key: cord-0108564-pyipwmxf
authors: Ralko, Arnaud; Merino, Jaime
title: Novel chiral quantum spin liquids in Kitaev magnets
date: 2019-10-22
journal: nan
DOI: nan
sha: 2745f0a9afb6ebf36e9c504f59bea38144df4c66
doc_id: 108564
cord_uid: pyipwmxf

Mott insulators under sufficiently strong spin-orbit coupling can display quantum spin liquid phases with topological order and fractional excitations. Quantum magnets with pure Kitaev spin exchange interactions can host a gapped quantum spin liquid with a single Majorana edge mode propagating in the counter-clockwise direction when a small positive magnetic field is applied. Here, we show how under a sufficiently strong positive magnetic field a topological transition into a gapped quantum spin liquid with two Majorana edge modes propagating in the clockwise direction occurs. The Dzyaloshinskii-Moriya interaction is found to turn the non-chiral Kitaev's gapless quantum spin liquid into a chiral one with equal Berry phases at the two Dirac points. Thermal Hall conductance experiments can provide evidence of the novel topologically gapped quantum spin liquid states predicted.

A quantum spin liquid (QSL) is an exotic state of matter in which localized spins do not order even at zero temperature in contrast to magnetic ordering observed in conventional insulating magnets. QSL's are highly entangled states which cannot be characterized by a Landau local order parameter. Exhibiting topological order, emergent gauge fields and fractional excitations [1, 2] , they are at the heart of an intense research activity. The first concrete example of a two-dimensional QSL has been the resonating valence bond (RVB) state envisioned by Anderson [3] to describe the ground state of triangular antiferromagnets and as the parent insulating phase of high-T c superconductors. Due to the spin correlations, a spin-flip in an RVB state fractionalizes into two neutral spin-1/2 particles (spinons) which can propagate freely around the lattice. In spite of intense experimental efforts, there is no unambiguous observation of fractionalization -such as the expected spinon continuum in the spin excitation spectra -in real materials.

The interplay between strong Coulomb interaction and spin-orbit coupling [4] in the honeycomb magnets such as A 2 IrO 3 [5] (with A=Li, Na), H 3 LiIr 2 O 6 [6] , α-RuCl 3 , and organometallic frameworks [7] can lead to special compass interactions which frustrate the magnetic order of the S = 1/2 pseudospins. The exact QSL ground sate of the Kitaev [8] model has opened the possibility of finding fractionalized excitations in spin-orbit coupled Mott insulators on honeycomb lattices. From the decomposition of the spin operators onto four non-interacting Majorana fermions, Kitaev showed that the elementary spin excitations of the Kitaev QSL (KQSL) are fractionalized into itinerant Majorana fermions with Dirac dispersion, and localized ones giving Z 2 gauge fluxes. Recent observations on α-RuCl 3 [9, 10] and H 3 LiIr 2 O 6 [6] have been interpreted in terms of the existence of such two types of excitations.

However, a realistic description of honeycomb mate-rials requires including additional spin interactions not included in the Kitaev model as well as considering large magnetic fields beyond the perturbative regime discussed so far. Since there is no exact solution in these physically relevant situations new theoretical approaches are required to properly describe the system. For instance, exact numerical and slave fermion approaches [11] [12] [13] of the Kitaev model have found a transition to a gapless U (1) spin liquid phase under sufficiently strong applied magnetic fields [9, 15] . On the other hand, the effect of Heisenberg and symmetric spin exchange terms needs have been considered [2, 3] in order to accurately describe the magnetically ordered phases [18] observed in Na 2 IrO 3 and α-RuCl 3 . Finally, the next-nearestneighbor Dzyaloshinskii-Moriya (DM) has been invoked as being relevant for the description of real materials [19] but its effect on the Kitaev model remains little explored so far [8] .

Here, we report on two novel topological QSLs arising when either a strong magnetic field or a DM interaction are considered in the pure Kitaev model. We have discovered that the gapped QSL state with Chern number ν = ±1 (depending on the direction of the field) predicted at low tilted magnetic fields [8] , undergoes a novel topological transition to a different, topologically gapped QSL with ν = ±2. Such topological transition occurs in a regime in which the Dirac cones disappear due to strong hybridisation between itinerant and localized Majorana fermions. We also predict the presence of a novel gapless chiral QSL induced by the DM interaction and that is characterized by equal Berry phases at the two Dirac cones (φ K = φ K = ±π) in contrast to the opposite Berry phases found in the pure Kitaev model (φ K = −φ K = ±π). The novel topological gapped QSL states found here could be tested through thermal Hall experiments.

Since we are interested in the description of com-peting topological phases starting from the KQSL, the H K = 2 ij ,γ K γ S γ i S γ j -where the three nearestneighbor bonds ij of the honeycomb lattice are denoted by γ = x, y, z -is the most relevant starting point [21] . The next-nearest-neighbor DM interaction: H DM = ij D ij · S i × S j is also important for describing the magnetic orders observed in Iridates [19, 23] , and it is known that, combined with the magnetic field, can open a non-trivial topological gap. For this purpose, we also consider the term H B = − i B · S i . Hence, the final hamiltonian reads:

and we will consider an isotropic Kitaev interaction i. e. K γ = K in the rest of the paper, as well as a the magnetic field B and the DM interaction D ij = D parametrized in function of tilt parameters t and d as B = B(t, t, 1)/ √ 1 + 2t 2 and D = D(d, d, 1)/ √ 1 + 2d 2 respectively, ranging from the pure z direction to the case perpendicular to the honeycomb plane. 

Phase diagram of Kitaev model with DM interaction under magnetic field. The full phase diagram of the Kitaev model in the presence of a magnetic field pointing in the [1, 1, 1] direction (t = 1) is shown. Empty regions indicate gapless phases and the size of the hexagonal symbols indicate the size of the gap. Gray areas correspond to the gapped fully polarized (FP) phase with Chern number ν = 0, the red to a gapped QSL with ν = +1 denoted as gQSL+1. A gapped QSL with an unconventional Chern number of ν = −2, termed gQSL−2 (dark blue) occurs between the FP and the gQSL+1. An ungapped QSL, uQSL, with equal Berry phases at the two Dirac points occurs for B = 0 and K = 1 at a non-zero DM, D < 0.5, which becomes the gQSL+1 around D ∼ 0.5 − 0.65, to finally become gapless and non-topological at a larger D (dark gray). In this region, CR refers to classical regimes, beyond the accessibility of the present theory. The left inset shows the Majorana decomposition of the model considered while the right inset shows the first Brillouin zone and symmetry points of the honeycomb lattice.

The full model is treated using Kitaev's Majorana de-composition of the spin S α = 1 4 (ib α c − i 1 2 α,β,δ b β b δ ) [8] where greek letters span the space dimensions (x, y, z) and α,β,δ is the Levi-Civita symbol (Fig. 1 ). In this notation, b operators correspond to flux band variables while c describes the itinerant Majorana fermions. These are in the presence of the three other localized Majoranas which act as gauge fluxes (see Fig. 3 (a) for the dispersion relation of the pure Kitaev model).

Going away the specific Kitaev point requires the proper consideration of the constraint on the number of fermions -four Majoranas per spin -that can be only achieved in average by introducing Lagrange multipliers {λ i }, as defined below. Explicitly implementing the fermion constraint -crucial for the physics to be well described -allows for non trivial hybridization between the matter Majorana band and flux degrees of freedom, at the heart of the novel topological phases reported in the present work. Then, any four-body term of the Hamiltonian is mean-field decoupled in the three possible channels hence leading to a quadratic model [1] . Solving the one electron Hamiltonian reexpressed in the Fourier space at each k vector allows us to construct a Slater determinant from which self-consistent equations (SCE) are derived. We then solve these SCE without imposing any restrictions on the mean-field solutions up to a desired tolerance on the set of mean field parameters, the energy and the constraint. Calculations are performed up to 60 × 60 clusters which are sufficiently large to describe the behavior of the model in the thermodynamic limit.

The complete phase diagram of model (1) can be nicely represented in a ternary plot as displayed in Fig. 1 for B in the [1, 1, 1] direction (t = 1) and D along the zdirection (d = 0) for simplicity, realizing that the case d = 1 is qualitatively similar. On this graph, the full parameter range of the model onto the plane fulfills the constraint K + D + B = 1, providing K > 0, D > 0 and B > 0, the area of the hexagons is proportional to the gap at this point, and the color refers to different states of matter. In this (K, D, B) space, the vertex defined by (0,0,1) corresponds to the topologically trivial (ν = 0) fully polarized (FP) state, (1,0,0), to the gapless Kitaev QSL (KQSL) and (0,1,0) to a gapless classical state whose magnetic properties remain yet to be determined. Three different topological phases characterized by their Chern numbers can be distinguished in the phase diagram. A gapped topological QSL with ν = +1 denoted by gQSL +1 is topologically equivalent to the QSL found by Kitaev at weak magnetic fields. The novel gapped QSL with unconventional Chern number of ν = −2 is denoted by gQSL −2 . The gray areas correspond to the FP state with ν = 0. The mechanisms driving these topological states are explained below, but we emphasize here the presence of the two novel gapped topological QSL with large Chern numbers, ν = ±2. These are the result of the strong competition between the three terms entering the hamiltonian and occur in intermediate regions between the FP and the gQSL +1 phases. These phases are chiral QSLs since time reversal symmetry is broken either explicitly by the applied magnetic field or spontaneously for B = 0 and D = 0. Interestingly, at zero field, B = 0, a small but finite DM of D 0.5 (with K = 1) leads to equal Berry phases, φ k , around the Dirac nodes (see Fig. 1 for the first Brillouin zone of the lattice) φ K = φ K = ±π in contrast to the opposite Berry phases found in the pure Kitaev model, φ K = −φ K = ±π. Hence, we have unraveled a new ungapped QSL with non-zero chirality which we denote by uQSL.

Applying a tilted magnetic field is found to open a gap in the Majorana fermion spectrum, consistent with perturbation theory [8] . The gap is opened symmetrically with respect to the zero energy and the resulting gapped QSL is topological with a nonzero Chern number ν = ±1, the sign depending on the direction of the magnetic field. For instance, for a [111] magnetic field, we recover a Chern number of ν = +1 in agreement with the exact solution of Kitaev. In contrast to previous mean field analysis [1] , the gap opening we find is due to the constraint we explicitly impose on the Majorana fermions of our MMFT since it is not exactly verified away from the pure Kitaev model [24] . Implementing such constraint on average over the spins through Lagrange multipliers λ α leads to onsite contributions of the type:

where summation over repeated indices is assumed. Thus, we can associate the opening of the gap to the nonzero Lagrange multipliers required to properly recover the original spin operators from the Majorana fermions. Concomitantly with the gap opening, the magnetic field leads to non-zero chiral currents of Majorana fermions between the n.n.n sites as shown in Fig. 2 (a). Only when the three components of the magnetic field are non-zero, all λ α are simultaneously non-zero as well. This leads to hybridization among the four Majoranas at each site which ultimately leads to the gap opening. The size of the gap depends on t, the actual orientation of the magnetic field. When the field is parallel to one of the natural axis (say t = 0 for B = Bu z ), the gap is zero and we have a gapless QSL. At a critical B we find that the ±π Berry phases at the Dirac points can switch their signs as found earlier [1] .

One key feature of our MMFT stands in these newly implemented fermion constraint. Indeed, we note that previous MMFT analyses [1] needed to add explicitly by hand the three-spin term to the Kitaev hamiltonian (1) in order to recover the gap found by Kitaev. In our case the gap opens after imposing the constraint in a consistent way, even at small fields. Indeed, by relaxing the constraint on the Majorana operators without adding the three-spin terms, we would find a gapless QSL instead. Hence, our MMFT is capable of describing correctly the exact KQSL as well as the gapped QSL under low magnetic fields without making any extra assumptions. We now discuss the effect of DM on the pure Kitaev model (B = 0). As stated above, as the DM is increased the Berry phases around the Dirac points become equal: φ K = φ K = ±π, indicating a change in the nature of the KQSL which characterizes the uQSL. As the D parameter is further increased above a critical value, D 0.5 (with B = 0 and K = 1), the system opens up a gap leading to a topologically gapped chiral QSL with ν = ±1 for either d = 0 or 1, gQSL ±1 . The origin of the non-zero Chern number may be associated with the occurrence of anisotropic chiral amplitudes between n.n.n. sites as shown in Fig. 2 (b) for d < 1. This lattice nematicity induced by anisotropy of D is completely restored for d = 1 at which the n.n.n. chiral amplitudes are the ones displayed in (a). In any case, the DM interaction induces both an ungapped and a gapped phase. This uQSL is a novel QSL which breaks TRS spontaneously in contrast with the gapped chiral QSL reported on the decorated honeycomb lattice [26] due to its gapless character. When a [001] magnetic field (t = 0) is applied in the presence of a nonzero DM, a gapless QSL with the two Dirac cones shifted in opposite directions by the same amount occurs. The sublattice symmetry respected by the DM interaction protects Dirac cones from opening a gap. By tilting the magnetic field to t = 1, a gap opens up as found in the case of zero DM leading to a gQSL ±1 . But unlike opening symmetrically around zero, the gap centers are equally shifted in opposite directions at the two cones. In Fig. 3 netic field increases up to about B ∼ 0.6, (with K = 1) a gap opens up at the Dirac points while the flux bands remain almost flat. This gQSL +1 phase -since it is gapped and ν = +1 -is adiabatically connected to the gapped QSL found by Kitaev [8] . As the magnetic field is increased up to B c 1.43 the flux bands are gradually distorted becoming dispersive and strongly hybridized with the Majorana matter bands. Note also how the bands are becoming closer at the M points for B ∼ B c . As the field is increased beyond B c , a gap opens up at this newly formed band touching points with Berry flux magnitudes larger than π in contrast to typical Dirac cones. This can be attributed to the fact that the matter and flux Majoranas are now forming unseparable composite objects due to the strong hybridization in this magnetic field regime. Hence, we find a gapped topological QSL with Chern number ν = −2 emerging in the range B ∈ [1.43, 1.52]. At larger magnetic fields (B > 1.52) there is a transition to a gapped and fully polarized insulator with a trivial topology (ν = 0) -the full polarization is an artefact of the method. The Majorana dispersions are also strongly modified by D even for B = 0. Recent numerical studies [11]- [10] suggest the existence of a gapless intermediate phase in a somewhat similar parameter range. In spite of the different features found (gapless vs. gapped), our gap closing around B c can be associated with the large enhancement of low energy excitations developing near the PL phase. Interestingly, we have supported by ED the fact that D and B combined possibly lead to a gap opening. All these points are detailed in [24] .

To conclude, we discuss our results in the context of Kitaev materials. Although they are mostly believed to have FM couplings [4, 19, 29] , some works suggest [28, 30] AFM couplings as mostly considered here. QSL behavior has recently been observed in the honeycomb magnet H 3 LiIr 2 O 6 [6] . Although α-RuCl 3 is magnetically ordered, there is experimental evidence for its proximity to a QSL phase [27, 28] . Under high pressure above 1 GPa [31] or applying a magnetic field destroys AF order giving way to a gapped QSL [32, 33] . Strikingly, recent thermal Hall conductivity experiments find fractional quantization of the thermal conductance which is attributed to the Majorana edge modes in a gQSL +1 . [9] NMR experiments find a spin gap ∆ ∝ B 3 at small fields [10] due to the fractionalization of the spin into two gauge fluxes and a gapped Majorana fermion as predicted by Kitaev. [8] Fig. 4 shows that a similar spin gap opening with a [111] magnetic field should be observed in AFM Kitaev materials. taev materials, the gQSL +1 (under a positive magnetic field) would survive up to an applied tilted magnetic field of B B c , way beyond the perturbative regime. The Majorana edge states in this QSL will contribute to the thermal Hall conductance [8, 34] , κ xy /T = π 12 k 2 B d , as re-cently observed. [9] This is half and opposite in sign to the thermal Hall conductance observed in an integer quantum Hall effect experiment [35] associated with electronic charge. Strikingly, since we find a distinct gapped QSL with ν = −2 in the range B ∼ 1.43 − 1.52, our analysis predicts a sudden jump of κ xy /T , the thermal Hall coefficient, from π 12 k 2 B d to − π 6 k 2 B d around B c . This signals a novel topological transition in AFM Kitaev magnets that could be searched experimentally.

J. M. acknowledges financial support from (MAT2015-66128-R) MINECO/FEDER, Unión Europea and from (mobility program: "Salvador de Madariaga": PRX18/00070) Ministerio de Educación, Cultura y Deporte in Spain and the hospitality from Néel Institute in Grenoble. This Hamiltonian can now be mean-field decoupled following three different channels of binilears:

and corresponding constant terms. The mean field Hamiltonian is then Fourier transformed with the specific form:

where n s is the number of Bravais sites in the lattice and the extra √ 2 allows for recovering standard fermionic commutation relations in q-space:

where c a −q can be identified to the creation operator c a+ q . Thus, in the reciprocal space, the Majoranas follow the rules {c a q , c b q } = {c a+ q , c b+ q } = 0 and {c a q , c b+ q } = δ q,q δ a,b as standard fermions.

Then, one can simply diagonalize the mean field Hamiltonian with all terms decoupled as previously shown in the single particle basis. The set of Lagrange multipliers {λ i } are fixed by imposing the single occupancy constraint given by Eqns. (5) , at the mean-field level only, using a least square minimization. This done, we obtain the mean field many-body wave function by constructing a Slater determinant up to half-filling. The operators of the type i 2 c a i c b j are then computed in this ground state and re-injected in the mean field hamiltonian leading to a set of self-consistent equations. We repeat this procedure until convergence of the mean field parameters and the energy up to a desired tolerance δ. In our case, we have fixed this tolerance to at least δ = 10 −10 on the parameters. All mean field parameters are in the ground state Slater determinant of the Majoranas, providing an efficient and stable way to converge towards the saddle points. Our construction easily deals with enlarged unitcells (we have checked our results up to 12 site unit-cell) which allows for higher symmetry breaking mean-field solutions which could be overlooked in too small clusters. This is for instance the case of the zig-zag and stripy phases reported in the next section when the Heisenberg exchange term is added to the system.

In contrast with other Majorana mean-field theories, our approach imposes the single occupancy constraint on average throughout the full phase diagram. The differences between our approach and a close mean-field theory [1] which uses a similar four-Majorana representation is provided in Section. V of the present Supplementary Material. A similar single occupancy constraint has been considered [3] but using Abrikosov fermions on a model containing pairing terms. It has been shown [1] that the four-Majorana construction indeed corresponds to this type of Bogolioubov-deGennes model. However, the possibility of explicitly having the four bands as in our MMFT, three associated with the gauge fluxes and one with the matter provides more insights about the nature of the different phases. An augmented parton construction has also been proposed [2] , in a model in which the constraint has not been violated. This approach is designed for the description of the dynamics of the excitations above the ground state energy in the Kitaev +Γ term.

As mentioned in the main text, the magnetic orders observed in candidate materials of the Kitaev model indicate the presence of Heisenberg like interactions and even other types of exchanges. Before studying in detail the phase diagram under a magnetic field and the DM interaction, we have verified that our self-consistent MMFT with constraints can reproduce the phase diagram of the Heisenberg-Kitaev model found through exact numerical techniques [4, 9] . Thus, we have considered the hamiltonian H = H K + H J with H K = 2K i,j ,γ S γ i S γ j and H J = J i,j S i · S j . We then introduce the extrapolation parameter θ defined as tan θ = J/K and solve the self-consistent equations of our theory for θ ∈ [0, 2π]. The corresponding phase diagram is displayed in Fig.5 . Note that we have considered theories up to four sites per unit cell, neces-sary to stabilize large unit-cell magnetic orders such as the Zig-Zag and the Stripy phases [5] .

The good agreement with other techniques indicates the relevance of the method and its capability of treating magnetic orders and QSL phases on equal footing. In addition, we see that the pure Kitaev limit (θ = 0) lies deep in the domain of the Kitaev QSL (1) (KQSL 1 ), and since we are interested in the effects of the magnetic field and Dzyaloshinskii-Moriya interactions on this QSL, we can restrict our analysis to this θ = 0 starting point without any loss of generality. 

As discussed in the main text, the DM interaction has an important effect on the ground state of the pure Kitaev model i. e. the KQSL. Even with no applied magnetic field, a small but finite DM can induce a transition from a KQSL to an uQSL. The uQSL is a chiral QSL which is characterized by having the two Berry phases around the Dirac cones equal to +π or to −π in contrast to the KQSL. At sufficiently large D and with B = 0 a transition to a gQSL −1 (see the phase diagram shown in Fig. 1 of the main text) occurs. In order to analyze this transition we show in Fig. 6 the change in the Majorana spectrum with increasing D.

The Berry phases and Chern numbers determining the topological properties of the model are obtained numerically using a multiband approach. This is necessary since the itinerant and flux Majorana bands are occupied. This is done by defining the overlap matrix M The Chern number is immediately obtained from this expression by discretising the first Brillouin zone in elementary four-site plaquettes [7] (k j → k j + u 1 → k j + u 1 + u 2 → k j + u 2 → k j ) and defining the Berry flux matrix F (k j ) = Im ln |U (k j )| on each of them. The total Chern number ν of the system is then nothing else but the sum of all the Berry fluxes over the whole BZ as ν = 1 2π j F (k j ) and so the thermal Hall coefficient is half-quantized as:

We now discuss the effect of the single occupancy constraint, needed to recover spin operators from the Majorana fermions, on the magnetic properties of the Kitaev model under an external magnetic field. In our MMFT we have imposed (on average) the condition of single occupancy leading to the following constraints:

This is in contrast with previous MMFT that do not impose these constraints (on average) explicitly [1] . In previous theories, the spin operators are defined as S α = ib α c, which implicitly assume that the single occupancy constraint given by the conditions over the Majorana

However, as we discuss below, the above constraints are only satisfied in the Kitaev limit i.e. when no magnetic field nor other exchange interactions are applied to the Kitaev model.

In Fig. 7 we show results obtained within our MMFT with the constraints (11) fulfilled and compared with the case in which they have been removed. The MMFT without constraints recovers the dependence of the magnetic moment with a [001] magnetic field reported previously in the literature [1] . Namely a transition from a gapless phase to another gapless phase at which the ±π Berry phases associated with the two Dirac cones switch their sign occurs around B = 0.42 and a second transition to a gapped polarized phase occurs at B = 0.52. When the constraints are imposed though, these three different phases are present but their critical transition values are shifted. Also we find that the polarized phase is fully saturated in contrast to the case when no constraints are imposed.

Let's discuss the effect of the constraints when a magnetic field is applied in the [111] direction. In Fig. 8 we compare the magnetic moment and the gap with and without the constraints imposed. The first important difference between the two approaches is that while our MMFT (with the constraints imposed) naturally recovers the topological gapped phase (with Chern number, ν = ±1) even at small magnetic fields B, as found by Kitaev. In contrast, the system is gapless up to B ∼ 1.4 in the no-constraint MMFT. In order to recover Kitaev's gapped topological phase, three spin terms arising in perturbation theory to O(B 3 ) need to be added by hand to the Kitaev model [1] . Hence, imposing the constraint is readily essential for recovering the correct behavior of the Kitaev model under a [111] magnetic field in any regime of the magnetic field. Importantly, no topological phases are found when the constraints are not imposed. While the two Berry phases around the Dirac cones have op-posite signs, ±π, in the ungapped phase for B 1.4, the gapped phase found for B 1.4 is trivial, ν = 0, as shown in Fig. 8 . Finally, without the constraints, there are n signatures of the ν = −2 topological phase found with our MMFT (Fig. 4 of the main text) . 

In order to challenge our gQSL −2 phase, we have computed the gap spectra of the model by exact diagonalizations (ED) on a 18 site cluster, for 4 different values of D, in function of B along the [111] direction and using the translational symmetries (Fig. 9 ). As found in previous studies [9, 10] , the intermediate phase defined by the change of symmetry in the ground state, is expected to be gapless, even though larger cluster sizes should be analyzed to reach a definitive conclusion about the gap opening in the thermodynamic limit. However, for a cluster of 32 sites [9] , a significant shrinking of the energy excitations lets suggest that the gap might be vanishing at the thermodynamic limit.

We are here interested on the behavior of the spectrum of the intermediate phase under the presence of the DM term. As we have obtained in the MMFT, the gQSL −2 extends in the parameter space for finite K, D and B, and is always gapped. In our ED results, starting from D = 0 (upper-left panel) and slowly increasing D, a clear increasing of the gap between the ground state energy and the first excited state is observed. A finite size scaling would be necessary for being definitely conclusive about a possible gap opening, but we clearly show here that the combined effect of D and B tends to increase the gap which is in good agreement with our MMFT.

The second effect of this combination is the location of the boundaries of the intermediate phase. As D in-creases, they are pushed to higher magnetic field, a tendency that is also reproduced by our MMFT.

Since the gap we have found in the gQSL −2 when D = 0 is very small in comparison with the one in the gQSL +1 phase or the polarized phase, it is not inconsistent with ED calculations, as these cannot exclude the presence of a tiny gap in the thermodynamic limit. Moreover, the behavior of the intermediate phase (change of symmetry, D vs. B, gap increasing) in ED agrees with our phase diagram of Fig. 1 in our manuscript. This gives further support to the existence of our intermediate gQSL −2 phase, which should be gapped when D and B are combined [8] . As pointed out above, we cannot exclude that, for D = 0, the gap indeed closes in the thermodynamic limit as suggested by exact numerical treatments on small systems [9, 10] . direction for various D obtained by ED on a 18 site cluster. The first 10 levels are displayed in the three inequivalent symmetry sectors (see inset) at the Γ (light blue), X (dark blue) and K (red) points. As D increases from 0 to 0.05, the gap in the intermediate phase, defined by the change of the ground state symmetry sector, grows, in agreement with our MMFT results. Hence, gap opening in the thermodynamic limit cannot be excluded for any finite D.

Quantum spin liquid signatures in Kitaevlike frustrated magnets

Challenges in design of Kitaev materials: Magnetic interactions from competing energy scales

Perturbed Kitaev model: excitation spectrum and longranged spin correlations

We have mapped out the phase diagram of the Heisenberg-Kitaev model confirming the presence of an extended KQSL phase (see Supplementary material [24]). Without loss of generality

Kitaev-Heisenberg Model on a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides A2IrO3

Interplay of many-body and single-particle interactions in iridates and rhodates

(i) details of the Majorana Mean Field Theory, (ii) the extension to the Heisenberg-Kitaev model (iii) the effect of the DM interaction on the Majorana spectrum, (iv) the computation of the multiband Berry phases and Chern numbers

Succesive Majorana topological transitions driven by a magnetic field in the Kitaev model

Exact chiral spin liquid with non-abelian anyons

Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet

Neutron scattering in the proximate quantum spin liquid α-RuCl3

Models and materials for generalized Kitaev magnetism

Kitaev magnetism in honeycomb RuCl3 with intermediate spin-orbit coupling

Pressure-induced melting of magnetic order and emergence of a new quantum state in α-RuCl3

Unusual phonon heat transport in α-RuCl3: strong spin-phonon scattering and field-induced spin gap

Evidence for a field-induced quantum spin liquid in α-RuCl3

Cross-correlated responses of topological superconductors and superfluids

Quantized thermal transport in the fractional quantum Hall effect

Chern numbers and discretized Brillouin zone: efficient method of computing (spin) Hall conductances

Two-Magnon Bound States in the Kitaev model in a

Successive Majorana topological transitions driven by a magnetic field in the Kitaev model

Dynamics of a quantum spin liquid beyond integrability: The Kiatev-Heisemnberg-Γ model in an augmented parton mean-field theory

Quantum phase transition in Heisenberg-Kitaev model

Kitaev-Heisenberg Model on a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides A2IrO3

Phase diagram and spin correlations of the Kitaev-Heisenberg model: Importance of quantum effects

Possible proximity of the Mott insulating iridate Na2IrO3 to a topological phase: Phase diagram of the Heisenberg-Kitaev model in a magnetic field

Chern numbers and discretized Brillouin zone: efficient method of computing (spin) Hall conductances

Perturbed Kitaev model: excitation spectrum and longranged spin correlations

Emergence of a field-driven U(1) spin liquid in the Kitaev honeycomb model

Two-Magnon Bound States in the Kitaev model in a

As mentioned in the main text, mapping the spins 1/2 onto four Majorana fermions can be seen as a two-step procedure. First we proceed to the standard parton construction by introducing two Abrikosov fermions:These two fermion species carry the magnetic moment of the spin, and for each of them, two Majorana fermions can be introduced as:let's keep the c i notation for simplicity. Defined like this, the spins read:and the single occupation constraints f + i↑ f i↑ + f + i↓ f i↓ = 1, f + i↑ f + i↓ = 0 and f i↑ f i↓ = 0 (see below for discussion) yield to three additional terms to be introduced with corresponding Lagrange multipliers {λ i } in the model:Let's now focus on the pure isotropic Kitaev model for illustrating how the method is implemented. Injecting the above defined mapping and considering the constraints to be fulfilled, the Hamiltonian can be cast as: