Monte Carlo Calculation of Transfer Matrix Eigenvalues

Tohru Koma

Masuo Suzuki

T. Tsuzuki

An improved transfer matrix method is proposed for calculating the partition function of quantum spin systems, which is expressed by the M-th approximant of the Suzuki-Trotter transformation with Trotter number M. The point is to incorporate the symmetry property of the Trotter subsystem in calculation. The size of our transfer matrix is reduced very much. Thermodynamics is described by the maximum eigenvalue of the transfer matrix. Our method is applied to the XY model with spin 1/2. We consider the cases of pair and tri spin decompositions with Trotter number up to 4 (M≤4). We can do our calculation analytically in those cases. In the pair spin decomposition we find results which have been obtained by the Monte Carlo simulation and by a transfer matrix method by computer. We find nice results in the tri spin decomposition. Our result in the case M=4 is as good as the result in M=8 by the previous Monte Carlo simulation on comparing them with the exact result of Katsura.

Hiroshi Katsumori

Kramers and Wannier's successive screw technique is applied to the three-dimensional Ising lattice gauge theory. The largest eigenvalue of the transfer matrix is directly calculated on a computer, independently of Monte Carlo simulation. Anomalous behavior of the specific heat implies the existence of a phase transition.

Smita Krishnaswamy, Igor L. Markov, John P. Hayes