** Density Matrix Renormalization Group**

**一**

The density-matrix renormalization group (DMRG) is the most powerful numerical method for simulating one-dimenional quantum lattice models.

1992 Steven R White published the first paper on the DMRG.

Some extensions of the DMRG we proposed:

· Momentum space DMRG: This is the approach for performing DMRG calculations in momentum space, or more generally in other basis space with non-local and off-diagonal interactions, such as the angular momentum space and the Hartree-Fock basis space used in the Quantum Chemistry calculations

· Quantum transfer-matrix renormalization group [Refs 1 & 2]: this is the most accurate numerical method for evaluating thermodynamic quantities of 1D quantum lattice models.

· Pace-keeping DMRG: this is the method for solving a time-dependent Schrodinger equation of many interacting particles.

**一**

The first work on the momentum space DMRG was published in 1996:

This paper removes the barrier in the extension of the DMRG from the real space to the momentum space, or more generally to any other basis space, at 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 the 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.

**一**

Free TMRG code is available upon request

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.

It was then further improved by introducing the non-symmetric density-matrix in

· 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.

The non-symmetric density matrix is defined by tracing out the environment degrees of freedom from the (non-symmetric) transfer matrix, i.e.

_{P}_{sys} = Tr_{env}_{} T .

where T is the transfer matrix. The partition function is proportional to

Z ~ T*r*_{sys} *P*_{sys} .

A thorough application of the TMRG to the quantum spin chains is given in

· T. Xiang, “Thermodnamics of Heisenberg spin chains in a magnetic field”, Physical Review B 58 (1998) 9142-9149. Cond-mat/9808179.

· J. Lou, T. Xiang, and Z. B. Su, “Thermodynamics of the bilinear-biquadratic spin one Heisenberg chain”, Physical Review Letters 85 (2000) 2380-2383. Cond-mat/0003102.

· H. T. Lu, Y. H. Su, L. Q. Sun, J. Chang, C. S. Liu, G. H. Luo, T. Xiang, “Thermodynamic properties of tetrameric bond-alternating spin chains”, Physical Review B 71 (2005) 144426-1-7. cond-mat/0412275.

H. T. Lu, Y. J. Wang, Shaojin Qin, T. Xiang, “Zigzag spin chains with antiferromagnetic-ferromagnetic interactions: Transfer-matrix renormalization group study” Physical Review B 74, 134425 (2006). cond-mat/0603519.

The Fermi momentum in the 1D Kondo lattice can be accurately determined by the TMRG. This is demonstrated in

· Y. H. Su, Q. H. Xiao, T. Xiang, X. Q. Wang, and Z. B. Su. “Instability of the Fermi surface in the one-dimensional Kondo lattice”, Journal of Physics: Condensed Matter 16 (2004) 5163-5169.

** Pace-Keeping DMRG**

**一**

Pace-keeping DMRG is the method for solving a time-dependent Schrodinger equation of many interacting particles. This method was first introduced in this short paper:

· H. G. Luo, T. Xiang, and X. Q. Wang, Comment on “Time-Dependent Density-Matrix Renormalization Group: A Systematic Method for the Study of Quantum Many-Body Out-of- Equilibrium Systems”, Physical Review Letters 91 (2003) 049701-1-1. cond-mat/0212580.

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