Generic nearest-neighbor kagome model: XYZ and Dzyaloshinskii-Moriya couplings with comparison to the pyrochlore-lattice case

The kagome lattice is a paragon of geometrical frustration, long-studied for its association with novel ground states including spin liquids. Many recently synthesized kagome materials feature rare-earth ions, which may be expected to exhibit highly anisotropic exchange interactions. The consequences of this combination of strong exchange anisotropy and extreme geometrical frustration are yet to be fully understood. Here, we establish a general picture of the interactions and resulting ground states arising from nearest-neighbor exchange anisotropy on the kagome lattice. We determine a generic anisotropic exchange Hamiltonian from symmetry arguments. In the high-symmetry case where reflection in the kagome plane is a symmetry of the system, the generic nearest-neighbor Hamiltonian can be locally defined as an XYZ model with out-of-plane Dzyaloshinskii-Moriya interactions. We proceed to study its phase diagram in the classical limit, making use of an exact reformulation of the Hamiltonian in terms of irreducible representations (irreps) of the lattice symmetry group. This reformulation in terms of irreps naturally explains the threefold mapping between three families of models supporting spin liquids, as recently studied by the present authors [Nat. Commun. 7, 10297 (2016)]. In addition, a number of unusual states are stabilized in the regions where different forms of ground-state order compete, including a stripy phase with a local ${\mathbb{Z}}_8$ symmetry and a classical analog of a chiral spin liquid. As a peculiar property of the kagome lattice, the generic model turns out to be a fruitful hunting ground for the coexistence, in the same ground-state configuration, of multiple forms of long-range magnetic orders. In exotic instances, partial long-range order may also coexist in the ground state with a finite fraction of disordered spin degrees of freedom. These results are compared and contrasted with those obtained on the pyrochlore lattice, and connection is made with recent progress in understanding quantum models with $S=1/2$.

Figure 1

The up and down triangles of the kagome lattice are respectively colored in violet (A) and orange (B). The $x$ and $y$ axes are in plane, while the $z$ axis is out of plane, pointing towards the reader.

Figure 2

Graphical representations of the ${\sigma }_v$ reflections and $C_3$ rotations in the $C_{3v}$ symmetry group. The sublattices on a triangle, $\{S_0,S_1,S_2\}$, are labeled in the clockwise convention. Because of the antisymmetric Dzyaloshinskii-Moriya interaction [Eq. (23)], it is important to conserve the same clockwise convention for all triangles.

Figure 3

Local bases ${\mathcal{B}}^k$ used in Eq. (23) where the generic nearest-neighbor Hamiltonian on kagome takes the form of an XYZ model with Dzyaloshinskii-Moriya (DM) interactions. All $z_k$ axes are pointing out of plane. All $x_k$ ($y_k$) axes are pointing along (orthogonal to) their local bond.

Figure 4

Spin configurations corresponding to different irreducible representations, as expressed in Eqs. (28, 29, 30, 31, 32). $E_{\mathrm{FM}1},E_{\mathrm{FM}2}$, and $A_{2z}$ have saturated magnetization respectively along the $x,y$, and $z$ axes. $A_1$ and $A_{2\perp }$ have maximum negative vector chirality, while $E_{\mathrm{AF}}$ has maximum positive vector chirality [Eq. (79)]. When considered together, the $A_1$ and $A_{2\perp }$ irreps are labeled $A$ for convenience.

Figure 5

Zero-temperature phase diagram of the XXZ model with Dzyaloshinskii-Moriya interactions (XXZDM). The phase diagram can be divided into five continuous regions when represented on a sphere (a), with spherical coordinates $(\theta ,\phi )$: $(J_x,D,J_z)=(cos\phi sin\theta ,-sin\phi sin\theta ,cos\theta )$. The threefold symmetry of the XXZDM kagome model [52] is transparent in this representation. The left/right hemispheres correspond to ferromagnetic/antiferromagnetic coupling $J_z$, which can be projected onto two planar phase diagrams for a quantitative comparison, respectively, (b) and (c). The $E_z$ region corresponds to the multiphase ground state discussed in Secs. 3f and 4b2. $A_{2z}$ and $E_z$ regions take the form of triangles centered at the “poles” of each hemisphere. The latter is noticeably smaller because of its comparatively high antiferromagnetic frustration, making it less energetically stable when competing with the surrounding ordered phases. The name of specific models with extensive degeneracy are written in white, as defined in Ref. [52] and given in Table 2. For convenience, the corresponding spin configurations are reproduced in (d). The $A$ region corresponds to $A_1$ and $A_{2\perp }$ (Fig. 4), which have the same energy in the XXZDM model.

Figure 6

All spin configurations generated by the $(A,E_{\mathrm{AF}}),(E_{\mathrm{AF}},E_{\mathrm{FM}}),$ or $(E_{\mathrm{FM}},A)$ pair of irreps correspond to a three-color mapping on the kagome sites, or equivalently on the honeycomb bonds. While this coloring is straightforward in the former case [76] (top right), it is less so if the ferromagnetic $E_{\mathrm{FM}}$ states are involved (top left). The mapping is given by the matrix at the bottom. For example for a spin on sublattice 1 (respectively, 0 or 2) in a ferromagnetic triangle of violet color, the corresponding color is orange (respectively, violet, or cyan).

Figure 7

Regions of the XYZDM parameter space $(J_x,J_y$) where the lowest eigenvalue ${\lambda }_{I_0}$ corresponds to a given irrep $I_0\in \{A_1$ (blue), $A_{2\perp }$ (red), $A_{2z}$ (green), $E_{\min }$ (beige), $E_z$ ($\mathrm{white}\left)\right\}$. The top and bottom panels correspond to $J_z=1$ and $-0.5$, respectively. The values of $D$ are given at the top of the figure. For these values of $J_z$, the $E_z$ and $A_{2z}$ regions appear as ground states of the XYZDM model for $\left|D\right|<\sqrt{3}/2$. The $J_x=J_y$ line acts as a mirror in parameter space (Sec. 5a).

Figure 8

Ground states for an arbitrary value of coupling parameters sitting at the frontier between $A_{2\perp }$ and $E_{\min }$; $0<J_x=-\frac{J_y}{3},D=0,J_y/2<J_z<-J_y$, which corresponds to $\eta =-0.5$. See Eqs. (89, 90, 91, 92) for the spin configurations. The two states on the bottom right correspond to $A_{2\perp }$. Each of the $A_{2\perp }$ states can be “paired” with three other states, having one spin in common with the same orientation; see, for example, the violet spin of the two colored triangles. On the other hand, none of the six other states can be paired together. Hence the ${\mathbb{Z}}_8$ degeneracy is divisible between two exclusive groups of four states, centered around each of the $A_{2\perp }$ states.

Figure 9

Example of a stripe order emerging on the frontier between $E_{\min }$ and $A_{2\perp }$ irreps. The magenta and cyan triangles correspond, for example, to the two coloured states of Fig. 8. Spins on the “B” lines are long-range ordered all over the lattice. As for spins on the “A” lines, they are long-range ordered in one direction (horizontal) but disordered in the other direction (vertical) where they can randomly take one out of two possible orientations. By symmetry, the stripes can also be diagonal.

Figure 10

In the XYZDM model, when the $E_{\min }\in \{E_{\alpha },E_{\beta }\}$ irrep meets one of the one-dimensional antiferromagnetic irreps, $A_1$ or $A_{2\perp }$, there is a local ${\mathbb{Z}}_8$ degeneracy for each triangle [Fig. 8]. For special values of parameters on this frontier, there is an enhancement of the symmetry, where every ground state shares a common spin orientation with three other ground states. The four different sets of high-symmetry ground states are displayed here, together with their parameter region. Injecting the corresponding value of $\eta$ in Eqs. (89, 90, 91, 92) and (93)–(96) gives the expression of the spin configurations. The values of $\eta$ are uniquely determined by the coupling parameters $\{J_x,J_y,J_z,D\}$ [Eqs.(57) and (87)]. The two states on the right correspond to either $A_1$ or $A_{2\perp }$.

Figure 11

Example of a configuration with crossing stripes, a possible ground state for the parameters given in Eqs. (98) and (99). Each color corresponds to one of the eight degenerate ground states of Fig. 10. The grey lines are a guide to the eye for the position of the stripes. For each of the eight colors, the three edges of the triangle are covered by stripes in a different way.

Figure 12

Five-step illustration on how to calculate the entropy of the ensemble of crossing-stripe configurations. For the sake of clarity, the size of the lattice on step (5) is $L_x=16$ and $L_y=2$, which gives $N_{\mathrm{\Delta }}=L_x{L}_y=32$ triangles.

Figure 13

Projection (in green) of the ground-state configurations in the kagome plane, at the frontier between $E_{\min }$ and $E_z$, for $\eta =\pm 1$ [Eqs. (104) and (105)]. When expressed in the local bases ${\mathcal{B}}^{k=0,1,2}$ of Fig. 3, the O(2) invariance takes the form of a circle in spin space which lies entirely in the local ${(y,z)}_{k=0,1,2}$ ($\eta =+1$, left) or ${(x,z)}_{k=0,1,2}$ ($\eta =-1$, right) planes.

Figure 14

Sextets of ground states at the frontier between $E_{\min }$ and $E_z$ for $\eta =+1$ (top) and $\eta =-1$ (bottom) [Eqs. (104, 105, 106, 107)]. Each spin orientation appears in two different ground states. The paving of the kagome lattice by these six states is equivalent to a tricolouring problem, and the spins have been coloured accordingly. Here $s$ has been arbitrarily fixed to a value of $\pi$.

Figure 15

(a) Pyrochlore lattice whose minimal unit cell is a tetrahedron, made of four sublattices. Each sublattice possesses an easy axis connecting the center of the two neighboring tetrahedra. The spin component along this easy axis is noted $S^z$. [(b) and (c)] Ground states of Hamiltonian (113) for classical Heisenberg spins assuming a negative (b) and positive (c) value of $J_{zz}$. (b) is derived from the results of Ref. [12], together with the irrep decomposition illustrated in (d)–(h). (c) has been obtained in Ref. [102], while semiclassical and quantum mean field versions thereof were derived in Refs. [66, 103] and Ref. [104], respectively. (d) “all in all out” states transforming like the $A_2^{\mathrm{pyro}}$ irrep. (e) ${\mathrm{\Gamma }}_5$ states [105] transforming like the $E^{\mathrm{pyro}}$ irrep. (f) Palmer-Chalker states [106] transforming like the $T_2^{\mathrm{pyro}}$ irrep. (g) Canted, easy plane, ferromagnetic states transforming like the $T_1^{\mathrm{pyro}}$ irrep. We label the basis vector corresponding to this state as $T_{1,\text{pl}}^{\mathrm{pyro}}$. (h) Spin-ice state, with two spins pointing inside and two pointing outside the tetrahedron, which transforms like the $T_1^{\mathrm{pyro}}$ irrep. We label the basis vector corresponding to this state as $T_{1,\text{ice}}^{\mathrm{pyro}}$. Spins configurations in panels (d,h) and (e,g,h) lie respectively along their local easy axes and within their local easy planes. All results in this figure can be found in Refs. [12, 56, 66, 102, 103].