arXiv:1608.07154 · Physical review. B./Physical review. B
Coupling quantum Monte Carlo and independent-particle calculations: Self-consistent constraint for the sign problem based on the density or the density matrix
Quantum Monte Carlo (QMC) methods are one of the most important tools for studying interacting quantum many-body systems. The vast majority of QMC calculations in interacting fermion systems require a constraint to control the sign problem. The constraint involves an input trial wave function which restricts the random walks. We introduce a systematically improvable constraint which relies on the fundamental role of the density or one-body density matrix. An independent-particle calculation is coupled to an auxiliary-field QMC calculation. The independent-particle solution is used as the constraint in QMC, which then produces the input density or density matrix for the next iteration. The constraint is optimized by the self-consistency between the many-body and the independent-particle calculations. The approach is demonstrated in the two-dimensional Hubbard model by accurately determining the ground state when collective modes separated by tiny energy scales are present in the magnetic and charge correlations. Our approach also provides an ab initio way to predict effective interaction parameters for independent-particle calculations.
Quantum Monte Carlo (QMC) methods are one of the most important tools for studying interacting quantum many-body systems. The vast majority of QMC calculations in interacting fermion systems require a constraint to control the sign problem. The constraint involves an input trial wave function which restricts the random walks. We introduce a systematically improvable constraint which relies on the fundamental role of the density or one-body density matrix. An independent-particle calculation is coupled to an auxiliary-field QMC calculation. The independent-particle solution is used as the constraint in QMC, which then produces the input density or density matrix for the next iteration. The constraint is optimized by the self-consistency between the many-body and the independent-particle calculations. The approach is demonstrated in the two-dimensional Hubbard model by accurately determining the ground state when collective modes separated by tiny energy scales are present in the magnetic and charge correlations. Our approach also provides an ab initio way to predict effective interaction parameters for independent-particle calculations.
$A\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ computations of finite-temperature properties are important in fermion systems such as correlated models, ultracold Fermi atomic gases, and optical lattices, and realistic materials. Here, the authors present a highly accurate approach that eliminates the sign problem in standard determinantal quantum Monte Carlo (DQMC) using a gauge constraint in auxiliary-field space. The constraint can be systematically improved by a self-consistent procedure coupling the many-body computation with an independent-particle calculation. A benchmark is performed in the Hubbard model across the entire temperature range.
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 shell model provides a powerful framework for nuclear structure calculations. The nucleons beyond an inert magic core are confined in a valence shell and interact through an effective two-body potential generally determined from the G-matrix method. However, the applicability of the shell model remains limited by the exponential growth of the many-body space with the number of valence nucleons and with the size of the one-body space. The quantum Monte Carlo (QMC) methods may thus be considered as a potentially attractive alternative to the direct diagonalization of the Hamiltonian matrix. The many-body problem is indeed reduced to a set of stochastic one-body problems describing independent particles in a fluctuating external field. Up to date, the QMC approaches of the shell model lead to the ground-state and the thermodynamic properties, but a detailed spectroscopy of nuclei is not reachable. Furthermore, they suffer from a common pathology of fermionic QMC methods, namely the sign/phase problem, which strongly contaminates the convergence of the calculation. In this context, we will present a new QMC method, providing the "yrast spectroscopy" of nuclei, and in which the sign/phase problem is managed through the constrained path approximation that is widely used in the ab initio nuclear calculations and in condensed matter physics.
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 describe and discuss a recently proposed quantum Monte Carlo algorithm to compute the ground-state properties of various systems of interacting fermions. In this method, the ground state is projected from an initial wave function by a branching random walk in an overcomplete basis of Slater determinants. By constraining the determinants according to a trial wave function |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉 is exact. We illustrate the method by describing in detail its implementation for the two-dimensional one-band Hubbard model. We show results for lattice sizes up to 16\ifmmode\times\else\texttimes\fi{}16 and for various electron fillings and interaction strengths. With simple single-determinant wave functions as |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, the method yields accurate (often to within a few percent) estimates of the ground-state energy as well as correlation functions, such as those for electron pairing. We conclude by discussing possible extensions of the algorithm.
We introduce an efficient approach to implement correlated many-body trial wave functions in auxiliary-field quantum Monte Carlo (AFQMC). To control the sign/phase problem in AFQMC, a constraint is derived from an exact gauge condition but is typically imposed approximately through a trial wave function or trial density matrix, whose quality can affect the accuracy of the method. Furthermore, the trial wave function can also affect the efficiency through importance sampling. The most natural form of the trial wave function has been single Slater determinants or their linear combinations. More sophisticated forms, for example, with the inclusion of a Jastrow factor or other explicit correlations, have been challenging to use, and their implementation is often assumed to require a quantum computer. In this work, we demonstrate that a large class of correlated wave functions, written in the general form of multi-dimensional integrals over hidden or auxiliary variables times Slater determinants, can be implemented as a trial wave function by coupling the random walkers to a generalized Metropolis sampling. We discuss the fidelity of AFQMC with stochastically sampled trial wave functions, which are relevant to both quantum and classical algorithms. We illustrate the method and show that an efficient implementation can be achieved, which preserves the computational scaling of AFQMC. We test our method in molecules under bond stretching and in transition metal diatomics. Significant improvements are seen in both accuracy and efficiency over typical trial wave functions, and the method yields total ground-state energies systematically within chemical accuracy. The method can be useful for incorporating other advanced wave functions, for example, neural quantum state wave functions optimized from machine learning techniques, or for other forms of fermion quantum Monte Carlo.