Eleven Competing Phases in the Heisenberg-Gamma (J$\Gamma$) Ladder

The spin-orbit generated $\Gamma$ interaction is known to induce strong frustration and to be significant in realistic models of materials. To gain an understanding of the possible phases that can arise from this interaction, it is of considerable interest to focus on a limited part of parameter space in a quasi one-dimensional model where high precision numerical results can be obtained. Here we study the Heisenberg-Gamma (J$\Gamma$) ladder, determining the complete zero temperature phase diagram by analyzing the entanglement spectrum (ES) and energy susceptibility. A total of 11 different phases can be identified. Two of the phases, the antiferromagnetic Gamma (A$\Gamma$) and ferromagnetic Gamma (F$\Gamma$) phases, have previously been observed in the Kitaev-Gamma ladder, demonstrating that the A$\Gamma$-phase is a symmetry protected topological phase (SPT) protected by $TR\times \mathcal{R}_{b}$ symmetry, the product of time-reversal ($TR$) and $\pi$ rotation around the $b$-axis ($\mathcal{R}_{b}$), while the F$\Gamma$-phase is related to a rung-singlet phase through a local unitary transformation. Three other phases, $\Upsilon$, $\Omega$ and $\delta$ show no conventional order, a doubling of the entanglement spectrum and for the $\Upsilon$ and $\Omega$-phases a gap is clearly present. The $\delta$-phase has a significantly smaller gap and displays incommensurate correlations, with a peak in the static structure factor, $S(k)$ continuously shifting from $k/\pi\mathord{=}2/3$ to $k\mathord{=}\pi$. In the $\Omega$-phase we find pronounced edge-states consistent with a SPT phase protected by the same $TR\times \mathcal{R}_{b}$ symmetry as the A$\Gamma$-phase. The precise nature of the $\Upsilon$ and $\delta$-phases is less clear.