We present a multiscale approach to efficiently embed an ab initio correlated chemical fragment described by its energy-weighted density matrices and entangled with a wider mean-field many-electron system. This approach, first presented by Fertitta and Booth [Phys. Rev. B 98, 235132 (2018)], is here extended to account for realistic long-range interactions and broken symmetry states. The scheme allows for a systematically improvable description in the range of correlated fluctuations out of the fragment into the system, via a self-consistent optimization of a coupled auxiliary mean-field system. It is discussed that the method has rigorous limits equivalent to the existing quantum embedding approaches of both dynamical mean-field theory and density matrix embedding theory, to which this method is compared, and the importance of these correlated fluctuations is demonstrated. We derive a self-consistent local energy functional within the scheme and demonstrate the approach for hydrogen rings, where quantitative accuracy is achieved despite only a single atom being explicitly treated.
We present a multiscale approach to efficiently embed an ab initio correlated chemical fragment described by its energy-weighted density matrices and entangled with a wider mean-field many-electron system. This approach, first presented by Fertitta and Booth [Phys. Rev. B 98, 235132 (2018)], is here extended to account for realistic long-range interactions and broken symmetry states. The scheme allows for a systematically improvable description in the range of correlated fluctuations out of the fragment into the system, via a self-consistent optimization of a coupled auxiliary mean-field system. It is discussed that the method has rigorous limits equivalent to the existing quantum embedding approaches of both dynamical mean-field theory and density matrix embedding theory, to which this method is compared, and the importance of these correlated fluctuations is demonstrated. We derive a self-consistent local energy functional within the scheme and demonstrate the approach for hydrogen rings, where quantitative accuracy is achieved despite only a single atom being explicitly treated.
Self-consistent quantum embedding forms a route to bridge local descriptions of electronic structure methods and the electronic structure of very large systems and condensed phases. Existing examples of quantum embedding methods include the density functional embedding of Cortona, Wesolowski and Warshel, based on the single-particle density and Carter, and the dynamical mean-field theory, based on the single-particle Greens function. In this talk, however, I will describe a new formalism for quantum embedding that is based on the single-particle density matrix, and which is based on a self-consistent approximation to the exact entanglement embedding of a subsystem in an environment. Unlike density functional embedding, this formalism can practically describe strongly correlated systems, and unlike dynamical mean-field theory, it is easily combined with traditional ground-state electronic structure methods, such as quantum chemical wavefunction methods and projector and variational Monte Carlo approaches. I will demonstrate the performance of the method on the phase diagram of the 1D and 2D Hubbard models and discuss applications to ab-initio systems.
We extend our density matrix embedding theory (DMET) [Phys. Rev. Lett.2012, 109, 186404] from lattice models to the full chemical Hamiltonian. DMET allows the many-body embedding of arbitrary fragments of a quantum system, even when such fragments are open systems and strongly coupled to their environment (e.g., by covalent bonds). In DMET, empirical approaches to strong coupling, such as link atoms or boundary regions, are replaced by a small, rigorous quantum bath designed to reproduce the entanglement between a fragment and its environment. We describe the theory and demonstrate its feasibility in strongly correlated hydrogen ring and grid models; these are not only beyond the scope of traditional embeddings but even challenge conventional quantum chemistry methods themselves. We find that DMET correctly describes the notoriously difficult symmetric dissociation of a 4 × 3 hydrogen atom grid, even when the treated fragments are as small as single hydrogen atoms. We expect that DMET will open up new ways of treating complex strongly coupled, strongly correlated systems in terms of their individual fragments.
Quantum embedding approaches involve the self-consistent optimization of a local fragment of a strongly correlated system, entangled with the wider environment. The `energy-weighted' density matrix embedding theory (EwDMET) was established recently as a way to systematically control the resolution of the fragment-environment coupling and allow for true quantum fluctuations over this boundary to be self-consistently optimized within a fully static framework. In this work, we reformulate the algorithm to ensure that EwDMET can be considered equivalent to an optimal and rigorous truncation of the self-consistent dynamics of dynamical mean-field theory (DMFT). A practical limitation of these quantum embedding approaches is often a numerical fitting of a self-consistent object defining the quantum effects. However, we show here that in this formulation, all numerical fitting steps can be entirely circumvented, via an effective Dyson equation in the space of truncated dynamics. This provides a robust and analytic self-consistency for the method, and an ability to systematically and rigorously converge to DMFT from a static, wave function perspective. We demonstrate that this improved approach can solve the correlated dynamics and phase transitions of the Bethe lattice Hubbard model in infinite dimensions, as well as one- and two-dimensional Hubbard models where we clearly show the benefits of this rapidly convergent basis for correlation-driven fluctuations. This systematically truncated description of the effective dynamics of the problem also allows access to quantities such as Fermi liquid parameters and renormalized dynamics, and demonstrates a numerically efficient, systematic convergence to the zero-temperature dynamical mean-field theory limit.
Quantum embedding is an appealing route to fragment a large interacting quantum system into several smaller auxiliary ‘cluster’ problems to exploit the locality of the correlated physics. In this work we critically review approaches to recombine these fragmented solutions in order to compute non-local expectation values, including the total energy. Starting from the democratic partitioning of expectation values used in density matrix embedding theory, we motivate and develop a number of alternative approaches, numerically demonstrating their efficiency and improved accuracy as a function of increasing cluster size for both energetics and non-local two-body observables in molecular and solid state systems. These approaches consider the N-representability of the resulting expectation values via an implicit global wave function across the clusters, as well as the importance of including contributions to expectation values spanning multiple fragments simultaneously, thereby alleviating the fundamental locality approximation of the embedding. We clearly demonstrate the value of these introduced functionals for reliable extraction of observables and robust and systematic convergence as the cluster size increases, allowing for significantly smaller clusters to be used for a desired accuracy compared to traditional approaches in ab initio wave function quantum embedding.
In recent studies by Yalouz et al. [J. Chem. Phys. 157, 214112 (2022)] and Sekaran et al. [Phys. Rev. B 104, 035121 (2021) and Computation 10, 45 (2022)], density matrix embedding theory (DMET) has been reformulated through the use of the Householder transformation as a novel tool to embed a fragment within extended systems. The transformation was applied to a reference non-interacting one-electron reduced density matrix to construct fragments' bath orbitals, which are crucial for subsequent ground state calculations. In the present work, we expand upon these previous developments and extend the utilization of the Householder transformation to the description of multiple electronic states, including ground and excited states. Based on an ensemble noninteracting density matrix, we demonstrate the feasibility of achieving exact fragment embedding through successive Householder transformations, resulting in a larger set of bath orbitals. We analytically prove that the number of additional bath orbitals scales directly with the number of fractionally occupied natural orbitals in the reference ensemble density matrix. A connection with the regular DMET bath construction is also made. Then, we illustrate the use of this ensemble embedding tool in single-shot DMET calculations to describe both ground and first excited states in a Hubbard lattice model and an ab initio hydrogen system. Finally, we discuss avenues for enhancing ensemble embedding through self-consistency and explore potential future directions.
Quantum embedding is an appealing route to fragment a large interacting quantum system into several smaller auxiliary `cluster' problems to exploit the locality of the correlated physics. In this work we critically review approaches to recombine these fragmented solutions in order to compute non-local expectation values, including the total energy. Starting from the democratic partitioning of expectation values used in density matrix embedding theory, we motivate and develop a number of alternative approaches, numerically demonstrating their efficiency and improved accuracy as a function of increasing cluster size for both energetics and non-local two-body observables in molecular and solid state systems. These approaches consider the N-representability of the resulting expectation values via an implicit global wave~function across the clusters, as well as the importance of including contributions to expectation values spanning multiple fragments simultaneously, thereby alleviating the fundamental locality approximation of the embedding. We clearly demonstrate the value of these introduced functionals for reliable extraction of observables and robust and systematic convergence as the cluster size increases, allowing for significantly smaller clusters to be used for a desired accuracy compared to traditional approaches in ab initio wave~function quantum embedding.