The bond-dependent Kitaev model on the honeycomb lattice with anyonic excitations has recently attracted considerable attention. However, in solid-state materials other spin interactions are present, and among such additional interactions, the off-diagonal symmetric Gamma interaction, another type of bond-dependent term, has been particularly challenging to fully understand. A minimal Kitaev-Gamma model has been investigated by various numerical techniques under a magnetic field, but definite conclusions about field-induced spin liquids remain elusive. One reason for this may lie in the limited sizes of the two-dimensional geometry it is possible to access numerically, and missed incommensurately ordered states may be interpreted as a spin liquid. Here we focus on the Kitaev-Gamma ladder model as a guide to the phase space of disordered states which could potentially become a spin liquid in the two-dimensional limit. We determine the entire phase diagram in the presence of a magnetic field along the [111] direction. Because of the competition between the interactions and the field, an extremely rich phase diagram emerges with 15 distinct phases. Focusing near the antiferromagnetic Kitaev region, we identify nine different phases solely within this region: several incommensurate magnetically ordered phases, spin-nematic, and two chiral phases with enhanced entanglement. Of particular interest is a highly entangled phase with staggered chirality with zero-net flux occurring at intermediate field, which along with its companion phases outlines a heart-shaped region of high entanglement, the heart of entanglement. We compare our results for the ladder with a $C_3$ symmetric cluster of the two-dimensional honeycomb lattice, and offer insight into possible spin liquids in the two-dimensional limit.

中文翻译：

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纠缠之心：Kitaev-Gamma梯子中的手性，向列相和不相称相

具有非离子激发的蜂窝晶格上依赖键的Kitaev模型最近引起了相当大的关注。但是，在固态材料中，存在其他自旋相互作用，并且在此类附加相互作用中，对角对称的Gamma相互作用（另一种依赖于键的术语）对充分理解尤其具有挑战性。在磁场下，已通过各种数值技术研究了最小的Kitaev-Gamma模型，但关于场感应自旋液体的确切结论仍然难以捉摸。造成这种情况的一个原因可能是二维几何形状的大小有限，可以通过数字方式访问，并且错位的无序排列状态可以解释为自旋液体。在这里，我们将重点放在Kitaev-Gamma阶梯模型上，以指导无序状态的相空间，该状态可能在二维范围内成为自旋液体。我们确定在[111]方向上存在磁场的情况下的整个相位图。由于相互作用和磁场之间的竞争，出现了具有15个不同相位的极其丰富的相位图。着眼于反铁磁Kitaev区域附近，我们仅在该区域内识别出九个不同的相：几个不对称的磁有序相，自旋向列相和 出现了具有15个不同阶段的极其丰富的阶段图。着眼于反铁磁Kitaev区域附近，我们仅在该区域内识别出九个不同的相：几个不对称的磁有序相，自旋向列相和 出现了具有15个不同阶段的极其丰富的阶段图。着眼于反铁磁Kitaev区域附近，我们仅在该区域内识别出九个不同的相：几个不对称的磁有序相，自旋向列相和两个手性相增强了纠缠。特别令人感兴趣的是具有交错手征的高度纠缠相，中间电场出现零净通量，该相及其伴随相勾勒出一个高纠缠心形区域，即纠缠的心脏。我们将阶梯的结果与$C_3$ 二维蜂窝网格的对称簇，并提供对二维极限中可能的自旋液体的洞察。