Quantum Phase Transitions In Topological Quantum Spin Systems

Phase transitions are present in life everyday and most of the phase transitions that human beings can feel are caused by thermodynamic fluctuations. There are tremen-dous interesting phenomena unveiled by people's quest on the nature of phase tran-sitions, including the very attractive critical phenomenon in, e.g., the transition from liquid to gas. In contrast, there is another kind of phase transitions purely caused by quantum fluctuations:the quantum phase transitions (QPTs). Here 'quantum' means that we need no control on temperature, pressure, or any other classic quantity to induce a QPT and quantum fluctuations are solely explained by the uncertainty relation, which is a basic quantum mechanical rule. Many phenomena in condensed matter physics are related with QPTs occurred therein.Most QPTs can be well understood in the framework of symmetry breaking. How-ever, there are some transitions that cannot be characterized using a symmetry-breaking theory. One example is the so-called topological QPT. A topological QPT represents a QPT that changes the topological property of the system. Topology is related with the overall property of the quantum system, and a topological order parameter is different from a local order parameter in the Laudau theory of symmetry breaking. Signatures of topological order in many-body quantum systems include, e.g., the existence of ex-citations obeying fractional statistics, ground-state degeneracy related to the topology of the system instead of the symmetry, and topological entanglement entropy.A spin model on a two-dimensional honeycomb lattice first introduced by A. Kitaev at Caltech has attracted much interest in recent years because it contains topological or-der and can be exactly solvable in the vertex-free sector. This Kitaev model can exhibit both Abelian anyons and non-Abelian anyons, which can be explicitly demonstrated. A topological quantum system provides a potential candidate for topological quantum computing. Also, many unknown physical properties are expected to exist in the Ki-taev spin models. Here we study two typical kinds of Kitaev spin models in the aspect of QPTs; one is the extended Kitaev spin model on a honeycomb lattice and the other is the Kitaev spin model on a triangle-honeycomb lattice. These topological spin sys-tems exhibit excitations obeying non-Abelian statistics and can be useful in topological quantum computation.We solve the extended Kitaev spin model on a honeycomb lattcie that is placed on a torus, and mainly focus on the QPT between the phase with Abelian anyons and the phase with non-Abelian anyons. We first apply the Jordan-Wigner transformation to the spin operators and then introduce Majorana fermions to obtain its ground state in the vortex-free sector. We show that the third directional derivative of the ground-state energy is discontinuous at each point on. the critical line separating the Abelian and non-Abelian phases, while its first and second directional derivatives are continuous at this point. This implies that the topological quantum phase transition is continuous in this extended Kitaev model. Moreover, at this critical point, we also study the non-analyticity of the entanglement (i.e., the von Neumann entropy) between two nearest neighbor spins and the rest of the spins in the system. We find that the second direc-tional derivative of the von Neumann entropy is closely related to the third directional derivative of the ground-state energy and it is also discontinuous at the critical point. Our approach directly reveals that both the entanglement measure and the ground-state energy can be used to characterize the topological QPT in the extended Kitaev model.Recently, QPTs in the Kitaev spin models on honeycomb and triangle-honeycomb lattices were investigated. The obtained QPTs include the transition between a gapped Abelian phase and a gapless phase, the transition between Abelian and non-Abelian phases, and the transition between two non-Abelian phases with different Chern num-bers. Nevertheless, to the best of our knowledge, the QPT between two topological phases of the same Chern number (which belong to the same topological class) has not yet been found. Here we show that in the Kitaev spin model on a triangle-honeycomb lattice, a QPT can indeed happen between two gapped phases in. the same topologi-cal class, in addition to the ordinary topological QPT between two phases of different Chern numbers. This is due to the exotic ground-state phase diagram which has either critical curves between two gapped phases with the same or different Chern numbers or critical points where several different gapped phases terminate. These results reveal that the Kitaev spin model on a triangle-honeycomb lattice can exhibit novel topologi-cal properties. Moreover, we find that these QPTs result from the singular behaviors of the nonlocal spin-spin correlations at the critical points.