In this paper, we study quantum phase transitions and magnetic properties of a one-dimensional spin-1/2 Gamma model, which describes the off-diagonal exchange interactions between edge-shared octahedra with strong spin-orbit couplings along the sawtooth chain. The competing exchange interactions between the nearest neighbors and the second neighbors stabilize the semimetallic ground state in terms of spinless fermions, and give rise to a rich phase diagram, which consists of three gapless phases. We find distinct phases are characterized by the number of Weyl nodes in the momentum space, and such changes in the topology of the Fermi surface without symmetry breaking produce a variety of Lifshitz transitions, in which the Weyl nodes situating at $k=\pi$ change from type I to type II. A coexistence of type-I and type-II Weyl nodes is found in phase II. The information measures including concurrence, entanglement entropy, and relative entropy can effectively signal the second-order transitions. The results indicate that the Gamma model can act as an exactly solvable model to describe Lifshitz phase transitions in correlated electron systems.

中文翻译：

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一维Gamma模型中的Lifshitz相变

在本文中，我们研究了一维自旋1/2 Gamma模型的量子相变和磁性，该模型描述了沿锯齿链具有强自旋轨道耦合的边缘共享八面体之间的非对角交换相互作用。最近邻和第二邻之间的竞争性交换相互作用以无旋费米子稳定了半金属基态，并产生了一个富相图，该相图由三个无间隙相组成。我们发现，动量空间中Weyl节点的数量是不同相的特征，费米表面拓扑结构的这种变化而没有对称性破坏会产生各种Lifshitz跃迁，其中Weyl节点位于$ķ=\pi$从类型I更改为类型II。在阶段II中发现了I型和II型Weyl节点共存。包括并发，纠缠熵和相对熵在内的信息度量可以有效地表示二阶跃迁。结果表明，Gamma模型可以作为描述相关电子系统中Lifshitz相变的完全可解模型。