Session A58: Quantum Embedding Methods: Methods

Dynamical Downfolding and Construction of Effective Hamiltonians for Correlated Systems

There is longstanding interest in reducing the computational cost for electronic structure calculations of correlated systems. Recently, Romanova et. al have developed a method of dynamical downfolding to map a large correlated problem onto a small (treatable) subspace [Romanova et al., npj  Computational Materials 9 (1), 126, 2023]. The method entails the construction of an effective correlated quasiparticle Hamiltonian, which requires solving auxiliary one- and two-body propagator problems. This work succeeded in finding quantitative agreement with the experimental spectrum of an NV center. The limitations of this approach have, however, yet to be explored. Using a solvable model system, we investigate how the renormalization of one-body and two-body terms arises in the downfolded Hamiltonian and the correspondence to the aforementioned dynamical downfolding procedure.  Further, we explore the effectiveness of the downfolding procedure as the coupling between the correlated subspace and the rest space is increased.   

Determinantal quantum Monte Carlo solver for cluster perturbation theory

Cluster perturbation theory (CPT) is a technique for computing the spectral function of fermionic models with local interactions. By combining the solution of the model on a finite cluster with perturbation theory on intracluster hoppings, CPT provides access to single-particle properties with arbitrary momentum resolution while incurring low computational cost. Here, we introduce determinantal quantum Monte Carlo (DQMC) as a solver for CPT. Compared to the standard solver, exact diagonalization (ED), the DQMC solver reduces finite size effects through utilizing larger clusters, allows study of temperature dependence, and enables large-scale simulations of a greater set of models. We discuss the implementation of the DQMC solver for CPT and benchmark the CPT + DQMC method for the attractive and repulsive Hubbard models, showcasing its advantages over standard DQMC and CPT + ED simulations.   

Low-energy effective model for H4 square molecule derived from ab initio calculations

Effective model Hamiltonians are the workhorse of the study of complex electronic states. Typically these models are some degree of phenomenological, in that they are justified based on their agreement with experimental results. However, as models become more complex, this comparison becomes difficult to make in a systematic way. On the other hand, it is now possible to perform many-body quantum calculations to high accuracy for systems with ab initio Hamiltonians and up to hundreds of electrons. These calculations give key insights into correlated electron physics at that scale.

We rigorously examine effective model Hamiltonians for the purely theoretical H₄ square molecule, using exact ab initio calculations as a data source. We explicitly construct a 1-1 mapping between a single-band model and the infinite dimensional low-energy ab initio space. Renormalization of operators such as double occupancy are derived as a part of the process. We find that such operator renormalization is key to obtaining simple and accurate models from first principles.   

Equilibrium Quantum Impurity Problems via Matrix Product State Encoding of the Retarded Action

In the 0+1 dimensional imaginary-time path integral formulation of quantum impurity problems, the retarded action encodes the hybridization of the impurity with the bath. We explore the computational power of representing the retarded action as matrix product state (RAMPS).

We focus on the challenging Kondo regime of the single-impurity Anderson model, where non-perturbative strong-correlation effects arise at very low energy scales. We demonstrate that the RAMPS approach reliably reaches the Kondo regime for a range of interaction strengths $U$, with a numerical error scaling as a weak power law with inverse temperature. We investigate the convergence behavior of the method with respect to bond dimension and time discretization by analyzing the error of local observables in the full interacting problem and find polynomial scaling in both parameters.

Our results suggest that the RAMPS approach offers an alternative avenue for exploring quantum impurity problems, thereby setting the stage for future advancements in the method's capability to address more complex quantum impurity scenarios. Overall, our study contributes to the development of efficient and accurate non-wavefunction-based tensor-network methods for quantum impurity problems.   

A route to systematically improveable effective Hamiltonians in condensed matter

Effective models of complex systems are fundamental to our understanding of matter. Traditionally, these models are developed either through direct phenomenological modeling of experiments, through heuristic interpretation of first principles calculations, or a combination of both. Typically the latter are based on density functional theory calculations, which contain inherent approximations that are difficult to overcome systematically. Fortunately, over much work in the past few decades, it has become possible to obtain highly accurate many-body solutions of first principles models of matter using techniques like quantum chemistry and quantum Monte Carlo. However, connecting these solutions to lower energy models has been a challenge. 

I will present work on constructing a framework to allow for two main advances in the construction of effective Hamiltonians from first principles. (1) As the first principles solutions approach exact ones, the effective Hamiltonian should approach exact. (2) The effective Hamiltonians should be as simple as possible, including renormalization of variables in order to achieve simplicity. I will also go over several toy examples in which we benchmark standard embedding techniques using highly accurate data, including the derivation of spin models and embedded defect models.   

Derivation of the Ghost Gutzwiller Approximation from Quantum Embedding principles: the Ghost Density Matrix Embedding Theory

We outline the derivation of the Ghost Gutzwiller Approximation (gGA) from principles closely related to those in Density Matrix Embedding Theory (DMET). This derivation emphasizes the crucial role of ghost degrees of freedom and provides a conceptual connection between DMET and gGA. We also outline recent progress in exploiting the quantum-embedding algorithmic structure of this framework to reduce its computational cost.   

Capturing Long-Range Electron Correlation in Ab Initio Green's Function Embedding

We present an ab initio interacting-bath dynamical embedding theory (ibDET) for accurate treatment of local and long-range electron interactions in periodic systems. The dynamical mean-field theory (DMFT) and its cluster extension has achieved success in simulating correlated materials, but it remains a significant challenge to capture long-range electron correlation and preserve transitional symmetry. In this work, instead of fitting fictitious non-interacting bath orbitals from hybridization function, we derive a set of interacting bath orbitals including cluster-specific natural orbitals using a projective approach. The self-energy calculated from the embedding problem can then be rotated back to the full system to capture long-range physics more accurately. Using the coupled-cluster Green's function method as the impurity solver, we apply the GW+ibDET approach to study spectral properties of two-dimensional boron nitride, magnesium oxide, and bulk sodium, and obtain good agreement with experimental measurements.   

Ab-initio downfolding to interacting model Hamiltonians: comprehensive analysis and benchmarking

Although regularly used to derive simplified model Hamiltonians and to describe correlated matter, ab-initio downfolding procedures have not been comprehensively analyzed.

We fill this gap with a systematic benchmark study in the vanadocene molecule, where we are able to first establish reference data for ground and charge-neutral excited state energies and charge densities using state-of-the-art first-principles methods (the equation-of-motion coupled-cluster method, fixed-node diffusion Monte Carlo, and auxiliary field quantum Monte Carlo). The downfolding procedure is assessed afterwards based on comparisons to these first-principles references. We 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 correction diminishes the quality. Background screening to 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.    

Inchworm quasi Monte Carlo for quantum impurities

The inchworm expansion is a promising approach to solving strongly correlated quantum impurity models due to its reduction of the sign problem in real and imaginary time. We show that the imaginary time integration is amenable to quasi Monte Carlo, with enhanced N^-1 convergence, compared to standard inchworm Monte Carlo calculations with N^-1/2 convergence. This extends the applicability of the inchworm method to, e.g., multi-orbital Anderson impurity models with off-diagonal hybridization, relevant for materials simulation, where continuous time hybridization expansion Monte Carlo has a severe sign problem. We also present an open source implementation of our Inchworm quasi Monte Carlo approach: QInchworm.jl, implemented in the Julia programming language.   

Low-scaling algorithms for many-body electronic structure and downfolding for quantum embedding

Quantum embedding has become one of the most successful techniques for materials simulations due to its multilayer treatments for different physical scales. 

Aside from the need of efficient solvers for the strongly correlated subspace, constructing a material-specific low-energy Hamiltonian beyond density functional theory (DFT) requires an efficient many-body method for the weakly correlated environment. In this talk, I will report our recent efforts on the low-scaling algorithms for many-body electronic structure and downfolding based on tensor hypercontraction (THC). THC is a systematically controlled compression technique for generic many-body Hamiltonians in any canonical basis. The resulting representation of the many-body Hamiltonians exhibits desired separability in the orbital and k-points indices which leads to the constructions of low-scaling algorithms for the following many-body calculations. We demonstrate the applicability and efficiency of THC in the context of downfolding procedures, including GW electronic structure and constrained RPA. The efficiency of these THC-based many-body methods provides a route for constructing low-energy Hamiltonian beyond DFT for large-scale systems.