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The early years of quantum Monte Carlo (1): the ground state

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Abstract

The history of the development of Monte Carlo methods to solve the many-body problem in quantum mechanics is presented. The relation starts with the early attempts on first available computers just after the war and extends until the years 80s with the celebrated calculation of the electron gas by Ceperley and Alder. Usage is made of an interview of David Ceperley by the author.

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Notes

  1. The copy of the letter can be obtained from the web site of the archives of the US government, under reference LAMS551. It is also reproduced in Hurd (1985).

  2. See for example a special issue of the Los Alamos Science, number 15 (1987) dedicated to Stan Ulam: several papers are personal memories and refer to the period (Metropolis 1987). This review can be searched and downloaded from the Los Alamos library web site.

  3. In the following, we will write t rather than \(\tau \) for the imaginary time to simplify the notation.

  4. A million random digits with 100,000 Normal Deviates, Rand Corporation. Those numbers were obtained from measurements in a physical experiment. They were made available to the community on punched cards. Nowadays they are printed in a free ebook which can be downloaded from the following web: https://www.rand.org/pubs/monograph_reports/MR1418.html. Interesting historical details are provided in the introduction.

  5. See for example Battimelli et al. (2020) or Mareschal (2018) and Levesque and Hansen (2019).

  6. In a book preface (Monte Carlo Methods, by Malvin H. Kalos and Paula A. Whitlock, J. Wiley, 1986, Gerald Goertzel was thanked for “having introduced one of us to the mixed joys of Monte Carlo on primitive computers”.

  7. Mal Kalos often points out in a joking manner that his career’s bifurcations were very much like a random walk.

  8. The index n refers to both the spatial and the spin wave function; in order to make reading easier, we only keep one index.

  9. One could think to vary the boundaries in a variational way, but this appears complicated in a highly dimensional space, it only works in low-dimensional systems like a two-particle system.

  10. A 128 fermions systems necessitated 5 seconds on the CDC 6600 for a single pass (each particle is moved once on average). This appeared costly in 1977. Petaflops computers, which are common nowadays, are 10 orders of magnitude faster than the 6600!

  11. APS News, June 2003 (Volume 12, 6).

  12. It would be a constant if the wave function would be the exact ground state wave function.

  13. It is worth mentioning that Eq. (5.3) bears some similarity with the Smoluchowski equation in chemical physics: the Smoluchowski equation has no branching term and it describes Brownian particles undergoing an acceleration due to an external force. It is also simulated by a stochastic method known as Brownian Dynamics. The branching term is essential in the present context as it describes all the interaction part of the N-body problem. Moreover, it is also the term which allows to introduce importance sampling.

  14. An interview of Mary Ann Mansigh by Daan Frenkel can be viewed among the Cecam lectures, on the Cecam web site (https://www.cecam.org/cecam-lectures/).

  15. Besides, the Alder group had the privilege, through a friendly and totally unofficial agreement with the computer centre, to use freely the idle time. Whenever the large hydrodynamic codes of the bomb designers would crash, and that was happening sufficiently frequently, the computer would restart a QMC code from the last written configuration.

  16. It ranked number three in the list of most cited papers published by PRL, a ranking made in 2002 by the APS-while PRL started in 1958. Number 1 was S. Weinberg 1967 paper on the electro-weak interactions, number 2 was M. K. Wu 1987 letter on high Tc superconductors, while the 4th was Binnig’s paper on Atomic Force Microscopy and number 5 was 1985 Car-Parrinello’s seminal paper on Density Functional Theory.

  17. It is worth mentioning that David Ceperley never wrote the long PR paper which was originally planned on electron gas: David Ceperley confesses that it was harder than previously thought to improve on it. And also as Berni Alder puts it: “after I finish a problem, I like to move on”.

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Acknowledgements

The idea to undertake this study was a suggestion by Giovanni Ciccotti whom I wish to thank for a constant support, a generous availability and many useful suggestions. I wish also to thank David Ceperley for his hospitality during my visit to him in Illinois and for later correspondance. Stimulating discussions with Benoit Roux and Carlo Pierleoni are gratefully acknowledged. David Ceperley’s interview was made possible through financial support provided by the Neubauer’s Collegium of the University of Chicago. During the completion of this article came the sad news of the death of Berni Julian Alder the day preceding his 95th anniversary. I wish to acknowledge here his constant, generous and enthusiastic availability for sharing his memories as well as his views on the field he has contributed to create.

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  1. Physics Department, ULB, Bruxelles, Belgium

    Michel Mareschal

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Mareschal, M. The early years of quantum Monte Carlo (1): the ground state. EPJ H 46, 11 (2021). https://doi.org/10.1140/epjh/s13129-021-00017-6

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