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The spin-orbit generated $\Gamma$ interaction is known to induce strongfrustration and to be significant in realistic models of materials. To gain anunderstanding of the possible phases that can arise from this interaction, itis of considerable interest to focus on a limited part of parameter space in aquasi one-dimensional model where high precision numerical results can beobtained. Here we study the Heisenberg-Gamma (J$\Gamma$) ladder, determiningthe complete zero temperature phase diagram by analyzing the entanglementspectrum (ES) and energy susceptibility. A total of 11 different phases can beidentified. Two of the phases, the antiferromagnetic Gamma (A$\Gamma$) andferromagnetic Gamma (F$\Gamma$) phases, have previously been observed in theKitaev-Gamma ladder, demonstrating that the A$\Gamma$-phase is a symmetryprotected 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 arung-singlet phase through a local unitary transformation. Three other phases,$\Upsilon$, $\Omega$ and $\delta$ show no conventional order, a doubling of theentanglement spectrum and for the $\Upsilon$ and $\Omega$-phases a gap isclearly present. The $\delta$-phase has a significantly smaller gap anddisplays incommensurate correlations, with a peak in the static structurefactor, $S(k)$ continuously shifting from $k/\pi\mathord{=}2/3$ to$k\mathord{=}\pi$. In the $\Omega$-phase we find pronounced edge-statesconsistent 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.

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