Renormalization group analysis of competing quantum phases in the J1-J2 Heisenberg model on the kagome lattice

In this section we elaborate on various technical details of the pseudofermion functional renormalization group method, which we apply to map out the phase diagram of the KHM. As mentioned in the paper, we first express the spin operators in terms of pseudofermions, using the representation S = 1/2 ∑ αβ f † ασ μ αβfβ , (α, β =↑, ↓, μ = x, y, z) with fermionic operators f↑ and f↓ and Paulimatrices σ. The introduction of pseudofermions leads to an artificial enlargement of the Hilbert space and therefore requires the fulfillment of an occupancy constraint, excluding empty and doubly occupied states. Since an unphysical occupation acts as a vacancy in the spin lattice associated with an excitation energy on the order of the exchange coupling, particle number fluctuations are suppressed at zero temperature such that the constraint is naturally fulfilled in the ground state. The fermionic representation allows one to apply standard diagrammatic many-body techniques. In particular, the free propagator in Matsubara-space is simply given by G0(iω) = 1 iω , where due to the absence of quadratic terms in the fermionic Hamiltonian, G0(iω) does not contain any self-energy contributions. As a consequence of this property, the propagator is strictly local in real space. The FRG procedure first amounts to introducing an infrared frequency cutoff in the propagator, replacing G0(iω) → G0 (iω) = Θ(|ω|−Λ) iω . Here, we use a sharp cutoff implemented by a Θ-function, however, the FRG does not rely on this particular choice. As a consequence of this replacement, all m-particle vertex functions acquire a Λ-dependence, which is described by the FRG flow equations. A schematic form of these equations is shown in Figs. 4a and 4b. The flow of the self energy (Fig. 4a) couples to itself (via fully dressed propagators) and to the two-particle vertex. In turn, the flow of the two-particle vertex (Fig. 4b) couples to the self energy, to itself and to the three-particle vertex. This way the FRG generates an infinite hierarchy of coupled RG equations for all m-particle vertices. For an actual solution, a truncation procedure needs to be performed. While for most FRG applications, it is sufficient to neglect the threeparticle vertex completely, here we keep certain threeparticle contributions as indicated in Fig. 4c. This truncation leads to a better fulfillment of Ward-identities [1] and, moreover, ensures that via better feedback of the self energy into the two-particle vertex flow, spin-mean field as well as RVB-mean field limits are included in an entirely self-consistent way. Magnetic ordering tendencies and quantum fluctuations can, hence, be described within the same RG framework. The spin susceptibility, which is the central quantity to be investigated in the pseudofermion FRG, can be computed by fusing the external legs of the two-particle vertex, see Fig. 4d. The RG flow starts in the limit Λ→∞ where the two-particle vertex is given by the bare interaction ∼ J and ends at Λ = 0 when the cutoff is completely removed. The RG equations in Fig. 4 are solved in the frequency domain (keeping all frequency dependencies of the vertex functions) and in real space. For a finite set of differential equations the frequency dependencies need to be discretized, typically by choosing a logarithmic mesh of discrete frequency points. Furthermore, the real space dependence needs to be truncated. While formally, we treat the system as an infinite lattice and exploit translation invariance, spin-spin correlations (i.e., two-particle vertices) are only taken into account up to maximal length. Consequently, the effective system size, i.e., the area in which all spins are correlated among each other, becomes finite (in our studies up to 317 lattice sites).