Anisotropic exchange and noncollinear antiferromagnets on a noncentrosymmetric fcc half-Heusler structure

One of the signatures of the face-centered cubic (fcc) antiferromagnet as a typical example of a geometrically frustrated system is the large ground state degeneracy of the classical nearest-neighbor and next-nearest-neighbor Heisenberg (isotropic) model on this lattice. In particular, collinear states are degenerate with noncollinear and noncoplanar ones: this degeneracy is accidental and is expected to be lifted by anisotropic exchange interactions. In this work, we derive the most general nearest and next-nearest-neighbor exchange model allowed by the space-group symmetry of the noncentrosymmetric half-Heusler compounds, which includes three anisotropic terms: the so-called Kitaev, Gamma, and Dzyaloshinskii-Moriya interactions—most notably, the latter is allowed by the breaking of inversion symmetry in these materials and has not been previously been studied in the context of the fcc lattice. We compute the resulting phase diagram, show how the different terms lift the ground state degeneracy of the isotropic model, and lay emphasis on finding regimes where multi-$q$ (noncollinear/noncoplanar) states are selected by anisotropy. We then discuss the role of quantum fluctuations and the coupling to a magnetic field in the ground state selection and show that these effects can stabilize noncoplanar (triple-$q$) states. These results suggest that some half-Heusler antiferromagnets might host rare noncollinear/noncoplanar orders, which may, in turn, explain the unusual transport properties detected in these semimetals.

Figure 1

[(a) and (b)] Two enantiomer crystal structures of the half-Heusler compound ABC. They differ by the location of the C sites which break the inversion symmetry. The nearest-neighbor bonds are shown in green, blue, and red according to their $\gamma$ axes: $x$, $y$, and $z$, respectively. [(c) and (d)] Dzyaloshinskii-Moriya vectors ${\mathbf{D}}_{ij}$ for the bonds of an elementary tetrahedron. The four sites are labeled by $i,j\in \{0,1,2,3\}$ and the bonds are oriented as $i\to j$ with the convention $i<j$. The two inequivalent configurations of DM vectors [(c) and (d)] are related by ${\mathbf{D}}_{ij}\to -{\mathbf{D}}_{ij}$ and can be associated with the two enantiomers [(a) and (b)].

Figure 2

Luttinger-Tisza phase diagrams for the anisotropic model with $J_1>0$. We consider the $J_1\text{−}{J}_2$ Heisenberg interactions and we vary separately the relative strength of (a) the Kitaev coupling, at $\mathrm{\Gamma },D=0$, (b) the Gamma coupling, at $K,D=0$, and (c) the Dzyaloshinskii-Moriya coupling, at $K,\mathrm{\Gamma }=0$. In some regions of the diagrams, anisotropy lifts the degeneracy between single-$q$ and multi-$q$ states (as described in Sec. 4). In this case, we indicate the corresponding ground state configuration ($1q$, $2q$,...). (d) Luttinger-Tisza wave vectors associated with the different phases: the commensurate AFM orders I, II, III, and the incommensurate spiral phases (labeled i1,..., i5). Here $q$ and $p$ are fixed and unique (up to lattice symmetries) but vary continuously within the incommensurate phases with the parameters of the model. All wave vectors are expressed in units of $2/a$, where $a$ is the cubic unit cell parameter.

Figure 3

Selection of cuts of the Luttinger-Tisza phase diagram in the five-dimensional parameter space: $(K,\mathrm{\Gamma })$ plane (upper row), $(K,D)$ plane (middle row), and in the $(\mathrm{\Gamma },D)$ plane (bottom row). For the $(K,\mathrm{\Gamma })$ plane, several values of the DM coupling $D$ were considered. For the $(K,D)$ and $(\mathrm{\Gamma },D)$ planes, three different values of $J_2$ were chosen, namely ferromagnetic (left), vanishing (middle), antiferromagnetic (right). The phases are labeled like in Fig. 2.

Figure 4

Schematic representation of the ground states and degeneracies of the model, in the presence of anisotropy. In each case, the $1q$, $2q$, $3q$, and $4q$ levels represents only the lowest-energy state of the equal-weight single-$q$, double-$q$, triple-$q$, and quadruple-$q$ manifold, respectively.

Figure 5

Transition from a single-$q$ type-I state to a noncollinear single-$q$ type-III state upon increasing the strength of the DM interactions $D$ (with fixed $J_2<0$). (a) Path (dotted line) followed in the phase diagram upon showing the states in (b). In the intermediate state i2, the spiral wave vector is $\mathbf{q}=(\pi ,0,Q)$ with $Q=arcsin(D/J_2)$. (b) The I-III phase transition can be seen as a continuous rotation of the spins in the ground state, $z$ layer by $z$ layer. $Q$ is the angle between neighboring layers.

Figure 6

Minimal energy of the type-III single-$q$, double-$q$, and triple-$q$ equal-weight manifolds for $D\ne 0,K\ne 0$.

Figure 7

Properties of the type-I ground state in the presence of a magnetization and Kitaev coupling $K<0$, as a function of the strength $m$ and the direction of the magnetization $\mathbf{m}$. (a) Spin configurations in the canted double-$q$, triple-$q$, and single-$q$ states, stabilized when the magnetization is along [001], [111], and [110], respectively. Top row: $m=0$. Bottom row: $m=0.4$. (b) Polar plot of the chirality $\kappa$ [whose square is defined in Eq. (26)] per site. (a) Polar plot of the indicator $\eta$ [defined in Eq. (29)].

Figure 8

Luttinger-Tisza phase diagram of the model in the $(K,\mathrm{\Gamma })$ plane (with $J_2=0$ and $D$ taking three different values), in which the quantum fluctuations were taken into account for the type-I phase. The stability regions of the fluctuation-driven triple-$q$ type-I state are indicated by white stripes in the phase diagram.

Figure 9

Phase diagram of the Heisenberg $J_1\text{−}{J}_2$ model on the fcc lattice. It contains a ferromagnetic phase (FM), three commensurate antiferromagnetic (AFM) phases labeled type-I, type-II, and type-III orders, as well as an incommensurate (inc) phase with wave vector $(\pi ,q,0)$ ($q$ is arbitrary) on the semi-infinite line $J_2=0,J_1>0$. Adapted from Ref. [1].

Figure 10

Examples of spin arrangements for the fcc antiferromagnetic orders I, II and III stabilized in the $J_1\text{−}{J}_2$ phase diagram. The multi-$q$ states shows here are ‘equal-weight’ states. Below each spin configuration, the corresponding ordering wave vectors are written in units of $2/a$, where $a$ is the cubic unit cell parameter, as well as the number of sites in the magnetic unit cell, $N_{\text{u.c.}}$.