Short (inner) bars of sub-kiloparsec radius have been hypothesized to be an important mechanism for driving gas inflows to small scales, thus feeding central black holes. Recent numerical simulations have shown that the growth of central black holes in galaxies can destroy short bars, when the black hole reaches a mass of $sim 0.1%$ of the total stellar mass of the galaxy. We study $N$-body simulations of galaxies with single and double bars to track the long-term evolution of the central stellar mass distribution. We find that the destruction of the short bar contributes significantly to the growth of the bulge. The final bulge mass is roughly equal to the sum of the masses of the initial pseudo bulge and short bar. The initially boxy/peanut-shaped bulge of Sersic index $nlesssim1$ is transformed into a more massive, compact structure that bears many similarities to a classical bulge, in terms of its morphology ($n approx 2$), kinematics (dispersion-dominated, isotropic), and location on standard scaling relations (Kormendy relation, mass-size relation, and correlations between black hole mass and bulge stellar mass and velocity dispersion). Our proposed channel for forming classical bulges relies solely on the destruction of short bars without any reliance on mergers. We suggest that some of the less massive, less compact classical bulges were formed in this manner.
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