Probing the stability of the spin-liquid phases in the Kitaev-Heisenberg model using tensor network algorithms

We study the extent of the spin liquid phases in the Kitaev-Heisenberg model using infinite projected entangled-pair states tensor network ansatz wave functions directly in the thermodynamic limit. To assess the accuracy of the ansatz wave functions, we perform benchmarks against exact results for the Kitaev model and find very good agreement for various observables. In the case of the Kitaev-Heisenberg model, we confirm the existence of six different phases: Néel, stripy, ferromagnetic, zigzag, and two spin liquid phases. We find finite extents for both spin liquid phases and discontinuous phase transitions connecting them to symmetry-broken phases.

Figure 1

Top: four-site unit cell iPEPS ansatz on the brick-wall lattice. Introducing trivial (dashed) bonds in a square lattice leads to a so-called brick-wall lattice reproducing the connectivity of the honeycomb lattice. The blue, red, green, and purple tensors (circles) in the unit cell (highlighted cluster) provide the variational parameters. Bottom: effective mapping used for the evaluation of observables. On the left, the ket portion of the network prior to contraction. On the right, a contracted bra-ket network is shown from the top with tensors labeled by lowercase letters representing a bra-ket pair contracted. The orange tensors (squares) labeled by E represent environment tensors, obtained, e.g., via the CTM algorithm, accounting for an effectively infinite environment. Thick lines (bonds) correspond to indices of dimension $\chi$, solid lines to indices of dimension $D$, and dashed lines represent trivial bonds. Bond labeling convention adopted is given by the $x,y,z$ labels.

Figure 2

(a) Energy per site and (b) magnetization as a function of inverse bond dimension. Errors in the energy for the largest value studied $\left(D=7\right)$ are of the order of ${10}^{-4}$. Magnetization values are normalized to 1.

Figure 3

Nearest-neighbor correlators for corresponding bond types in the ferromagnetically coupled case (a) and antiferromagnetically coupled case (b). All remaining correlators are at most of the order of ${10}^{-3}$ for the largest value of $D$ and exhibit similar improvement with increasing $D$.

Figure 4

Regions spanned by the different phases found using iPEPS. Four different magnetically ordered (collinear) phases are found: Néel (orange/top right), zigzag (yellow/top left), ferromagnetic (dark green/bottom left), and stripy (light green/bottom right). Magnetically ordered phases are characterized using the order parameters in (4, 5, 6, 7, 8). Blue regions correspond to the spin liquid phases. Phase boundary angles correspond to $D=6$ estimates with the spin liquid regions increasing their extent as the bond dimension $D$ is increased.

Figure 5

Order parameters as a function of the angle $\phi$ in the vicinity of the points $\phi =\pm {90}^{\circ }$, normalized to 1. Regions of strongly suppressed symmetry breaking are clearly visible in both cases with finite discontinuities in the order parameters. The shaded regions indicate the estimated extents of the regions covered by the QSL phases. All order parameters not shown remain very close to zero.

Figure 6

Top: energy crossings for the FSL to stripy phase transition. Dotted (purple), dashed-dotted (red), and dashed (orange) lines correspond to the location of the phase transition for $D=4,5,6$, respectively. The inset shows the maximum deviation of the bond energies from the mean value normalized to the mean value. Bottom: $O_{\mathrm{mag}}$ within each phase (green circles: stripy/blue squares: FSL) and the reconstructed ground-state curve (red diamonds). The inset shows the behavior of $O_{\mathrm{stripy}}$ over the same range of angles. Order parameters are normalized to one. Dashed orange lines indicate the estimated location of the phase transition for $D=6$.

Figure 7

Left: energy crossings (top) and $O_{\mathrm{mag}}$ (bottom) for Néel and spin liquid phases. Center: energy crossings (top) and magnetization (bottom) for zigzag and spin liquid phases. Right: energy crossings (top) and magnetization (bottom) for ferromagnetic and spin liquid phases.