Deformed triangular lattice antiferromagnets in a magnetic field: Role of spatial anisotropy and Dzyaloshinskii-Moriya interactions

Recent experiments on the anisotropic spin-1/2 triangular antiferromagnet Cs${}_2$CuBr${}_4$ have revealed a remarkably rich phase diagram in applied magnetic fields, consisting of an unexpectedly large number of ordered phases. Motivated by this finding, we study the role of three ingredients—spatial anisotropy, Dzyaloshinskii-Moriya interactions, and quantum fluctuations—on the magnetization process of a triangular antiferromagnet, coming from the semiclassical limit. The richness of the problem stems from two key facts: (1) the classical isotropic model with a magnetic field exhibits a large accidental ground-state degeneracy and (2) these three ingredients compete with one another and split this degeneracy in opposing ways. Using a variety of complementary approaches, including extensive Monte Carlo numerics, spin-wave theory, and an analysis of Bose-Einstein condensation of magnons at high fields, we find that their interplay gives rise to a complex phase diagram consisting of numerous incommensurate and commensurate phases. Our results shed light on the observed phase diagram for Cs${}_2$CuBr${}_4$ and suggest a number of future theoretical and experimental directions that will be useful for obtaining a complete understanding of this material's interesting phenomenology.

Figure 1

Spatially anisotropic triangular lattice with exchange $J$ along horizontal bonds and $J^{\prime }$ along diagonal bonds. We define axes such that the sites lie in the $(x,y)$ plane as shown. Vectors ${\mathit{\delta }}_{1,2,3}$ connect nearest-neighbor sites of the lattice. In the isotropic limit where $J=J^{\prime }$ and Dzyaloshinskii-Moriya coupling is absent, the Hamiltonian (1) exhibits a highly degenerate classical ground-state manifold wherein spins order in an underconstrained three-sublattice pattern. We label the three sublattices by $A,B$, and $C$.

Figure 2

Important members of the classical ground-state manifold exhibited by Eq. (1) when $J=J^{\prime }$ and Dzyaloshinskii-Moriya coupling is absent: (a) Y, (b) up-up-down (UUD), (c) V, (d) distorted V, (e) inverted Y, and (f) umbrella states. The arrows denote the spin orientations on the $A$, $B$, and $C$ sublattices of Fig. 1.

Figure 3

Phase diagram of the classical isotropic model. Amongst the large set of accidentally degenerate classical ground states, thermal fluctuations select the Y, UUD, and V states illustrated in Figs. 2a through 2(c).

Figure 4

(a) Magnetization $M$ and (b) its field derivative $dM/dh$ in the vicinity of one-third magnetization. The data were obtained with Monte Carlo simulations of the classical isotropic model at temperature $T=0.2J$. The collinear UUD state underlies the drop in magnetic susceptibility near $h=2.5J$, visible in both (a) and (b). The dashed line in (a) shows the $T=0$ magnetization of the classical triangular antiferromagnet, $M=h/\left(9J\right)$.

Figure 5

Leading $1/S$ quantum energy splittings between the Y/UUD/V states and the inverted-Y (dashed line) and umbrella states (solid line). These curves illustrate an important trend—quantum fluctuations are most effective at splitting the accidental degeneracy near 1/3 magnetization where low-energy collinear states are available.

Figure 6

Energy differences between various states using the leading $1/S$ result from spin-wave theory (solid lines) and the biquadratic classical approximation (dashed lines). Green lines represent the difference between Y/UUD/V and inverted-Y energies, red between Y/UUD/V and umbrella, and black between inverted-Y and umbrella.

Figure 7

$dM/dh$ (blue circles) and coplanarity $K$ (green squares), defined in Eq. (11), for the classical “isotropic-plus-biquadratic” model at $T=0.027J$, obtained by annealing from high temperature.

Figure 8

Phase diagram of the quantum Heisenberg model with DM interactions, as a function of magnetic field $h$. The magnetic field orients along the DM axis, $\mathbf{D}\parallel \mathbf{h}$, in (a) and orthogonal to the DM axis, $\mathbf{D}\perp \mathbf{h}$, in (b).

Figure 9

Phase diagram of the model in Eq. (29). The fully polarized state corresponds to $\rho =0$, while all other states exhibit a finite boson density, $\rho >0$. Umbrella states correspond to $\phi =0$ and $\pi /2$, while the mixed state with $0<\phi <\pi /2$ represents a distorted-V phase. Observe that the origin of the phase diagram represents a tetracritical point.

Figure 10

Phase diagram of the classical anisotropic model with $J^{\prime }/J=0.765$. Interestingly, at this anisotropy strength the entropically favored planar phases are absent (within the resolution of our numerics), save for the collinear UUD state, which now requires intermediate temperatures to appear.

Figure 11

$dM/dh$ versus $h$ at $T=0.168J$ (green circles) and $T=0.12J$ (blue squares) for the classical anisotropic model with $J^{\prime }/J=0.765$.

Figure 12

Chiralities $\kappa$ (blue squares), ${\kappa }_{\perp }$ (red triangles), and ${\kappa }_z$ (green circles) versus $T$ at $h=2.3J$ for the classical anisotropic model with $J^{\prime }/J=0.765$. The discontinuous jumps near $T=0.13J$ signify a first-order transition between umbrella and UUD phases.

Figure 13

Ground-state phase diagram for the spatially anisotropic triangular antiferromagnet with quantum fluctuations modeled via the biquadratic approximation discussed in Sec. 3e. All data points were obtained with Monte Carlo simulated annealing numerics. Dashed lines represent phase boundaries, which are absent due to finite-size effects in our simulations but are expected on general grounds for an infinite system.

Figure 14

Classical phase diagram for spins with spatially anisotropic exchange interactions, DM coupling, and a magnetic field applied perpendicular to the DM vector. Phase boundaries were computed analytically to leading order in DM strength $D/J$ and anisotropy strength ${(J-J^{\prime })}^2/J^2$ as described in Appendix .

Figure 15

Magnetization vs field at temperature $T=0.001J$ for the spatially anisotropic antiferromagnet with $J^{\prime }=0.765J$, DM interactions satisfying $\mathbf{D}\perp \mathbf{h}$, and quantum effects modeled via a biquadratic interaction. The curves shown correspond to different values of the DM strength: $D=0$ (blue circles), $D=0.01J$ (green triangles), $D=0.02J$ (red stars), $D=0.04J$ (cyan squares), and $D=0.05J$ (pink hexagons). The quick rounding of the lower edge of the plateau reflects the symmetry equivalence of the Y and UUD states when DM interactions are present in this field orientation.

Figure 16

Sketch of the phase diagram for the quantum spatially anisotropic model with varying DM coupling strength, $J^{\prime }/J=0.765$, and a magnetic field $\mathbf{h}$ oriented perpendicular to the DM vector. Data points indicate phase boundaries determined using simulating annealing numerics with quantum fluctuations modeled via a biquadratic interaction. Dashed lines interpolate between these data points and are drawn for convenience. The location of the inverted-Y phase is approximate as discussed in the main text.