  Symmetry-broken electronic phases support neutral collective excitations. For
example, monolayer graphene in the quantum Hall regime hosts a nearly ideal
ferromagnetic phase at filling factor $\nu=1$ that spontaneously breaks spin
rotation symmetry. This ferromagnet has been shown to support spin-wave
excitations known as magnons which can be generated and detected electrically.
While long-distance magnon propagation has been demonstrated via transport
measurements, important thermodynamic properties of such magnon
populations--including the magnon chemical potential and density--have thus far
proven out of reach of experiments. Here, we present local measurements of the
electron compressibility under the influence of magnons, which reveal a
reduction of the $\nu=1$ gap by up to 20%. Combining these measurements with
estimates of the temperature, our analysis reveals that the injected magnons
bind to electrons and holes to form skyrmions, and it enables extraction of the
free magnon density, magnon chemical potential, and average skyrmion spin. Our
methods furnish a novel means of probing the thermodynamic properties of
charge-neutral excitations that is applicable to other symmetry-broken
electronic phases.
