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A continuum compass model on the honeycomb lattice: phase diagram from tensor networks

Haiyuan Zou, Bo Liu, Erhai Zhao, W. Vincent Liu

Published Dec 7, 2015• 0 citations
OpenAlex

Abstract

Quantum spin models with spatially dependent interactions, known as compass models, play an important role in study of frustrated quantum magnetism and Mott insulators with orbital degrees of freedom. One example is Kitaev model on honeycomb lattice with spin liquid ground states and anyonic excitations. It is highly desirable to go beyond this exactly solvable limit by generalizing Kitaev model to connect with specific materials or other known models. A conventional way to achieve this is through hybridization with other models. For example, Kitaev-Heisenberg model was proposed to describe iridates. Here we pursue an alternative direction and propose a new dubbed the tripod model, which contains a continuum of compass-type models. It smoothly interpolates Ising Kitaev and quantum $120^\circ$ model by tuning a single parameter $\theta'$, angle between three legs of a tripod in spin space. Not only does tripod model establish a natural connection between three paradigmatic models, but it also enables study of quantum phase transitions between them. We solve this model numerically by tensor networks in thermodynamic limit and find a spin-disordered (spin liquid) phase between an antiferromagnetic phase and a dimer phase. Our results unambiguously establish long-range order of geometrically frustrated $120^\circ$ model and provide an intuitive picture for phase transition between spin-liquid phase and dimer phase. The tripod model proposed here could be realized in cold atoms experiments by introducing non-Abelian artificial gauge fields for fermionic Hubbard model.

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Quantum spin models with spatially dependent interactions, known as compass models, play an important role in the study of frustrated quantum magnetism. One example is the Kitaev model on the honeycomb lattice with spin-liquid (SL) ground states and anyonic excitations. Another example is the geometrically frustrated quantum 120° model on the same lattice whose ground state has not been unambiguously established. To generalize the Kitaev model beyond the exactly solvable limit and connect it with other compass models, we propose a new model, dubbed 'the tripod model', which contains a continuum of compass-type models. It smoothly interpolates the Ising model, the Kitaev model, and the quantum 120° model by tuning a single parameter , the angle between the three legs of a tripod in the spin space. Hence it not only unifies three paradigmatic spin models, but also enables the study of their quantum phase transitions. We obtain the phase diagram of the tripod model numerically by tensor networks in the thermodynamic limit. We show that the ground state of the quantum 120° model has long-range dimer order. Moreover, we find an extended spin-disordered (SL) phase between the dimer phase and an antiferromagnetic phase. The unification and solution of a continuum of frustrated spin models as outline here may be useful to exploring new domains of other quantum spin or orbital models.

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The Kitaev model, which hosts a quantum spin liquid (QSL) in the ground state, was originally defined on a two-dimensional honeycomb lattice, but can be straightforwardly extended to any tricoordinate lattices in any spatial dimensions. In particular, the three-dimensional (3D) extensions are of interest as a realization of 3D QSLs, and some materials like $β$-Li$_{2}$IrO$_{3}$, $γ$-Li$_2$IrO$_3$, and $β$-ZnIrO$_{3}$ were proposed for the candidates. However, the phase diagrams of the models for those candidates have not been fully elucidated, mainly due to the limitation of numerical methods for 3D frustrated quantum spin systems. Here we study the Kitaev-Heisenberg model defined on a 3D hyperhoneycomb lattice, by using the pseudofermion functional renormalization group method. We show that the ground-state phase diagram contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases, similar to the two-dimensional honeycomb case. Our results respect the four-sublattice symmetry inherent in the model, which was violated in the previous study. Moreover, we also show how the phase diagram changes with the anisotropy in the interactions. The results provide a reference for the search of the hyperhoneycomb Kitaev materials.

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Quantum spin models with spatially dependent interactions, known as compass models, play an important role in study of frustrated quantum magnetism and Mott insulators with orbital degrees of freedom. One example is Kitaev model on honeycomb lattice with spin liquid ground states and anyonic excitations. It is highly desirable to go beyond this exactly solvable limit by generalizing Kitaev model to connect with specific materials or other known models. A conventional way to achieve this is through hybridization with other models. For example, Kitaev-Heisenberg model was proposed to describe iridates. Here we pursue an alternative direction and propose a new dubbed the tripod model, which contains a continuum of compass-type models. It smoothly interpolates Ising Kitaev and quantum 120∘120^\circ120∘ model by tuning a single parameter θ′\theta'θ′, angle between three legs of a tripod in spin space. Not only does tripod model establish a natural connection between three paradigmatic models, but it also enables study of quantum phase transitions between them. We solve this model numerically by tensor networks in thermodynamic limit and find a spin-disordered (spin liquid) phase between an antiferromagnetic phase and a dimer phase. Our results unambiguously establish long-range order of geometrically frustrated 120∘120^\circ120∘ model and provide an intuitive picture for phase transition between spin-liquid phase and dimer phase. The tripod model proposed here could be realized in cold atoms experiments by introducing non-Abelian artificial gauge fields for fermionic Hubbard model.
Quantum spin models with spatially dependent interactions, known as compass models, play an important role in the study of frustrated quantum magnetism. One example is the Kitaev model on the honeycomb lattice with spin-liquid (SL) ground states and anyonic excitations. Another example is the geometrically frustrated quantum 120° model on the same lattice whose ground state has not been unambiguously established. To generalize the Kitaev model beyond the exactly solvable limit and connect it with other compass models, we propose a new model, dubbed 'the tripod model', which contains a continuum of compass-type models. It smoothly interpolates the Ising model, the Kitaev model, and the quantum 120° model by tuning a single parameter , the angle between the three legs of a tripod in the spin space. Hence it not only unifies three paradigmatic spin models, but also enables the study of their quantum phase transitions. We obtain the phase diagram of the tripod model numerically by tensor networks in the thermodynamic limit. We show that the ground state of the quantum 120° model has long-range dimer order. Moreover, we find an extended spin-disordered (SL) phase between the dimer phase and an antiferromagnetic phase. The unification and solution of a continuum of frustrated spin models as outline here may be useful to exploring new domains of other quantum spin or orbital models.
Compass models are theories of matter in which the couplings between the internal spin (or other relevant field) components are inherently spatially (typically, direction) dependent. Compass-type interactions appear in diverse physical systems including Mott insulators with orbital degrees of freedom (where interactions sensitively depend on the spatial orientation of the orbitals involved), the low energy effective theories of frustrated quantum magnets, systems with strong spin-orbit couplings (such as the iridates), vacancy centers, and cold atomic gases. Kitaev's models, in particular the compass variant on the honeycomb lattice, realize basic notions of topological quantum computing. The fundamental inter-dependence between internal (spin, orbital, or other) and external (i.e., spatial) degrees of freedom which underlies compass models generally leads to very rich behaviors including the frustration of (semi-)classical ordered states on non-frustrated lattices and to enhanced quantum effects prompting, in certain cases, the appearance of zero temperature quantum spin liquids. As a consequence of these frustrations, new types of symmetries and their associated degeneracies may appear. These intermediate symmetries lie midway between the extremes of global symmetries and local gauge symmetries and lead to effective dimensional reductions. We review compass models in a unified manner, paying close attention to exact consequences of these symmetries, and to thermal and quantum fluctuations that stabilize orders via order out of disorder effects. We review non-trivial statistics and the appearance of topological quantum orders in compass systems in which, by virtue of their intermediate symmetry standard orders do not arise. This is complemented by a survey of numerical results. Where appropriate theoretical and experimental results are compared.
Compounds of transition metal ions with strong spin-orbit coupling recently attracted attention due to the possibility to host frustrated bond-dependent anisotropic magnetic interactions. In general, such interactions lead to complex phase diagrams that may include exotic phases, e.g., the Kitaev spin liquid. Here we report on our comprehensive analysis of the global phase diagram of the extended Kitaev-Heisenberg model relevant to honeycomb lattice compounds Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ and $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$. We have utilized recently developed method based on spin coherent states that enabled us to resolve arbitrary spin patterns in the cluster ground states obtained by exact diagonalization. Global trends in the phase diagram are understood in combination with the analytical mappings of the Hamiltonian that uncover peculiar links to known models---Heisenberg, Ising, Kitaev, or compass models on the honeycomb lattice---or reveal entire manifolds of exact fluctuation-free ground states. Finally, our study can serve as a methodological example that can be applied to other spin models with complex bond-dependent non-Heisenberg interactions.
We study a spin model on honeycomb lattice with two types of Heisenberg exchange couplings, JJJ and J~\tilde JJ~, where JJJ is for the conventional spin and and J~\tilde JJ~ for rotated spin. When J~=0\tilde J=0J~=0, this is the conventional Heisenberg model. When J=0, the system is either in a stripy antiferromagnetic order(J~0\tilde J 0J~0, antiferromagnetic for rotated spin). The competition between two ferromagnetic orders or two antiferromagnetic orders induces Kitaev's spin liquid phase characterized by the exactly solvable Kitaev model (J=J~J=\tilde JJ=J~). Our model can be applied to layered Mott insulators A2_22​IrO3_33​ (A=Li, Na). For a monolayer of Li2_22​IrO3_33​, we show that it is possible to tune the controlling parameter into the Kitaev spin liquid regime by a link-dependent Rashba spin-orbital coupling.
We study the ground-state phase diagram of the spin-1/21/21/2 Kitaev-Heisenberg model on the bilayer honeycomb lattice with large-scale tensor network calculations based on the infinite projected entangled pair state technique as well as high-order series expansions. We find that beyond various magnetically ordered phases, including ferromagnetic, zigzag, antiferromagnetic (AFM) and stripy states, two extended quantum spin liquid phases arise in the proximity of the Kitaev limit. While these ordered phases also appear in the monolayer Kitaev-Heisenberg model, our results further show that a valence bond solid state emerges in a relatively narrow range of parameter space between the AFM and stripy phases, which can be adiabatically connected to isolated Heisenberg dimers. Our results highlight the importance of considering interlayer interactions on the emergence of novel quantum phases in the bilayer Kitaev materials.
The Kitaev model, which hosts a quantum spin liquid (QSL) in the ground state, was originally defined on a two-dimensional honeycomb lattice, but can be straightforwardly extended to any tricoordinate lattices in any spatial dimensions. In particular, the three-dimensional (3D) extensions are of interest as a realization of 3D QSLs, and some materials like βββ-Li2_{2}2​IrO3_{3}3​, γγγ-Li2_22​IrO3_33​, and βββ-ZnIrO3_{3}3​ were proposed for the candidates. However, the phase diagrams of the models for those candidates have not been fully elucidated, mainly due to the limitation of numerical methods for 3D frustrated quantum spin systems. Here we study the Kitaev-Heisenberg model defined on a 3D hyperhoneycomb lattice, by using the pseudofermion functional renormalization group method. We show that the ground-state phase diagram contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases, similar to the two-dimensional honeycomb case. Our results respect the four-sublattice symmetry inherent in the model, which was violated in the previous study. Moreover, we also show how the phase diagram changes with the anisotropy in the interactions. The results provide a reference for the search of the hyperhoneycomb Kitaev materials.