The Schmidt decomposition of quantum many-body states, which is the basis of many powerful quantum many-body techniques, relies on a partition of the whole system in terms of orthogonal local basis functions. We propose in this Letter a nonorthogonal decomposition of Slater determinants, which reduces to the conventional Schmidt decomposition in the orthogonal limit. The new decomposition provides a natural tool for building local correlation spaces when the orbitals are overlapping, which can be used to extend some existing embedding methods based on the Schmidt decomposition like density matrix embedding theory (DMET) to overlapping subsystem partitions to achieve greater flexibility and efficiency. We also propose a new quantum embedding strategy that bridges the ab initio model potential (AIMP) embedding and DMET, which circumvents the necessarity of a mean-field calculation of the whole system like AIMP, and is able to capture quantum entanglement like DMET.
The Schmidt decomposition of quantum many-body states, which is the basis of many powerful quantum many-body techniques, relies on a partition of the whole system in terms of orthogonal local basis functions. We propose in this Letter a nonorthogonal decomposition of Slater determinants, which reduces to the conventional Schmidt decomposition in the orthogonal limit. The new decomposition provides a natural tool for building local correlation spaces when the orbitals are overlapping, which can be used to extend some existing embedding methods based on the Schmidt decomposition like density matrix embedding theory (DMET) to overlapping subsystem partitions to achieve greater flexibility and efficiency. We also propose a new quantum embedding strategy that bridges the ab initio model potential (AIMP) embedding and DMET, which circumvents the necessarity of a mean-field calculation of the whole system like AIMP, and is able to capture quantum entanglement like DMET.
The Schmidt decomposition of quantum many-body states, which is the basis of many powerful quantum many-body techniques, relies on a partition of the whole system in terms of orthogonal local basis functions. By carefully investigating the spectrum of the truncated density matrices in a non-orthogonal basis, we propose in this work a non-orthogonal decomposition of Slater determinants, which reduces to the conventional Schmidt decomposition in the orthogonal limit. The new decomposition provides a natural tool for building local correlation spaces when the orbitals are overlapping, and we are able to extend some existing embedding methods based on the Schmidt decomposition like density matrix embedding theory (DMET) to overlapping subsystem partitions to achieve greater flexibility and efficiency. We also propose a new quantum embedding strategy as another utility of the non-orthogonal decomposition that bridges the \textit{ab initio} model potential (AIMP) theory and DMET, which does not require a mean-field calculation of the whole system like AIMP, and is able to capture quantum entanglement like DMET.
We describe an efficient quantum embedding framework for realistic ab initio density matrix embedding theory (DMET) calculations in solids. We discuss in detail the choice of orbitals and mapping to a lattice, treatment of the virtual space and bath truncation, and the lattice-to-embedded integral transformation. We apply DMET in this ab initio framework to a hexagonal boron nitride monolayer, crystalline silicon, and nickel monoxide in the antiferromagnetic phase, using large embedded clusters with up to 300 embedding orbitals. We demonstrate our formulation of ab initio DMET in the computation of ground-state properties such as the total energy, equation of state, magnetic moment, and correlation functions.
Quantum embedding methods enable the study of large, strongly correlated quantum systems by (usually self-consistent) decomposition into computationally manageable subproblems, in the spirit of divide-and-conquer methods. Among these, Density Matrix Embedding Theory (DMET) is an efficient approach that enforces self-consistency at the level of one-particle reduced density matrices (1-RDMs), facilitating applications across diverse quantum systems. However, conventional DMET is constrained by the requirement that the global 1-RDM (low-level descriptor) be an orthogonal projector, limiting flexibility in bath construction and potentially impeding accuracy in strongly correlated regimes. In this work, we introduce a generalized DMET framework in which the low-level descriptor can be an arbitrary 1-RDM and the bath construction is based on optimizing a quantitative criterion related to the maximal disentanglement between different fragments. This yields an alternative yet controllable bath space construction for generic 1-RDMs, lifting a key limitation of conventional DMET. We demonstrate its consistency with conventional DMET in appropriate limits and exploring its implications for bath construction, downfolding (impurity Hamiltonian construction), low-level solvers, and adaptive fragmentation. We expect that this more flexible framework, which leads to several new variants of DMET, can improve the robustness and accuracy of DMET.
Matrix elements between nonorthogonal Slater determinants represent an essential component of many emerging electronic structure methods. However, evaluating nonorthogonal matrix elements is conceptually and computationally harder than their orthogonal counterparts. While several different approaches have been developed, these are predominantly derived from the first-quantized generalized Slater-Condon rules and usually require biorthogonal occupied orbitals to be computed for each matrix element. For coupling terms between nonorthogonal excited configurations, a second-quantized approach such as the nonorthogonal Wick's theorem is more desirable, but this fails when the two reference determinants have a zero many-body overlap. In this contribution, we derive an entirely generalized extension to the nonorthogonal Wick's theorem that is applicable to all pairs of determinants with nonorthogonal orbitals. Our approach creates a universal methodology for evaluating any nonorthogonal matrix element and allows Wick's theorem and the generalized Slater-Condon rules to be unified for the first time. Furthermore, we present a simple well-defined protocol for deriving arbitrary coupling terms between nonorthogonal excited configurations. In the case of overlap and one-body operators, this protocol recovers efficient formulas with reduced scaling, promising significant computational acceleration for methods that rely on such terms.
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.
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.