Order-by-disorder and magnetic field response in the Heisenberg-Kitaev model on a hyperhoneycomb lattice

We study the finite-temperature phase diagram of the Heisenberg-Kitaev model on a three-dimensional hyperhoneycomb lattice. Using semiclassical analysis and classical Monte Carlo simulations, we investigate quantum and thermal order-by-disorder, as well as the magnetic-ordering temperature. We find the parameter regime where quantum and thermal fluctuations favor different magnetic orders, which leads to an additional finite-temperature phase transition within the ordered phase. This transition, however, occurs at a relatively low temperature and the entropic effects may dominate most of the finite-temperature region below the ordering temperature. In addition, we explore the magnetization process in the presence of a magnetic field and discover spin-flop transitions which are sensitive to the applied-field direction. We discuss implications of our results for future experiments on a hyperhoneycomb lattice system such as the recently discovered $\beta$-Li${}_2$IrO${}_3$.

Figure 1

The $\left(111\right)$-skew-stripy phase on the hyperhoneycomb lattice with the coordinate system shown on the right. Each Ir site is described by a localized $j_{\text{eff}}=1/2$ pseudospin and is connected with three nearest neighbors. The $\alpha$ links are colored red, green, and blue for the $x$, $y$, and $z$ bonds, respectively. The red, orange, green, and blue pseudospins point in the $(1,1,1)$, $(-1,-1,1)$, $(-1,1,-1)$, and $(1,-1,-1)$ directions. Note that these pseudospins are noncoplanar and form an 8-sublattice structure, which is a generic feature of the skew-stripy and skew-zigzag phases. On the other hand, the $\left(100\right)$, $\left(010\right)$, and $\left(001\right)$ skew-stripy/skew-zigzag states (not shown) are collinear spin orders, while $\left(n_x{n}_{y}0\right)$, $\left(0n_y{n}_z\right)$, and $\left(n_{x}0n_z\right)$ skew-stripy/skew-zigzag states (not shown) are coplanar states.

Figure 2

Finite-temperature phase diagram of classical HK model obtained within MC simulation. For both cases of FM $J$ ($J<0$, lower horizontal axis) and AF $J$ ($J>0$, upper horizontal axis), there are two magnetic orders : FM and skew-zigzag phase for FM $J$, Néel and skew-stripy phase for AF $J$. The blue line repres-ents magnetic-ordering temperature and the red solid line is the phase boundary where the two distinct phases are degenerate. The two black dotted lines are for the exact solvable (anti)ferromagnetic Heisenberg model. Thermal ObD favors collinear FM/skew-zigzag phase (or Néel/skew-stripy phase) with spins pointing along (001) direction, when $J$ and $-K$ have opposite signs (i.e., frustrated case) in both unrotated and rotated basis. Otherwise, ordered phases have collinear spin states with spins pointing along (100) direction. The errors in the calculated ordering temperatures are smaller than the size of indicated data points.

Figure 3

Finite-temperature phase diagram obtained within the Holstein-Primakoff linear spin-wave approximation with $S=1/2$. The top and bottom horizontal axes are for the antiferromagnetic $(J>0)$ and ferromagnetic $(J<0)$ Heisenberg exchanges, respectively. These two axes are related by a four-sublattice rotation (see Sec. 2). The Néel and skew-stripy labeling of the left and right regions apply to the top axis $(J>0)$ while zigzag and ferromagnet (FM) apply to the bottom axis $(J<0)$. The first-order phase boundary between these regions is indicated by a solid vertical line. Phase boundaries between different states selected by thermal ObD within each region are indicated by dashed lines. The first-order phase boundary between the thermal ObD-selected and quantum ObD-selected states appears in the lower right of the phase diagram. The estimated ordering temperature is given by the curve between the paramagnetic (PM) region and the magnetically ordered regions. See main text for details on the ObD mechanism and the nature of the selected states.

Figure 4

Magnetization curve as a function of $h/J$ for $\mathbf{h}=h/\sqrt{3}\left(111\right)$, $K/J=4$, and $J>0$ [in (100)-skew-stripy phase]. Red, purple, blue points are MC results for $T/J=0.05,0.1,0.5$ (lines are drawn as a guide to the eye) and the black line is from numerical energy minimization of $H_{\mathbf{h}}$ for $T/J=0$. The spin-flop transition is present at the saturation field $h_{\mathrm{sat}}/J\approx 2.7$.

Figure 5

Plot of saturation field $h_{\mathrm{sat}}/J$ at zero temperature as a function of $K/J$ within the skew-stripy phase. Black and blue lines are the saturation field when the field is applied in the (100), (111) directions, respectively, based on the energy minimization of $H_{\mathbf{h}}$ at indicated points. The red line is the angle-averaged saturation field obtained by the linear spin-wave calculation.

Figure 6

Magnetization curve as a function of $h/J$ for $\mathbf{h}=h/\sqrt{3}\left(111\right)$, $K/\left|J\right|=-4$, and $J<0$ [in (100)-skew-zigzag phase]. Red, purple, blue points are MC results (lines are drawn as a guide to the eye) and the black line is from numerical energy minimization of $H_{\mathbf{h}}$ for $T/J=0$. The spin-flop transition is present at the saturation field $h_{\mathrm{sat}}/J\approx 4.3$.

Figure 7

Plot of saturation field $h_{\mathrm{sat}}/J$ at zero temperature as a function of $K/\left|J\right|$ within the skew-zigzag phase. Black and blue lines are the saturation field when the field is applied to the (100), (111) directions, respectively, based on the energy minimization of $H_{\mathbf{h}}$ at indicated points. The red line is the angle-averaged saturation field obtained by linear spin-wave calculation.

Figure 8

Field-direction dependence of saturation field ($h_{\text{sat}}$). The color maps indicate the fraction of the saturation field compared to the maximum attained in directions perpendicular to the cubic axes. Red contours are shown indicating $h_{\text{sat}}/max\left(h_{\text{sat}}\right)$ values of 0.9 and 0.8. Panels (a) and (b) are for the skew-stripy phase at $K=1.5\left|J\right|$ and $K=3.0\left|J\right|$, respectively. Notice the solid angle with low saturation field grows as $\left|K\right|$ increases. Panels (c) and (d) are for the skew-zigzag phase at $K=-2.0\left|J\right|$ and $K=-4.0\left|J\right|$. Notice the opposite trend: the solid angle with low saturation field decreases as $\left|K\right|$ increases.

Figure 9

Calculated free energy from the quadratic order of fluctuating fields $\varphi ,\chi$ for different FM ordering vectors: Red, green, blue, and purple colored lines are free energies for FM order $\overline{\mathbf{S}}//\left(001\right),\left(100\right),\left(110\right),\left(111\right)$, respectively. Free energy differences between $\overline{\mathbf{S}}=\left(001\right)$ and $\overline{\mathbf{S}}=\left(100\right)$ are replotted in Fig. 10 for a better resolution.

Figure 10

Plot of free-energy difference between the two different FM orders, $\Delta F/T=(F_{\overline{\mathbf{S}}=\left(001\right)}-F_{\overline{\mathbf{S}}=\left(100\right)})/T$. Thermal ObD selects FM order along $\left(001\right)$ for $K/\left|J\right|<0$, whereas, it favors FM order along $\left(100\right)$ for $K/\left|J\right|>0$.