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.
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.
The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.
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.
We introduce a method that combines the power of both the lattice Green' function Monte Carlo (LGFMC) with the auxiliary field quantum Monte Carlo (AFQMC) techniques, and allows us to compute exact ground-state properties of the Hubbard model for $U\ensuremath{\lesssim}4t$ on finite clusters. Thanks to LGFMC, one obtains unbiased zero temperature results, not affected by the so-called Trotter approximation of the imaginary time propagator ${e}^{\ensuremath{-}H\ensuremath{\tau}}$. At the same time, the AFQMC formalism yields a remarkably fast convergence in $\ensuremath{\tau}$ before the fermion sign problem becomes prohibitive. As an application we report ground-state energies of the Hubbard model at U/t=4 with up to 100 sites.
to develop quantum Monte Carlo (QMC) approaches for excited states and for more systematic and accurate treatment of ground states of strongly correlated sytems and heavier elements. We use the auxiliary-field quantum Monte Carlo (AFQMC) framework, which is a many-body total energy method that looks like many coupled mean-field calculations. I will discuss this method’s relations and dierences with traditional density-functional and quantum chemistry methods, and with other forms of QMC. The AFQMC method uses importance-sampled random walks in the space of nonorthogonal Slater determinants to project out the many-body ground state, or excited state. The random walks are constrained by a trial wave function (TWF) to remove the so-called sign problem, giving approximate (but non-perturbative) results. With singledeterminant TWFs from Hartree-Fock or density-functional theory (DFT), this approach can be thought of as a post-DFT method with no parameter. In this form, the accuracy is typically comparable to the quantum chemistry coupled-cluster CCSD(T) method near equilibrium geometries, and better than CCSD(T) when bonds are stretched. The computational cost of AFQMC scales as N 3 with characteristic size of the system. Its accuracy can be further improved at the cost of increased computing (releasing the constraint) or better TWF. With AFQMC, we can systematically target selected orbitals to correlate. This allows many-body calculations in solids with a frozen core, with down-folded Hamiltonians, or possibly embeded in a DFT calculations to extend scales. Results from a few on-going applications will be presented, including band structures in simple solids, magnetic phases in MnO, and Cobalt adsoption on graphene.
We present a quantum Monte Carlo (QMC) technique for calculating the exact finite-temperature properties of Bose-Fermi mixtures. The Bose-Fermi auxiliary-field quantum Monte Carlo (BFAFQMC) algorithm combines two methods, a finite-temperature AFQMC algorithm for bosons and a variant of the standard AFQMC algorithm for fermions, into one algorithm for mixtures. We demonstrate the accuracy of our method by comparing its results for the Bose-Hubbard and Bose-Fermi-Hubbard models against those produced using exact diagonalization for small systems. Comparisons are also made with mean-field theory and the worm algorithm for larger systems. As is the case with most fermion Hamiltonians, a sign or phase problem is present in the BFAFQMC algorithm. We discuss the nature of these problems in this framework and describe how they can be controlled with well-studied approximations to expand the BFAFQMC algorithm's reach. This algorithm can serve as an essential tool for answering many unresolved questions about many-body physics in mixed Bose-Fermi systems.
Ultracold-atom simulations of the Hubbard model provide insights into the character of charge and spin correlations in and out of equilibrium. The corresponding numerical simulations, on the other hand, remain a significant challenge. We build on recent progress in the quantum Monte Carlo (QMC) simulation of electrons in continuous space and apply similar ideas to the square-lattice Hubbard model. We devise and benchmark two discrete-time QMC methods, namely the fermionic-propagator QMC (FPQMC) and the alternating-basis QMC (ABQMC). In FPQMC, the time evolution is represented by snapshots in real space, whereas the snapshots in ABQMC alternate between real and reciprocal space. The methods may be applied to study equilibrium properties within the grand-canonical or canonical ensemble, external field quenches, and even the evolution of pure states. Various real-space/reciprocal-space correlation functions are also within their reach. Both methods deal with matrices of size equal to the number of particles (thus independent of the number of orbitals or time slices), which allows for cheap updates. We benchmark the methods in relevant setups. In equilibrium, the FPQMC method is found to have an excellent average sign and, in some cases, yields correct results even with poor imaginary-time discretization. ABQMC has a significantly worse average sign, but also produces good results. Out of equilibrium, FPQMC suffers from a strong dynamical sign problem. On the contrary, in ABQMC, the sign problem is not time-dependent. Using ABQMC, we compute survival probabilities for several experimentally relevant pure states.