The honeycomb lattice iridates $A_2$IrO$_3$ ($A =$ Na, Li) are candidates for realization of the Kitaev-Heisenberg model although their proximity to Kitaev's quantum Spin-Liquid (QSL) is still debated. We report on heat capacity $C$ and entropy $S_{mag}$ for $A_2$IrO$_3$ ($A =$ Na, Li) in the temperature range $0.075~{\rm K}\leq T \leq 155$~K\@. We find a well separated two-peak structure for the magnetic heat capacity $C_{mag}$ for both materials and $S_{mag}$ for Na$_2$IrO$_3$ shows a shoulder between the peaks with a value close to ${1\over 2}$Rln$2$. These features signal the fractionalization of spins into Majornana Fermions close to Kitaev's QSL [Phys. Rev. B {\bf 92}, 115122 (2015); Phys. Rev. B {\bf 93}, 174425 (2016).]. These results provide the first thermodynamic evidence that $A_2$IrO$_3$ are situated close to the Kitaev QSL. Additionally we measure the high temperature $T\leq 1000$~K magnetic susceptibility $χ$ and estimate the Weiss temperature $θ$ in the true paramagnetic state. We find $θ\approx -127$~K and $-105$~K, for Na$_2$IrO$_3$ and Li$_2$IrO$_3$, respectively.
We investigate a bilayer Kitaev model, where two honeycomb layers are coupled by the Heisenberg interactions, to discuss effects of an interlayer coupling against the Kitaev quantum spin liquids (QSLs). In this model, there exists a local conserved quantity, which results in no long-range spin correlations in the system. Using the exact diagonalization, bond operator mean-field theory, and cluster expansion techniques, we study ground state properties in the system. The obtained results suggest the existence of a first-order quantum phase transition between the Kitaev QSL and singlet-dimer sta…
Quantum spin liquids (QSLs) form an extremely unusual magnetic state in which the spins are highly correlated and fluctuate coherently down to the lowest temperatures, but without symmetry breaking and without the formation of any static long-range-ordered magnetism. Such intriguing phenomena are not only of great fundamental relevance in themselves, but also hold the promise for quantum computing and quantum information. Among different types of QSLs, the exactly solvable Kitaev model is attracting much attention, with most proposed candidate materials, e.g., RuCl$_3$ and Na$_2$IrO$_3$, havin…
Quantum spin liquids (QSLs) are topologically ordered exotic states of matter that host fractionalized excitations. Kitaev proposed a particular route towards a QSL via strongly bond-dependent interactions on the hexagonal lattice. A number of candidate Kitaev QSL materials have been pursued, but all have appreciable non-Kitaev interactions, which put these systems far from the QSL regime. Using time-domain terahertz spectroscopy (TDTS) we observed a broad magnetic continuum over a wide range of temperature and field in the honeycomb cobalt-based magnet, BaCo$_2$(AsO$_4$)$_2$, which has been p…
Topological states of matter such as quantum spin liquids (QSLs) are of great interest because of their remarkable predicted properties including protection of quantum information and the emergence of Majorana fermions. Such QSLs, however, have proven difficult to identify experimentally. The most promising approach is to study their exotic nature via the wave-vector and intensity dependence of their dynamical response in neutron scattering. A major search has centered on iridate materials which are proposed to realize the celebrated Kitaev model on a honeycomb lattice - a prototypical topolog…
We investigate the quantum spin liquid (QSL) ground state of anisotropic Kitaev model with antiferromagnetic (AFM) coupling under the $[001]$ magnetic field with the finite-temperature Lanczos method (FTLM). In this anisotropic AFM Kitaev model with $K_{X}=K_{Y}$, $K_{X}+K_{Y}+K_{Z}=-3K$, and $K_{Z}<-K$, with magnetic field increasing, the gapped QSL experiences a transition to a gapless QSL at $h_{c1}=gμ_{B}H_{z1}/K$, to another gapless QSL with $C_{6}$ rotational symmetry at $h_{c2}$, and to a new $U(1)$ gapless QSL between $h_{c3}$ and $h_{c4}$, respectively. These indicate that magnetic fi…
The main question we address is how to probe the fractionalized excitations of a quantum spin liquid (QSL), for example, in the Kitaev honeycomb model. By analyzing the energy spectrum and entanglement entropy, for antiferromagnetic couplings and a field along either [111] or [001], we find a gapless QSL phase sandwiched between the non-Abelian Kitaev QSL and polarized phases. Increasing the field strength towards the polarized limit destroys this intermediate QSL phase, resulting in a considerable reduction in the number of frequency modes and the emergence of a beating pattern in the local d…
Significant efforts have focused on the magnetic excitations of relativistic Mott insulators, predicted to realize the Kitaev quantum spin liquid (QSL). This exactly solvable model involves a highly entangled state resulting from bond-dependent Ising interactions that produce excitations which are non-local in terms of spin flips. A key challenge in real materials is identifying the relative size of the non-Kitaev terms and their role in the emergence or suppression of fractional excitations. Here, we identify the energy and temperature boundaries of non-Kitaev interactions by direct compariso…
The exactly-solvable Kitaev model of two-dimensional honeycomb magnet leads to a quantum spin liquid (QSL) characterized by Majorana fermions, relevant for fault-tolerant topological quantum computations. In the high-field paramagnetic state of $α$-RuCl3, half-integer quantization of thermal Hall conductivity has been reported as a signature of edge current, but the bulk nature of this state remains elusive. Here, from high-resolution heat capacity measurements under in-plane field rotation, we find strongly angle-dependent low-energy excitations in bulk $α$-RuCl3. The excitation gap has a sex…
The quantum spin liquid (QSL) state has been searched intensively in Kitaev-like materials, such as the Iridium oxides $A_2$IrO$_3$ and $α$-RuCl$_3$. The half-filled Kitaev-Hubbard model with bond dependent hopping terms is used to describe the Kitaev-like materials, which is calculated using the state-of-the-art fermionic projected entangled pair states (fPEPS) method. We find a QSL phase near the Mott insulator transition, which has a strong first-order transition to the semi-metal phase with the decrease of Hubbard $U$. We suggest that a promising routine to find the QSL is to find the Irid…
The ground state of the bilayer Kitaev model with the Heisenberg-type interlayer exchange interaction is investigated by means of the exact diagonalization. Calculating the ground-state energy, local quantity defined on each plaquette, and dynamical spin structure factor, we obtain results suggesting the existence of a quantum phase transition between the Kitaev quantum spin liquid (QSL) and dimer singlet states when the interlayer coupling is antiferromagnetic. On the other hand, increasing the ferromagnetic interlayer coupling, there exists no singularity in the physical quantities, suggesti…
The Kitaev model, whose ground state is a quantum spin liquid (QSL), was originally conceived for spin $S=1/2$ moments on a honeycomb lattice. In recent years, the model has been extended to higher $S$ from both theoretical and experimental interests, but the stability of the QSL ground state has not been systematically clarified for general $S$, especially in the presence of other additional interactions, which inevitably exist in candidate materials. Here we study the spin-$S$ Kitaev-Heisenberg models by using an extension of the pseudofermion functional renormalization group method to gener…
A decade ago, Alexei Kitaev proposed an exactly solvable $S$ = 1/2 model on a two-dimensional honeycomb lattice, where the spins fractionalize into Majorana fermions and form a topological quantum spin liquid (QSL) in the ground state. It was soon recognized that a family of complex iridium oxides, as well as ruthenium chloride, with honeycomb structure are magnetic insulators and accommodate essential ingredients of the Kitaev model, due to the interplay of electron correlation and spin-orbit coupling. This initiated a race to materialize the Kitaev QSL and to capture the signature of Majoran…
The pursuit of quantum spin liquid (QSL) in the Kitaev honeycomb magnets has drawn intensive attention recently. In particular, $α$-RuCl$_3$ has been widely recognized as a promising candidate for the Kitaev QSL. Although the compound exhibits an antiferromagnetic order under zero field, it is believed to be endowed with fractionalized excitations, and can be driven to the QSL phase by magnetic fields. Here, based on a realistic $K$-$J$-$Γ$-$Γ'$ model for $α$-RuCl$_3$ [1], we exploit the exponential tensor renormalization group approach to explore the phase diagram of the compound under magnet…
Recent experimental evidence for a field-induced quantum spin liquid (QSL) in $α$-RuCl$_3$ calls for an understanding for the ground state of honeycomb Kitaev model under a magnetic field. In this work we address the nature of an enigmatic gapless paramagnetic phase in the antiferromagnetic Kitave model, under an intermediate magnetic field perpendicular to the plane. Combining theoretical and numerical efforts, we identify this gapless phase as a $U(1)$ QSL with spinon Fermi surfaces. We also reveal the nature of continuous quantum phase transitions involving this $U(1)$ QSL, and obtain a pha…
Kitaev magnets are materials with bond-dependent Ising interactions between localized spins on a honeycomb lattice. Such interactions could lead to a quantum spin-liquid (QSL) ground state at zero temperature. Recent theoretical studies suggest two potential signatures of a QSL at finite temperatures, namely a scaling behavior of thermodynamic quantities in the presence of quenched disorder, and a two-step release of the magnetic entropy. Here, we present both signatures in Ag$_{3}$LiIr$_{2}$O$_{6}$ which is synthesized from $α$-Li$_{2}$IrO$_{3}$ by replacing the inter-layer Li atoms with Ag a…
The S=3/2 Kitaev honeycomb model (KHM) is a quantum spin liquid (QSL) state coupled to a static Z$_2$ gauge field. Employing an SO(6) Majorana representation of spin 3/2's, we find an exact representation of the conserved plaquette fluxes in terms of static Z$_2$ gauge fields akin to the S=1/2 KHM which enables us to treat the remaining interacting matter fermion sector in a parton mean-field theory. We uncover a ground-state phase diagram consisting of gapped and gapless QSLs. Our parton description is in quantitative agreement with numerical simulations, and is furthermore corroborated by th…
Recently H$_3$LiIr$_2$O$_6$ has been reported as a spin-orbital entangled quantum spin liquid (QSL) [K. Kitagawa et al., Nature {\bf 554}, 341 (2018)], albeit its connection to Kitaev QSL has not been yet identified. To unveil the related Kitaev physics, we perform the first Raman spectroscopy studies on single crystalline H$_3$LiIr$_2$O$_6$ samples. We implement a soft chemical replacement of Li$^+$ with H$^+$ from $α$-Li$_2$IrO$_3$ single crystals to synthesize the single crystal samples of the iridate second generation H$_3$LiIr$_2$O$_6$. The Raman spectroscopy can be used to diagnose the Q…
It is now well established that the Kitaev honeycomb model in a magnetic field along the $[111]$-direction harbors an intermediate gapless quantum spin liquid (QSL) phase sandwiched between a gapped non-abelian QSL at low fields $H< H_{c1}$ and a partially polarized phase at high fields $H> H_{c2}$. Here, we analyze the low field and high field phases and phase transitions in terms of single- and two-magnon excitations using exact diagonalization (ED) and density matrix renormalization group (DMRG) methods. We find that the energy to create a bound state of two-magnons $Δ_p$ becomes lower than…
We present the electron tunneling transport and spectroscopic characters of a superconducting {\it Josephson} junction with a barrier of single anisotropic Kitaev quantum spin liquid (QSL) layer. We find that the dynamical spin correlation features are well reflected in the direct-current differential conductance $dI^{c}/dV$ of the single-particle tunneling, including the unique spin gap and dressed itinerant Majorana dispersive band, in addition to an energy shift $2Δ$ of two-lead superconducting gaps. From the spectral characters, we identify different topological quantum phases of the aniso…
A new quantum spin liquid (QSL) candidate material H$_{3}$LiIr$_{2}$O$_{6}$ was synthesized recently and was found not to show any magnetic order or phase transition down to low temperatures. In this work, we study the quantum dynamics of the hydrogen ions, i.e., protons, in this material by combining first-principles calculations and theoretical analysis. We show that each proton and its adjacent oxygen ions form an electric dipole. The dipole interactions and the proton tunneling are captured by a transverse-field Ising model with a quantum disordered paraelectric ground state. The dipole ex…
The Kitaev model predicts that quantum spin liquids (QSLs) will form at low temperatures under certain special conditions, and materials hosting the QSL state are frequently sought. The layered honeycomb lattice material α-RuCl3 has emerged as a prime candidate for displaying the Kitaev QSL state. Here we describe a new polymorph of RuI3 with a layered honeycomb lattice structure, synthesized at moderately high pressures and stable under ambient conditions. Preliminary characterization reveals metallic, paramagnetic behavior, the absence of long-range magnetic order down to 0.35 K and an unusu…
In recent years, several magnetic Mott insulators with strong spin-orbit couplings were suggested to be proximate to the Kitaev quantum spin liquid (QSL) whose one of the most exciting features is the fractionalization of spin excitations into itinerant Majorana fermions and static $Z_2$ fluxes. Motivated by the emergence of this plethora of 4d and 5d transition metal Kitaev materials and by the fact that some level of disorder is inevitable in real materials, here we study how the Kitaev QSL responds to various forms of disorder, such as vacancies, impurities, and bond randomness. First, we a…
Motivated by recent synthesis of the hyper-honeycomb material $β$-$\mathrm{Li_2 Ir O_3}$, we study the dynamical structure factor (DSF) of the corresponding 3D Kitaev quantum spin-liquid (QSL), whose fractionalised degrees of freedom are Majorana fermions and emergent flux-loops. Properties of this 3D model are known to differ in important ways from those of its 2D counterpart -- it has finite-temperature phase transition, as well as distinct features in Raman response. We show, however, that the qualitative behaviour of the DSF is broadly dimension-independent. Characteristics of the 3D DSF i…
Contrary to the original expectation, Na$_2$IrO$_3$ is not a Kitaev's quantum spin liquid (QSL) but shows a zig-zag-type antiferromagnetic order in experiments. Here we propose experimental clues and criteria to measure how a material in hand is close to the Kitaev's QSL state. For this purpose, we systematically study thermal and spin excitations of a generalized Kitaev-Heisenberg model studied by Chaloupka $et$ $al$. in Phys. Rev. Lett. 110, 097204 (2013) and an effective ab initio Hamiltonian for Na$_2$IrO$_3$ proposed by Yamaji $et$ $al$. in Phys. Rev. Lett. 113, 107201 (2014), by employin…
Searching for Kitaev quantum spin liquid (QSL) is a fascinating and challenging problem. Much effort has been devoted to honeycomb lattice candidates with strong spin-orbit coupling in 5d-electron iridates and 4delectron RuCl3. Recently, theoretical studies suggested that the 3d7 Co-based honeycomb materials with high spin state S=3/2 and effective orbital angular momentum L=1 could also be promising candidates of Kitaev QSL. One of the candidates, BaCo2(AsO4)2, was revisited recently. The long range magnetic order in BaCo2(AsO4)2 can be suppressed by very weak in-plane magnetic field, suggest…
We identify fingerprints of a proximate quantum spin-liquid (QSL), observable by finite-temperature dynamical thermal transport within a minimal version of the idealized Kitaev model on a two-leg ladder, if subjected to inevitably present Heisenberg couplings. Using exact diagonalization and quantum typicality, we uncover (i) an insulator-conductor crossover induced by fracton recombination at infinitesimal Heisenberg coupling, (ii) low- and high-energy signatures of fractons, which survive far off the pure QSL point, and (iii) a non-monotonous current life-time versus Heisenberg couplings. Gu…
The Kitaev model on a honeycomb lattice predicts a paradigmatic quantum spin liquid (QSL) exhibiting Majorana Fermion excitations. The insight that Kitaev physics might be realized in practice has stimulated investigations of candidate materials, recently including alpha-RuCl3. In all the systems studied to date, non-Kitaev interactions induce magnetic order at low temperature. However, in-plane magnetic fields of roughly 8 Tesla suppress the long-range magnetic order in alpha-RuCl3 raising the intriguing possibility of a field-induced QSL exhibiting non-Abelian quasiparticle excitations. Here…
Recently it was realized that the zigzag magnetic order in Kitaev materials can be stabilized by small negative off-diagonal interactions called the $Γ'$ terms. To fully understand the effect of the $Γ'$ interactions, we investigate the quantum $K$-$Γ$-$Γ'$ model on the honeycomb lattice using the variational Monte Carlo method. Two multinode Z$_2$ quantum spin liquids (QSLs) are found at $Γ'>0$, one of which is the previously found proximate Kitaev spin liquid called the PKSL14 state which shares the same projective symmetry group (PSG) with the Kitaev spin liquid. A remarkable result is that…
We investigate signatures of topological quantum phase transitions (TQPTs) between the $Z_2$ quantum spin liquids (QSLs). In two spatial dimensions, $Z_2$ QSLs and their TQPTs are only well defined at zero temperature ($T=0$), and it is imperative to clarify their observable signatures under nonzero temperatures. Here, we present the vestiges of TQPTs between $Z_2$ QSLs with Majorana fermions in terms of thermal Hall conductivity $κ_{xy}$ at nonzero temperatures. The $κ_{xy}/T$ shows characteristic temperature dependences around TQPTs. We argue that an exponential upturn near $T=0$ and the pea…
As one of the most promising Kitaev quantum-spin-liquid (QSL) candidates, $α$-RuCl$_3$ has received a great amount of attention. However, its ground state exhibits a long-range zigzag magnetic order, which defies the QSL phase. Nevertheless, the magnetic order is fragile and can be completely suppressed by applying an external magnetic field. Here, we explore the evolution of magnetic excitations of $α$-RuCl$_3$ under an in-plane magnetic field, by carrying out inelastic neutron scattering measurements on high-quality single crystals. Under zero field, there exist spin-wave excitations near th…
We study the spin transport through the quantum spin liquid (QSL) by investigating the real-time and real-space dynamics of the Kitaev spin system with a zigzag structure in terms of the time-dependent Majorana mean-field theory. After the magnetic field pulse is introduced to one of the edges, the spin moments are excited in the opposite edge region although no spin moments are induced in the Kitaev QSL region. This unusual spin transport originates from the fact that the $S=1/2$ spins are fractionalized into the itinerant and localized Majorana fermions in the Kitaev system. Although both Ma…
The exploration of quantum spin liquids (QSLs) has been guided by different approaches including the resonating valence bond (RVB) picture, deconfined lattice gauge theories and the Kitaev model. More recently, a spin liquid ground state was numerically established on the ruby lattice, inspired by the Rydberg blockade mechanism. Here we unify these varied approaches in a single parent Hamiltonian, in which local fluctuations of anyons stabilize deconfinement. The parent Hamiltonian is defined on kagomé triangles -- each hosting four RVB-like states -- and includes only Ising interactions and s…
Honeycomb iridates such as $γ$-Li$_2$IrO$_3$ are argued to realize Kitaev spin-anisotropic magnetic exchange, along with Heisenberg and possibly other couplings. While systems with pure Kitaev interactions are candidates to realize a quantum spin liquid ground state, in $γ$-Li$_2$IrO$_3$ it has been shown that the balance of magnetic interactions leads to the incommensurate spiral spin order at ambient pressure below 38 K. We study the fragility of this state in single crystals of $γ$-Li$_2$IrO$_3$ using resonant x-ray scattering (RXS) under applied hydrostatic pressures of up to 3.0 GPa. RXS …
$α$-RuCl$_3$ is a leading candidate material for theobservation of physics related to the Kitaev quantum spin liquid (QSL). By combined susceptibility, specific-heat, and nuclear-magnetic-resonance measurements, we demonstrate that $α$-RuCl$_3$ undergoes a quantum phase transition to a QSL in a magnetic field of 7.5 T applied in the $ab$ plane. We show further that this high-field QSL phase has gapless spin excitations over a field range up to 16 T. This highly unconventional result, unknown in either Heisenberg or Kitaev magnets, offers insight essential to establishing the physics of $α$-RuC…
$α$-RuCl$_3$ has been studied extensively because of its proximity to the Kitaev quantum-spin-liquid (QSL) phase and the possibility of approaching it by tuning the competing interactions. Here we present the first polarized inelastic neutron scattering study on $α$-RuCl$_3$ single crystals to explore the scattering continuum around the $Γ$ point at the Brillouin zone center, which was hypothesized to be resulting from the Kitaev QSL state but without concrete evidence. With polarization analyses, we find that while the spin-wave excitations around the M point vanish above the transition tempe…
