** Numerical Renormalization Group**

** DMRG**

** Momentum Space DMRG and Quantum Chemistry Calculation**

In 1996 the first work on the momentum space DMRG was published. This paper removes the barrier in the extension of the DMRG to the momentum space, or more generally to any other basis space, in which the interaction is non-local and off-diagonal. It sets the foundation for carrying the DMRG calculation not only in the momentum space, but also in the Quantum Chemistry study. In real space, the interaction is generally local and diagonal, and the effort in evaluating and storing these interaction terms scales linearly with the lattice size N. However, in the momentum space (or in the quantum chemistry calculation), the interaction contains at least N3 terms. This obstructs the application of the DMRG if all these terms have to be handled independently.

** Quantum Transfer-Matrix Renormalization Group**

** (free TMRG code available)**

TMRG is the most accurate numerical method for evaluating thermodynamic quantities of 1D quantum lattice models. This method was first introduced in

· R. J. Bursill, T. Xiang, and G. A. Gehring, “The density matrix renormalization group for a quantum spin chain at non-zero temperature”, Journal of Physics: Condensed Matter 8 (1996) L583-L590. Cond-mat/9609001.

· X. Wang and T. Xiang, “Transfer matrix DMRG for thermodynamics of one-dimensional quantum systems”, Physical Review B 56 (1997) 5061-5064. Cond-mat/9705301.

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