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 physica
l 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.
We have proposed a density-matrix renormalization group (DMRG) scheme to optimize the one-electron basis states of molecules. It improves significantly the accuracy and efficiency of the DMRG in the study of quantum chemistry or other many-fermion sy
stem with nonlocal interactions. For a water molecule, we find that the ground state energy obtained by the DMRG with only 61 optimized orbitals already reaches the accuracy of best quantum Monte Carlo calculation with 92 orbitals.
