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arXiv:1007.4050 · Physical Review B

Exotic phase diagram of a topological quantum system

Xiaofeng Shi, Yan Chen, J. Q. You

Published Nov 9, 2010• 9 citations
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Abstract

We study the quantum phase transitions (QPTs) in the Kitaev spin model on a triangle-honeycomb lattice. In addition to the ordinary topological QPTs between Abelian and non-Abelian phases, we find new QPTs which can occur between two phases belonging to the same topological class, namely, either two non-Abelian phases with the same Chern number or two Abelian phases with the same Chern number. Such QPTs result from the singular behaviors of the nonlocal spin-spin correlation functions at the critical points.

DOI
10.1103/physrevb.82.174412

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We study the quantum phase transitions (QPTs) in the Kitaev spin model on a triangle-honeycomb lattice. In addition to the ordinary topological QPTs between Abelian and non-Abelian phases, we find new QPTs which can occur between two phases belonging to the same topological class, namely, either two non-Abelian phases with the same Chern number or two Abelian phases with the same Chern number. Such QPTs result from the singular behaviors of the nonlocal spin-spin correlation functions at the critical points.
We study the quantum phase transition between Abelian and non-Abelian phases in an extended Kitaev spin model on the honeycomb lattice, where the periodic boundary condition is applied by placing the lattice on a torus. Our analytical results show that this spin model exhibits a continuous quantum phase transition. Also, we reveal the relationship between bipartite entanglement and the ground-state energy. Our approach directly shows that both the entanglement and the ground-state energy can be used to characterize the topological quantum phase transition in the extended Kitaev spin model.
Thermodynamic properties of chiral spin liquids are investigated for a variant of the Kitaev model defined on a decorated honeycomb lattice. Using the quantum Monte Carlo simulation, we find that the model exhibits a finite-temperature phase transition associated with the time reversal symmetry breaking, in both topologically trivial and nontrivial regions. Numerical results for the Chern number and the thermal Hall conductivity indicate that the phase transition changes from a continuous to a discontinuous transition as we vary the coupling constants to reach the non-Abelian phase coming from the Abelian phase of the model. In addition, we find as a diagnostic of the chiral spin liquids, successive crossovers with multistage entropy release above the critical temperature, which indicates that the hierarchical fractionalization of a quantum spin occurs differently between the two regions.
We have proposed an exactly solvable model defined on 2D decorated lattices of two types. The ground-state phase diagram of the system includes different topological phases with gapless chiral edge states. We show that two types of chiral spin liquid with gapless edge modes are realized on lattices with different symmetry. The phase transition between the topological phase with chiral gapped (Chern number zero) and the topological phase with chiral gapless edge modes (Chern number ) occurs in the model on the square (symmetric) decorated lattice. On the rectangular (asymmetric) decorated lattice the topological phase is defined by a chiral gapless (gapped) edge mode in the x ( y ) direction and a chiral gapped (gapless) edge mode in another y ( x ) direction. We show that a Kitaev model on a decorated asymmetric square lattice exhibits the quantum phase transition between topological phases with equal Chern numbers.
In extended Kitaev models on the honeycomb lattice, off-diagonal interactions (e.g., the $\mathrm{\ensuremath{\Gamma}},\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ terms) can give rise to non-Kitaev quantum spin liquids and several magnetically ordered phases. In the present paper, we dope holes to the system and study the resultant $t\text{\ensuremath{-}}K\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ model using mean-field theory. The interplay between the charge and spin degrees of freedom results in a rich phase diagram. Similar to doped cuprates, superconductors, pseudogap phases, Fermi liquid, strange metal and paramagnetic phase are generated. What is different is that, we obtain more than one superconducting phase (including a topological one) and more than one pseudogap phase no matter what the original spin state is. The Chern number of the topological superconductor is either $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}2$ or $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}1$, depending on the ratio $\mathrm{\ensuremath{\Gamma}}/|K|$ in the spin channel. We further find that an intermediate in-plane magnetic field can slightly enlarge the size of the topological superconducting phase.
Exactly solvable lattice models for spins and non-interacting fermions provide fascinating examples of topological phases, some of them exhibiting the localized Majorana fermions that feature in proposals for topological quantum computing. The Chern invariant ν is an important characterization of such phases. Here we look at the square–octagon variant of Kitaev's honeycomb model. It maps to spinful paired fermions and enjoys a rich phase diagram featuring distinct Abelian and non-Abelian phases with ν = 0,±1,±2,±3 and ±4. The ν = ±1 and ν = ±3 phases all support localized Majorana modes and are examples of Ising and SU(2) 2 anyon theories, respectively.
We extend the Yao-Kivelson decorated honeycomb lattice Kitaev model [H. Yao and S. A. Kivelson, Phys. Rev. Lett. 99, 247203 (2007)] of an exactly solvable chiral spin liquid by including disordered exchange couplings. We have determined the phase diagram of this system and found that disorder enlarges the region of the topological non-Abelian phase with finite Chern number. We study the energy level statistics as a function of disorder and other parameters in the Hamiltonian, and show that the phase transition between the non-Abelian and Abelian phases of the model at large disorder can be associated with pair annihilation of extended states at zero energy. Analogies to integer quantum Hall systems, topological Anderson insulators, and disordered topological Chern insulators are discussed.