Functional renormalization group analysis of Dzyaloshinsky-Moriya and Heisenberg spin interactions on the kagome lattice

We investigate the effects of Dzyaloshinsky-Moriya (DM) interactions on the frustrated $J_1-J_2$ kagome-Heisenberg model using the pseudofermion functional renormalization group (PFFRG) technique. In order to treat the off-diagonal nature of DM interactions, we develop an extended PFFRG scheme. We benchmark this approach in parameter regimes that have previously been studied with other methods and find good agreement of the magnetic phase diagram. Particularly, finite DM interactions are found to stabilize all types of noncollinear magnetic orders of the $J_1-J_2$ Heisenberg model ($\mathbf{q}=0, \sqrt{3}\times \sqrt{3}$, and cuboc orders) and shrink the extents of magnetically disordered phases. We discuss our results in the light of the mineral herbertsmithite which has been experimentally predicted to host a quantum spin liquid at low temperatures. Our PFFRG data indicate that this material lies in close proximity to a quantum critical point. In parts of the experimentally relevant parameter regime for herbertsmithite, the spin-correlation profile is found to be in good qualitative agreement with recent inelastic-neutron-scattering data.

Figure 1

(a) Illustration of the DM interaction on the kagome lattice: arrows on the nearest-neighbor bonds indicate the bond orientation of the DM term $\sim {\mathbf{D}}_{ij}·\left({\mathbf{S}}_i\times {\mathbf{S}}_j\right)$. Each arrow starts at site $i$ and ends at site $j$. For up-pointing (down-pointing) triangles, the DM vector ${\mathbf{D}}_{ij}$ is oriented parallel (antiparallel) to the $z$ axis. (b) Classical phase diagram of the $J_1-J_2$ Heisenberg model featuring $\mathbf{q}=0$ Néel order, cuboc order, ferromagnetic order, and $\sqrt{3}\times \sqrt{3}$ Néel order: the transition between the cuboc and the ferromagnetic phase occurs at $J_2=-J_1/3$ with $J_1<0$ while all other phase transitions coincide with the $J_1$ or $J_2$ axis, respectively [41].

Figure 2

Diagrammatic representation of the FRG equations in Eqs. (5) and (6): the $m$-particle vertices are illustrated as gray shaded disks or polygons. Arrows with (without) a slash denote the single-scale propagator $S^{\mathrm{\Lambda }}$ (fully dressed propagator $G^{\mathrm{\Lambda }}$). The flow of the self-energy ${\mathrm{\Sigma }}^{\mathrm{\Lambda }}$ couples to itself via $S^{\mathrm{\Lambda }}$ and to the two-particle vertex ${\mathrm{\Gamma }}^{\mathrm{\Lambda }}$ (first line). In a similar fashion, the flow of the two-particle vertex ${\mathrm{\Gamma }}^{\mathrm{\Lambda }}$ couples to all $m$-particle vertices with $m\le 3$ (second line). This scheme repeats for higher vertices up to infinite order (not shown).

Figure 3

Flowing susceptibilities of the $J_1-D$ model treated within an RPA scheme: shown are the susceptibilities for $\sqrt{3}\times \sqrt{3}$ order (blue circles) and $\mathbf{q}=0$ order (green squares). Upper panel: at vanishing DM coupling, $\sqrt{3}\times \sqrt{3}$ order is found to be preferred over $\mathbf{q}=0$ order (dashed red lines illustrate the critical $\mathrm{\Lambda }$ scale at which the flow diverges first). Lower panel: at $D=0.01J_1$, the $\mathbf{q}=0$ order is dominant, indicating a change of magnetization at infinitesimal $D$. For the peak positions of these types of order in reciprocal space, see Fig. 4.

Figure 4

(a) Dominant peak positions for the four types of classical order in the $J_1-J_2$ kagome-Heisenberg model: ferromagnetic order (purple triangle), $\sqrt{3}\times \sqrt{3}$ order (blue circles), $\mathbf{q}=0$ order (green squares), and cuboc order (orange crosses) are shown within the boundaries of the extended Brillouin zone (black hexagon). The edges of the first Brillouin zone are depicted gray. (b) Competition between the $\mathbf{q}=0$ (green squares) and $\sqrt{3}\times \sqrt{3}$ (blue circles) susceptibilities as a function of $D$. The data correspond to $\mathrm{\Lambda }\approx 0.19$. Small symbols indicate that the flow has entered the symmetry-broken regime below the critical $\mathrm{\Lambda }$ where the $\text{SU}\left(2\right)$-invariant PFFRG approach is no longer valid. (c)–(f) Static spin susceptibilities ${\chi }^{xx,\mathrm{\Lambda }}\left(\mathbf{k}\right)$ for various DM interaction strengths. The black hexagon denotes the boundaries of the extended Brillouin zone. Note that, in magnetically disordered regimes such as (c), the plot corresponds to $\mathrm{\Lambda }=0$ while in (d), (e), and (f) the susceptibility is shown at a $\mathrm{\Lambda }$ value right above the $\mathbf{q}=0$ instability (indicated by arrows in Fig. 5).

Figure 5

$\mathrm{\Lambda }$ dependence of the static susceptibility ${\chi }^{xx,\mathrm{\Lambda }}\left(\mathbf{k}\right)$ for various DM-interaction strengths taken at the maximum in momentum space. Crosses (circles) indicate that the maximum in $\mathbf{k}$ space at the respective $\mathrm{\Lambda }$ and $D$ values is located at the $\sqrt{3}\times \sqrt{3}$ ($\mathbf{q}=0$) position. Instability features associated with critical $\mathrm{\Lambda }$ scales are marked by arrows. Below those scales, the susceptibilities are plotted by dashed lines.

Figure 6

Phase diagram of the $J_1-J_2-D$ model as a function of $\theta \in [0,2\pi )$ and various values of $D$: colored regions are the classically ordered phases of Fig. 1 while the white regimes are magnetically disordered. Uncertainties in the phase boundaries between magnetically ordered and nonmagnetic phases are indicated by light-colored stripes. The $\theta$ values of the phase boundaries are also listed in Table 1.

Figure 7

(a) Section of the phase diagram in the $J_2-D$ plane relevant for herbertsmithite: large (small) icons denote that the RG flow does (does not) signal long-range order. Black circles indicate regions of numerical uncertainties where we cannot reliably determine the magnetic properties. The ordered phase is dominated by $\mathbf{q}=0$ order [confer Fig. 4], whereas, in the paramagnetic regime, we find dominant $\mathbf{q}=0$ (green squares), $\sqrt{3}\times \sqrt{3}$ (blue circles), as well as incommensurate (red triangles) spin fluctuations. (b) RG flow of the maximal susceptibility for $(J_1,J_2,D)=(1,0.02,0.1)$ [see red circle in (a)] showing a weak instability feature (arrow): as in Fig. 5, crosses (circles) indicate that the maximum in momentum space resides at the $\sqrt{3}\times \sqrt{3}$ ($\mathbf{q}=0$) position. The inset shows the susceptibility in $\mathbf{k}$ space. Small maxima at the midpoints of the extended Brillouin zone's edges (corresponding to $\mathbf{q}=0$ correlations) are in agreement with low-energy inelastic neutron-scattering data [30].

Figure 8

(a) Approximation scheme for the $\mathrm{\Lambda }$-dependent susceptibility: we divide a certain interval into two sectors, I and II, which lie between two adjacent pairs of susceptibility peaks (or susceptibility upturns). Within each sector, ${\chi }^{xx,\mathrm{\Lambda }}\left(\mathbf{k}\right)$ is approximated by a tangent connecting neighboring peaks. While the angle between both tangents is negligible for $D/J_1=0$, we find a finite value for $D/J_1=0.2$ indicating that both curves belong to different phases. A more quantitative measure for the size of the kink is obtained by considering tangents between additional pairs of adjacent peaks in a fixed $\mathrm{\Lambda }$ interval and calculating the average angle $\overline{\epsilon }$. (b) Averaged angle $\overline{\epsilon }$ as a function of $D/J_1$. While $\overline{\epsilon }$ is almost constant for $D/J_1<0.1$, there is a pronounced increase for higher values. Based on this behavior, we estimate the phase transition of the $J_1-D$ model to be at $D=\left(0.12\pm 0.02\right)J_1$ (blue shaded area).