Correlated spin liquids in the quantum kagome antiferromagnet at finite field: a renormalization group analysis

We analyze the antiferromagnetic spin-1/2 XXZ model on the kagome lattice at finite external magnetic field with the help of a non-perturbative zero-temperature renormalization group (RG) technique. The exact nature of the ground and excited state properties (e.g. gapped or gapless spectrum etc) of this system are still debated. Approximate methods have typically been adopted towards understanding the low-energy spectrum. Following the work of Kumar et al (2014 Phys. Rev. B 90 174409), we use a Jordan–Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbor interactions. While the work of Kumar et al was confined mostly to the plateau at 1/3-filling (magnetization per site) in the XY regime, we analyze the role of inter-spinon interactions in shaping the phases around this plateau in the entire XXZ model. The RG phase diagram obtained contains three spin liquid phases whose position is determined as a function of the exchange anisotropy and the energy scale for fluctuations arising from spinon scattering. Two of these spins liquids are topologically ordered states of matter with gapped, degenerate states on the torus. The gap for one of these phases corresponds to the one-spinon band gap of the Azbel–Hofstadter spectrum for the XY part of the Hamiltonian, while the other arises from two-spinon interactions. The Heisenberg point of this problem is found to lie within the interaction gapped spin liquid phase, in broad agreement with a recent experimental finding. The third phase is an algebraic spin liquid with a gapless Dirac spectrum for spinon excitations, and possess properties that show departures from the Fermi liquid paradigm. The three phase boundaries correspond to critical theories, and meet at a SU(2)-symmetric multicritical point. This special critical point agrees well with the gap-closing transition point predicted by Kumar et al. We discuss the relevance of our findings to various recent experiments, as well as results obtained from other theoretical analyses.


Introduction
Geometrically frustrated spin systems have drawn considerable attention ever since Anderson's seminal work highlighted the connection between the resonating valence bond (RVB) ground states and high temperature superconductivity [1]. These systems are known to show exotic ground and excited state properties, such as liquid like ground states, fractional excitations etc [2,3]. The nearest-neighbor (n.n.) S=1/2 kagome antiferromagnet (KA) is one such promising model system in the search for unconventional states of matter. Despite sustained interest, the precise nature of the ground state and excitations of this system remain uncertain. Some analytical and numerical approaches have predicted a spin liquid ground state for the S=1/2 Heisenberg KA model Hamiltonian with a gapped excitation spectrum. These include studies using exact diagonalization (ED) [4], density-matrix-renormalization group (DMRG) [5][6][7][8][9], quantum Monte Carlo (QMC) [10], Schwinger fermion mean-field method [11], slave fermion approach [12] and tensor network states (TNS) [13,14]. On the other hand, some other studies predict a gapless or critical spin liquid ground state. This includes a Gutzwiller purely by a one-spinon gap. The computed dispersion spectrum in the reduced MBZ shows the gapped plateaux that are robust in the thermodynamic limit. section 3 will be devoted towards the construction of an effective pseudospin problem enabling the treatment of spinon interactions via a RG analysis. In section 4, we present the RG flow equations for single-spinon gap and unveil the mechanism that leads to the closing of this gap. We extend the RG analysis to also find the physics responsible for the opening of a two-spinon gap opening, as well as a phase that supports gapless spinons with a Dirac dispersion. Finally, by computing the effective Hamiltonians for the critical phase boundaries, we reveal the entire phase diagram obtained from our RG analysis. In section 5, we define the different topological quantities that are employed in distinguishing the different phases. In section 6, we calculate the self-energy, quasiparticle residue and lifetime for spinon excitations of the gapless phase. We conclude in section 7 with some discussions. Finally, we present detailed calculations for various sections in the appendices.

Fermionised Kagome XXZ model at finite field
The S=1/2 antiferromagnetic XXZ Hamiltonian for the kagome lattice in the presence of an external magnetic field (h) can be written as [15] å å where J is the exchange constant and λ is the magnetic anisotropy between Ising and XY terms (λ=1 is the Heisenberg point). The sum is taken over nearest neighbor sites and Î   ( ) r R i , , where = + ˆR n a n a 1 1 2 2 (n 1 , n 2 are integer) corresponds to the lattice vector for three sub-lattice unit cell (up triangles). Further,â 1 andâ 2 are the basis vectors and Î ( ) i a b c , , denote the three sub-lattices (see figure 1). The Hamiltonian is invariant under lattice translation along any basis vector of the lattice.
Recently, we developed a non-perturbative approach based on twist operators towards obtaining firm criteria for the nature of the ground state of the kagome Heisenberg quantum antiferromagnet at zero as well as non-zero field [39]. In the present work, we aim to further develop a detailed understanding of the kagome problem at finite magnetic field via a complementary non-perturbative RG analysis. In this way, we will carefully assess the role played by the Ising part of inter-spin interactions in determining the stability, as well as the properties, of the many-body ground state. In doing so, we rely on the results obtained by Fradkin and collaborators, who showed that CS topological gauge-field theories can be obtained for gapped states corresponding to certain rational fractions of the magnetization [44]. Thus, for a microscopic approach to the origin of different ground state properties at finite magnetization, we begin by mapping the problem of interacting quantum spins onto a system of fermionic spinons coupled to a CS gauge field (with statistical angle θ=1/2π) via a JW transformation in two spatial dimensions [46,44]. For details of the JW transformation, we refer the reader to the detailed discussion in Kumar et al [44].
Upon performing the JW transformation, the fermionic Hamiltonian (1) , , is the fermion density operator,  ( ) A r t , is the spatial part of the CS statistical gauge field employed in mapping spins to fermionic spinons. The magnetic field h now tunes the fermionic filling. We first obtain the dispersion spectra of the non-interacting spinons on the kagome lattice from the XY- 2 , such that the distance between nearest neighbor sites is fixed to half of the length of a basis vector, say, |ˆ| a 1 . Every triangular unit cell has three different sites marked by a, b and c indices. part of the Hamiltonian by putting λ=0. It is important, however, to remember that the spinons remain coupled to the statistical gauge field in equation (2). A mean-field ansatz then involves considering a uniform flux in every plaquettes with the gauge choice as shown in figure 2, and ignoring the fluctuations of the gauge field. One then obtains plateaux of the magnetization corresponding to 1/3, 2/3, 5/9 etc. Indeed, the plateaux are associated with a uniform flux of f p = 2 p q with  Î p q , , and correspond to an average filling of á ñ = n p q (where q is the periodicity of magnetic unit cell). From the associated Hofstadter spectrum for the free spinon problem, the plateaux at 1/3 is found to be the most robust (i.e. it is stabilized by the largest single-spinon gap), and corresponds to 1/3-filling (average site occupancy) of each sublattice in the unit cell [44]. One can obtain the dispersion spectrum for the á ñ = n 1 3 state by solving the associated Harper's equation numerically for the three sub-lattices [44,52,53]. Hereafter, we assume a value of the exchange coupling J=1. As the periodicity of the magnetic unit cell is q=3 (i.e. an enlarged unit cell of nine sites), a spinon band associated with a given sublattice is further split into three bands, giving a total of nine bands in the spinon spectrum. For a filling of 1/3, the lowest three bands are filled and the rest empty, i.e. the effective chemical potential (equivalent to the magnetic field (h) of the original spin problem) is placed midway between the third and fourth spinon bands. Therefore, in analyzing the effects of spinon scattering at low energy scales via a RG formalism, it is sufficient to focus on the 3rd (completely filled) and 4th (empty) bands. Figure 3 shows the dispersion spectrum at 1/3-filling in the 3rd and 4th bands, clearly indicating the onespinon spectral gap of the free spinon problem. From a finite size scaling analysis, we find that the minimum gap between the third and fourth bands saturates at a value of Δ 0 =1.355±0.001 (in units of J) in the thermodynamic limit at the corresponding momentum coordinates of  figure 3). We recall that the effect of Ising interactions on the one-particle spectrum was analyzed in [44] at the level of mean-field theory together with the effects of saddle-point fluctuations of the gauge degrees of freedom. This analysis concluded that the gap would close at * l = 0.6. It, however, was unable to reach any firm conclusions on the nature of Ising dominated phase lying at couplings * l l > . In the following sections, we conduct a RG analysis of the quantum fluctuations arising from the interplay of the XY and Ising terms This RG phase diagram obtained will clarify the nature of various gapless and gapped phases as well as the transitions between them.
1 (x 1 is the coordinate vector alongâ 1 direction) gives a flux of 1/3 unit flux quantum within every plaquette. 3. Effective two-patch problem for 1/3-plateau Above, we have identified the existence of a minimum gap between the third and fourth bands at two points in kspace. Although the dispersion spectrum in figure 3 is not symmetric about these two minimum-gap points, we can nevertheless consider the immediate vicinity of these points in constructing an effective two-patch problem. Figure 4 shows the schematic diagram of two-patch problem, where a and b are the two minimum energydifference patch-center points in the lower band with momentum indices  k a 1 and  k b 1 respectively, where the suffix 1 denotes the 3rd band in figure 3. In the same way, the momentum indices  k a 2 and  k b 2 represent the other two minimum energy-difference patch-center points in the upper band, where suffix 2 denotes the 4th band in figure 3. These four momenta are connected: In what follows, Δ 0 represents the onespinon energy gap between two bands, hereafter referred to as the hybridization gap.
The states present at the patch-centers (labeled by the symbols (1, 2) and (a, b)) are connected via interband particle-particle (PP) or particle-hole (PH) scattering events between the 3rd and 4th bands. This is induced by the n.n. (Ising) interaction term in the Hamiltonian (1). In this way, the scattering processes in the vicinity of the two patch-centers form a two patch model described in figure 4. A general pair of electronic states taken from these two bands are marked by momentum eigenvalues ( ) (blue and red dot in figure 4) and ( ) (green and orange dots), where Λ and Λ−δΛ are the momentum displacements from the two patch-centers. δΛ is the momentum asymmetry of pair ( . The two-spinon interaction terms in the Hamiltonian, equation (2), involve the scattering of such pairs of states between the 3rd and 4th bands. These scattering processes come with an additional energy cost above the bare band gap . This energy mismatch then leads to a logarithmic singularity in the second order term of the associated T-matrix for the resonant pairs chosen symmetrically (i.e. with δΛ=0) from the two patch-centers ( Recall that similar log-divergences appear in the calculation of the T matrix for the Kondo problem [54], and required a careful RG analysis for further insight. For these symmetrically chosen resonant pair states, there exists PP and PH scattering channels acting on the low energy subspaces: . The creation/annihilation operators satisfy the usual fermion anti-commutation relations. The selection procedure for the low energy subspaces employed above is analogous to BCS's construction [55,56] used to reach the BCS reduced Hamiltonian starting from a general electronic problem. The subspaces equations (4), (5) are spanned by the basis states asymmetrically placed about the lower/upper band center.
respectively. Putting Λ=0 gives the basis states of the patch centers. The PP pseudospin vector acting on the subspace equation (6) is given as follows [56] and the pseudo-spin vector in the PH channel equation (7) as We will now present the form of the projected patch-center Hamiltonian: Here, H is given in equation (2), and the projection operator Q h is given by where h indexes the states at the patchcenter Λ=0. Q h projects onto the subspace (Λ=0 in equations (4), (5)) of the four patch-center states 1 2 2 in the PP and PH scattering channels (see appendix A for detailed derivation). The contribution to the effective Hamiltonian  from the PP channel is given by Similarly, the contribution from the PH channel is In H PP , ò 1 and ò 2 are the kinetic energies for the patch-centers in the lower (3rd) and upper (4th) bands respectively. As we take the middle of the two bands as the zero energy surface, ò 1 =−ò 2 . Further, a b are the k-space spinon number operators at the lower and upper patchcenters respectively. V q=0 and ¹ V q 0 are the zero and non-zero momentum scattering amplitude. In the PH channel, the two spinons scatter from separate bands. Their kinetic energy terms cancel one other, ensuring the absence of kinetic terms from H PH . The origin of various scattering terms in the PP and PH channel are described in detail in appendix A.
We set Λ=0 in order to obtain the pseudospins precisely at the patch centers in equations (8) and ( . By first writing the Hamiltonians (10) and (11) in terms of pseudospins and then combining both, we have (as the momentum transfer is π/2Δ k2π/3). In the next section, we will consider the effects of scattering between these patch-centers and the states in their vicinity.

RG study for two-patch problem
In the section above, we saw the logarithmic instabilitites in the low energy subspace of the resonant pairs (i.e. with vanishing kinetic energy) due to PP or PH scattering processes. Based on the calculation equation (3), we confine our interest solely to the resonant pair subspace and project out the off-resonant pair scattering terms from the Hamiltonian equation (2). Akin to the poor man's scaling approach to the Kondo problem [57], a treatment of such logarithmic singularities arising at leading order in the coupling Jλ demands a RG theory for the resonant pairs of the two-patch problem. As the gap between the patch centers of the 3rd and 4th band governs the low energy physics, the PP or PH scattering processes from them to a high energy resonant pair subspace (equations (4), (5), with the projection operator Q Λ projecting onto the states L 2 , ) allow us to formulate a minimal RG scaling theory. Within the pseudospin language, these spinon scattering processes correspond to quantum fluctuations of the z-component of the PP and PH pseudospins (equations (8) and (9)) that are induced by pseudospin-flip scattering processes [48]. As we will see below, this Hamiltonian RG formulation will enable us in reaching some firm conclusions on the existence and nature of various ground states.
The RG procedure involves the iteration of two steps. As shown in figure 4, the first step is the partitioning of the Hamiltonian into a low energy resonant pair subspace (Λ=0 in equations (4) and (5)) via a patch-center projection operator Q h , and the boundary subspace via a projection operator Q Λ . The boundary pseudospins are displaced by Λ in energy-momentum space from the patch centers of the 3rd and 4th bands. In the second step, off-diagonal terms ( ) associated with PP or PH scattering between the boundary (Λ) and the patch-center (h) states are removed via Gauss-Jordan block diagonalization, leading to decoupling of the boundary pseudospin. As presented in detail in appendix B, this procedure involves the elimination of pseudospin terms of the kind + - represents the pseudo-spin for upper band in the PP channel with momentum at L  k 2 etc (see figure 4). This RG scheme does not involve a perturbative expansion in any coupling, and is similar to Aoki's non-perturbative RG [58] procedure of Gaussian elimination of single particle states employed for the Anderson disorder problem. The RG is also connected to Glazek and Wilson's non-pertubative Hamiltonian RG procedure [59] employed for quantum mechanical Hamiltonians, and has been applied recently by some of us to the study of Mott-Hubbard transitions in the 2D Hubbard model [48].
The iterative removal of one PP or PH pseudospin at every RG step leads to a RG flow equation for the lowenergy two-patch subspace Hamiltonian As defined above, L H h X n is the off-diagonal term coupling states at the energy-momentum boundary (−Λ and Λ) and two patch-centers ( h 1 and h 2 ). The diagonal operator  L H n contains the Ising interaction between boundary pseudospins and all Zeeman-like terms involving the pseudospins. The quantum fluctuation scale ω is the undetermined energy eigenvalue of the system containing contributions from the quantum dynamics of the inter-pseudospin correlations, as well as the pseudospin self-energy induced by the off-diagonal terms. This quantum dynamics is manifested in the Heisenberg equation of motion for the diagonal operator (  L H n , ), arising from the non-commutativity of the diagonal and off-diagonal terms.
By constructing the two-patch problem, we can now write the PP subspace projected (with projection operator L P D ) Hamiltonian H projected into three parts: , and a patch-center-boundary coupling ( We reiterate the fact that, in reaching the Hamiltonians equation (14) for the resonant pair subspace, we have ignored the sub-dominant contributions from all non-resonant scattering terms . Further, we have ignored the PH pseudospin scattering processes here, as they include all intraband scattering processes and do not contribute to the RG flow of the hybridization gap. Note that H Λ contains an interaction between the two boundaries, while H h contains interactions between the patch-center states. Further, as the hybridization energy gap term receives no contribution from the PH channel, we will ignore contributions from the PH channel when analyzing the effect of interactions in leading to a quantum critical point (QCP) (i.e. in closing the single particle gap) in section 4.1. Instead, in section 4.2, we will find their contributions as being critical to the opening of a many-body gap beyond the QCP.

Renormalization of hybridization gap
The inter-band scattering processes between the patch-centers and the boundaries leads to renormalization of the single particle terms in the two patch Hamiltonian where ω quantifies the energy cost for pseudospin fluctuations arising from inter-band spin-flip scattering processes. The renormalization equation (15) implies that a pseudospin scattering between the lower band patch-center ( h 1 ) and the boundary of the upper band involves the excitation to an intermediate configuration . In the de-excitation process, the pseudospins return to their original configuration from this intermediate configuration (see figure 5(a)). We note that the numerator of the flow equations support a oneloop form similar to that obtained from the poor man scaling RG for the Kondo problem [57]. However, the denominator contains pseudospin (i.e. two-particle) self energies that are also renormalized in the process. This leads to the non-perturbative nature of the flow equations. We note that a similar feedback in the renormalization of the two particle vertices is observed in the functional RG formalism as arising from the single-particle self energy [60].  (14) simplifies to = U V 2 0 0 , where V 0 is the bare interaction strength. V n is the renormalised interaction strength/ two-particle self-energy obtained from the nth step of the RG process. The intermediate boundary state's eigenenergy is given by The action of the pseudospin-flip operators on the intermediate state allows determination of the numerator in equation (15). Thus, the renormalization equation takes the form Similarly, the pseudospin scattering between upper band patch-center and the boundary of the lower band (see figure 5(b)) leads to Together, the complete contribution to the Hamiltonian renormalization from the above processes is given by . By taking a trace of equation (20) with respect to the final pseudospin state   ñ L | D 1 2 , the renormalized hybridization term of the patch-center Hamiltonian (H h in equation (14)) attains the form This leads to the RG flow equation for the hybridization gap dD = D -D + n n n 1 and the interaction strength d = - 2 0 is the sign of the non-interacting two particle Green function. In the RG equation given above, for w = D 0 , the quantity c will change sign from +Ve to -Ve for D < V 2 0 0 . This makes the RG flows for both the hybridization gap and inter-band scattering term irrelevant. This marks (2)-symmetric QCP across which the hybridization gapclosing transition takes place, revealing a singular Fermi surface with two Dirac points. This is shown in figure 6 as a black filled circle. The critical value * = V 0.68 0 obtained is close to the mean-field value of 0.6 obtained by Kumar et al in [44]. As we will see in section 4.3 below, this special point lies at the meeting of three quantum critical lines, two of which possess SU(2)-symmetry (line-2 and line-3 in figure 6) and the third a U(1) symmetry (line-1 in figure 6). Indeed, this point corresponds to a multicritical point as it is reached at a special value of Δ 0 , V 0 and ω.
On the other hand, in the regime < D V 0 2 0 (marked as I in figure 6), the interband scattering terms assists the hybridization gap and both the RG flows are relevant (as can be seen from equation (22) We can now detail some of the properties of the gapped ground state on this magnetization plateau. Following [39], applying the twist operator å å twice on the ground state wave function returns the ground state wave function to itself, yielding a gapped twofold ground state degeneracy on the torus [61][62][63]. Here, the indices b and c of the operatorŜ z represent the sub-lattices of the kagome system (see figure 1). This gap opening is an outcome of the two particle scattering across the two-patch centers (see figure 4). This scattering process is described by the pseudospin-flip part of the Hamiltonian equation (12), and can be connected to the PP-projected twist operator as follows (for details see appendix D) 0 . The topological phase π that arises out of this non-commutativity allows us to extract a fractional charge = q 1 2 associated with the spectral flow between the two ground states [64,65]. Further, we will also see in section 5 below that the many-body gap originating from the operatorÔ 2 protects the degenerate ground state manifold, resulting in a topological Chern number associated with a spin Hall conductivity. Our findings are consistent with the CS gauge field theory developed in Kumar et al [44].

Gap opening beyond the QCP
We will now study the putative gapped phase lying beyond the QCP   respectively, the corresponding boundary energies are computed from  as a function of ω and Δ 0 . Thus the latter forms a gap in the PH pseudospin Hilbert space, and determines the resultant stable fixed point Hamiltonian In the fixed point theory equation (31), the PH pseudospin subspace equation (5) condenses at low energies, making the PP pseudospin vector magnitude vanish: . This leads to the vanishing of the Zeeman term in equation (28). Similar to equation (25) for the PH interband scattering channel, one can show that the spin-flip part in the fixed point Hamiltonian can be written in terms of the doubled twist operator projected onto the PH pseudospin space (see appendix D for details). This allows us, once again, to predict a gapped, two-fold degenerate ground state on the torus [61]. Further, following the arguments presented earlier, spectral flow between the two degenerate ground states is associated with a fractional charge of = q  (31) above, the two-particle gapped phase corresponds to a gapped SU(2) symmetric phase comprised of electron-hole pairs from the 3rd and 4th bands. This effective Hamiltonian was reached by an RG analysis respecting the SU(2) symmetry of quantum fluctuation terms (equation (30)) arising from the non-commutativity of various terms in equation (28) ). This indicates that we could have equivalently studied a SU(2) non-Abelian lattice gauge theory on the kagome lattice associated with such quantum fluctuations [66]. In the continuum limit, such a SU(2)-symmetric non-abelian gauge theory possesses a disordered ground state with a dynamically generated mass gap. Such a gauge theory can be obtained from a fermionic nonlinear sigma model of massive Dirac fermions in (2+1) dimensions coupled to a SU(2) order parameter [67], and possesses a topological Hopf term. We will further quantify the topological invariants of the gapped phases in regions I and II in the next section.

Gapless phase and phase boundaries
In the regime w > D -V 0 0 1 2 , both PP and PH scattering processes are irrelevant, leading to a state with robust gapless Dirac spinons. This is indicated as region III of the phase diagram in figure 6, and corresponds to an algebraic spin liquid [15,24]. Here, the RG flows in equation (30)  . This corresponds to the appearance of the Dirac point Fermi surface, with a vanishing effective patch-center Hamiltonian, H III =0. We will further elucidate the properties of this gapless phase in section 6.
Finally, we can define fixed point patch-center Hamiltonians for the various phase boundaries in the phase diagram. The fixed point Hamiltonian for the boundary separating phase-I from phase-II (line-1 in figure 6) is,  (27)). H 1 thus corresponds to a line of quantum critical theories with U(1)-symmetry, and possesses a topological theta-term [68]. On the other hand, for the boundaries separating phase-I from phase-III (line-2) and that separating phase-II from phase-III (line-3) corresponding to SU(2)-symmetric critical theories: H 2 =H 3 =0. The vanishing effective Hamiltonian for such gapless Dirac cones on line-2, line-3 and phase III indicates an emergent PH symmetry. This leads to a SU(2)-symmetric Wess-Zumino-Novikov-Witten topological term with coefficient = S 1 2 [68] in the theory.

Topological quantum numbers and spin Hall conductivity
We now use various topological quantum numbers to distinguish the different phases in the phase diagram figure 6. We begin by rewriting the RG equation (22) as The value of N Λ is then evaluated as (for details see appendix C) We then find that L N is non-trivial in phase II (the two-spinon gapped phase) and trivial otherwise w w Finally, by mapping the problem in phase I close to the gap-closing point to that of massive Dirac spinons, we find the Chern number C=1 (see appendix C). Then, following Kumar et al [44], we can connect this Chern number with an induced CS term in the corresponding gauge-field theory (θ F ), θ F =C/2π=1/2π. Together with the original CS statistical angle θ=1/2π, this leads to an effective CS coupling [44] q q q q q p = + = ( )

Properties of gapless phase in the RG phase diagram
Finally, we turn to a detailed analysis of region III in the phase diagram figure 6. As discussed above, we find here a gapless phase with Dirac spinons, constituting a singular Fermi surface of two Dirac-points. The PP and PH scattering processes between the lower and upper Dirac cones that participate in gap opening were earlier found to be RG irrelevant in this phase (see section 4.2). Nevertheless, the velocity of the Dirac spinons can well undergo a non-trivial renormalization due to residual intercone scattering processes about the Dirac points where we have shown the case of L¢ = L 0 (the boundary state) and Λ=Λ 0 as process I and II respectively in figure 7. In turn, such a velocity renormalization will affect the spinon self-energy (Σ), quasiparticle residue (Z) and lifetime (τ) of the gapless spinons. An investigation of these properties is presented below.
To begin with, we define the energy bandwidth(W) in the problem  7). The renormalization of the spinon dispersion (and hence the spinon velocity) takes place via the scattering of the pair of spinons in the PP channel. This is shown in figure 7 as processes that lead from the lower Dirac cone boundary  L 1 0 to the upper Dirac cone window (Λ<Λ 0 ). The PH pseudospins have zero dispersion magnitude due to patch-center (a, b) interchange symmetry, as can be seen from the absence of terms involving  L L A S z 1 in Hamiltonian  equation (12). Thus, the PH pseudospins do not take part in the spinon velocity renormalization. The process indicated as II in figure 7 (via dotted arrows) proceeds in the same way, but for the holes in the upper Dirac cone. The RG flow proceeds as the bandwidth is lowered iteratively from  L  relation, equations (47) and (48). Recall that in the RG relation for the stability of the gapped phases I and II, the intermediate configuration energy involved the interaction cost of a pseudospin from the lower band and another from upper band (equation (14)). Thus, the fluctuation scale ω for the RG gapped phases is double that of the scale in the gapless phase: ω=2ω 1 . Further, note that in the denominator of equation (48), the intermediate configuration energy is negative and the states at Dirac points are fixed at zero energy, such that the fluctuation scale ω 1 for the renormalized lower band states is negative: ω 1 <0. Thus, in order to put the RG of the spinon dispersion on the same footing as the RG relations that led to the phase diagram figure 6, we will write the RG equation (48) in terms of the fluctuation scale ω used earlier , as can be seen from figure 3 (by noting that the Dirac points will appear energy E=−1.0). Further, in order to stay within the boundaries of the gapless region III in the phase diagram figure 6, the maximum value of w - . From this, we can see that 1 n . This leads to the RG flow being irrelevant, and to a reduction of the dispersion magnitude until it stops at a stable fixed point value.
At the stable fixed point In the continuum limit, the difference RG equation (49) is replaced by a differential RG flow equation From the above, we can now derive the renormalization of the Fermi velocity v F . In the bare dispersion at a scale * L from the Dirac points, the energy expression is * where v 0 F is the bare Fermi velocity. From the above discussion, we find that corrections to the dispersion are RG irrelevant, such that * *   < L L 1 1 0 , i.e. the overall magnitude of the dispersion at the stable fixed point is lower in comparison to the bare one. This leads us to conclude that the renormalized Fermi velocity is lowered from its bare value, 0 F 0 , and the Dirac cones are flattened. Below, we will see the effects of velocity renormalization on the quasiparticle residue as well as the lifetime of the spinon excitations. This will offer further insight on the nature of the gapless spinon liquid phase.

Spinon self-energy, residue and lifetime
From the stable fixed point single-spinon energy, we compute the real part of the self energy for the gapless phase from the renormalization of the spinon dispersion (i.e. the difference between final fixed point energy and initial energy) , the exponent can be simplified to 0 , the self energy can be written as where Θ is the Heaviside step theta function. We can now compute the quasi-particle residue Z(Λ, ω) for lowenergy excitations w  D + The quasiparticle residue  Z 1 upon approaching the gapless Dirac-point Fermi surface (L  0), indicating non-interacting spinon quasiparticles in the gapless phase. However, in order to learn whether this result indicates a Landau-Fermi liquid or not, we should also compute the spinon lifetime τ. Thus, from the Kramers-Kronig relation, we compute the imaginary part of the self-energy and thereby the spinon lifetime τ(Λ, ω) The Fourier transformation of w L ( ) G , shows that the two-point correlation function decays algebraically in real space, supporting the formation of an algebraic spin liquid ground state with likely long-ranged entanglement in phase III.

Discussion and outlook
The RG phase diagram shown in figure 6 encapsulates the major findings of this work. We conclude with a discussion on the relevance of our findings, as well as an outlook on future prospects. Experiments on the Herbertsmithite and Volborthite materials are thought to be described by the physics of the S=1/2 Heisenberg KA. Indeed, recent experiments on Volborthite at high magnetic fields (~-20 160 T) have revealed the existence of a robust plateau in the magnetization per site of 1/3 [31]. From our RG analysis, such a gapped ground state for the Heisenberg KA corresponds to a quantum spin liquid driven by spinon correlations (phase-II in figure 6). It will, therefore, be interesting to search for experimental signatures of topological order within the plateau, e.g. fractional excitations above the gap. Indeed, claims of the observation of such fractional excitations in Herbetsmithite at zero-field from neutron scattering measurements have appeared recently [27]. A magnetization plateau at 1/3 in the Heisenberg KA at finite field has also been proposed from numerical methods [9,70].
It is worth noting the liquid-like nature of the ground state obtained by us in the two gapped phases. Given that RG transformations necessarily preserve all symmetries of the Hamiltonian, the spin liquid ground states of Phases-I and II do not exhibit the further breaking of any lattice translation, rotation or spin symmetries. For such topologically ordered spin liquid ground states, we expect the presence of short-ranged RVB type ordering [68]. This can, for instance, be compared with the finding of competing lattice Nematic and VBC orders for the 1/3 plateau in the XXZ [43] and Heisenberg [37] kagome antiferromagnets with nine-site unit cells obtained from tensor network methods. It is tempting, but not straightforward, to relate such symmetry broken orders to instabilities of the liquid ground states obtained by us for an enlarged nine-site unit cell (see section 2). This will involve the incorporation of symmetry broken orders into the RG scheme [48], and we leave this to further investigations.
Equally interesting would be a search for the gapless algebraic spin liquid found by us (phase-III in figure 6). For this, it may be possible to melt the gapped spin liquid in phase-II by enhancing quantum fluctuations through the application of pressure [71,72]. We note that the existence of a gapless algebraic phase in the KA has just recently been proposed as being accessible via thermal melting of the finite-field ordered gapped phase [22]. It will be interesting to compare such a thermally induced gapless phase with that reached purely from critical quantum fluctuations (line-2 in figure 6). Theoretical proposals of a gapless Dirac spin liquid in the Heisenberg KA at zero field have appeared in the literature [15, 17-19, 21, 24]. It is pertinent to check whether the gapless phase observed by us at finite field is connected in any way to these results. Given that we find the Heisenberg KA at 1/3 magnetization to be in a topological spin liquid phase, it appears plausible that the zero-field algebraic spin liquid found from various analyses is unstable at finite field. A more subtle possibility involves an adiabatic connection between the zero-field gapless spin liquid and that observed by us in Phase III. Confirming this needs, however, an investigation that lies well beyond the purview of the present study. Further, it is promising to investigate whether such a 2D algebraic Dirac spin liquid can exists at the boundaries of a topologically ordered ground state of the three-dimensional pyrochlore S=1/2 Heisenberg antiferromagnet [73,74].
The non-trivial multicritical point lying at the intersection of the three spin liquid phases (black circle in figure 6) represents an intriguing result. The experimental observation of such an exotic QCP would be highly valuable, as it is likely to be masked by the formation of a symmetry-broken ground state. Finally, given that critical theories describing line-2 and line-3 relate to relativistic fermionic criticality in two-dimensions, it appears pertinent to employ similar RG methods to the investigation of Mott criticality in graphene and its analogs [75][76][77][78]), as well as topological transitions in quantum Hall systems [79] and other topological insulators [80].

Appendix A. Scattering processes in PP and PH channels
The processes labeled as (1-8) and (9-24) in figure A1 present all possible intra-and inter-band scattering responsible for the instability in the PP for PH channels respectively.
The pseudo-spinors for the PP channel are define as  Similarly, the pseudo-spinors for PH channel are defined as   where, in the last step, we have used = + + - in the Hamiltonian for scattering in the PH channel, equation (11), to obtain (upto a constant) Then, the total Hamiltonian for the system is (equation (12) in the main text) On the other hand, in a de-excitation process, a pseudospin scatters between the boundary of the lower band and patch-center of the upper band, yielding   In order to compute the Chern number related to the one-spinon gap, we can write an effective two-level Hamiltonian in the low-energy neighborhood of the gapped two-band problem as follows where the σʼs are Pauli matrices and p is the momentum (with respect to the Fermi momentum). It is well known that the two-level problem (with a structure of the Hamiltonian given above) possesses a geometric (Berry) phase γ=Ω/2, where Ω is the solid angle created by the loop on the Bloch sphere. If we take an integral over the entire Bloch sphere (with a total solid angle of 4π), we obtain the topological Chern number (C) [50] p p = = ( ) C 1 4 4 1. C4

Appendix D. Twist operators
In section 4.1, we have defined the twist operator projected onto the PP subspace (see equations (24) and (25)).   k 2 are the momenta of the two Dirac-points. In the above, we used the fact that º   k k 2 1 , i.e. these two points are connected by a reciprocal lattice vector of the magnetic Brillouin zone for the spinon system. The indices 1 and 2 represent, as always, the lower and upper bands respectively. Under the action of the doubled twist operator on the center of mass momentum, p « -  k k N 2 1 2 1 , which is equivalent to the scattering of a spinon between the two patch-centers (i.e. the two Dirac-points). This scattering process is captured by the effective Hamiltonian  where L P S is the projection onto the same bands mentioned above, but for the PH channel. In the projected space of the 3rd and 4th bands the Hamiltonians   have a band-inversion symmetry, as the chemical potential is pinned at the middle of the two bands.
Appendix E. RG relations in the gapless phase