Beating Monte Carlo: The Polynomial Method in Lattice Problems

Christian Scullard | 19-DR-013

Project Overview

Phase transitions occur in a wide range of physical systems, such as ferromagnets, Bose-Einstein condensates, and even collapsing black holes. Theoretically, computing the location of these critical points is often very challenging, even in heavily simplified models. This project sought to extend some very recent major advances in the calculation of the critical point of the two-dimensional Potts model, a simple model of ferromagnetism, to higher dimensions and to a wider variety of systems. The ultimate aim was to learn if a version of this new technique, called the method of critical polynomials, could have any application to the very difficult computations done in lattice quantum chromodynamics (QCD), and particularly to the quark–gluon deconfinement transition.

We began the project by completing calculations conducted on supercomputers, of percolation critical points of two-dimensional planar lattices. This was an international collaboration that utilized both LLNL's supercomputing capability and that of France's IDRIS. The accuracy of our results placed these values out of reach of Monte Carlo and other traditional techniques, likely permanently, thus establishing that our method is truly the state of the art in this field. Next, we wanted to understand exactly what made the method of critical polynomials so accurate in two dimensions; it outperforms standard techniques, like Monte Carlo, by many orders of magnitude. This had been somewhat mysterious previously, but by studying a model system called the kagome hypergraph we were able to understand this issue better. We then generalized the method, which was devised for two-dimensional planar systems, to non-planar graphs and three-dimensional slabs. Finding that this generalization was very computationally expensive, we devised a hybrid critical-polynomial and Monte Carlo scheme that was very effective and an improvement on existing techniques.

In all, we found that while it was possible to use the method of critical polynomials in higher dimensions, the computational costs were prohibitive. We learned that the accuracy in two dimensions stemmed from the fact that there exist many exact solutions there, and furthermore that the accuracy of the method increases the "closer" one is, defined in a particular way, to one of these exact points. In three dimensions, there are no exact solutions and the method suffers in accuracy for that reason. We found that for non-planar systems, a hybrid method where the critical polynomial is used to locate the critical point and Monte Carlo is used to sample the system is a better option. However, we learned during the course of the project that in four dimensions, where lattice QCD is done, there are once again some exact systems. This is a surprising fact and, while we did not have the time to explore this fully, we have some hope that it will open future research avenues and that the method may indeed have some application to lattice-gauge theories after all.

Mission Impact

This project was funded in the inaugural round of the Disruptive Research (DR) LDRD program. This was a high-risk and high-reward project with the goal of generalizing a revolutionary new technique in the field of lattice statistical mechanics to the lattice QCD computations conducted at LLNL. What this project has made clear is that this is still possible but will require an additional breakthrough to make it a reality. This work resulted in three publications that substantially contributed to the field of lattice critical models. One of these used the computational capabilities at LLNL to set the standard for accuracy of percolation critical points that will likely hold for decades, helping to solidify LLNL's reputation as a leader in supercomputing.

Publications, Presentations, and Patents

Bell, T. H. et al., 2022. "Critical Points of the Random Cluster Model with Newman-Ziff Sampling." Journal of Physics A: Mathematical and Theoretical 55: 044001 (January 2022).

Scullard, Christian R., and Jesper L. Jacobsen. 2020. "Bond Percolation Thresholds on Archimedean lattices from Critical Polynomial Roots." Physical Review Research 2: 012050.

Scullard, C. R. et al., 2021. "Critical Percolation on the Kagome Hypergraph." Journal of Physics A: Mathematical and Theoretical 54(5): 055006.

Scullard, C. R. "The Method of Critical Polynomials on Supercomputers." Stochastic Processes and Their Applications, Northwestern University, Evanston IL. July 8-12, 2019.