arXiv:1608.03316 · Physical review. B./Physical review. B
Cluster size convergence of the density matrix embedding theory and its dynamical cluster formulation: A study with an auxiliary-field quantum Monte Carlo solver
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 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 convergence towards the thermodynamic limit. We use our two formulations to produce new accurate estimates for the energy and local moment of the two-dimensional Hubbard model for $U/t=2,4,6$. These results compare favorably with the best data available in the literature, and help resolve earlier uncertainties in the moment for $U/t=2$.
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 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 convergence towards the thermodynamic limit. We use our two formulations to produce new accurate estimates for the energy and local moment of the two-dimensional Hubbard model for U/t=2,4,6. These results compare favorably with the best data available in the literature, and help resolve earlier uncertainties in the moment for U/t=2.
Two of the primary sources of error in the Cluster dynamical mean field theory (CDMFT) technique arise from the use of finite size clusters and finite size baths, which makes the development of impurity solvers that can treat larger systems an essential goal. In this work we introduce an impurity solver based on the recently developed dynamical variational Monte Carlo (dVMC) method. Variational Monte Carlo possesses a favorable scaling as a function of system size, which enables the treatment of systems beyond the reach of current exact diagonalization solvers. To benchmark the technique, we perform a systematic set of CDMFT calculations on the single-band one-dimensional Hubbard model. We compare to results obtained with an exact diagonalization solver for small clusters, and against the exact solution in the thermodynamic limit obtained by Lieb and Wu for larger clusters. The development of improved impurity solvers will help extend the reach of quantum cluster methods, which can be applied to a wide range of strongly correlated electron systems, promising new insights into their emergent behavior.
We introduce density matrix embedding theory (DMET), a quantum embedding theory for computing frequency-independent quantities, such as ground-state properties, of infinite systems. Like dynamical mean-field theory, DMET maps the bulk interacting system to a simpler impurity model and is exact in the noninteracting and atomic limits. Unlike dynamical mean-field theory, DMET is formulated in terms of the frequency-independent local density matrix, rather than the local Green's function. In addition, it features a finite, algebraically constructible bath of only one bath site per impurity site, with no bath discretization error. Frequency independence and the minimal bath make DMET a computationally simple and efficient method. We test the theory in the one-dimensional and two-dimensional Hubbard models at and away from half filling, and we find that compared to benchmark data, total energies, correlation functions, and metal-insulator transitions are well reproduced, at a tiny computational cost.
The dynamical cluster approximation (DCA) is a systematic extension beyond the single-site approximation in dynamical mean field theory, to include spatially nonlocal correlations in quantum many-body simulations of strongly correlated systems. We extend the DCA with a continuous lattice self-energy in order to achieve better convergence with cluster size. This method, which we call DCA+, cures the cluster-shape dependence problems of the DCA, without suffering from causality violations of previous attempts to interpolate the cluster self-energy. A practical approach based on standard inference techniques is given to deduce the continuous lattice self-energy from an interpolated cluster self-energy. We study the pseudogap region of a hole-doped two-dimensional Hubbard model and find that, in the DCA+algorithm, the self-energy and pseudogap temperature T∗ converge monotonously with cluster size. Introduction of a continuous lattice self-energy eliminates artificial long-range correlations and thus significantly reduces the sign problem of the quantum Monte Carlo cluster solver in the DCA+algorithm compared to the normal DCA. Simulations with much larger cluster sizes thus become feasible, which, along with the improved convergence in cluster size, raises hope that precise extrapolations to the exact infinite cluster size limit can be reached for other physical quantities as well.
We describe the use of coupled-cluster theory as an impurity solver in dynamical mean-field theory (DMFT) and its cluster extensions. We present numerical results at the level of coupled-cluster theory with single and double excitations (CCSD) for the density of states and self-energies of cluster impurity problems in the one- and two-dimensional Hubbard models. Comparison to exact diagonalization shows that CCSD produces accurate density of states and self-energies at a variety of values of U/t and filling fractions. However, the low cost allows for the use of many bath sites, which we define by a discretization of the hybridization directly on the real frequency axis. We observe convergence of dynamical quantities using approximately 30 bath sites per impurity site, with our largest 4-site cluster DMFT calculation using 120 bath sites. We suggest that coupled-cluster impurity solvers will be attractive in ab initio formulations of dynamical mean-field theory.
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.