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arXiv:2312.17359 · Physical review. B./Physical review. B

Topological transitions in the Yao-Lee spin-orbital model and effects of site disorder

Vladislav Poliakov, Wen-Han Kao, Natalia Borisovna Perkins

Published Aug 8, 2024• 11 citations
arXivHTMLPDF

Abstract

The Yao-Lee model is an example of exactly solvable spin-orbital models that are generalizations of the original Kitaev honeycomb model with extra local orbital degrees of freedom. Similarly to the Kitaev model, both spin and orbital degrees of freedom are effectively represented using sets of three-flavored Majorana fermions. The Yao-Lee model exhibits a quantum spin liquid ground state with gapped and immobile ${Z}_{2}$ fluxes and threefold degenerate itinerant Majorana fermions. Our work demonstrated that by introducing different time-reversal-symmetry- (TRS) breaking fields one can split the degeneracy of Majorana fermions and close the gap for some of the bands, thus changing its topology. We calculated a comprehensive topological phase diagram for the Yao-Lee model by considering various combinations of TRS-breaking fields. This investigation revealed the emergence of distinct topological regions, each separated by nodal lines, signifying an evolution in the model's topological properties. We also investigated the impact of vacancies in the system. Our findings revealed that while vacancies modify the low-energy spectrum of the model, their presence has a limited impact on the topological properties of the model, at least for small-enough concentrations.

DOI
10.1103/physrevb.110.054418

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The Yao-Lee model is an example of exactly solvable spin-orbital models that are generalizations of the original Kitaev honeycomb model with extra local orbital degrees of freedom. Similarly to the Kitaev model, both spin and orbital degrees of freedom are effectively represented using sets of three-flavored Majorana fermions. The Yao-Lee model exhibits a quantum spin liquid ground state with gapped and immobile Z2{Z}_{2}Z2​ fluxes and threefold degenerate itinerant Majorana fermions. Our work demonstrated that by introducing different time-reversal-symmetry- (TRS) breaking fields one can split the degeneracy of Majorana fermions and close the gap for some of the bands, thus changing its topology. We calculated a comprehensive topological phase diagram for the Yao-Lee model by considering various combinations of TRS-breaking fields. This investigation revealed the emergence of distinct topological regions, each separated by nodal lines, signifying an evolution in the model's topological properties. We also investigated the impact of vacancies in the system. Our findings revealed that while vacancies modify the low-energy spectrum of the model, their presence has a limited impact on the topological properties of the model, at least for small-enough concentrations.
Spin-orbital generalization of Kitaev model provides a robust extension to the original Kitaev model. However, real materials often exhibit competing interactions that break exact solvability which can give rise to new phases. Motivated by recent microscopic proposals of coexisting Yao-Lee and Kitaev couplings, we investigate the fate of the ground state when two independent exactly solvable spin liquid Hamiltonians each originally formulated on different lattice geometries are combined on a common lattice environment. We first focus on the hybrid Kitaev's honeycomb and square-lattice model. Using self-consistent mean-field analysis and perturbative calculation, we show that the strong-Kitaev regime yields magnetic order in the spin sector, while the orbital sector retains its topological order. We further analyze the hybridization of the Yao-Lee and square-lattice models and find that the model exhibits a rich evolution of Majorana Dirac bands and Lifshitz transitions. Remarkably, when the Yao-Lee and square-lattice couplings are equal and opposite, the model restores its exact solvability with a single itinerant Majorana flavor. These results demonstrate that hybrid spin liquid platforms may host various emergent phases beyond conventional exactly solvable limits.
The exactly solvable spin-1/2 Kitaev model on a honeycomb lattice has drawn significant interest, as it offers a pathway to realizing the long-sought-after quantum spin liquid. Building upon the Kitaev model, Yao and Lee introduced another exactly solvable model on an unusual star lattice featuring non-abelian spinons. The additional pseudospin degrees of freedom in this model could provide greater stability against perturbations, making this model appealing. However, a mechanism to realize such an interaction in a standard honeycomb lattice remains unknown. Here, we provide a microscopic theory to obtain the Yao-Lee model, on a honeycomb lattice by utilizing strong spin-orbit coupling of anions edge-shared between two e g ions in the exchange processes. This mechanism leads to the desired bond-dependent interaction among spins rather than orbitals, unique to our model, implying that the orbitals fractionalize into gapless Majorana fermions and fermionic octupolar excitations emerge. Since the conventional Kugel-Khomskii interaction also appears, we examine the phase diagram, including these interactions, using classical Monte Carlo simulations and exact diagonalization techniques. Our findings reveal a broad region of disordered states that break rotational symmetry in the bond energy, suggesting intriguing behavior reminiscent of a spin-orbital liquid.
Spin-orbital generalizations of Kitaev model, such as Yao-Lee model, have attracted recent attention due to their enhanced stability of spin liquid phases against perturbations. Motivated by microscopic calculations for the realization of Yao-Lee model showing additional interactions, we study the phase diagram of the Yao-Lee model with added Kitaev and Heisenberg terms. While the plaquette operator is conserved even in the presence of added perturbations, the model becomes no longer exactly solvable. Using perturbation and Majorana mean-field theory, we find magnetic order can arise in the spin sector while the orbital sector remains a liquid for dominant Kitaev interactions, whereas both sectors form liquid phases when Yao-Lee interactions dominate. Additional Heisenberg exchange can enhance or suppress the magnetic order, revealing a rich coexistence of magnetic and topological phases.
We present an exactly solvable spin-orbital model based on the Gamma-matrix generalization of a Kitaev-type Hamiltonian. In the presence of small magnetic fields, the model exhibits a critical phase with a spectrum characterized by topologically protected Fermi points. Upon increasing the magnetic field, Fermi points carrying opposite topological charges move toward each other and annihilate at a critical field, signaling a phase transition into a gapped phase with trivial topology in three dimensions. On the other hand, by subjecting the system to a staggered magnetic field, an effective time-reversal symmetry essential to the existence of three-dimensional topological insulators is restored in the auxiliary free fermion problem. The nontrivial topology of the gapped ground state is characterized by an integer winding number and manifests itself through the appearance of gapless Majorana fermions confined to the two-dimensional surface of a finite system.
We consider the effect of vacancies on the low-energy excitation spectrum of a quantum spin liquid realized in the exactly solvable Yao–Lee model [H. Yao and D.-H. Lee, Phys. Rev. Lett. 107, 087205 (2011)]. Physically, vacancies can appear for different reasons (e.g., because of zero magnetic moments on the lattice, or the presence of nonmagnetic impurities, or a random reduction of local bonds of magnetic moments with the remaining lattice). It is shown numerically that the finite density of random vacancies in this model leads to the accumulation of states near zero energy, which can be detected from the change of the behavior of heat capacity at low temperatures. Moreover, it is shown that the low-energy modes are localized more strongly than remaining eigenmodes. This effect is illustrated using the inverse participation ratio (IPR). In the case of time reversal symmetry breaking (e.g., due to the presence of a magnetic field), a gap is opened in the fermion spectrum of the model, and vacancy-induced localized states appear. The energies of these states depend on the structure of the interactions responsible for the time inversion symmetry breaking.
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.