Quantum Transfer Matrix Renormalization Group (TMRG)
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.
¡¡
rsys = Trenv T .
where T is the transfer matrix. The partition function is proportional to
Z ~ Trsys rsys .
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.
¡¡