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arXiv:2604.00815 · Physical review. B./Physical review. B

Stable determinant Monte Carlo simulations at low temperatures

Thomas Luu, Johann Ostmeyer, Petar Sinilkov, Finn L. Temmen

Published Oct 27, 2026• 0 citations
arXivHTMLPDF

Abstract

At low temperatures $T$ where $1/T=β\gg1$ the naïve implementation of determinant quantum Monte Carlo (DQMC) methods suffers from loss of precision and numerical instabilities when evaluating the fermion determinant. This instability propagates into the calculation of observables that rely on the evaluation of the inverse of the fermion matrix, or the Greens function. For DQMC methods that rely on the Hamiltonian Monte Carlo (HMC) algorithm, an additional complication comes from evaluating the force terms required for integrating Hamilton's equations of motion, since here loss of precision and numerical instabilities are also prevalent. We show how to address all these issues using various choices of matrix decompositions, allowing us to simulate at $β\gtrsim 90$, which corresponds to room temperature for graphene structures. Furthermore, our implementation has numerical costs that scale similarly to the naïve implementation, namely as $\mathcal{O}(N_x^3N_t)$, where $N_x$ ($N_t$) is the number of spatial (temporal) sites.

DOI
10.1103/52s8-n486

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At low temperatures TTT where 1/T=β≫11/T=β\gg11/T=β≫1 the naïve implementation of determinant quantum Monte Carlo (DQMC) methods suffers from loss of precision and numerical instabilities when evaluating the fermion determinant. This instability propagates into the calculation of observables that rely on the evaluation of the inverse of the fermion matrix, or the Greens function. For DQMC methods that rely on the Hamiltonian Monte Carlo (HMC) algorithm, an additional complication comes from evaluating the force terms required for integrating Hamilton's equations of motion, since here loss of precision and numerical instabilities are also prevalent. We show how to address all these issues using various choices of matrix decompositions, allowing us to simulate at β≳90β\gtrsim 90β≳90, which corresponds to room temperature for graphene structures. Furthermore, our implementation has numerical costs that scale similarly to the naïve implementation, namely as O(Nx3Nt)\mathcal{O}(N_x^3N_t)O(Nx3​Nt​), where NxN_xNx​ (NtN_tNt​) is the number of spatial (temporal) sites.
We assess numerical stabilization methods employed in fermion many-body quantum Monte Carlo simulations. In particular, we empirically compare various matrix decomposition and inversion schemes to gain control over numerical instabilities arising in the computation of equal-time and time-displaced Green's functions within the determinant quantum Monte Carlo (DQMC) framework. Based on this comparison, we identify a procedure based on pivoted QR decompositions which is both efficient and accurate to machine precision. The Julia programming language is used for the assessment and implementations of all discussed algorithms are provided in the open-source software library StableDQMC.jl.
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