Diagram and Hamiltonian of $S=1/2$ Kagome Antiferromagnets: Bridging Theory and Experiment

Abstract

Spin-$1/2$ kagome antiferromagnets are leading candidates for realizing quantum spin liquid (QSL) ground states. While QSL ground states are predicted for the pure Heisenberg model, understanding the robustness of the QSL to additional interactions that may be present in real materials is a forefront question in the field. Here we employ large-scale density-matrix renormalization group simulations to investigate the effects of next-nearest neighbor exchange couplings $J_2$ and Dzyaloshinskii-Moriya interactions $D$, which are relevant to understanding the prototypical kagome materials herbertsmithite and Zn-barlowite. By utilizing clusters as large as XC12 and extrapolating the results to the thermodynamic limit, we precisely delineate the scope of the QSL phase, which remains robust across an expanded parameter range of $J_2$ and $D$. Direct comparison of the simulated static and dynamic spin structure factors with inelastic neutron scattering reveals the parameter space of the Hamiltonians for herbertsmithite and Zn-barlowite, and, importantly, provides compelling evidence that both materials exist within the QSL phase. These results establish a powerful convergence of theory and experiment in this most elusive state of matter.

Figures (5)

• Figure 1: (a) Heisenberg $J_1$-$J_2$-$D$ model in Eq.(\ref{['Eq:Ham']}) on a kagome cylinder. Periodic and open boundary conditions are imposed, respectively, along the directions specified by the lattice basis vectors $e_2$ and $e_1$. The small triangle in the shaded region denotes a unit cell. $J_1$ and $J_2$ are NN and NNN spin exchange couplings, and $D$ is the DM interaction. (b) The first and extended Brillouin zones, the reciprocal lattice vectors $b_1$ and $b_2$, and the high-symmetry points: $\Gamma$, $\Gamma'$, $\rm K^\prime$, P (the midpoint of $\Gamma'$, $\rm K^\prime$), and $G=5b_2/6$. (c) Ground state phase diagram of the system as a function of $J_2$ and $D$. The solid symbols with error bars label the phase boundaries determined by DMRG calculations.
• Figure 2: Static spin structure factors $S(\mathbf{q})$ obtained from ground state DMRG simulations on XC12-9 cylinders for three characteristic sets of parameters: (a) $J_2=-0.1$ and $D=0$, the system exhibits the $\sqrt{3}\times\sqrt{3}$ magnetic order with sharp peaks in $S(\mathbf{q})$ at $\rm K'$; (b)$J_2=0$ and $D=0$, the system has a QSL ground state with diffuse $S(\mathbf{q})$; (c) $J_2=0.1$ and $D=0.1$, the system has $q=0$ order with sharp peaks in $S(\mathbf{q})$ at $\rm \Gamma'$; The first and extended Brillouin zones are indicated by the dashed hexagons. The results have been $D_6$ symmetrized. The color scale has an upper cutoff of 2.
• Figure 3: Squared $\mathbf{q}$=0 magnetic order parameter $m^2(\mathbf{Q}=\Gamma')$ as a function of (a) $J_2$ at $D=0$ and (b) $D$ at $J_2=0$. The shaded regions label the phase transitions between the QSL and $\mathbf{q}=0$ magnetically ordered phase. (c) Examples of finite-size extrapolations of $m^2(\mathbf{Q}=\Gamma')$ for different $D$ at $J_2=0$ using second-order polynomials in $1/\sqrt{N}$.
• Figure 4: Contour plot of the deviation $\delta R(J_2,D)$ (defined in Eq. \ref{['Eq:NeuSca']}) between DMRG simulations as a function of $J_2$ and $D$ on XC12 cylinders and neutron scattering data on (a) Zn-barlowite Breidenbach2025 and (b) herbertsmithite Han2012HBS_unpublished. The white dashed lines denote phase boundaries in Fig. \ref{['Fig:latt_and_phd']}. The best-fit regions of both materials fall within the QSL phase. The contour is plotted with interpolation and an upper cutoff of 1.5 in the color scale.
• Figure 5: The dynamic spin structure factor $S(\mathbf{q},\omega)$, comparing results from DDMRG simulations (form factor adjusted) and measured inelastic neutron scattering data in the $(HK0)$ plane for Zn-barlowite Breidenbach2025 for (a) $\hbar\omega=0.15J$ and (b) $\hbar\omega=0.4J$ at $T=1.7$ K. For the comparisons, the results are $D_6$-symmetrized and shown on relative intensity scales with the structural Brillouin zones overlaid.