  We present the first stellar velocity dispersion measurement of a massive
quenching galaxy at z=4.01. The galaxy is first identified as a massive z>~4
galaxy with suppressed star formation from photometric redshifts based on deep
multi-band data in the UKIDSS Ultra Deep Survey field. A follow-up
spectroscopic observation with MOSFIRE on Keck revealed strong multiple
absorption features, which are identified as Balmer absorption lines, giving a
secure redshift of z=4.01. Thanks to the high S/N of the spectrum, we are able
to estimate the stellar velocity dispersion, sigma=268+/-59 km/s. This velocity
dispersion is consistent with that of massive galaxies today, implying no
significant evolution in stellar velocity dispersion over the last 12 Gyr.
Based on an upper limit on its physical size from deep optical images
(r_eff<1.3 kpc), we find that its dynamical mass is consistent with the stellar
mass inferred from photometry. Furthermore, the galaxy is located on the mass
fundamental plane extrapolated from lower redshift galaxies. Combining all
these results, we find that the velocity dispersion does not significantly
evolve with redshift, although the size and mass of massive quenched galaxies
do. This suggests that the mass in the core of massive galaxies does not evolve
significantly, while most of the mass growth occurs in the outskirts of the
galaxies, which also increases the size. This picture is consistent with a
two-phase formation scenario in which mass and size growth is due to accretion
in the outskirts of galaxies via mergers.
