Recent discoveries in neutron scattering experiments for Kapellasite and Herbertsmithite as well as theoretical calculations of possible spin liquid phases have revived interest in magnetic phenomena on the kagome lattice. We study the quantum phase diagram of the S = 1/2 Heisenberg kagome model as a function of nearest neighbor coupling J1 and second neighbor coupling J2. Employing the pseudofermion functional renormalization group, we find four types of magnetic quantum order (q = 0 order, cuboc order, ferromagnetic order, and √ 3 × √ 3 order) as well as extended magnetically disordered regions by which we specify the possible parameter regime for Kapellasite. In the disordered regime1, the flatness of the magnetic susceptibility at the zone boundary which is observed for Herbertsmithite can be reconciled with the presence of small J2 > 0 coupling. In particular, we analyze the dimer susceptibilities related to different valence bond crystal (VBC) patterns, which are strongly inhomogeneous indicating the rejection of VBC order in the RG flow. Introduction. Frustrated magnetism is a focus of contemporary research in condensed matter physics, combining a plethora of experimental scenarios and diverse theoretical approaches to describe them. One of the most fascinating challenges of the field has been to investigate and understand the interplay of magnetic quantum order and disorder on the kagome lattice. A major reason why this lattice of corner-sharing triangles yields such a complicated structure of quantum phases is already evident from the classical kagome Heisenberg model (KHM): As a function of nearest neighbor and next nearest neighbor Heisenberg couplings J 1 and J 2 , many different magnetic orders are present [1], where an infinite number of degenerate ground states can be found [2]. From a theoretical perspective, not many rigorous results about the quantum phase diagram are known so far. Advanced mean field theories have provided important guidance as to what type of ordered and disordered quantum phases could possibly be found [3][4][5][6][7][8], but cannot give unambiguous information about which phase will eventually be stabilized in the microscopic model. A peculiar feature of the KHM which is known since early exact numerical calculations of finite size clusters [9] is the large amount of singlet states at low energy. This suggests a plethora of competing quantum-disordered phases and is probably one of the main reasons why the interpretation of present results for the J 1 KHM from microscopic numerical approaches is not yet settled [10][11][12][13].

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