Static and Dynamic Correlations in Many-Electron Systems

Yue Wang

Andreas Deppeler, Andrew J. Millis

We use the dynamical mean-field theory to develop a systematic and computationally tractable method for studying electron-phonon interactions in systems with arbitrary electronic correlations. The method is formulated as an adiabatic expansion around the limit of static phonons. No specific electronic ground state is assumed. We derive an effective low-frequency phonon action whose coefficients are static local correlation functions of the underlying electron system. We identify the correct expansion parameters. At a critical electron-phonon interaction strength the system undergoes a transition to a polaronic state. We determine the location of this polaronic instability in the presence of electron-electron interactions, doping, and quantum lattice fluctuations and present the formalism needed for study of the electron self-energy and effective mass.

Anton Pauli, Akshat Mishra, Malte Rösner +1 more

Correlated electron physics is intrinsically a multiscale problem, since high-energy electronic states screen the interactions between the correlated electrons close to the Fermi level, thereby reducing the magnitude of the interaction strength and dramatically shortening its range. Thus the handling of screening is an essential ingredient in the first-principles modeling of correlated electron systems. Screening is an intrinsically dynamic process and the corresponding downfolding methods such as the constrained random phase approximation indeed produce a dynamic interaction. However, many low-energy methods require an instantaneous interaction as input, which makes it necessary to map the fully dynamic interaction to an effective instantaneous interaction strength. It is not clear if and when such an effective model can capture the physics of the one with dynamic interaction and how to best perform the mapping. Here, we provide a systematic benchmark relevant to correlated materials, in the form of the Anderson impurity model. Overall, we find that a static approximation can be valid and that the moment-based approach recently proposed by Scott and Booth can be a good tool to find the value of the static interaction. We also identify physical regimes, especially under doping, where an instantaneous interaction cannot capture all of the relevant physics.

J. J. Quinn, George Simion

Wei Liang Huang, Yifan Cheng, Haibo Ma

In recent years, numerous novel materials have emerged in fields such as high-temperature superconductivity and quantum communication, including unconventional superconductors, topological insulators, and spin liquids. These materials exhibit significant quantum phenomena such as quantum interference, tunneling, fluctuations, and entanglement, often arising from strong correlation effects among electrons within molecules. In the community of quantum chemistry, strong correlation effects are typically classified into dynamic electron correlation and static electron correlation. Dynamic electron correlation, caused by the instantaneous movement of electrons leading to charge density fluctuations, presents computational challenges due to the necessity of introducing a multi-particle form of the wave function. This departure from the mean-field single-particle perspective results in a substantial increase in the computational cost for calculating electron correlation levels. On the other hand, static electron correlation arises from the near-degeneracy of numerous frontier molecular orbitals, resulting in a quantum superposition of electron configurations. It requires the use of multi-configuration methods and faces the well-known “exponential wall” challenge, where computational complexity exponentially grows with the system size. Therefore, due to the involvement of both dynamic and static electron correlations in strongly correlated systems, achieving precise electronic structure calculations is exceptionally difficult, representing a significant challenge in theoretical chemistry and computational physics. Traditionally, to describe the static correlation, quantum chemists often define crucial correlated orbitals and their electrons within the complete active space (CAS). Full configuration interaction (FCI) calculations are performed within the CAS, while the region outside the CAS is described by mean-field methods. While these multi-configuration methods allow for arbitrary electron arrangement within the CAS, effectively capturing the static correlation of the system, they struggle to handle the dynamic correlations beyond the CAS. Therefore, combining them with multi-reference configuration interaction (MRCI), multi-reference perturbation theory (MRPT), multi-reference coupled cluster theory (MRCC), and other multi-reference methods becomes essential to accurately handle strong correlation systems. Constructing multi-configuration wave functions to obtain static correlation by multi-configuration methods, and then generating excitation configurations based on these multi-configuration wave functions to obtain dynamic correlation has become the mainstream of the development of current wave function methods. Challenges in this direction include the exponential growth in computational complexity with an increase in CAS size and the precision of multi-reference methods affecting the accuracy of capturing dynamic correlations beyond the active space. Over the last decade, various multi-configuration methods, such as selected configuration interaction (sCI), Density matrix renormalization group (DMRG), full configuration interaction quantum Monte Carlo (FCIQMC), and neural-network quantum states (NQS), have successfully expanded the computable active space by explicitly or implicitly extracting and compressing complete configuration space information. Furthermore, combining these multi-configuration methods with traditional multi-reference approaches, such as MRCI, MRPT and MRCC, enables capturing dynamic correlations caused by electronic instantaneous interactions beyond the active space. This achieves higher precision and quantitative descriptions for mid-sized strongly correlated molecular systems (around 50 active orbitals). Additionally, novel methods like Block-Correlated Coupled Cluster Theory (BCCC) have made progress in attempting to discuss static and dynamic correlations within a unified framework. This article primarily introduces the advancements in wave function theory electronic structure methods for strongly correlated molecular systems in recent years. The first section details multi-configuration methods that have expanded the CAS size, including sCI, DMRG, FCIQMC, NQS, and outlines their developmental trajectories. The second section delves into the combination of these multi-configuration methods with traditional multi-reference approaches, leveraging larger active spaces to address dynamic correlations and achieve higher precision results. Finally, the article discusses the prospects of these methods in three directions: Handling static and dynamic correlations within a unified framework, integrating with relativistic effects, and combining with machine learning.

Werner Kutzelnigg