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arXiv:2403.18246 · Communications Physics

Sampling electronic Fock states using determinant quantum Monte Carlo

Shuhan Ding, Shaozhi Li, Yao Wang

Published Jan 30, 2025• 4 citations
arXivHTMLPDF

Abstract

Analog quantum simulation based on ultracold atoms in optical lattices has catalyzed significant breakthroughs in the study of quantum many-body systems. These simulations rely on the statistical sampling of electronic Fock states, which are not easily accessible in classical algorithms. In this work, we modify the determinant quantum Monte Carlo by integrating a Fock-state update mechanism alongside the auxiliary field. This method enables efficient sampling of Fock-state configurations. The Fock-state restrictive sampling scheme further enables the pre-selection of multiple ensembles at no additional computational cost, thereby broadening the scope of simulation to more general systems and models. Employing this method, we analyze static correlations of the Hubbard model up to the fourth order and achieve quantitative agreement with cold-atom experiments. The simulations of dynamical spectroscopies of the Hubbard and Kondo-lattice models further demonstrate the reliability and advantage of this method. Ultracold atoms in optical lattices have advanced the study of quantum many-body systems but are challenging to replicate with classical algorithms. The authors propose a modified quantum Monte Carlo method which integrates Fock-state updates, enabling efficient simulations that match experimental results and extend to broader systems and models.

DOI
10.1038/s42005-025-01963-z

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Analog quantum simulation based on ultracold atoms in optical lattices has catalyzed significant breakthroughs in the study of quantum many-body systems. These simulations rely on the statistical sampling of electronic Fock states, which are not easily accessible in classical algorithms. In this work, we modify the determinant quantum Monte Carlo by integrating a Fock-state update mechanism alongside the auxiliary field. This method enables efficient sampling of Fock-state configurations. The Fock-state restrictive sampling scheme further enables the pre-selection of multiple ensembles at no additional computational cost, thereby broadening the scope of simulation to more general systems and models. Employing this method, we analyze static correlations of the Hubbard model up to the fourth order and achieve quantitative agreement with cold-atom experiments. The simulations of dynamical spectroscopies of the Hubbard and Kondo-lattice models further demonstrate the reliability and advantage of this method. Ultracold atoms in optical lattices have advanced the study of quantum many-body systems but are challenging to replicate with classical algorithms. The authors propose a modified quantum Monte Carlo method which integrates Fock-state updates, enabling efficient simulations that match experimental results and extend to broader systems and models.
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.
The quantum Monte Carlo algorithm is arguably one of the most powerful computational many-body methods, enabling accurate calculation of many properties in interacting quantum systems. In the presence of the so-called sign problem, the algorithm typically relies on trial wave functions to eliminate the exponential decay of signal-to-noise ratio, usually at the expense of a bias in the result. The quality of the trial state therefore is critical for accurate simulations. In this work, benchmark results of the ground state auxiliary-field quantum Monte Carlo method are reported for the Hubbard model on several geometries. We demonstrate that when multi-determinant generalized Hartree-Fock states are used as trial wave functions, the systematic errors can be systematically reduced to a low level and the results compare favorably with the exact diagonalization data.
Spurred by recent technological advances, there is a growing demand for computational methods that can accurately predict the dynamics of correlated electrons. Such methods can provide much-needed theoretical insights into the electron dynamics probed via time-resolved spectroscopy experiments and observed in non-equilibrium ultracold atom experiments. In this article, we develop and benchmark a numerically exact Auxiliary Field Quantum Monte Carlo (AFQMC) method for modeling the dynamics of correlated electrons in real time. AFQMC has become a powerful method for predicting the ground state and finite temperature properties of strongly correlated systems mostly by employing constraints to control the sign problem. Our initial goal in this work is to determine how well AFQMC generalizes to real-time electron dynamics problems without constraints. By modeling the repulsive Hubbard model on different lattices and with differing initial electronic configurations, we show that real-time AFQMC is capable of accurately capturing long-lived electronic coherences beyond the reach of mean field techniques. While the times to which we can meaningfully model decrease with increasing correlation strength and system size as a result of the exponential growth of the dynamical phase problem, we show that our technique can model the short-time behavior of strongly correlated systems to very high accuracy. Crucially, we find that importance sampling, combined with a novel adaptive active space sampling technique, can substantially lengthen the times to which we can simulate. These results establish real-time AFQMC as a viable technique for modeling the dynamics of correlated electron systems and serve as a basis for future sampling advances that will further mitigate the dynamical phase problem.
Determinant quantum Monte Carlo is a method for studying magnetic, transport and thermodynamic properties of interacting fermions on a lattice. It is widely used to explore the physics of strongly correlated quantum systems, from cuprate superconductors to ultracold atoms trapped on optical lattices. This paper contains a description of recent algorithmic advances in the determinant quantum Monte Carlo technique. Focus will be on algorithms developed for hybrid multicore processor and GPU platforms. The resulting speed-up of the simulations will be quantified. Simulations’ results will also be presented, with an emphasis on physical quantities that can now be computed for large numbers of sites.
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The auxiliary-field quantum Monte Carlo (AFQMC) method is a general numerical method for correlated many-electron systems, which is being increasingly applied in lattice models, atoms, molecules, and solids. Here, we introduce the theory and algorithm of the method specialized for real materials and present several recent developments. We give a systematic exposition of the key steps of AFQMC, closely tracking the framework of a modern software library we are developing. The building of a Monte Carlo Hamiltonian, projecting to the ground state, sampling two-body operators, phaseless approximation, and measuring ground state properties are discussed in detail. An advanced implementation for multi-determinant trial wave functions is described, which dramatically speeds up the algorithm and reduces the memory cost. We propose a self-consistent constraint for real materials, and discuss two flavors for its realization, either by coupling the AFQMC calculation to an effective independent-electron calculation or via the natural orbitals of the computed one-body density matrix.