Three-dimensional quantum spin liquids in models of harmonic-honeycomb iridates and phase diagram in an infinite-$D$ approximation

Motivated by the recent synthesis of two insulating ${\mathrm{Li}}_2{\mathrm{IrO}}_3$ polymorphs, where ${\mathrm{Ir}}^{4+}$ $S_{\mathrm{eff}}=1/2$ moments form 3D (“harmonic”) honeycomb structures with threefold coordination, we study magnetic Hamiltonians on the resulting $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ hyperhoneycomb lattice and $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ stripyhoneycomb lattice. Experimentally measured magnetic susceptibilities suggest that Kitaev interactions, predicted for the ideal ${90}^{\circ }$ Ir-O-Ir bonds, are sizable in these materials. We first consider pure Kitaev interactions, which lead to an exactly soluble 3D quantum spin liquid (QSL) with emergent Majorana fermions and ${\mathrm{Z}}_2$ flux loops. Unlike 2D QSLs, the 3D QSL is stable to finite temperature, with $T_c\approx \left|K\right|/100$. On including Heisenberg couplings, exact solubility is lost. However, by noting that the shortest closed loop $\ell$ is relatively large in these structures, we construct an $\ell \to \infty$ approximation by defining the model on the Bethe lattice. The phase diagram of the Kitaev-Heisenberg model on this lattice is obtained directly in the thermodynamic limit, using tensor network states and the infinite-system time-evolving-block-decimation (iTEBD) algorithm. Both magnetically ordered and gapped QSL phases are found, the latter being identified by an entanglement fingerprint.

Figure 1

The stripyhoneycomb lattice of iridium in $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. The recently synthesized stripyhoneycomb lattice (space group No. 66 Cccm) has threefold coordinated sites, which form hexagons arranged in stripes of alternating orientation. It is the $n=1$ member of the harmonic-honeycomb [11] series of structures; the distinct hyperhoneycomb lattice of $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ (Fig. 2) has $n=0$. Parent orthorhombic coordinate system and unit cell (boxed) are shown. In the limit of superexchange via ideal oxygen octahedra, the magnetic Hamiltonian is dominated by Kitaev-type couplings $(x,y,z$ labels at bottom), leading to an exactly solvable model of a 3D quantum spin liquid.

Figure 2

The hyperhoneycomb lattice of Ir in $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. The hyperhoneycomb lattice (space group is No. 70 Fddd) has threefold coordinated sites and is the $n=0$ member of the harmonic-honeycomb structural series. Its shortest loops are decagons, motivating the large-$\ell$ loop length approximation for solving the frustrated quantum Hamiltonian on the 3D lattice. (Right) ${\mathbb{Z}}_2$ flux loops in the QSL phase. Selected bonds (dark orange) of type $z$ (top right) or $x,y$ (bottom right) are chosen to host nonzero vector potential $u_{i,j}=-1$ within the QSL ${\mathbb{Z}}_2$ gauge sector, producing a ${\mathbb{Z}}_2$ closed flux loop excitation, which encircles these bonds.

Figure 3

Strong Kitaev exchanges capturing $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ anisotropic susceptibility. Magnetic susceptibility (inset: inverse susceptibility) along principal axes $(z=b,x\pm y=a,c)$, measured [11] for a $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ crystal (bright lines) and theoretical mean-field fit (dark lines). Susceptibility is fitted by the minimal Hamiltonian (1) with parameters $(K_c,K_d,J_c,J_d)$ at $(-17,-7,6.3,0.8)$ meV; magnetic order (recently found [22] to be a noncoplanar spiral) is captured [22] by Eq. (1) at $(-15,-12,5,2.5)$ meV, supplemented by $c$-axis Ising exchange on $c$ bonds and $J_2$ Heisenberg exchange on second neighbors. In both cases, large Kitaev exchanges $K_c\mathrm{and}{K}_d$ are necessary to describe the material.

Figure 4

Quantum phase diagram in the large-$\ell$ limit. Phase diagram of the frustrated quantum Hamiltonian (1), computed via tensor network states within an infinite-$D$ or large-$\ell \to \infty$ approximation to the hyperhoneycomb's $\ell =10$. Except for quantitative extent of QSLs (not to scale), we expect it to describe the stripyhoneycomb and hyperhoneycomb lattices of $\gamma$- and $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$, for which we argue this is a physical model. The two-parameter space shown here [Eqs. (1) and (11)] has polar axes $r$ tuning symmetry allowed Kitaev bond anisotropy and $\varphi$ setting relative strength of Kitaev and Heisenberg interactions. The QSL phases, successfully stabilized on the Bethe lattice by the algorithm's finite entanglement cutoff $\chi$, were identified by an entanglement fingerprint.

Figure 5

Edge sharing ${\mathrm{IrO}}_6$ octahedra in the 3D lattices. Iridium (purple sphere) is coordinated by six oxygens forming vertices of an octahedra. When octahedra share edges as shown, the exchange pathways (dotted purple lines) give rise to Kitaev interactions, coupling a spin component $\gamma \in \{x,y,z\}$ normal to that Ir-Ir bond and to the shared edge (shown in corresponding color $\{\text{red}$, green, $\text{blue}\left\}\right)$. Octahedron at left shows the relation between the Ir-O $x,y,z$ axes and the crystallographic parent orthorhombic $a,b,c$ axes of the 3D lattices. The symmetry-distinguished $c$ axis is also a preferred axis for Ir-Ir bonds (thick blue bond shown); the perpendicular edge (X'ed out gray lines) is not shared by any two ${\mathrm{IrO}}_6$ octahedra. The $c$ bonds host $z=b$ Kitaev exchange.

Figure 6

Stripy-X magnetic pattern on the stripyhoneycomb lattice. The classical magnetic pattern associated with the stripy-X/Y AF ordered phase, here shown for stripy-X correlations. Spins, collinear along $S^x$ (white/gray sites correspond to $S^x$ up/down spins), are aligned along $x$-type (red) bonds and anti-aligned along $y$-type (green) and $z$-type (blue) bonds. The unit cell is doubled [ordering wave vector $(\pi ,\pi ,0)\text{]}$ to form the full unit cell of the parent orthorhombic $a,b,c$ axes. These stripy-X/Y correlations are predicted by the strong FM Kitaev exchanges necessary for the mean-field fit to the $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ magnetic susceptibility.

Figure 7

Nodal contour of Majorana fermions in the 3D QSL. In the gapless 3D Kitaev spin liquid phase, the emergent Majorana fermions are gapless at momentum points which form this 1D contour within the 3D momentum space. The contour is identical for the QSLs on the stripyhoneycomb lattice and on the hyperhoneycomb lattice; it is set by $\vec{k}·\vec{c}=0$ and $cos(\vec{k}·\vec{a}/2)+cos(\vec{k}·\vec{b}/2)=1/2$. with $a,b,c$ the parent orthorhombic axes (shown). Introducing coupling-strength bond anisotropy shrinks this contour until it collapses to a point and then gaps out, yielding the gapped spin liquid phase.

Figure 8

Kitaev spin liquid with bond anisotropy. The phase diagram of the 3D Kitaev spin liquids as a function of bond anisotropy, designating whether the emergent Majorana fermions are gapped or gapless, is identical on the 3D lattices and the 2D honeycomb model, and is independent of whether the anisotropy breaks or preserves lattice symmetry. Here the magnitudes of Kitaev coupling $|K_x|,|K_y|,\mathrm{and}|K_z|$ are given by the distance to the respective edges of the triangle. The vertical line corresponds to the symmetry-allowed bond-anisotropy, modifying $|K_z|$ by the parameter $a$ [Eq. (11)] with particular values shown.

Figure 9

Infinite-$D$ $z=3$ tree with eight-site iTEBD cell. Taking the $\ell \to \infty$ limit of the hyperhoneycomb lattice $\ell =10$ results in the $z=3$ tree, a lattice in infinite dimensions. Cutting any bond gives an entanglement bipartition (red dashed line), enabling entanglement-controlled computations with tensor network states. Bonds are labeled by Kitaev coupling; site labels show the unit cell used for the iTEBD computation. This eight-site unit cell (sites and bond Kitaev labels shown) used in the thermodynamic-limit iTEBD computation admits the Klein duality [24, 39, 54], and is thus expected to capture all the symmetry-breaking patterns in the phase diagram.

Figure 10

Kitaev-Heisenberg isotropic phase diagram via iTEBD. The magnetic order parameters of the four phases are directly observed by iTEBD, working in the thermodynamic limit. The stripy and FM phases surround an exact solution with saturated magnetic order parameter $m=\hslash /2$; Neel and zigzag phases exhibit a moment reduced by quantum fluctuations. The energy per bond (purple) also provides the phase transitions, as well as benchmarking (see Fig. 15). The QSL phases exist around the Kitaev points at $\theta /\pi =0,1$ but here are gapless and cannot be numerically captured with finite entanglement.

Figure 11

Gapped spin liquid at $K<0$ and surrounding magnetic phases, via $\chi =12$ iTEBD. Sufficient bond anisotropy, here $a=1/2$, gaps the QSL fermion spectrum and enables a tensor product state representation. The QSL phase, here for $K<0$, is identified by the vanishing magnetic order parameters (here the stripy and ferromagnet) as well as by its entanglement entropies on the various bonds. The entanglement on the two weak bonds peaks at the transition (with slight spurious symmetry breaking), and that on the strong bond rises sharply in the QSL, matching the exact solution's entropy of Fig. 12.

Figure 12

Entanglement entropies as QSL fingerprints in iTEBD and exact majorana solutions. Entanglement entropies from the exact solution with Majorana fermions and ${\mathbb{Z}}_2$ gauge fields (dotted lines) and from iTEBD computations (solid lines). The Kitaev bond strength anisotropy parameter $a$ is varied across the gapped phase $a>1/4$ for which the iTEBD algorithm can capture the quantum spin liquid. The minimally entangled states of the QSL on the tree carry ${\mathbb{Z}}_2$ gauge sector entanglement only on strong bonds.

Figure 13

Transitions between magnetic orders. Comparison between iTEBD (solid blue line) and classical mean-field theory (dashed red line), of the location in $\varphi$ as a function of anisotropy $a$ of the first-order transition between two adjacent magnetic orders. (Top left) Transition between zigzag and FM phases. (Top right) Transition between stripy and Neel phases. (Bottom) Transitions shown on the radial plot corresponding to Fig. 4.

Figure 14

Magnetic orders on the hyperhoneycomb lattice within the Kitaev-Heisenberg phase diagram. Three magnetic configurations are shown: clockwise from top left these are Neel, stripy-Z, and zigzag-Z. Blue spheres denote up spins and red spheres denote down spins in these collinear antiferromagneic orders. Stripy-Z is dual to a $z$-oriented ferromagnet, zigzag-Z is dual to a $z$-oriented Neel order.

Figure 15

Kitaev-Heisenberg magnetic phases with $a=1/2$ anisotropy. See description for Fig. 10. The energy measured in iTEBD is always found to be bounded from above by classical product states and from below by considering a maximally entangled cluster, providing a check on the algorithm. The QSL phases here are gapped and shown in Figs. 11 and 16, but their extent is not visible in this scale due to the strong anisotropy.

Figure 16

Gapped spin liquid for $K>0$ and surrounding magnetic phases within iTEBD. See the description in Fig. 11. Here we show the QSL at $a=1/2,K>0$, which competes with the zigzag and Neel orders. Different entropy curves occur here compared to the $K<0$ QSL since while both the FM and stripy orders are effectively ferromagnets with nearly saturated ordered moments, the Neel and zigzag phases involve substantial quantum fluctuations.

Figure 17

Isotropic $a=0$ Kitaev-Heisenberg phase diagram via iTEBD: correlators and entanglement. The four magnetically ordered phases can be identified by various measures in addition to their direct order parameters. These include signatures of the transitions in energy derivatives (not shown here), spin-spin correlators (top), and entanglement entropies on the various bonds in the unit cell (bottom). The entanglement entropies vanish at the exactly solvable points (shown by yellow lines) where the ground state, a (hidden) ferromagnet, is a simple product state.

Figure 18

Further benchmarks of iTEBD for the QSLs. The iTEBD spin correlators match the expected result for the pure Kitaev model, vanishing except for nearest neighbor Kitaev-matched spins; all magnetic order parameters vanish (shown), and the iTEBD ground-state energy per bond (cyan) matches the energy computed from the majorana fermion spectrum (black), in the gapped QSL phase.