  We have developed a different quantum transfer matrix method to accurately
determine thermodynamic properties of the Hofstadter model. This method
resolves a technical problem which is intractable by other methods and makes
the calculation of physical quantities of the Hofstadter model in the
thermodynamic limit at finite temperatures feasible. It is shown that the
quantum correction to the de Haas-van Alphen (dHvA) oscillation of
magnetization bears the energy structure of Hofstadter butterfly. The
measurement of this quantum correction, which can be materialized on the
superlattice or cold atom systems, can reveal unambiguously the Hofstadter
fractal energy spectrum.
