# Finite Temperature Auxiliary Field Quantum Monte Carlo in the Canonical Ensemble

**Authors:** Tong Shen, Yuan Liu, Yang Yu, Brenda Rubenstein  
**arXiv:** [2010.09813](https://arxiv.org/abs/2010.09813)  
**Date:** 2020-10-19  
**Summary page:** https://www.summarizepaper.com/en/paper/2010.09813v1/

## Key points

- Finite Temperature Auxiliary Field Quantum Monte Carlo (AFQMC) and Determinant Quantum Monte Carlo (DQMC) are commonly used in quantum Monte Carlo methods to investigate finite temperature phase diagrams of multidimensional lattice models and materials.
- These techniques are typically formulated in the grand canonical ensemble, which makes them difficult to apply to condensates like superfluids and benchmark against alternative methods that are formulated in the canonical ensemble.
- A new recursive approach for performing AFQMC simulations in the canonical ensemble has been developed by Tong Shen, Yuan Liu, Yang Yu, and Brenda Rubenstein.
- This approach does not require knowledge of chemical potentials and exploits the fact that AFQMC solves many-body problems by decoupling many-body propagators into integrals over one-body problems to which non-interacting theories can be applied.
- The accuracy of this technique has been benchmarked on illustrative Bose and Fermi Hubbard models, demonstrating its ability to converge more quickly to the ground state than grand canonical AFQMC simulations.
- The authors believe that their novel use of HS-transformed operators will motivate the development of a variety of other methods.
- Their technique will enable direct performance comparisons against other many-body approaches formulated in the canonical ensemble.
- This work presents a significant advancement in quantum Monte Carlo methods by providing a new approach for performing AFQMC simulations in the canonical ensemble without requiring knowledge of chemical potentials.

## Summary

In the field of quantum Monte Carlo methods, Finite Temperature Auxiliary Field Quantum Monte Carlo (AFQMC) and Determinant Quantum Monte Carlo (DQMC) have been widely used to investigate the finite temperature phase diagrams of various multidimensional lattice models and materials. However, these techniques are typically formulated in the grand canonical ensemble, which makes them difficult to apply to condensates like superfluids and benchmark against alternative methods that are formulated in the canonical ensemble. Additionally, working in the grand canonical ensemble requires determining chemical potentials that produce desired fillings, leading to increased overhead. To address these challenges, a new recursive approach for performing AFQMC simulations in the canonical ensemble has been developed by Tong Shen, Yuan Liu, Yang Yu, and Brenda Rubenstein. This approach does not require knowledge of chemical potentials and exploits the fact that AFQMC solves many-body problems by decoupling many-body propagators into integrals over one-body problems to which non-interacting theories can be applied. The accuracy of this technique has been benchmarked on illustrative Bose and Fermi Hubbard models, demonstrating its ability to converge more quickly to the ground state than grand canonical AFQMC simulations. The authors believe that their novel use of HS-transformed operators to implement algorithms originally derived for non-interacting systems will motivate the development of a variety of other methods. They anticipate that their technique will enable direct performance comparisons against other many-body approaches formulated in the canonical ensemble. Overall, this work presents a significant advancement in quantum Monte Carlo methods by providing a new approach for performing AFQMC simulations in the canonical ensemble without requiring knowledge of chemical potentials. This development has important implications for investigating condensates like superfluids and benchmarking against alternative methods formulated in the canonical ensemble.