Abstract
We propose a new Monte Carlo method for calculating eigenvalues of transfer matrices leading to free energies and to correlation lengths of classical and quantum many-body systems. Generally, this method can be applied to the calculation of the maximum eigenvalue of a nonnegative matrix  such that all the matrix elements of Âk are strictly positive for an integerk. This method is based on a new representation of the maximum eigenvalue of the matrix  as the thermal average of a certain observable of a many-body system. Therefore one can easily calculate the maximum eigenvalue of a transfer matrix leading to the free energy in the standard Monte Carlo simulations, such as the Metropolis algorithm. As test cases, we calculate the free energies of the square-lattice Ising model and of the spin-1/2XY Heisenberg chain. We also prove two useful theorems on the ergodicity in quantum Monte Carlo algorithms, or more generally, on the ergodicity of Monte Carlo algorithms using our new representation of the maximum eigenvalue of the matrixÂ.
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References
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller,J. Chem. Phys. 21:1087 (1953).
K. Binder,J. Comp. Phys. 59:1 (1985).
K. Binder, ed.,Monte Carlo Methods in Statistical Physics, 2nd ed. (Springer, Berlin, 1986).
K. Binder, ed.,Applications of the Monte Carlo Method in Statistical Physics (Springer, Berlin, 1984).
H. De. Raedt and A. Lagendijk,Phys. Rep. 127:233 (1985).
M. Suzuki, ed.,Quantum Monte Carlo Methods in Equilibrium and Nonequilibrium (Springer, Berlin, 1986).
M. Suzuki,J. Stat. Phys. 43:883 (1986).
J. P. Hansen and L. Verlet,Phys. Rev. 184:151 (1969).
M. P. Nightingale and H. W. Blöte,Phys. Rev. Lett. 60:1562 (1988).
D. M. Ceperly and M. H. Kalos, inMonte Carlo Methods in Statistical Physics, 2nd ed., K. Binder, ed. (Springer, Berlin, 1986).
J. H. Hetherington,Phys. Rev. A 30:2713 (1984).
R. J. Baxter,Exactly Solved Models in Statistical Mechanics (Academic Press, 1982).
L. Onsager,Phys. Rev. 65:117 (1944).
A. G. Schlijper, A. R. D. van Bergen, and B. Smit,Phys. Rev. A 41:1175 (1990); A. G. Schlijper and B. Smit,J. Stat. Phys. 56:247 (1989).
H. Betsuyaku,Phys. Rev. Lett. 53:629 (1984).
M. Suzuki,Phys. Rev. B 31:2957 (1985).
M. Suzuki and M. Inoue,Prog. Theor. Phys. 78:787 (1987).
M. Inoue and M. Suzuki,Prog. Theor. Phys. 79:645 (1988).
T. Koma,Prog. Theor. Phys. 78:1213 (1987).
T. Koma,Prog. Theor. Phys. 81:783 (1989).
T. Koma,Prog. Theor. Phys. 83:655 (1990).
M. Marcu, inQuantum Monte Carlo Methods in Equilibrium and Nonequilibrium, M. Suzuki, ed. (Springer, Berlin, 1986).
S. Miyashita,J. Phys. Soc. Jpn. 57:1934 (1988).
K. Kubo, T. A. Kaplan, and J. R. Borysowicz,Phys. Rev. B 38:11550 (1988).
A. Wiesler,Phys. Lett. 89A:359 (1982).
Y. Okabe and M. Kikuchi,Phys. Rev. B 34:7896 (1986).
A. B. Bortz, M. H. Kalos, and J. L. Lebowitz,J. Comp. Phys. 17:10 (1975).
R. Takahashi, T. Koma, T. Irisawa, and T. Kan, Annual Meeting of Physical Society of Japan, Osaka, Japan, March 1990.
H. Akaike,IEEE Trans. Automatic Control AC-19:716 (1974).
M. Suzuki,Phys. Lett. 113A:299 (1985).
E. Lieb, T. Schultz, and D. Mattis,Ann. Phys. (NY)16:407 (1961).
S. Katsura,Phys. Rev. 127:1508 (1962).
M. Yamada,J. Phys. Soc. Jpn. 59:848 (1990).
T. Koma, Annual Meeting of Physical Society of Japan, Osaka, Japan, March 1990.
K. Nomura and M. Yamada,Phys. Rev. B 43:8217 (1991).
M. Takahashi,Phys. Rev. B 43:5788 (1991).
H. Tsunetsugu,J. Phys. Soc. Jpn. 60:1460 (1991).
R. J. Glauber,J. Math. Phys. 4:294 (1963).
K. Kawasaki, inPhase Transitions and Critical Phenomena, Vol. 2, C. Domb and M. S. Green, eds. (Academic Press, New York, 1972).
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Koma, T. A new Monte Carlo power method for the eigenvalue problem of transfer matrices. J Stat Phys 71, 269–297 (1993). https://doi.org/10.1007/BF01048100
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DOI: https://doi.org/10.1007/BF01048100