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arXiv:1303.5922

Compass and Kitaev models -- Theory and Physical Motivations

Zohar Nussinov, Jeroen van den Brink

Published Mar 24, 2013• 12 citations
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Abstract

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.

DOI
10.48550/arxiv.1303.5922

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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.
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. A simple illustrative example is furnished by the 90\ifmmode^\circ\else\textdegree\fi{} compass model on a square lattice in which only couplings of the form ${\ensuremath{\tau}}_{i}^{x}{\ensuremath{\tau}}_{j}^{x}$ (where ${{\ensuremath{\tau}}_{i}^{a}{}}_{a}$ denote Pauli operators at site iii) are associated with nearest-neighbor sites iii and jjj separated along the xxx axis of the lattice while ${\ensuremath{\tau}}_{i}^{y}{\ensuremath{\tau}}_{j}^{y}$ couplings appear for sites separated by a lattice constant along the yyy axis. Similar compass-type interactions can appear in diverse physical systems. For instance, compass models describe Mott insulators with orbital degrees of freedom where interactions sensitively depend on the spatial orientation of the orbitals involved as well as the low-energy effective theories of frustrated quantum magnets, and a host of other systems such as vacancy centers, and cold atomic gases. The fundamental interdependence 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 nonfrustrated 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. In this article, compass models are reviewed 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. This is complemented by a survey of numerical results. In addition to reviewing past works, a number of other models are introduced and new results established. In particular, a general link between flat bands and symmetries is detailed.
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.
Quantum spin liquids have fascinated condensed matter physicists for decades because of their unusual properties such as spin fractionalization and long-range entanglement. Unlike conventional symmetry breaking, the topological order underlying quantum spin liquids is hard to detect experimentally. Even theoretical models are scarce for which the ground state is established to be a quantum spin liquid. The Kitaev honeycomb model and its generalizations to other tricoordinated lattices are chief counterexamples—they are exactly solvable, harbor a variety of quantum spin liquid phases, and are also relevant for certain transition metal compounds including the polymorphs of (Na,Li) 2 IrO 3 iridates and RuCl 3 . In this review, we give an overview of the rich physics of the Kitaev model, including two-dimensional and three-dimensional fractionalization as well as dynamic correlations and behavior at finite temperatures. We discuss the different materials and argue how the Kitaev model physics can be relevant even though most materials show magnetic ordering at low temperatures.
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.
A large part of modern condensed matter concerns non-local or global properties of systems under the broad term of topology. Closely intertwined with certain senses of topology are symmetries of the system, which physically constrain our description and offer non-perturbative insights into material properties. The range of possible phases of matter is greatly enriched by quantum mechanics, including certain types of non-local entanglement that cannot be understood in the usual symmetry-breaking paradigm of local order parameters. Beyond weakly interacting topology we find the stronger notion of \textit{topological order} in correlated systems, a prominent example being the S=12S = \tfrac{1}{2}S=21​ Kitaev honeycomb model in a weak magnetic field.Through a remarkable combination of lattice geometry, strong spin-orbit coupling, and electron interactions, several material candidates have been put forward to realize Kitaev physics, including the 4d4d4d Mott insulator α\alphaα-RuCl3_{3}3​ which offers tantalizing experimental hints of topological order. In this thesis we explore several aspects of Kitaev candidates and related systems under two broad themes. One involves symmetry-informed and constrained observables, including a proposed singularity in a thermodynamic quantity accompanying phase transitions between two Ising topological orders (ITOs). The ITO physics realized by the Kitaev model requires time-reversal symmetry to be broken, and much of the non-trivial physics is found at finite magnetic field. To this end, we study a microscopic spin model appropriate for α\alphaα-RuCl3_{3}3​ and show that it exhibits an interesting field-induced paramagnetic state between the low- and high-field magnetic orders, consistent with recent experimental findings. We determine several properties of a lower-dimensional spin-SSS analogue of the Kitaev model, including a multi-critical point at finite field separating different topological phases. Finally, we derive quantum state purities of entangled photons emitted from a spin-triplet superconducting heterostructure and discuss its use in a new paradigm of entangled spectroscopy.