Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2302.02899 · Physical review. B./Physical review. B

Renormalized density matrix downfolding: A rigorous framework in learning emergent models from ab initio many-body calculations

Yueqing Chang, Sonali Joshi, Lucas K. Wagner

Published Nov 4, 2024• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.110.195103

Related papers

Noneigenstate Ab Initio Density Matrix Downfolding for Constructing Model Hamiltonians in Quantum Chemistry

David Wilian Oliveira de Sousa, Elvira R. Sayfutyarova

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.

Journal of Chemical Theory and Computation•Aug 28, 2025•0 citations

Non-eigenstate Ab Initio Density Matrix Downfolding to produce Model Hamiltonians in Quantum Chemistry

David Wilian Oliveira de Sousa, Elvira R. Sayfutyarova

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.

•Apr 21, 2025•0 citations

From Real Materials to Model Hamiltonians With Density Matrix Downfolding

Huihuo Zheng, Hitesh J. Changlani, Kiel T. Williams +2 more

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).

arXiv:1712.00477•Frontiers in Physics•May 11, 2018•33 citations

Requirements for building effective Hamiltonians using quantum-enhanced density matrix downfolding

Shivesh Pathak, Antonio Russo, Stefan Seritan +4 more

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.

arXiv:2403.01043•Mar 2, 2024•1 citation

Downfolding from ab initio to interacting model Hamiltonians: comprehensive analysis and benchmarking of the DFT+cRPA approach

Yueqing Chang, Erik G. C. P. van Loon, Brandon Eskridge +8 more

Abstract Model Hamiltonians are regularly derived from first principles to describe correlated matter. However, the standard methods for this contain a number of largely unexplored approximations. For a strongly correlated impurity model system, here we carefully compare a standard downfolding technique with the best possible ground-truth estimates for charge-neutral excited-state energies and wave functions using state-of-the-art first-principles many-body wave function approaches. To this end, we use the vanadocene molecule and analyze all downfolding aspects, including the Hamiltonian form, target basis, double-counting correction, and Coulomb interaction screening models. We find that the choice of target-space basis functions emerges as a key factor for the quality of the downfolded results, while orbital-dependent double-counting corrections diminish the quality. Background screening of the Coulomb interaction matrix elements primarily affects crystal-field excitations. Our benchmark uncovers the relative importance of each downfolding step and offers insights into the potential accuracy of minimal downfolded model Hamiltonians.

arXiv:2311.05987•npj Computational Materials•Jun 21, 2024•28 citations

Density-matrix based determination of low-energy model Hamiltonians from ab initio wavefunctions

Hitesh J. Changlani, Huihuo Zheng, Lucas K. Wagner

We propose a way of obtaining effective low energy Hubbard-like model Hamiltonians from ab initio quantum Monte Carlo calculations for molecular and extended systems. The Hamiltonian parameters are fit to best match the ab initio two-body density matrices and energies of the ground and excited states, and thus we refer to the method as ab initio density matrix based downfolding. For benzene (a finite system), we find good agreement with experimentally available energy gaps without using any experimental inputs. For graphene, a two dimensional solid (extended system) with periodic boundary conditions, we find the effective on-site Hubbard U(∗)/t to be 1.3 ± 0.2, comparable to a recent estimate based on the constrained random phase approximation. For molecules, such parameterizations enable calculation of excited states that are usually not accessible within ground state approaches. For solids, the effective Hamiltonian enables large-scale calculations using techniques designed for lattice models.

arXiv:1504.03704•The Journal of Chemical Physics•Aug 4, 2015•32 citations
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.
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.
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).
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 Model Hamiltonians are regularly derived from first principles to describe correlated matter. However, the standard methods for this contain a number of largely unexplored approximations. For a strongly correlated impurity model system, here we carefully compare a standard downfolding technique with the best possible ground-truth estimates for charge-neutral excited-state energies and wave functions using state-of-the-art first-principles many-body wave function approaches. To this end, we use the vanadocene molecule and analyze all downfolding aspects, including the Hamiltonian form, target basis, double-counting correction, and Coulomb interaction screening models. We find that the choice of target-space basis functions emerges as a key factor for the quality of the downfolded results, while orbital-dependent double-counting corrections diminish the quality. Background screening of the Coulomb interaction matrix elements primarily affects crystal-field excitations. Our benchmark uncovers the relative importance of each downfolding step and offers insights into the potential accuracy of minimal downfolded model Hamiltonians.
We propose a way of obtaining effective low energy Hubbard-like model Hamiltonians from ab initio quantum Monte Carlo calculations for molecular and extended systems. The Hamiltonian parameters are fit to best match the ab initio two-body density matrices and energies of the ground and excited states, and thus we refer to the method as ab initio density matrix based downfolding. For benzene (a finite system), we find good agreement with experimentally available energy gaps without using any experimental inputs. For graphene, a two dimensional solid (extended system) with periodic boundary conditions, we find the effective on-site Hubbard U(∗)/t to be 1.3 ± 0.2, comparable to a recent estimate based on the constrained random phase approximation. For molecules, such parameterizations enable calculation of excited states that are usually not accessible within ground state approaches. For solids, the effective Hamiltonian enables large-scale calculations using techniques designed for lattice models.