Theory of magnetic phase diagrams in hyperhoneycomb and harmonic-honeycomb iridates

Motivated by recent experiments, we consider a generic spin model in the $j_{\mathrm{eff}}=1/2$ basis for the hyperhoneycomb and harmonic-honeycomb iridates. Based on microscopic considerations, the effect of an additional bond-dependent anisotropic spin exchange interaction $\left(\Gamma \right)$ beyond the Heisenberg-Kitaev model is investigated. We obtain the magnetic phase diagrams of the hyperhoneycomb and harmonic-honeycomb $\left(\mathcal{H}\text{−}1\right)$ lattices via a combination of the Luttinger-Tisza approximation, $\text{single}-\mathbf{Q}$ variational ansatz, and classical Monte Carlo simulated annealing. The resulting phase diagrams on both systems show the existence of incommensurate, noncoplanar spiral magnetic orders as well as other commensurate magnetic orders. The spiral orders show counterpropagating spiral patterns, which may be favorably compared to recent experimental results on both iridates. The parameter regimes of various magnetic orders and ordering wave vectors are quite similar in both systems. We discuss the implications of our work for recent experiments and also compare our results to those of the two-dimensional honeycomb iridate systems.

Figure 1

Ideal iridium ion network in the hyperhoneycomb and $\mathcal{H}\text{−}1$ lattices. The $x$, $y$, and $z$ bonds are colored red, green, and blue, respectively. The bridging sites are colored cyan in panel (b), and the $z$ bonds connecting adjacent bridging sites are the bridging $z$ bonds. The gray dashed boxes are the conventional unit cells. The conventional lattice vectors—$a$, $b$, and $c$—are marked accordingly.

Figure 2

Classical phase diagrams for the $J\text{−}K\text{−}\Gamma$ pseudospin model with $\Gamma \ge 0$. The parametrization of the exchange interactions can be found in Eq. (4). A detailed description of the phases can be found in Sec. 5 while a summary can be found in Table 1. The color contours are guides for the eye: in the case of spiral (SP) states, they represent the length of the $\mathbf{Q}$ vector, whereas in the case of nonspiral states, they represent properties relevant to that particular phase; see Sec. 5 for details.

Figure 3

Real-space spin configurations of the ${\mathrm{SP}}_{a^{}}$ spiral states obtained from simulated annealing with spins projected on to the $ac$ plane. The chosen parameter points yield $\mathbf{Q}=(0.33,0,0)$; however, we note that the $\mathbf{Q}$ vectors in these phases are generally incommensurate and are not the same as the peak positions of the structure factor (see main text and Figs. 9 and 10). The black dashed boxes enclose the conventional unit cells. Identical colors indicate that the sublattices share spiral planes. Shades of blue indicate that the spiral planes are aligned with the honeycomb planes while shades of red indicate that the spiral planes are not aligned with the honeycomb planes. The handedness of adjacent sites can be readily verified as being opposite: the spirals counterpropagate. Examples of the preserved rotation symmetry are indicated by the dotted blue lines.

Figure 4

Real-space spin configurations of the ${\mathrm{SP}}_{b^{}}$ spiral states obtained from simulated annealing with spins projected onto the $bc$ plane. The chosen parameter points yield $\mathbf{Q}=(0,0.33,0)$; however, we note that the $\mathbf{Q}$ vectors in these phases are generally incommensurate. See caption in Fig. 3 for further details. In the case of the ${\mathrm{SP}}_{b^{-}}$ phase, only the spiral planes and not the spins preserve the $C_2^b$ rotation symmetry.

Figure 5

Classical phase diagrams for the $J\text{−}K\text{−}\Gamma$ pseudospin model with $\Gamma \le 0$. The details of this phase diagram can be understood via a classical mapping that relates $(J,K,\Gamma )\to (-J,-K,-\Gamma )$; see Sec. 5c for details. The color contours are guides for the eye: in the case of spiral (SP) states, they represent the length of the $\mathbf{Q}$ vector, whereas in the case of nonspiral states, they represent properties relevant to that particular phase; see Sec. 5 for details.

Figure 6

Real-space spin configurations of the ${\overline{\text{SP}}}_{a^{}}$ spiral states obtained from simulated annealing with spins projected onto the $ac$ plane. The chosen parameter points yield $\mathbf{Q}=(0.33,0,0)$; however, we note that the $\mathbf{Q}$ vectors in these phases are generally incommensurate and are not the same as the peak positions of the structure factor (see main text and Figs. 11 and 12). The black dashed box encloses the conventional unit cell. Identical colors indicate that the sublattices share spiral planes. Shades of blue indicate that the spiral planes are aligned with the honeycomb planes while shades of red indicate that the spiral planes are not aligned with the honeycomb planes. The handedness of adjacent sites can be readily verified as being opposite: the spirals counterpropagate. Examples of the preserved rotation followed by time-reversal operation $({\overline{C}}_2^a=\Theta ·C_2^a$ where $\Theta$ is the time-reversal operation) are indicated by the dotted blue lines.

Figure 7

Real-space spin configurations of the ${\overline{\text{SP}}}_{b^{}}$ spiral states obtained from simulated annealing with spins projected onto the $bc$ plane. The chosen parameter points yield $\mathbf{Q}=(0,0.33,0)$; however, we note that the $\mathbf{Q}$ vectors in these phases are generally incommensurate. See caption in Fig. 6 for further details. ${\overline{C}}_2^b=\Theta ·C_2^b$ where $\Theta$ is the time-reversal operation. In the case of the ${\overline{\text{SP}}}_{b^{-}}$ phase, only the spiral planes and not the spins preserve the ${\overline{C}}_2^b$ operation.

Figure 8

Relative signs of $\Gamma$ chosen for the hyperhoneycomb and $\mathcal{H}\text{−}1$ lattices. The plus/minus sign on each bond corresponds to the sign in front of $\Gamma$ that appears in the Hamiltonian in Eq. (1). With this choice of relative signs in $\Gamma$, we are still free to choose either a positive or a negative value for $\Gamma$. The $x$, $y$, and $z$ bonds are colored red, green, and blue, respectively. The bridging sites are colored cyan. The gray dashed boxes are the conventional unit cells.

Figure 9

Static structure factors along the $\left(h0l\right)$ plane for the ${\mathrm{SP}}_{a^{}}$ phases presented in Fig. 3. These states are characterized by $\mathbf{Q}=(0.33,0,0)$. The darkness of the dots indicate the peaks' relative intensities. The $+$ spiral phases have peaks at $\mathbf{q}=({\mathbf{1}}_a+\mathbf{Q})=(1.33,0,0)$ while the $-$ spiral phases have peaks at $\mathbf{q}=({\mathbf{1}}_a-\mathbf{Q})=(0.66,0,0)$, where ${\mathbf{1}}_a=\left(100\right)$.

Figure 10

Static structure factors along the $\left(0kl\right)$ plane for the ${\mathrm{SP}}_{b^{}}$ phases presented in Fig. 4. These states are characterized by $\mathbf{Q}=(0,0.33,0)$. The darkness of the dots indicate the peaks' relative intensities. The $+$ spiral phases have peaks at $\mathbf{q}=({\mathbf{1}}_b+\mathbf{Q})=(0,1.33,0)$ while the $-$ spiral phase has a peak at $\mathbf{q}=({\mathbf{1}}_b-\mathbf{Q})=(0,0.66,0)$, where ${\mathbf{1}}_b=\left(010\right)$.

Figure 11

Static structure factors along the $\left(h0l\right)$ plane for the $\Gamma \le 0$ ${\overline{\text{SP}}}_{a^{}}$ phases presented in Fig. 6. These states are characterized by $\mathbf{Q}=(0.33,0,0)$. The darkness of the dots indicate the peaks' relative intensities.

Figure 12

Static structure factors along the $\left(0kl\right)$ plane for the $\Gamma \le 0$ ${\overline{\text{SP}}}_{b^{}}$ phases presented in Fig. 7. These states are characterized by $\mathbf{Q}=(0,0.33,0)$. The darkness of the dots indicates the peaks' relative intensities.