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Physical Review B

Fast update algorithm for the quantum Monte Carlo simulation of the Hubbard model

Phani K. V. V. Nukala, Thomas Maier, Michael S. Summers, Gonzalo Álvarez, T. C. Schulthess

Published Nov 17, 2009• 29 citations
Publisher

Abstract

This paper presents an efficient algorithm for computing the transition probability in auxiliary field quantum Monte Carlo simulations of strongly correlated electron systems using a Hubbard model. This algorithm is based on a low rank updating of the underlying linear algebra problem, and results in significant computational savings. The computational complexity of computing the transition probability and Green's function update reduces to $\mathcal{O}({k}^{2})$ during the $k\text{th}$ step, where $k$ is the number of accepted spin flips, and results in an algorithm that is faster than the competing delayed update algorithm. Moreover, this algorithm is orders of magnitude faster than traditional algorithms that use naive updating of the Green's function matrix.

DOI
10.1103/physrevb.80.195111

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This paper presents an efficient algorithm for computing the transition probability in auxiliary field quantum Monte Carlo simulations of strongly correlated electron systems using a Hubbard model. This algorithm is based on a low rank updating of the underlying linear algebra problem, and results in significant computational savings. The computational complexity of computing the transition probability and Green's function update reduces to O(k2)\mathcal{O}({k}^{2})O(k2) during the kthk\text{th}kth step, where kkk is the number of accepted spin flips, and results in an algorithm that is faster than the competing delayed update algorithm. Moreover, this algorithm is orders of magnitude faster than traditional algorithms that use naive updating of the Green's function matrix.
We present an efficient low-rank updating algorithm for updating the trial wave functions used in quantum Monte Carlo (QMC) simulations. The algorithm is based on low-rank updating of the Slater determinants. In particular, the computational complexity of the algorithm is O(kN) during the kth step compared to traditional algorithms that require O(N(2)) computations, where N is the system size. For single determinant trial wave functions the new algorithm is faster than the traditional O(N(2)) Sherman-Morrison algorithm for up to O(N) updates. For multideterminant configuration-interaction-type trial wave functions of M+1 determinants, the new algorithm is significantly more efficient, saving both O(MN(2)) work and O(MN(2)) storage. The algorithm enables more accurate and significantly more efficient QMC calculations using configuration-interaction-type wave functions.
Ground-state auxiliary-field quantum Monte Carlo is a powerful tool for interacting fermions but is often limited by the bottleneck of costly local update. Here, the authors develop two acceleration schemes: the delayed update, which replaces multiple vector outer products by one matrix-matrix multiplication, and the block force-bias update, which retains high efficiency while tuning acceptance via block size. Applied to 2D Hubbard models, they yield up to ~ 8 times speedup in 1600-site systems. These update schemes can have broad applicability to general correlated fermion systems.
Determinant Quantum Monte Carlo (DQMC) provides numerically exact solutions for strongly correlated fermionic systems but faces significant computational challenges with increasing system size. While submatrix updates were originally developed for Hirsch-Fye QMC with onsite interactions at finite temperatures, their comprehensive application in DQMC has remained unexplored despite noted algorithmic similarities. We present the first comprehensive application of submatrix updates in DQMC, significantly extending beyond the original scope by enabling simulations with extended interactions and at zero temperature. Building upon conventional fast updates and delay updates, our generalized implementation achieves an order-of-magnitude improvement in computational efficiency, enabling simulations of the half-filled Hubbard model on lattices up to 8,000 sites - a scale previously challenging with standard DQMC implementations. This enhanced computational capability allows us to accurately determine the finite-temperature phase diagram of the 3D Hubbard model at half-filling. Our findings not only shed light on the phase transitions within these complex systems but also pave the way for more effective simulations of strongly correlated electrons, potentially guiding experimental efforts in cold atom simulations of the 3D Hubbard model.
Abstract In this work for an interacting electrons system that obeys the Hubbard model, a new quantum Monte Carlo algorithm introduces for calculation of average Green function. This algorithm is applied to investigate the effects of electrons onsite Coulomb repulsion on the band structure of a square lattice in both single-site approximations such as dynamical mean field theory (DMFT) and multi-site approximations such as effective medium supercell approximation (EMSCA). The advantages of our algorithm in comparison to the Hirsch-Fye algorithm and also the Blankenbecler, Scalapino, and Sugar (BSS) algorithm are the elimination of instabilities resulting from the Metropolis algorithm in the accepting and rejecting configurations, stability at low temperatures, the elimination of systematic errors resulting from the update of the Green's function in the quantum Monte Carlo process, and considering different probabilities for each possible configuration. Finally, by using our algorithm, it is possible to calculate the interacting three-dimensional system's band structure and the density of states that obey the Hubbard model. We have applied our algorithm to an interactive two-dimensional square lattice. As a result, phase transition boundaries can be easily recognized through calculated bands for different repulsions. Our results show that critical Coulomb repulsion values for Mott transition are uc=9.05t and uc=2.4t for DMFT and BEMSCA respectively. This means that DMFT significantly overestimates band splitting due to electrons' Coulomb repulsion. We found by starting at low repulsions and then increasing electrons' Coulomb repulsion, a partially flatted valance band around the center of the first Brillouin zone appears, but disappears at high repulsions.
Quantum Monte Carlo simulations provide a powerful tool to compute static properties of quantum many-particles models and realistic systems. A vast array of ground-state energies and properties defined by equal-time correlation functions can be computed with high accuracy. On the other hand, the extensions needed to study time-displaced correlation functions, which provide access to excited states, have been challenging, especially for fermionic strongly correlated systems. The authors present a methodology that allows a high-accuracy calculation of the dynamical Green functions of many-fermions systems in the framework of the auxiliary-field quantum Monte Carlo method. They extract the the charge gap of the repulsive two-dimensional Hubbard model at half-filling as a function of the interaction strength and propose strategies for applying their methodology to more realistic systems.
The calculation of imaginary-time-displaced correlation functions with the auxiliary-field projector quantum Monte Carlo algorithm provides valuable insight (such as spin and charge gaps) into the model under consideration. One of the authors and M. Imada proposed a numerically stable method to compute those quantities [J. Phys. Soc. Jpn. 65, 189 (1996)]. Although precise, this method is expensive in CPU time. Here we present an alternative approach which is an order of magnitude quicker, just as precise, and very simple to implement. The method is based on the observation that for a given auxiliary field the equal-time Green-function matrix G is a projector: G2=G.{G}^{2}=G.G2=G.