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arXiv:2306.07641 · Physical review. B./Physical review. B

Unitary transformations within density matrix embedding approaches: A perspective on the self-consistent scheme for electronic structure calculation

Quentin Marécat, Benjamin Lasorne, Emmanuel Fromager, Matthieu Saubanère

Published Oct 12, 2023• 5 citations
arXivHTMLPDF

Abstract

In this work, we introduce a self-consistent scheme based on the one-body reduced density matrix ($\ensuremath{\gamma}$) formalism. A significant feature of this methodology is the utilization of an optimal unitary transformation of the Hamiltonian, determined through a self-consistently determined, unitary reflection $\mathbf{R}[\ensuremath{\gamma}]$. This enables the extraction of all reduced properties of the system from a smaller, accurately solved embedding cluster, and the systematic reconstruction of the reduced density matrix of the system. This process ensures that both extended and embedded systems satisfy the local virial-like relation, providing quantitative insight into the correspondence between the fragment in the extended system and its embedded analog. The performance and convergence of the method, as well as the N-representability of the resulting correlated density matrix, are evaluated and discussed within the context of the one-dimensional Hubbard model, which provides exact results for a comprehensive comparison.

DOI
10.1103/physrevb.108.155119

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In this work, we introduce a self-consistent scheme based on the one-body reduced density matrix (\ensuremathγ\ensuremath{\gamma}\ensuremathγ) formalism. A significant feature of this methodology is the utilization of an optimal unitary transformation of the Hamiltonian, determined through a self-consistently determined, unitary reflection R[\ensuremathγ]\mathbf{R}[\ensuremath{\gamma}]R[\ensuremathγ]. This enables the extraction of all reduced properties of the system from a smaller, accurately solved embedding cluster, and the systematic reconstruction of the reduced density matrix of the system. This process ensures that both extended and embedded systems satisfy the local virial-like relation, providing quantitative insight into the correspondence between the fragment in the extended system and its embedded analog. The performance and convergence of the method, as well as the N-representability of the resulting correlated density matrix, are evaluated and discussed within the context of the one-dimensional Hubbard model, which provides exact results for a comprehensive comparison.
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The N-representability problem of reduced density matrices represents a fundamental challenge in electronic structure theory. In this work, we focus on the N-representability of transition reduced density matrices and propose a practical approach to embed a p-body transition reduced density matrix (RDM) of an N-particle system into a (p + 1)-body RDM of an (N + 1)-particle system. This embedding allows us to apply a unitary evolution algorithm based on a recently developed adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to determine the N-representability of reduced density matrices. The algorithm works by effectively applying a sequence of unitary transformations to a given (N + 1)-particle initial state in such a way that the distance of a projection of its corresponding (p + 1)-body RDM (embedded p-body transition RDM) to a target is minimized. Our methodology enables the purification of transition reduced density matrices and the reconstruction of approximate wave functions involved in the studied transitions, providing an effective strategy to correct and refine transition density matrices. We validate our approach with numerical simulations on N-particle systems, including a three-electron model system and the H3 molecule, demonstrating its robustness and accuracy.
Density matrix embedding theory [G. Knizia and G. K.-L. Chan, Phys. Rev. Lett. 109, 186404 (2012)] and density embedding theory [I. W. Bulik, G. E. Scuseria, and J. Dukelsky, Phys. Rev. B 89, 035140 (2014)] have recently been introduced for model lattice Hamiltonians and molecular systems. In the present work, the formalism is extended to the ab initio description of infinite systems. An appropriate definition of the impurity Hamiltonian for such systems is presented and demonstrated in cases of 1, 2, and 3 dimensions, using coupled cluster theory as the impurity solver. Additionally, we discuss the challenges related to disentanglement of fragment and bath states. The current approach yields results comparable to coupled cluster calculations of infinite systems even when using a single unit cell as the fragment. The theory is formulated in the basis of Wannier functions but it does not require separate localization of unoccupied bands. The embedding scheme presented here is a promising way of employing highly accurate electronic structure methods for extended systems at a fraction of their original computational cost.
Quantum embedding based on the (one-electron-reduced) density matrix is revisited by means of the unitary Householder transformation. While being exact and equivalent to (but formally simpler than) density matrix embedding theory (DMET) in the noninteracting case, the resulting Householder-transformed density matrix functional embedding theory (Ht-DMFET) preserves, by construction, the single-particle character of the bath when electron correlation is introduced. In Ht-DMFET, the projected ``impurity+bath\mathrm{impurity}+\mathrm{bath}impurity+bath'' cluster's Hamiltonian (from which approximate local properties of the interacting lattice can be extracted) becomes an explicit functional of the density matrix. In the spirit of single-impurity DMET, we consider in this work a closed (two-electron) cluster constructed from the full-size noninteracting density matrix. When the (Householder-transformed) interaction on the bath site is taken into account, per-site energies obtained for the half-filled one-dimensional Hubbard lattice match almost perfectly the exact Bethe ansatz results in all correlation regimes. In the strongly correlated regime, the results deteriorate away from half-filling. This can be related to the electron number fluctuations in the (two-site) cluster which are neither described in Ht-DMFET nor in regular DMET. As expected, the per-site energies dramatically improve when increasing the number of embedded impurities. Formal connections with density/density matrix functional theories have been briefly discussed and should be explored further. Work is currently in progress in this direction.
The N-representability problem for reduced density matrices remains a fundamental challenge in electronic structure theory. Following our previous work that employs a unitary-evolution algorithm based on an adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to probe pure-state N-representability of reduced density matrices [J. Chem. Theory Comput. 2024, 20, 9968], in this work we propose a practical framework for determining the ensemble N-representability of a p-body matrix. This is accomplished using a purification strategy consisting of embedding an ensemble state into a pure state defined on an extended Hilbert space, such that the reduced density matrices of the purified state reproduce those of the original ensemble. By iteratively applying variational unitaries to an initial purified state, the proposed algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target p-body matrix, which serves as a measure of the N-representability of the target. This methodology facilitates both error correction of defective ensemble reduced density matrices, and quantum-state reconstruction on a quantum computer, offering a route for density-matrix refinement. We validate the algorithm with numerical simulations on systems of two, three, and four electrons in both, simple models as well as molecular systems at finite temperature, demonstrating its robustness.
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.