  The pseudofermion functional renormalization group (PFFRG) method has proven
to be a powerful numerical approach to treat frustrated quantum spin systems.
In its usual implementation, however, the complex fermionic representation of
spin operators introduces unphysical Hilbert space sectors which renders an
application at finite temperatures inaccurate. In this work, we formulate a
general functional renormalization group approach based on Majorana fermions to
overcome these difficulties. We, particularly, implement spin operators via an
$SO(3)$ symmetric Majorana representation which does not introduce any
unphysical states and, hence, remains applicable to quantum spin models at
finite temperatures. We apply this scheme, dubbed pseudo Majorana functional
renormalization group (PMFRG) method, to frustrated Heisenberg models on small
spin clusters as well as square and triangular lattices. Computing the finite
temperature behavior of spin correlations and thermodynamic quantities such as
free energy and heat capacity, we find good agreement with exact
diagonalization and the high-temperature series expansion down to moderate
temperatures. We, particularly, observe a significantly enhanced accuracy of
the PMFRG compared to the PFFRG at finite temperatures. More generally, we
conclude that the development of functional renormalization group approaches
with Majorana fermions considerably extends the scope of applicability of such
methods.
