Volume 25, Issue 2
Real-Time Diffusion Monte Carlo Method

Ilkka Ruokosenmäki & Tapio T. Rantala

Commun. Comput. Phys., 25 (2019), pp. 347-360.

Published online: 2018-10

[An open-access article; the PDF is free to any online user.]

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  • Abstract

Direct sampling of multi-dimensional systems with quantum Monte Carlo methods allows exact account of many-body effects or particle correlations. The most straightforward approach to solve the Schrödinger equation, Diffusion Monte Carlo, has been used in several benchmark cases for other methods to pursue. Its robustness is based on direct sampling of a positive probability density for diffusion in imaginary time. It has been argued that the corresponding real time diffusion can not be realised, because the corresponding oscillating complex valued distribution can not be used to drive diffusion. Here, we demonstrate that this can be done by turning the distribution piecewise positive and normalizable, and also, by using four types of walkers. This study is a proof of concept demonstration using the well-known and transparent case: one-dimensional harmonic oscillator. Furthermore, we show that our novel method can be used to find not only the ground state but also excited states and even the time evolution of a given wave function. Considering fermionic systems, this method may turn out to be feasible for finding the wave function nodes for other approaches.

  • Keywords

Path integral, quantum dynamics, first-principles, Monte Carlo, real-time.

  • AMS Subject Headings

81-08

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-25-347, author = {}, title = {Real-Time Diffusion Monte Carlo Method}, journal = {Communications in Computational Physics}, year = {2018}, volume = {25}, number = {2}, pages = {347--360}, abstract = {

Direct sampling of multi-dimensional systems with quantum Monte Carlo methods allows exact account of many-body effects or particle correlations. The most straightforward approach to solve the Schrödinger equation, Diffusion Monte Carlo, has been used in several benchmark cases for other methods to pursue. Its robustness is based on direct sampling of a positive probability density for diffusion in imaginary time. It has been argued that the corresponding real time diffusion can not be realised, because the corresponding oscillating complex valued distribution can not be used to drive diffusion. Here, we demonstrate that this can be done by turning the distribution piecewise positive and normalizable, and also, by using four types of walkers. This study is a proof of concept demonstration using the well-known and transparent case: one-dimensional harmonic oscillator. Furthermore, we show that our novel method can be used to find not only the ground state but also excited states and even the time evolution of a given wave function. Considering fermionic systems, this method may turn out to be feasible for finding the wave function nodes for other approaches.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0048}, url = {http://global-sci.org/intro/article_detail/cicp/12754.html} }
TY - JOUR T1 - Real-Time Diffusion Monte Carlo Method JO - Communications in Computational Physics VL - 2 SP - 347 EP - 360 PY - 2018 DA - 2018/10 SN - 25 DO - http://doi.org/10.4208/cicp.OA-2018-0048 UR - https://global-sci.org/intro/article_detail/cicp/12754.html KW - Path integral, quantum dynamics, first-principles, Monte Carlo, real-time. AB -

Direct sampling of multi-dimensional systems with quantum Monte Carlo methods allows exact account of many-body effects or particle correlations. The most straightforward approach to solve the Schrödinger equation, Diffusion Monte Carlo, has been used in several benchmark cases for other methods to pursue. Its robustness is based on direct sampling of a positive probability density for diffusion in imaginary time. It has been argued that the corresponding real time diffusion can not be realised, because the corresponding oscillating complex valued distribution can not be used to drive diffusion. Here, we demonstrate that this can be done by turning the distribution piecewise positive and normalizable, and also, by using four types of walkers. This study is a proof of concept demonstration using the well-known and transparent case: one-dimensional harmonic oscillator. Furthermore, we show that our novel method can be used to find not only the ground state but also excited states and even the time evolution of a given wave function. Considering fermionic systems, this method may turn out to be feasible for finding the wave function nodes for other approaches.

Ilkka Ruokosenmäki & Tapio T. Rantala. (2020). Real-Time Diffusion Monte Carlo Method. Communications in Computational Physics. 25 (2). 347-360. doi:10.4208/cicp.OA-2018-0048
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