The quantum Hall effect in two-dimensional electron gases involves the flow of topologically protected dissipationless charge currents along the edges of a sample. Integer or fractional electrical conductance is associated with edge currents of electrons or quasiparticles with fractional charges, respectively. It has been predicted that quantum Hall phenomena can also be created by edge currents with a fundamentally different origin: the fractionalization of quantum spins. However, such quantization has not yet been observed. Here we report the observation of this type of quantization of the Hall effect in an insulating two-dimensional quantum magnet, α-RuCl, with a dominant Kitaev interaction (a bond-dependent Ising-type interaction) on a two-dimensional honeycomb lattice. We find that the application of a magnetic field parallel to the sample destroys long-range magnetic order, leading to a field-induced quantum-spin-liquid ground state with substantial entanglement of local spins. In the low-temperature regime of this state, the two-dimensional thermal Hall conductance reaches a quantum plateau as a function of the applied magnetic field and has a quantization value that is exactly half of the two-dimensional thermal Hall conductance of the integer quantum Hall effect. This half-integer quantization of the thermal Hall conductance in a bulk material is a signature of topologically protected chiral edge currents of charge-neutral Majorana fermions (particles that are their own antiparticles), which have half the degrees of freedom of conventional fermions. These results demonstrate the fractionalization of spins into itinerant Majorana fermions and Z fluxes, which is predicted to occur in Kitaev quantum spin liquids. Above a critical magnetic field, the quantization disappears and the thermal Hall conductance goes to zero rapidly, indicating a topological quantum phase transition between the states with and without chiral Majorana edge modes. Emergent Majorana fermions in a quantum magnet are expected to have a great impact on strongly correlated quantum matter, opening up the possibility of topological quantum computing at relatively high temperatures.

Finite-temperature (T ) properties of a Kitaev model defined on a honeycomb lattice are investigated by a quantum Monte Carlo simulation, from the viewpoint of fractionalization of quantum S = 1/2 spins into two types of Majorana fermions, itinerant and localized. In this system, the entropy is released successively at two well-separated T scales, as a clear indication of the thermal fractionalization. We show that the high-T crossover, which is driven by itinerant Majorana fermions, is closely related with the development of nearestneighbor spin correlations. On the other hand, the low-T crossover originates in thermal fluctuations of fluxes composed of localized Majorana fermions, by which the spectrum of itinerant Majorana fermions is significantly disturbed. As a consequence, in the intermediate-T range between the two crossovers, the system exhibits Tlinear behavior in the specific heat and coherent transport of Majorana fermions, which are unexpected for the Dirac semimetallic spectrum in the low-T limit. We also show that the flux fluctuations tend to open an energy gap in the Majorana spectrum near the gapless-gapped phase boundary. Our results indicate that the fractionalization is experimentally observable in the specific heat, spin correlations, and transport properties.

Conventionally ordered magnets possess bosonic elementary excitations, called magnons. By contrast, no magnetic insulators in more than one dimension are known whose excitations are not bosons but fermions. Theoretically, some quantum spin liquids (QSLs) [1] -new topological phases which can occur when quantum fluctuations preclude an ordered state -are known to exhibit Majorana fermions [2] as quasiparticles arising from fractionalization of spins [3]. Alas, despite much searching, their experimental observation remains elusive. Here, we show that fermionic excitations are remarkably directly evident in experimental Raman scattering data [4] across a broad energy and temperature range in the two-dimensional material α-RuCl 3 . This shows the importance of magnetic materials as hosts of Majorana fermions. In turn, this first systematic evaluation of the dynamics of a QSL at finite temperature emphasizes the role of excited states for detecting such exotic properties associated with otherwise hard-to-identify topological QSLs.The Kitaev model has recently attracted attention as a canonical example of a QSL with emergent fractionalized fermionic excitations [2,5]. The model is defined for S = 1/2 spins on a honeycomb lattice with anisotropic bond-dependent interactions, as shown in Fig. 1a [2]. Recent theoretical work -by providing access to properties of excited states -has predicted signs of Kitaev QSLs in the dynamical response at T = 0 [6,7] and in the T dependence of thermodynamic quantities [8,9]. However, the dynamical properties at finite T have remained a theoretical challenge as it is necessary to handle quantum and thermal fluctuations simultaneously. Here, by calculating dynamical correlation functions over a wide temperature range we directly identify signatures of fractionalization in available experimental inelastic light scattering data.In real materials, Kitaev-type anisotropic interactions may appear through a superexchange process between j eff = 1/2 localized moments in the presence of strong spin-orbit coupling [10]. Such a situation is believed to be realised in several materials, such as iridates A 2 IrO 3 (A=Li, Na) [11,12] and a ruthenium compound α-RuCl 3 [4,[13][14][15]. These materials show magnetic ordering at a low T (∼ 10 K), indicating that some exchange interactions coexist with the Kitaev exchange and give rise to the magnetic order instead of the QSL ground state [16][17][18][19]. Nevertheless, evidence suggests that the Kitaev interaction is predominant (several tens to hundreds of Kelvin) [15,[18][19][20][21][22], which may provide an opportunity to observe the fractional excitations in a quantum paramagnetic state above the transition temperature as a proximity effect of the QSL phase.In particular, unconventional excitations were observed by polarized Raman scattering in α-RuCl 3 [4]. In this material, Néel ordering sets in only at T c ∼ 14K, while the Kitaev interaction appears to be much larger than the Heisenberg interaction [15,22], and hence finite-temperature signatures...

Quantum spin liquid is an exotic quantum state of matter in magnets. This state is a spin analogue of the liquid helium which does not solidify down to the lowest temperature due to strong quantum fluctuations. In conventional fluids, liquid and gas possess the same symmetry and adiabatically connect to each other by bypassing the critical end point. We find that the situation is qualitatively different in quantum spin liquids realizing in a three-dimensional Kitaev model; both gapless and gapped quantum spin liquid phases at low temperatures are always distinguished from the high-temperature paramagnet (spin gas) by a phase transition. The results challenge common belief that the absence of thermodynamic singularity down to the lowest temperature is a symptom of a quantum spin liquid. Tremendous efforts have been devoted to the realization of QSL, and several candidates were recently discovered in quasi two-dimensional (2D) and three-dimensional (3D) compounds [2][3][4][5][6]. In these compounds, QSL is usually identified by the absence of anomalies in the temperature (T ) dependence of physical quantities. Namely, it is implicitly supposed that a spin "gas" corresponding to the high-T paramagnet is adiabatically connected with QSL. This common belief lends itself to the fact that liquid and gas are adiabatically connected with each other in conventional fluids. In fact, the concept of QSL was originally introduced on the analogy of helium in which the liquid phase is retained down to the lowest T due to strong quantum fluctuations [7].In general, however, liquid and gas are distinguished by a discontinuous phase transition, while the adiabatic connection between them is guaranteed beyond the critical end point. Hence, a phase transition separating paramagnet and QSL is also expected. Nevertheless, the theory for thermodynamics of QSLs has not been seriously investigated thus far, and a thermodynamic phase transition for QSL has not ever been reported beyond the mean-field approximation. It is highly nontrivial whether a liquid-gas transition exists in quantum spin systems in a similar manner to that in conventional fluids. The issue is critical not only for theoretical understanding of QSLs but also for the interpretation of existing and forthcoming experiments.The lack of theoretical investigation of thermodynamics of QSLs is mainly due to the following two difficulties. One is the scarcity of well-identified QSLs. It is hard to characterize QSL because spatial quantum entanglement and many-body effects are essential for realizing QSL [8, 9]. The other difficulty lies in less choice of effective theoretical tools. Any biased approximation might be harmful for taking into account strong quantum and thermal fluctuations.In this Letter, we solve these difficulties by investigating a 3D extension of the Kitaev model [10], which sup- ports well-identified QSLs as the exact ground states [11] by applying an unbiased quantum Monte Carlo (MC) simulation without negative sign problem. By clarifying the phase diagram ...

Geometrical constraints to the electronic degrees of freedom within condensed-matter systems often give rise to topological quantum states of matter such as fractional quantum Hall states, topological insulators, and Weyl semimetals 1-3 . In magnetism, theoretical studies predict an entangled magnetic quantum state with topological ordering and fractionalized spin excitations, the quantum spin liquid 4 . In particular, the so-called Kitaev spin model 5 , consisting of a network of spins on a honeycomb lattice, is predicted to host Majorana fermions as its excitations. By means of a combination of specific heat measurements and inelastic neutron scattering experiments, we demonstrate the emergence of Majorana fermions in single crystals of α-RuCl 3 , an experimental realization of the Kitaev spin lattice. The specific heat data unveils a two-stage release of magnetic entropy that is characteristic of localized and itinerant Majorana fermions. The neutron scattering results corroborate this picture by revealing quasielastic excitations at low energies around the Brillouin zone centre and an hour-glass-like magnetic continuum at high energies. Our results confirm the presence of Majorana fermions in the Kitaev quantum spin liquid and provide an opportunity to build a unified conceptual framework for investigating fractionalized excitations in condensed matter 1,6-8 .Quantum spin liquids (QSLs) are an unconventional electronic phase of matter characterized by an absence of magnetic longrange order down to zero temperature. They are typically predicted to occur in geometrically frustrated magnets such as triangular, kagome, and pyrochlore lattices 4 , and typically display a macroscopic degeneracy that stabilizes a topologically ordered ground state. The Kitaev QSL state arises as an exact solution of the ideal two-dimensional (2D) honeycomb lattice with bond-directional Ising-type interactions (H = J γ K S γ i S γ j ; γ = x, y, z) on the three dis-