  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 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 $k=\pi$
interchange 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.
