Rigorous wave function embedding with dynamical fluctuations

The dynamical fluctuations in approaches such as dynamical mean-field theory (DMFT) allow for the self-consistent optimization of a local fragment, hybridized with a true correlated environment. We show that these correlated environmental fluctuations can instead be efficiently captured in a wave function perspective in a computationally cheap, frequency-independent, zero-temperature approach. This allows for a systematically improvable, short-time wave function analogue to DMFT, which entails a number of computational and numerical benefits. We demonstrate this approach to solve the correlated dynamics of the paradigmatic Bethe lattice Hubbard model, as well as detailing cluster extensions in the one-dimensional Hubbard chain where we clearly show the benefits of this rapidly convergent description of correlated environmental fluctuations

Spectral functions of strongly correlated extended systems via an exact quantum embedding

Density matrix embedding theory (DMET) [Phys. Rev. Lett., 109, 186404 (2012)], introduced a new approach to quantum cluster embedding methods, whereby the mapping of strongly correlated bulk problems to an impurity with finite set of bath states was rigorously formulated to exactly reproduce the entanglement of the ground state. The formalism provided similar physics to dynamical mean-field theory at a tiny fraction of the cost, but was inherently limited by the construction of a bath designed to reproduce ground state, static properties. Here, we generalize the concept of quantum embedding to dynamic properties and demonstrate accurate bulk spectral functions at similarly small computational cost. The proposed spectral DMET utilizes the Schmidt decomposition of a response vector, mapping the bulk dynamic correlation functions to that of a quantum impurity cluster coupled to a set of frequency dependent bath states. The resultant spectral functions are obtained on the real-frequency axis, without bath discretization error, and allows for the construction of arbitrary dynamic correlation functions. We demonstrate the method on the 1D and 2D Hubbard model, where we obtain zero temperature, thermodynamic limit spectral functions, and show the trivial extension to two-particle Green functions. This advance therefore extends the scope and applicability of DMET in condensed matter problems as a computationally tractable route to correlated spectral functions of extended systems, and provides a competitive alternative to dynamical mean-field theory for dynamic quantities.Comment: 6 pages, 6 figure

Non-linear biases, stochastically-sampled effective Hamiltonians and spectral functions in quantum Monte Carlo methods

In this article we study examples of systematic biases that can occur in quantum Monte Carlo methods due to the accumulation of non-linear expectation values, and approaches by which these errors can be corrected. We begin with a study of the Krylov-projected FCIQMC (KP-FCIQMC) approach, which was recently introduced to allow efficient, stochastic calculation of dynamical properties. This requires the solution of a sampled effective Hamiltonian, resulting in a non-linear operation on these stochastic variables. We investigate the probability distribution of this eigenvalue problem to study both stochastic errors and systematic biases in the approach, and demonstrate that such errors can be significantly corrected by moving to a more appropriate basis. This is lastly expanded to include consideration of the correlation function QMC approach of Ceperley and Bernu, showing how such an approach can be taken in the FCIQMC framework.Comment: 12 pages, 7 figure

Systematic improvability in quantum embedding for real materials

Quantum embedding methods have become a powerful tool to overcome deficiencies of traditional quantum modelling in materials science. However, while these are systematically improvable in principle, in practice it is rarely possible to achieve rigorous convergence and often necessary to employ empirical parameters. Here, we formulate a quantum embedding theory, building on the methods of density-matrix embedding theory combined with local correlation approaches from quantum chemistry, to ensure the ability to systematically converge properties of real materials with accurate correlated wave~function methods, controlled by a single, rapidly convergent parameter. By expanding supercell size, basis set, and the resolution of the fluctuation space of an embedded fragment, we show that the systematic improvability of the approach yields accurate structural and electronic properties of realistic solids without empirical parameters, even across changes in geometry. Results are presented in insulating, semi-metallic, and more strongly correlated regimes, finding state of the art agreement to experimental data

A framework for efficient ab initio electronic structure with Gaussian Process States

We present a general framework for the efficient simulation of realistic fermionic systems with modern machine learning inspired representations of quantum many-body states, towards a universal tool for ab initio electronic structure. These machine learning inspired ansatzes have recently come to the fore in both a (first quantized) continuum and discrete Fock space representations, where however the inherent scaling of the latter approach for realistic interactions has so far limited practical applications. With application to the 'Gaussian Process State', a recently introduced ansatz inspired by systematically improvable kernel models in machine learning, we discuss different choices to define the representation of the computational Fock space. We show how local representations are particularly suited for stochastic sampling of expectation values, while also indicating a route to overcome the discrepancy in the scaling compared to continuum formulated models. We are able to show competitive accuracy for systems with up to 64 electrons, including a simplified (yet fully ab initio) model of the Mott transition in three-dimensional hydrogen, indicating a significant improvement over similar approaches, even for moderate numbers of configurational samples.Comment: 15 pages, 5 figure

Explicitly correlated plane waves: Accelerating convergence in periodic wavefunction expansions

We present an investigation into the use of an explicitly correlated plane wave basis for periodic wavefunction expansions at the level of second-order M{\o}ller-Plesset perturbation theory (MP2). The convergence of the electronic correlation energy with respect to the one-electron basis set is investigated and compared to conventional MP2 theory in a finite homogeneous electron gas model. In addition to the widely used Slater-type geminal correlation factor, we also derive and investigate a novel correlation factor that we term Yukawa-Coulomb. The Yukawa-Coulomb correlation factor is motivated by analytic results for two electrons in a box and allows for a further improved convergence of the correlation energies with respect to the employed basis set. We find the combination of the infinitely delocalized plane waves and local short-ranged geminals provides a complementary, and rapidly convergent basis for the description of periodic wavefunctions. We hope that this approach will expand the scope of discrete wavefunction expansions in periodic systems.Comment: 15 pages, 13 figure