Emergent quantum phases in a frustrated $J_1\text{−}{J}_2$ Heisenberg model on the hyperhoneycomb lattice

We investigate possible quantum ground states as well as the classical limit of a frustrated $J_1\text{−}{J}_2$ Heisenberg model on the three-dimensional (3D) hyperhoneycomb lattice. Our study is inspired by the recent discovery of $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$, where ${\mathrm{Ir}}^{4+}$ ions form a 3D network with each lattice site being connected to three nearest neighbors. We focus on the influence of magnetic frustration caused by the second-nearest-neighbor spin interactions. Such interactions are likely to be significant due to the large extent of $5d$ orbitals in iridates or other $5d$ transition metal oxides. In the classical limit, the ground state manifold is given by line degeneracies of the spiral magnetic-order wave vectors when $J_2/J_1\gtrsim 0.17$ while the collinear stripy order is included in the degenerate manifold when $J_2/J_1=0.5$. Quantum order-by-disorder effects are studied using both the semiclassical $1/S$ expansion in the spin-wave theory and the Schwinger-boson approach. In general, certain coplanar spiral orders are chosen from the classical degenerate manifold for a large fraction of the phase diagram. Nonetheless quantum fluctuations favor the collinear stripy order over the spiral orders in an extended parameter region around $J_2/J_1=0.5$, despite the spin-rotation invariance of the underlying Hamiltonian. This is in contrast to the emergence of stripy order in the Heisenberg-Kitaev model studied earlier on the same lattice, where the Kitaev-type Ising interactions are important for stabilizing the stripy order. As quantum fluctuations become stronger, $U\left(1\right)$ and $Z_2$ quantum spin liquid phases are shown to arise via quantum disordering of the Néel, stripy, and spiral magnetically ordered phases. The effects of magnetic anisotropy and their relevance to future experiments are also discussed.

Figure 1

The hyperhoneycomb lattice structure with tricoordinated four sublattices (yellow, blue, green, and red spheres labeled by 0, 1, 2, and 3, respectively). ${\mathbf{a}}_i$ ($i=1,2,3$, red arrows) denote the primitive lattice vectors for the face-centered orthorhombic Bravais lattice. Black solid lines show the nearest-neighbor bonds and red dashed lines indicate the next-nearest neighbors that are connected via two nearest-neighbor bonds. Green dotted line has the same length as the distance between the next-nearest neighbors connected by red dashed lines, but two sites coupled by green dotted lines are not connected via two nearest-neighbor bonds.

Figure 2

Ordering wave vectors of the classical solutions obtained in the Luttinger-Tisza analysis with the soft spin constraint are shown as degenerate surfaces (blue) for (a) $J_2/J_1=0.2$, (b) $J_2/J_1=0.3$, (c) $J_2/J_1=0.5$, and (d) $J_2/J_1=0.7$. Black lines on the degenerate surfaces denote the ordering wave vectors of the classical spin states that satisfy the hard spin constraint. These degenerate lines of wave vectors are consistent with the results of the single-$\mathbf{Q}$ variational method (see Sec. 3b). Green spheres represent the selected ordering wave vectors in the presence of zero-point quantum fluctuations (see Sec. 4a for details).

Figure 3

Magnetic phase diagram of the $J_1\text{−}{J}_2$ Heisenberg model on the hyperhoneycomb lattice as a function of inverse spin magnitude $1/S$ and the ratio $J_2/J_1$, where zero-point quantum fluctuations are included in the linear spin-wave theory. Notice that the collinear orders such as the stripy and Néel phases become the ground states for a wider region of parameter space as quantum fluctuations become stronger or $1/S$ becomes bigger.

Figure 4

Classical spin configurations and Schwinger-boson mean-field parameters. Spin moments are represented by arrows. Mean-field parameters ${\eta }_n(n=1,2,3,4)$ in the Schwinger-boson theory are denoted by red, green, blue, and magenta lines, respectively. Solid lines correspond to antiferromagnetic correlations (${\eta }_n>0$) between two sites and dashed lines represent ferromagnetic arrangements (${\eta }_n=0$). For clarity, we show ${\eta }_{3,4}$ only on two $J_2$ bonds in each figure with the rest of them being omitted.

Figure 5

Mean-field phase diagram in the Schwinger-boson theory. At small $1/\kappa$, quantum order-by-disorder effects select the Néel, $\Gamma Z$ spiral, stripy, and $\Gamma XY$ spiral phases. When quantum fluctuations become stronger (or $1/\kappa$ becomes bigger), $U\left(1\right)$ and $Z_2$ quantum spin liquid phases arise. Thick solid (red thin) lines represent first (second) order phase transitions.

Figure 6

Mean-field parameters, $\{{\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4\}$, as functions of $J_2/J_1$ at $1/\kappa =2$ (magnetic-order regime) and $1/\kappa =7$ (spin-liquid regime).

Figure 7

Lower edge of the two-spinon spectrum in the $Z_2$ spin liquid. The spectrum is calculated at $J_2/J_1=0.84$ and $1/\kappa =7$ in the phase diagram of Fig. 5. Three plots show the spectra in three different planes in the first Brillouin zone: the $\Gamma XY$, $\Gamma YZ$, and $\Gamma XZ$ planes.

Figure 8

Directions of ${\widehat{\mathbf{e}}}_3$ chosen by possible magnetic anisotropies, which are normal to the spiral plane. (a) and (b) show the cases of the $\Gamma Z$ or $\Gamma Y$ spiral states and $\Gamma X$ spiral state, respectively (we take the limit $c_1=0$ for the latter case). The symbol $\perp \left[111\right]$ indicates that any ${\widehat{\mathbf{e}}}_3$ normal to $\left[111\right]$ is possible. The parameters $a$ and $b$ represent continuous change of ${\widehat{\mathbf{e}}}_3$ with $b>a$. The $\Gamma Z$ and $\Gamma Y$ spiral states select (001), (110), and $\left(1\overline{1}0\right)$ as spiral planes, whereas, for the $\Gamma X$ spiral states, the spiral plane is not completely determined and there are a number of possible choices depending on the anisotropy parameters $c_i$.

Figure 9

Red dots indicate high symmetric points in the Brillouin zone, and their coordinates are given by $\Gamma =(0,0,0)$, $Z=(-\frac{\pi }{6},-\frac{\pi }{6},0)$, $T=(-\frac{\pi }{6},-\frac{\pi }{6},-\frac{\pi }{2})$, $X_1=(-\frac{19\pi }{72},-\frac{5\pi }{72},-\frac{\pi }{2})$, $Y=(0,0,-\frac{\pi }{2})$, $A_1=(\frac{11\pi }{72},-\frac{11\pi }{72},-\frac{\pi }{2})$, $X=(\frac{29\pi }{72},-\frac{29\pi }{72},0)$, $L=(\frac{\pi }{6},-\frac{\pi }{3},-\frac{\pi }{4})$, and $A=(\frac{13\pi }{72},-\frac{37\pi }{72},0)$. Three primitive vectors of the reciprocal lattice are given by ${\mathbf{b}}_1=(\frac{\pi }{3},-\frac{2\pi }{3},\frac{\pi }{2})$, ${\mathbf{b}}_2=(-\frac{2\pi }{3},\frac{\pi }{3},-\frac{\pi }{2})$, and ${\mathbf{b}}_3=(\frac{2\pi }{3},-\frac{\pi }{3},-\frac{\pi }{2})$, respectively.