On the base of the diffusion Monte-Carlo method we develop the method allowing to simulate the quantum systems with complex wave function. The method is exact and there are no approximations on the simulations of the module and the phase of the system's wave function. In our method averaged value of any quantity have no direct contribution from the phase of distribution function but only from the phase of the Green function of diffusion equation. We test the method by the simulations of the ground state of fermions in two-dimensional parabolic well. Anyons are used for the representation of the two-dimensional (2D) fermions. We vary the number of fermions from two to ten and find a good agreement of the numerical results with analytical ones for the numbers of the particles N>4.
1. Ceperley D.M. and Kalos M.H. Monte Carlo Methods in Statistical Physics. Ed. by Binder K. Springer, Verlag, Berlin, Heidelberg, New York, (1979).
2. Kerbikov B.O., Polikarpov M.I., Shevchenko L.V. and Zamolodchikov A.B. Preprint ITEP, Moscow, 86–160 (1986).
3. Ortiz G., Ceperley D.M. and Martin R.M. New stochastic method for systems with broken time-reversal symmetry: 2D fermions in a magnetic field. Phys. Rev. Lett., Vol. 71, 2777 (1993).
4. Zhang L., Canright G. and Barnes T. Simulating complex problems with the quantum Monte Carlo method. Phys. Rev. B., Vol. 49, 12355(R) (1994).
5. Abdullaev B., Musakhanov M. and Nakamura A. Complex Diffusion Monte-Carlo Method for the systems with complex wave function: test by the simulation of 2$D$ electron in uniform magnetic field. cond-mat/0101330, pp. 12 (2001).
6. Polikarpov M.I. and Shevchenko L.V. Modified Monte-Carlo method for Green function. Preprint ITEP, Moscow, 43 (1987).
7. Laughlin R.B. Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations. Phys. Rev. Lett., Vol. 50, 1395 (1983).
8. Renolds P.J., Ceperley D.M., Alder B.J. and Lester W.A. Jr. Fixed-node quantum Monte Carlo for moleculesa. J. Chem. Phys., Vol. 77, 5593 (1982).
9. Schmidt K.E. and Kalos M.H. Monte Carlo Methods in Statistical Physics II. Ed. by Binder K. Springer, Berlin (1984); Schmidt K.E. and Ceperley D.M. The Monte Carlo Method in Condensed Matter Physics. Ed. by Binder K. Springer, Berlin (1991).
10. Zhang S., Carlson J. and Gubernatis J.E. Constrained Path Quantum Monte Carlo Method for Fermion Ground States. Phys. Rev. Lett., Vol. 74, 3652 (1995).
11. Carlson J., Gubernatis J.E., Ortiz G. and Zhang S. Issues and observations on applications of the constrained-path Monte Carlo method to many-fermion systems. Phys. Rev. B., Vol. 59, 12788 (1999).
12. Wilczek F. Magnetic Flux, Angular Momentum, and Statistics. Phys. Rev. Lett., Vol. 48, 1144 (1982).
13. Wu Y.S. Multiparticle Quantum Mechanics Obeying Fractional Statistics. Phys. Rev. Lett., Vol. 53, 111–115 (1984).
14. Laughlin R.B. Superconducting Ground State of Noninteracting Particles Obeying Fractional Statistics. Phys. Rev. Lett., Vol. 60, 2677 (1988).
15. Fetter A.L., Hanna C.B. and Laughlin R.B. Random-phase approximation in the fractional-statistics gas. Phys. Rev. B., Vol. 39, 9679 (1989).
16. Galicki V.M., Karnakov B.M. and Kogan V.I. The problems on the quantum mechanics. Moscow, "Nauka", 648 p. (1981) (in russian).
17. Abdullaev B., Musakhanov M. and Nakamura A. Approximate formula for the ground state energy of anyons in 2D parabolic well. cond-mat/0012423, pp. 7 (2000).
Abdullaev, Bakhodir; Musakhanov, Mirzayousuf; and Nakamura, Atsushi
"Complex diffusion Monte-Carlo method: test by the simulation of the 2D fermions,"
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 3
, Article 10.
Available at: https://uzjournals.edu.uz/mns_nuu/vol3/iss3/10