Due to advances in computer hardware and new algorithms, it is now possible to perform highly accurate many-body simulations of realistic materials with all their intrinsic complications. The success of these simulations leaves us with a conundrum: how do we extract useful physical models and insight from these simulations? In this article, we present a formal theory of downfolding–extracting an effective Hamiltonian from first-principles calculations. The theory maps the downfolding problem into fitting information derived from wave functions sampled from a low-energy subspace of the full Hilbert space. Since this fitting process most commonly uses reduced density matrices, we term it density matrix downfolding (DMD).
Due to advances in computer hardware and new algorithms, it is now possible to perform highly accurate many-body simulations of realistic materials with all their intrinsic complications. The success of these simulations leaves us with a conundrum: how do we extract useful physical models and insight from these simulations? In this article, we present a formal theory of downfolding–extracting an effective Hamiltonian from first-principles calculations. The theory maps the downfolding problem into fitting information derived from wave functions sampled from a low-energy subspace of the full Hilbert space. Since this fitting process most commonly uses reduced density matrices, we term it density matrix downfolding (DMD).
Model Hamiltonians represent a convenient way of reducing complex problems of many-electron quantum mechanics to much simpler problems: they can fully reproduce the core behaviors of a system of interest by encoding only the dominant physical interactions and using only a small number of associated parameters. Model Hamiltonians have been successfully applied to describe many chemical and physical phenomena. Density matrix downfolding (DMD) [ J. Chem. Phys. 2015, 143 (10), 102814 ] allows the derivation of model Hamiltonians of any form in a systematically improvable fashion by matching the energy spectrum of ab initio Hamiltonians with those of the model Hamiltonians. This method allows not only the improvement of existing models but also the construction of accurate and efficient physical models for various systems. While DMD looks like a promising approach, it has rarely been applied within chemistry, and neither its limits nor its practical performance is well-understood. In this work, we evaluated the performance of DMD, based on noneigenstates of ab initio Hamiltonians, for several realistic chemical systems: benzene, naphthalene, FeSe, and a prototypical Fe(IV)═O complex found in the active sites of 2-oxoglutarate-dependent oxygenases. Our results show that DMD is a reliable and computationally efficient tool for obtaining optimized model Hamiltonians in quantum chemistry. This not only opens the door to studying complex systems at reduced computational cost but also to isolating and understanding the physical core principles that dominate their behavior─this might offer new insights for tuning or even designing chemical systems for applications ranging from biochemistry to catalysis.
Model Hamiltonians represent a convenient way of reducing complex problems of many-electron quantum mechanics to much simpler problems: they can fully reproduce the core behaviors of a system of interest by encoding only the dominant physical interactions and using only a small number of associated parameters. Model Hamiltonians have been successfully applied to describe many chemical and physical phenomena. Density Matrix Downfolding (DMD) [J. Chem. Phys. 2015, 143 (10), 102814] allows the derivation of model Hamiltonians, of any form, in a systematically improvable fashion, by matching the energy spectrum of ab initio Hamiltonians with those of the model Hamiltonians. This method allows not only the improvement of existing models but also the construction of accurate and efficient physical models for various systems. While DMD looks like a promising approach, it has rarely been applied within chemistry, and neither its limits nor practical performance are well understood. In this work we evaluated the performance of DMD, based on non-eigenstates of ab initio Hamiltonians, for several realistic chemical systems: benzene, naphthalene, FeSe, and a prototypical Fe(IV)=O complex found in the active sites of 2-oxoglutarate-dependent oxygenases. Our results show that DMD is a reliable and computationally efficient tool for obtaining optimized model Hamiltonians in quantum chemistry. This not only opens the door to studying complex systems at reduced computational cost but also to isolating and understanding the physical core principles that dominate their behavior — this might offer new insights for tuning, or even designing, chemical systems for applications ranging from biochemistry to catalysis.
We present a generalized framework, renormalized density matrix downfolding (RDMD), to derive systematically improvable, highly accurate, and nonperturbative effective models from ab initio calculations. This framework moves beyond the common role of ab initio calculations as calculating the parameters of a proposed Hamiltonian. Instead, RDMD provides the capability to decide whether a given effective Hilbert space can be identified from the ab initio data and assess the relative quality of ansatz Hamiltonians. Any method of ab initio solution can be used as a data source, and as the ab initio solutions improve, the resultant model also improves. We demonstrate the framework in an application to the downfolding of a hydrogen chain to a spin model, in which we find the interatomic separations for which a nonperturbative mapping can be made even in the strong coupling regime where standard methods fail, and compute a renormalized spin model Hamiltonian that quantitatively reproduces the ab initio dynamics.
Density matrix downfolding (DMD) is a technique for regressing low-energy effective Hamiltonians from quantum many-body Hamiltonians. One limiting factor in the accuracy of classical implementations of DMD is the presence of difficult-to-quantify systematic errors attendant to sampling the observables of quantum many-body systems on an approximate low-energy subspace. We propose a hybrid quantum-classical protocol for circumventing this limitation, relying on the prospective ability of quantum computers to efficiently prepare and sample from states in well-defined low-energy subspaces with systematically improvable accuracy. We introduce three requirements for when this is possible, including a notion of compressibility that quantifies features of Hamiltonians and low-energy subspaces thereof for which quantum DMD might be efficient. Assuming that these requirements are met, we analyze design choices for our protocol and provide resource estimates for implementing quantum-enhanced DMD on both the doped 2-D Fermi-Hubbard model and an ab initio model of a cuprate superconductor.
Abstract The recently introduced coupled cluster (CC) downfolding techniques for reducing the dimensionality of quantum many-body problems recast the CC formalism in the form of the renormalization procedure allowing, for the construction of effective (or downfolded) Hamiltonians in small-dimensionality sub-space, usually identified with the so-called active space, of the entire Hilbert space. The resulting downfolded Hamiltonians integrate out the external (out-of-active-space) Fermionic degrees of freedom from the internal (in-the-active-space) parameters of the wave function, which can be determined as components of the eigenvectors of the downfolded Hamiltonians in the active space. This paper will discuss the extension of non-Hermitian (associated with standard CC formulations) and Hermitian (associated with the unitary CC approaches) downfolding formulations to composite quantum systems commonly encountered in materials science and chemistry. The non-Hermitian formulation can provide a platform for developing local CC approaches, while the Hermitian one can serve as an ideal foundation for developing various quantum computing applications based on the limited quantum resources. We also discuss the algorithm for extracting the semi-analytical form of the inter-electron interactions in the active spaces.
We explore the principles of many-body Hamiltonian complexity reduction via downfolding on an effective low-dimensional representation. We present a unique measure of fidelity between the effective (reduced-rank) description and the full many-body treatment for arbitrary (i.e., ground and excited) states. When the entire problem is mapped on a system of interacting quasiparticles [npj Computational Materials 9 (1), 126, 2023], the effective Hamiltonians can faithfully reproduce the physics only when a clear energy scale separation exists between the subsystems and its environment. We also demonstrate that it is necessary to include quasiparticle renormalization at distinct energy scales, capturing the distinct interaction between subsystems and their surrounding environments. Numerical results from simple, exactly solvable models highlight the limitations and strengths of this approach, particularly for ground and low-lying excited states. This work lays the groundwork for applying dynamical downfolding techniques to problems concerned with (quantum) interfaces.