At low temperatures
T where
1/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≳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), where
Nx (
Nt) is the number of spatial (temporal) sites.
Some algorithms for the numerically exact treatment of fermion determinants are summarised. This is not supposed to be a review, rather a concise handbook. The audience is expected to have a basic understanding of how to put fermions on a computer. We primarily discuss different ways to work with the fermion matrix in the "sausage" (Green's function) formulation for quantum Monte Carlo (QMC). We emphasise the need for varied approaches in different space-time volume regimes. In particular, for small spatial volumes we describe a numerically stable method based on dense matrix operations. It is designed specifically to deal with very low temperature regimes. On the other hand, for (relatively) large volumes we describe a highly efficient and scalable sparse matrix approach.
Accurate numerical simulations of a doped $t\ensuremath{-}J$ model on a two-leg ladder are presented for the particle number, chemical potential, magnetic susceptibility, and entropy in the limit of large exchange coupling on the rung using a finite-temperature density matrix renormalization group method. This required an improved algorithm to achieve numerical stability down to low temperatures. The thermal dissociation of hole pairs and of the rung singlets are separately observed, and the evolution of the hole pair binding energy and magnon spin gap with hole doping is determined.