ABSTRACT Quantum spin liquids (QSLs) are magnetically frustrated phases characterized by spin fractionalization, emergent gauge fields, and long‐range entanglement. Quantum spin–orbital liquids (QSOLs) form a subset of QSLs with fluctuating orbital degrees of freedom, normally adding a layer of complexity to an already involved research field. This topical review provides guidelines for understanding a specific type of QSOL in which the orbital operators facilitate the analysis of quantum liquids, thereby providing adequate starting points for /exploring these phases. Such models are extensions of the spin‐1/2 Kitaev honeycomb model (KHM), in the sense that their exact solutions depend on an extensive number of conserved quantities, combined with a mapping to a problem of Majorana fermions hopping on a static gauge field. The starting points to understand such models are classical Hamiltonians characterized by bond‐dependent Ising interactions. Such classical models are exactly solvable in spin basis thanks to their extensive symmetries and can be directly connected to classical spin ice (CSI) systems that satisfy a Gauss law. QSOLs are easily stabilized in these models by applying a transverse field or introducing other exchange mechanisms that preserve the conserved local operators. The theory of such Ising QSOLs bridges the KHM to specific types of CSIs, thus providing further insight into paradigmatic forms of spin liquids. Furthermore, bond‐dependent Ising models are related to minimal Hamiltonians describing Rydberg‐atom simulations, which provide experimental grounds for investigating them.
ABSTRACT Quantum spin liquids (QSLs) are magnetically frustrated phases characterized by spin fractionalization, emergent gauge fields, and long‐range entanglement. Quantum spin–orbital liquids (QSOLs) form a subset of QSLs with fluctuating orbital degrees of freedom, normally adding a layer of complexity to an already involved research field. This topical review provides guidelines for understanding a specific type of QSOL in which the orbital operators facilitate the analysis of quantum liquids, thereby providing adequate starting points for /exploring these phases. Such models are extensions of the spin‐1/2 Kitaev honeycomb model (KHM), in the sense that their exact solutions depend on an extensive number of conserved quantities, combined with a mapping to a problem of Majorana fermions hopping on a static gauge field. The starting points to understand such models are classical Hamiltonians characterized by bond‐dependent Ising interactions. Such classical models are exactly solvable in spin basis thanks to their extensive symmetries and can be directly connected to classical spin ice (CSI) systems that satisfy a Gauss law. QSOLs are easily stabilized in these models by applying a transverse field or introducing other exchange mechanisms that preserve the conserved local operators. The theory of such Ising QSOLs bridges the KHM to specific types of CSIs, thus providing further insight into paradigmatic forms of spin liquids. Furthermore, bond‐dependent Ising models are related to minimal Hamiltonians describing Rydberg‐atom simulations, which provide experimental grounds for investigating them.
Quantum spin liquids (QSLs) are strongly correlated systems displaying fascinating phenomena like long-range entanglement and fractionalized excitations. The research on these states has since its beginning followed trends generated by the synthesis of new compounds and the construction of new theoretical tools. In coherence with this history, a manifold of new results about QSLs were established during the past decade due to studies on the integrable Kitaev model on the honeycomb lattice. This j = 1/2 model displays bond-dependent and anisotropic exchanges that are essential to stabilize its QSL ground state with Majorana fermion excitations and emergent Z2 gauge field. Even more interestingly, this model is relevant to understand the magnetism of a certain class of 4/5d5 Mott insulators with specific lattice constraints, t2g orbital degeneracy and strong spin-orbit coupling (SOC). This mechanism defining these so-called Kitaev materials can be applied to similar compounds based on transition metal ions in different electronic configurations. In this thesis, I investigate minimal models for two types of 4/5d1 Mott insulators: the ones on the ordered double perovskite structure (ODP) and the ones isostructural to the Kitaev materials. Their effective models generically show bond-dependent and anisotropic interactions involving multipoles of an effective j = 3/2 angular momentum. Such degrees of freedom are conveniently written in terms of pseudospin s and pseudo-orbital τ operators resembling spin and orbital operators of Kugel-Khomskii models with twofold orbital degeneracy. Despite their anisotropy, the two realistic models display continuous global symmetries in the limit of vanishing Hund\'s coupling enhancing quantum fluctuations and possibly stabilizing a QSL phase. Parton mean-field theory was used to propose fermionic QSLs that will be called quantum spin-orbital liquids (QSOLs) due their dependence with s and τ. On ODPs, I studied a chiral QSOL with Majorana fermion excitations and a gapless spectrum characterized by nodal lines along the edges of the Brillouin zone. These nodal lines are topological defects of a non-Abelian Berry connection and the system exhibits dispersing surface states. Several experimental responses of the chiral QSOL within the mean-field approximation are compared with the experimental data available for the spin liquid candidate Ba2YMoO6. Moreover, based on a symmetry analysis, I discuss the operators involved in resonant inelastic X-ray scattering (RIXS) amplitudes for 4/5d1 Mott insulators and show that the RIXS cross sections allow one to selectively probe pseudospin and pseudo-orbital degrees of freedom. For the chiral spin-orbital liquid in particular, these cross sections provide information about the spectrum for different flavors of Majorana fermions. The model for materials isostructural to the Kitaev materials has an emergent SU(4) symmetry that is made explicit by means of a Klein transformation on pseudospin degrees of freedom. The model is known to stabilize a QSOL on the honeycomb lattice and instigated the investigation of QSOLs on a generalization of this lattice to three dimensions. Parton mean-field theory was used once again to propose the liquid states, and a variational Monte Carlo (VMC) method was used to compute the energies of the projected wave functions. The numerical results show that the lowest-energy QSOL corresponds to a zero-flux state with a Fermi surface of four-color fermionic partons. Further VMC computations also revealed that this state is stable against formation of plaquette ordering (tetramerization). The energy of this QSOL is highly competitive even when Hund\'s coupling induced perturbations are included, as shown by comparison with simple ordered states. Extensions and perspectives for future work are discussed in the end of this thesis.
Quantum spin liquids (QSLs) host a variety of fractionalized particles. In Kitaev's paradigmatic honeycomb model a spin-fractionalizes intoflux due to emergentgauge field and matter Majorana fermions. Although these excitations have well-defined dynamics in the integrable limit, their direct experimental identification is notoriously challenging: realistic materials inevitably host additional symmetry-allowed interactions that break integrability and hybridize gauge and matter sectors, while magnetic fields, which are often required to suppress competing order and stabilize a putative QSL regime, further entangle the responses of different fractionalized quasiparticles and may even drive the system into field-induced spin-liquid phases that are not adiabatically connected to the integrable limit. A prominent example is the quantum Majorana metal, in which the distinct dynamics of fractionalized Majorana fermions can become directly visible in scattering. This review highlights recent progress on these related questions: in which field-stabilized QSL regimes and nearby emergent phases, and under what conditions, can the response of a specific fractionalized quasiparticle be isolated and positively understood, thereby clarifying the existence and the experimental scope of putative spin liquids? We review the progress on these questions across Abelian, non-Abelian, and emergent quantum phases under magnetic field that are not perturbatively connected to the integrable limit. We connect these field-induced dynamical phenomena to concrete experimental observables, relevant for neutron scattering, resonant inelastic x-ray scattering, and pump-probe spectroscopy, techniques that are capable of identifying the nature of different magnetic-field-induced phases and resolving specific types of fractionalized particles, including Majoranas andfluxes.
In the field of quantum magnetism, the exactly solvable Kitaev honeycomb model serves as a paradigm for the fractionalization of spin degrees of freedom and the formation of [Formula: see text] quantum spin liquids. An intense experimental search has led to the discovery of a number of spin-orbit entangled Mott insulators that realize its characteristic bond-directional interactions and, in the presence of magnetic fields, exhibit no indications of long-range order. Here, we map out the complete phase diagram of the Kitaev model in tilted magnetic fields and report the emergence of a distinct gapless quantum spin liquid at intermediate field strengths. Analyzing a number of static, dynamical, and finite temperature quantities using numerical exact diagonalization techniques, we find strong evidence that this phase exhibits gapless fermions coupled to a massless U(1) gauge field. We discuss its stability in the presence of perturbations that naturally arise in spin-orbit entangled candidate materials.
The formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.
These lecture notes focus on the Kitaev honeycomb model, a two-dimensional quantumsystem with three distinct Ising-type interactions, depending on the direction of eachbond. This model has proven particularly important for describing quantum spin liquids,as well as for applications in topological quantum computing, due to its exotictopological properties. It constitutes the first exactly solvable theoretical model thatfully captures the behavior of a quantum spin liquid. We begin by presenting the geometryof the model, the Hamiltonian that describes the system, and Alexei Kitaev's effort toobtain an exact analytical solution, based on the decomposition of the Hilbert spaceinto flux sectors. We then present Ettore Majorana's observation that the Dirac equationalso admits real solutions, leading to the theoretical prediction of Majorana fermions-- particles that are identical to their antiparticles. We examine how the spatialseparation of Majorana operators leads to the rise of unbound Majorana fermions, andshowcase the exotic topological properties that manifest in physical systems such assuperconducting wires. Next, we represent the Hamiltonian in the basis of Majoranafermions, by expressing spin operators using four Majorana operators. The transitionfrom the extended space L~ to the physical subspace Lis achieved through a gauge transformation of the Z2 group. We thenreformulate the Hamiltonian in the basis of complex bond and matter fermions, transitionto momentum space, and assume that the system lies in the zero-flux sector. Through theBogoliubov--de Gennes transformation, we arrive at the final diagonalized Hamiltonianin the quasiparticle basis. Finally, we represent the possible phases -- the gaplessphase (phase B) and the gapped phases (phase Aαi) -- compute thecorresponding energy spectrum, and present the complete phase diagram. By employing thecorrelation function, we show that all phases are characterized by strong quantumentanglement and spin fractionalization, classifying the system as a quantum spin liquid.
Quantum spin orbital liquids (QSOLs) are a novel phase of matter, similar to quantum spin liquids, with quantum fluctuations in both spin and orbital degrees of freedom. We use nonequilibrium Green's function theory to study out-of-equilibrium spin transport in an exactly solvable QSOL model put forward by Yao and Lee. We find that the spin transport problem can be mapped to that of a free fermion problem with effective fermionic baths that have a rapidly varying density of states. In the gapless phase, the spin current ${I}_{s}\text{\ensuremath{-}}{V}_{s}$ relation is thus highly nonlinear, while in the chiral gapped phase, the spin current conductance is quantized to be $1/2\ensuremath{\pi}$ provided that the contacts are sufficiently wide. The quantized conductance is a signature of the topological nature of the chiral gapped QSOL.