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Europhysics Letters (EPL)

Quantum phase transitions in the Kitaev model on decorated lattices

Igor N. Karnaukhov

Published Jun 1, 2013• 3 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/102/57007

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We study an exactly solvable quantum spin model of Kitaev type on the kagome lattice. We find a rich phase diagram which includes a topological (gapped) chiral spin liquid with gapless chiral edge states, and a gapless chiral spin liquid phase with a spin Fermi surface. The ground state of the current model contains an odd number of electrons per unit cell, which qualitatively distinguishes it from previously studied exactly solvable models with a spin Fermi surface. Moreover, we show that the spin Fermi surface is stable against weak perturbations.
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.
We study a frustrated two-dimensional lattice spin model with Kitaev-type interaction. The lattice is obtained from the honeycomb lattice by replacing half of its sites with triangles. Using the SO(3) Majorana representation of spin, the model is exactly mapped into a Z2{Z}_{2}Z2​ lattice gauge theory of complex fermions with standard Gauss law constraint. We show that the ground state of the model is a chiral spin liquid, and it has gapless excitations. The Chern number of the ground state is $\ifmmode\pm\else\textpm\fi{}1$, implying the existence of a chiral edge mode. The vortex excitation of the model is bound with a Majorana zero mode, such mode behaves as non-Abelian anyons.
We extend the Kitaev model defined for the Pauli matrices to the Clifford algebra of $\ensuremath{\Gamma}$ matrices, taking the $4\ifmmode\times\else\texttimes\fi{}4$ representation as an example. On a decorated square lattice, the ground state spontaneously breaks time-reversal symmetry and exhibits a topological phase transition. The topologically nontrivial phase carries gapless chiral edge modes along the sample boundary. On the three-dimensional (3D) diamond lattice, the ground states can exhibit gapless 3D Dirac-cone-like excitations and gapped topological insulating states. Generalizations to even higher rank $\ensuremath{\Gamma}$ matrices are also discussed.
We show that the decorated honeycomb (star) lattice supports a number of topological insulating phases with a nontrivial Z2{Z}_{2}Z2​ invariant and time-reversal symmetry protected gapless edge modes. We investigate the stability of these phases with respect to various symmetry-breaking perturbations and demonstrate the connection to the recent discovery of an exactly solvable S=1/2S=1/2S=1/2 chiral spin-liquid model [Yao and Kivelson, Phys. Rev. Lett. 99, 247203 (2007)] with non-Abelian and Abelian excitations on the same lattice at strong interaction strength. Our work highlights the relationship between noninteracting topological band insulators and strongly interacting topologically ordered spin systems, and points to promising avenues for enlarging the number of known examples of both.
We study quantum phase transitions in the honeycomb Kitaev model under a magnetic field, focusing on the topological nature of Majorana fermion excitations. We find a gapless phase between the low-field gapless quantum spin liquid and the high-field gapped forced-ferromagnetic state for the antiferromagnetic Kitaev model in the [001] field by using the Majorana mean-field theory, in conjunction with the exact diagonalization and the spin-wave theory supporting the validity of this approach. The transition between the two gapless phases is driven by a topological change of the Majorana spectrum---line node formation interconnecting two Majorana cones. The peculiar change of the Majorana band topology is rationalized by a sign change of the effective Kitaev coupling by the magnetic field, which does not occur in the ferromagnetic Kitaev case. Upon tilting the magnetic field away from [001], the two gapless phases become gapped and topologically nontrivial, characterized by nonzero Chern numbers with different signs. The sign change of the Chern number leads to a reversal of the thermal edge current in the half-quantized thermal Hall effect.