The auxiliary field quantum Monte Carlo (AFQMC) method has been a workhorse in the field of strongly correlated electrons for a long time and has found its most recent implementation in the ALF package (https://alf.physik.uni-wuerzburg.de). The utilization of the Trotter decomposition to decouple the interaction from the non-interacting Hamiltonian makes this method inherently second order in terms of the imaginary time slice. We show that due to the use of the Hubbard-Stratonovich transformation (HST) a semigroup structure on the time evolution is imposed that necessitates the introduction of a new family of complex-hermitian splitting methods for the purpose of reaching higher order. We will give examples of these new methods and study their efficiency, as well as perform comparisons with other established second and higher order methods in the realm of the AFQMC method.
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 approximat…
We present a continuous-time Monte Carlo method for quantum impurity models, which combines a weak-coupling expansion with an auxiliary-field decomposition. The method is considerably more efficient than Hirsch-Fye and free of time discretization errors, and is particularly useful as impurity solver in large cluster dynamical mean field theory (DMFT) calculations.
We show how symmetry properties can be used to greatly increase the accuracy and efficiency in auxiliary-field quantum Monte Carlo (AFQMC) calculations of electronic systems. With the Hubbard model as an example, we study symmetry preservation in two aspects of ground-state AFQMC calculations, the Hubbard-Straonovich transformation and the form of the trial wave function. It is shown that significant improvement over state-of-the-art calculations can be achieved. In unconstrained calculations, the implementation of symmetry often leads to shorter convergence time and much smaller statistical e…
Auxiliary-field quantum Monte Carlo (AFQMC) has repeatedly demonstrated itself as one of the most accurate quantum many-body methods, capable of simulating both real and model systems. In this article we investigate the application of AFQMC to realistic strongly correlated materials in periodic Gaussian basis sets. Using nickel oxide (NiO) as an example, we investigate the importance of finite size effects and basis set errors on the structural properties of the correlated solid. We provide benchmark calculations for NiO and compare our results to both experiment measurements and existing theo…
The phaseless Auxiliary Field Quantum Monte Carlo method provides a well established approximation scheme for accurate calculations of ground state energies of many-fermions systems. Here we apply the method to the calculation of imaginary time correlation functions. We give a detailed description of the technique and we test the quality of the results for static and dynamic properties against exact values for small systems.
Auxiliary-field quantum Monte Carlo methods enable the calculation of thermal and ground state properties of correlated quantum many-body systems in model spaces that are many orders of magnitude larger than those that can be treated by conventional diagonalization methods. We review recent developments and applications of these methods in nuclei using the framework of the configuration-interaction shell model.
Quantum Monte Carlo (QMC) methods are some of the most accurate methods for simulating correlated electronic systems. We investigate the compatibility, strengths and weaknesses of two such methods, namely, diffusion Monte Carlo (DMC) and auxiliary-field quantum Monte Carlo (AFQMC). The multi-determinant trial wave functions employed in both approaches are generated using the configuration interaction using a perturbative selection made iteratively (CIPSI) technique. Complete basis set full configuration interaction (CBS-FCI) energies estimated with CIPSI are used as a reference in this compara…
Within the ground-state auxiliary-field quantum Monte Carlo technique, we introduce discrete Hubbard-Stratonovich transformations (HSTs) that are suitable also for spatially inhomogeneous trial functions. The discrete auxiliary fields introduced here are coupled to local spin or charge operators fluctuating around their Hartree-Fock values. The formalism can be considered as a generalization of the discrete HSTs by Hirsch [J. E. Hirsch, Phys. Rev. B 28, 4059 (1983)] or a compactification of the shifted-contour auxiliary-field Monte Carlo formalism by Rom et al. [N. Rom et al., Chem. Phys. Lett…
We describe the implementation of the frozen-orbital and downfolding approximations in the auxiliary-field quantum Monte Carlo (AFQMC) method. These approaches can provide significant computational savings compared to fully correlating all the electrons. While the many-body wave function is never explicit in AFQMC, its random walkers are Slater determinants, whose orbitals may be expressed in terms of any one-particle orbital basis. It is therefore straightforward to partition the full N-particle Hilbert space into active and inactive parts to implement the frozen-orbital method. In the frozen…
Quantum Monte Carlo simulations of fermions are hampered by the notorious sign problem whose most striking manifestation is an exponential growth of sampling errors with the number of particles. With the sign problem known to be an NP-hard problem and any generic solution thus highly elusive, the Monte Carlo sampling of interacting many-fermion systems is commonly thought to be restricted to a small class of model systems for which a sign-free basis has been identified. Here we demonstrate that entanglement measures, in particular the so-called Renyi entropies, can intrinsically exhibit a cert…
Auxiliary field quantum Monte Carlo methods for Hubbard models are generally based on a Hubbard-Stratonovitch transformation where the field couples to the z-component of the spin. This transformation breaks SU(2) spin invariance. The symmetry is restored only after summation over the auxiliary fields. Here, we analyze an alternative decomposition, which conserves SU(2) spin invariance, but requires the use of complex numbers. We show that this algorithm gets rid of the very large fluctuations observed in imaginary time displaced correlation functions of quantities which do not commute with th…
We outline how auxiliary-field quantum Monte Carlo (AFQMC) can leverage graphical processing units (GPUs) to accelerate the simulation of solid state sytems. By exploiting conservation of crystal momentum in the one- and two-electron integrals we show how to efficiently formulate the algorithm to best utilize current GPU architectures. We provide a detailed description of different optimization strategies and profile our implementation relative to standard approaches, demonstrating a factor of 40 speed up over a CPU implementation. With this increase in computational power we demonstrate the a…
The chromium dimer (Cr2) presents an outstanding challenge for many-body electronic structure methods. Its complicated nature of binding, with a formal sextuple bond and an unusual potential energy curve, is emblematic of the competing tendencies and delicate balance found in many strongly correlated materials. We present a near-exact calculation of the potential energy curve (PEC) and ground state properties of Cr2, using the auxiliary-field quantum Monte Carlo (AFQMC) method. Unconstrained, exact AFQMC calculations are first carried out for a medium-sized but realistic basis set. Elimination…
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) in the model under consideration. One of the authors and M. Imada [F.F. Assaad and M. Imada, J. Phys. Soc. Jpn. 65 189 (1996).] have proposed a numerically stable method to compute those quantities. 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 obs…
The auxiliary-field quantum Monte Carlo (AFQMC) method provides a computational framework for solving the time-independent Schroedinger equation in atoms, molecules, solids, and a variety of model systems by stochastic sampling. We introduce the theory and formalism behind this framework, briefly discuss the key technical steps that turn it into an effective and practical computational method, present several illustrative results, and conclude with comments on the prospects of ab initio computation by this framework.
We propose a new sampling method to calculate the ground state of interacting quantum systems. This method, which we call the adaptive sampling quantum monte carlo (ASQMC) method utilises information from the high temperature density matrix derived from the monte carlo steps. With the ASQMC method, the negative sign ratio is greatly reduced and it becomes zero in the limit $Δτ$ goes to zero even without imposing any constraint such like the constraint path (CP) condition. Comparisons with numerical results obtained by using other methods are made and we find the ASQMC method gives accurate res…
Numerically exact continuous-time Quantum Monte Carlo algorithm for finite fermionic systems with non-local interactions is proposed. The scheme is particularly applicable for general multi-band time-dependent correlations since it does not invoke Hubbard-Stratonovich transformation. The present determinantal grand-canonical method is based on a stochastic series expansion for the partition function in the interaction representation. The results for the Green function and for the time-dependent susceptibility of multi-orbital super-symmetric impurity model with a spin-flip interaction are pres…
We develop a new smoothing or extrapolating method, based on discrete Laguerre functions, for systematically analyzing the stochastic signal of shifted-contour auxiliary-field Monte Carlo. We study the statistical errors and extrapolation errors using full configuration-interaction energies for the doubly stretched water molecule. The only free parameter is the order N of the fit. We show that low N emphasizes stability while higher N enable improved extrapolation, at the cost of increased statistical errors. Typically, one should use low order for signals based on a small number of iterations…
We extend the recently introduced phaseless auxiliary-field quantum Monte Carlo (QMC) approach to any single-particle basis, and apply it to molecular systems with Gaussian basis sets. QMC methods in general scale favorably with system size, as a low power. A QMC approach with auxiliary fields in principle allows an exact solution of the Schrodinger equation in the chosen basis. However, the well-known sign/phase problem causes the statistical noise to increase exponentially. The phaseless method controls this problem by constraining the paths in the auxiliary-field path integrals with an appr…
We present an approach for ab initio many-body calculations of excited states in solids. Using auxiliary-field quantum Monte Carlo, we introduce an orthogonalization constraint to prevent collapse of the stochastic Slater determinants in the imaginary-time propagation. Trial wave functions from density-functional calculations are used for the constraints. Detailed band structures calculated for standard semiconductors are in good agreement with GW and experimental results. For the challenging ZnO wurtzite structure, we obtain a fundamental band gap of 3.26(16) eV, consistent with experiments.
Bond stretching mimics different levels of electron correlation and provides a challenging testbed for approximate many-body computational methods. Using the recently developed phaseless auxiliary-field quantum Monte Carlo (AF QMC) method, we examine bond stretching in the well-studied molecules BH and N$_2$, and in the H$_{50}$ chain. To control the sign/phase problem, the phaseless AF QMC method constrains the paths in the auxiliary-field path integrals with an approximate phase condition that depends on a trial wave function. With single Slater determinants from unrestricted Hartree-Fock (U…
We consider a generalization of the standard Metropolis algorithm acceptance/rejection decision rule and numerically explore its properties using auxiliary field quantum Monte Carlo. The generalization involves a free parameter which, given a criterion for proposing attempted moves, can be used to tune the average acceptance rate in a particular way. Such tuning can also potentially change Monte Carlo autocorrelation times, and the combination of the changing acceptance rate and autocorrelation times raises the possibility of more efficient simulations. We explore these issues using primarily …
Calculations of the binding energy of the transition metal oxide molecules TiO and MnO are presented, using a recently developed phaseless auxiliary field quantum Monte Carlo approach. This method maps the interacting many-body problem onto a linear combination of non-interacting problems by a complex Hubbard-Stratonovich transformation, and controls the phase/sign problem with a phaseless approximation relying on a trial wave function. It employs random walks in Slater determinant space to project the ground state of the system, and allows use of much of the same machinery as in standard dens…
A series of calculations for the first- and second-row post-d elements (Ga-Br and In-I) are presented using the phaseless auxiliary-field quantum Monte Carlo (AF QMC) method. This method is formulated in a Hilbert space defined by any chosen one-particle basis, and maps the many-body problem into a linear combination of independent-particle solutions with external auxiliary fields. The phase/sign problem is handled approximately by the phaseless formalism using a trial wave function, which in our calculations was chosen to be the Hartree-Fock solution. We used the consistent correlated basis s…
We introduce a method that combines the power of both the lattice Green function Monte Carlo (LGFMC) with the auxiliary field techniques (AFQMC), and allows us to compute exact ground state properties of the Hubbard model for U<~ 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 exp(- H t). On the other hand the AFQMC formalism yields a remarkably fast convergence in t before the fermion sign problem becomes prohibitive. As a first application we report ground state energies…
We address the computation of ground-state properties of chemical systems and realistic materials within the auxiliary-field quantum Monte Carlo method. The phase constraint to control the fermion phase problem requires the random walks in Slater determinant space to be open-ended with branching. This in turn makes it necessary to use back-propagation (BP) to compute averages and correlation functions of operators that do not commute with the Hamiltonian. Several BP schemes are investigated and their optimization with respect to the phaseless constraint is considered. We propose a modified BP …
Quantum Monte Carlo (QMC) algorithms have long relied on Jastrow factors to incorporate dynamic correlation into trial wave functions. While Jastrow-type wave functions have been widely employed in real-space algorithms, they have seen limited use in second-quantized QMC methods, particularly in projection methods that involve a stochastic evolution of the wave function in imaginary time. Here we propose a scheme for generating Jastrow-type correlated trial wave functions for auxiliary-field QMC methods. The method is based on decoupling the two-body Jastrow into one-body projectors coupled to…
We show that the recently developed phaseless auxiliary-field quantum Monte Carlo (AFQMC) method can be used to study excited states, providing an alternative to standard quantum chemistry methods. The phaseless AFQMC approach, whose computational cost scales as M^3-M^4 with system size M, has been shown to be among the most accurate many-body methods in ground state calculations. For excited states, prevention of collapse into the ground state and control of the Fermion sign/phase problem are accomplished by the approximate phaseless constraint with a trial wave function. Using the challengin…
Layered perovskite ruthenium oxides exhibit a striking series of metal-insulator and magnetic-nonmagnetic phase transitions easily tuned by temperature, pressure, epitaxy, and nonlinear drive. In this work, we combine results from two complementary state of the art many-body methods, Auxiliary Field Quantum Monte Carlo and Dynamical Mean Field Theory, to determine the low-temperature phase diagram of Ca$_2$RuO$_4$. Both methods predict a low temperature, pressure-driven metal-insulator transition accompanied by a ferromagnetic-antiferromagnetic transition. The properties of the ferromagnetic s…
We investigate the cluster size convergence of the energy and observables using two forms of density matrix embedding theory (DMET): the original cluster form (CDMET) and a new formulation motivated by the dynamical cluster approximation (DCA-DMET). Both methods are applied to the half-filled one- and two-dimensional Hubbard models using a sign-problem free auxiliary-field quantum Monte Carlo (AFQMC) impurity solver, which allows for the treatment of large impurity clusters of up to 100 sites. While CDMET is more accurate at smaller impurity cluster sizes, DCA- DMET exhibits faster asymptotic …
The Algorithms for Lattice Fermions package provides a general code for the finite temperature auxiliary field quantum Monte Carlo algorithm. The code is engineered to be able to simulate any model that can be written in terms of sums of single-body operators, of squares of single-body operators and single-body operators coupled to an Ising field with given dynamics. We provide predefined types that allow the user to specify the model, the Bravais lattice as well as equal time and time displaced observables. The code supports an MPI implementation. Examples such as the Hubbard model on the hon…
We describe an approach for many-body calculations with a finite-temperature, grand canonical ensemble formalism using auxiliary-field quantum Monte Carlo (AFQMC) with a self-consistent constraint to control the sign problem. The usual AFQMC formalism of Blankenbecler, Scalapino, and Sugar suffers from the sign problem with most physical Hamiltonians, as is well known. Building on earlier ideas to constrain the paths in auxiliary-field space [Phys. Rev. Lett. \textbf{83}, 2777 (1999)] and incorporating recent developments in zero-temperature, canonical-ensemble methods, we discuss how a self-c…
We propose a Monte Carlo method, which is a hybrid method of the quantum Monte Carlo method and variational Monte Carlo theory, to study the Hubbard model. The theory is based on the off-diagonal and the Gutzwiller type correlation factors which are taken into account by a Monte Carlo algorithm. In the 4x4 system our method is able to reproduce the exact results obtained by the diagonalization. An application is given to investigate the half-filled band case of two-dimensional square lattice. The energy is favorably compared with quantum Monte Carlo data.
Nuclear gradients from auxiliary-field quantum Monte Carlo and their application in geometry optimization and transition state search
University of Colorado, Boulder（科罗拉多大学博尔德分校）；Columbia University（哥伦比亚大学）；California Institute of Technology（加州理工学院）；Marcus Center for Theoretical Chemistry（马库斯理论化学中心）
We introduce a new numerical technique -- bosonic auxiliary-field Monte Carlo (bAFMC) -- which allows to calculate the thermal properties of large lattice-boson systems within a systematically improvable semiclassical approach, and which is virtually applicable to any bosonic model. Our method amounts to a decomposition of the lattice into clusters, and to an Ansatz for the density matrix of the system in the form of a cluster-separable state -- with non-entangled, yet classically correlated clusters. This approximation eliminates any sign problem, and can be systematically improved upon by us…
We have implemented recently developed multiple-projector pseudopotentials into the planewave based auxiliary-field quantum Monte Carlo (pw-AFQMC) method. Multiple-projector pseudopotentials can yield smaller planewave cut-offs while maintaining or improving transferability. This reduces the computational cost of pw-AFQMC, increasing its reach to larger and more complicated systems. We discuss the use of non-local pseudopotentials in the separable Kleinman-Bylander form, and the implementation in pw-AFQMC of the multiple-projector optimized norm-conserving pseudopotential ONCVPSP of Hamann. Th…
In the absence of a fermion sign problem, auxiliary field (or determinantal) quantum Monte Carlo (DQMC) approaches have long been the numerical method of choice for unbiased, large-scale simulations of interacting many-fermion systems. More recently, the conceptual scope of this approach has been expanded by introducing ingenious schemes to compute entanglement entropies within its framework. On a practical level, these approaches however suffer from a variety of numerical instabilities that have largely impeded their applicability. Here we report on a number of algorithmic advances to overcom…
We investigate the viability of the phaseless finite temperature auxiliary field quantum Monte Carlo (ph-FT-AFQMC) method for ab initio systems using the uniform electron gas as a model. Through comparisons with exact results and finite temperature coupled cluster theory, we find that ph-FT-AFQMC is sufficiently accurate at high to intermediate electronic densities. We show both analytically and numerically that the phaseless constraint at finite temperature is fundamentally different from its zero temperature counterpart (i.e., ph-ZT-AFQMC) and generally one should not expect ph-FT-AFQMC to a…
We present efficient algorithms for using selected configuration interaction (sCI) trial wave functions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC). These advancements, geared towards optimizing computational performance for longer CI expansions, allow us to use up to a million configurations feasibly in the trial state for ph-AFQMC. In one example, we found the cost of ph-AFQMC per sample to increase only by a factor of about $3$ for a calculation with $10^4$ configurations compared to that with a single one, demonstrating the tiny computational overhead due to a longer expans…
We present an extension of constrained-path auxiliary-field quantum Monte Carlo (CP-AFQMC) for the treatment of correlated electronic systems coupled to phonons. The algorithm follows the standard CP-AFQMC approach for description of the electronic degrees of freedom while phonons are described in first quantization and propagated via a diffusion Monte Carlo approach. Our method is tested on the one- and two-dimensional Holstein and Hubbard-Holstein models. With a simple semiclassical trial wavefunction, our approach is remarkably accurate for $ω/(2\text{d}tλ) < 1$ for all parameters in the Ho…
We present an \textit{ab initio} auxiliary field quantum Monte Carlo method for studying the electronic structure of molecules, solids, and model Hamiltonians at finite temperature. The algorithm marries the \textit{ab initio} phaseless auxiliary field quantum Monte Carlo algorithm known to produce high accuracy ground state energies of molecules and solids with its finite temperature variant, long used by condensed matter physicists for studying model Hamiltonian phase diagrams, to yield a phaseless, \textit{ab initio} finite temperature method. We demonstrate that the method produces interna…
Gauge Auxiliary-Field Quantum Monte Carlo Method for Many-Fermion Systems
Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University（中山大学中法核工程与技术学院）；Department of Physics, Graduate School of Science, The University of Tokyo（东京大学理学研究科物理学系）
Finite temperature auxiliary field-based Quantum Monte Carlo methods, including Determinant Quantum Monte Carlo (DQMC) and Auxiliary Field Quantum Monte Carlo (AFQMC), have historically assumed pivotal roles in the investigation of the finite temperature phase diagrams of a wide variety of multidimensional lattice models and materials. Despite their utility, however, these techniques are typically formulated in the grand canonical ensemble, which makes them difficult to apply to condensates like superfluids and difficult to benchmark against alternative methods that are formulated in the canon…
We discuss an alternative accurate Monte Carlo method to calculate the ground-state energy and related quantities for Laughlin states of the fractional quantum Hall effect in a disk geometry. This alternative approach allows us to obtain accurate bulk regime (thermodynamic limit) values for various quantities from Monte Carlo simulations with a small number of particles (much smaller than that needed with standard Monte Carlo approaches).
