Extended Many-Body Perturbation Theory for Strongly Correlated Systems Using Effective Hamiltonians

Core Concepts

This paper presents an extension of the many-body perturbation theory (MBPT) that enables the use of effective Hamiltonians containing beyond-mean-field correlations, addressing the challenge of overcounting correlations and improving the accuracy of calculations for strongly correlated quantum many-body systems.

Abstract

Bibliographic Information:

Photopoulos, R., & Boulet, A. (2024). Many-body perturbation theory for strongly correlated effective Hamiltonians using effective field theory methods. arXiv preprint arXiv:2402.17627v3.

Research Objective:

This paper aims to develop a systematic MBPT approach for strongly correlated effective Hamiltonians that avoids double-counting correlations and accurately reproduces ground state energies in both weak and strong coupling regimes.

Methodology:

The authors generalize the Rayleigh-Schrödinger perturbation theory by introducing free parameters adjusted to reproduce appropriate limits. They utilize the low-scale expansion in the bare weak-coupling regime and the strong-coupling limit to fix these parameters. This approach allows for the incorporation of beyond-mean-field correlations directly into the effective interaction.

Key Findings:

• The extended MBPT method successfully avoids double-counting of correlations when using beyond-mean-field effective Hamiltonians.
• The method accurately reproduces the ground state energy of various systems relevant to ultracold atomic, nuclear, and condensed matter physics, demonstrating its applicability across different domains.
• The results obtained can be interpreted within the framework of effective field theory, with the reorganized MBPT expansion providing an EFT expansion of observables.

Main Conclusions:

The proposed extended MBPT method offers a systematic and accurate approach for calculating ground state energies of strongly correlated quantum many-body systems using effective Hamiltonians. By incorporating beyond-mean-field correlations and avoiding double-counting, the method achieves improved accuracy compared to standard MBPT, particularly in the strong coupling regime.

Significance:

This research provides a valuable tool for studying strongly correlated quantum systems, which are ubiquitous in various fields of physics. The ability to accurately calculate ground state energies using effective Hamiltonians opens up new avenues for investigating these complex systems.

Limitations and Future Research:

The paper primarily focuses on ground state energy calculations. Future research could explore the application of this extended MBPT method to other observables and investigate its performance for systems with more complex low-scale expansions.

Stats

The Bertsch parameter for a spin-saturated Fermi gas is ξ = 0.376(4).

Quotes

Key Insights Distilled From

Many-body perturbation theory for strongly correlated effective Hamiltonians using effective field theory methods

by Raph... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2402.17627.pdf

Deeper Inquiries

How can this extended MBPT method be generalized to address time-dependent problems in quantum many-body systems?

Extending the presented extended MBPT method to time-dependent problems in quantum many-body systems presents a fascinating challenge with potentially significant implications. Here's a breakdown of potential strategies and considerations:
1. Time-Dependent Effective Hamiltonian:

The first step involves generalizing the effective Hamiltonian (Equation 4 in the paper) to incorporate time dependence. This could involve introducing time-dependent parameters or functions within the interaction term. The choice of parameterization would depend on the specific time-dependent phenomena being studied.
2. Time-Dependent Perturbation Theory:

Instead of the static Schrödinger equation, we'd employ the time-dependent Schrödinger equation. The time evolution operator, often expressed in terms of a time-ordered exponential, would become central to the formalism.
3. Diagrammatic Techniques:

Feynman diagrams, which provide a powerful visual and computational tool for MBPT, can be extended to the time domain. Time-dependent Green's functions would play a crucial role in this extension.
4. Adiabatic Switching:

One approach could involve adiabatically switching on the time-dependent part of the Hamiltonian. This would allow connecting the initial ground state to the time-evolved state.
5. Challenges and Considerations:

Complexity: Time-dependent MBPT is inherently more complex than its static counterpart. The number of diagrams and the difficulty of evaluating them increase significantly.
Choice of Parameterization: The success of the method would heavily rely on choosing an appropriate time-dependent parameterization of the effective Hamiltonian that captures the essential physics.
Non-Equilibrium Phenomena: This extension could provide insights into non-equilibrium phenomena in strongly correlated systems, such as dynamics after a quench or response to driving fields.

Could alternative parameterizations of the effective Hamiltonian further enhance the accuracy and applicability of this method?

Yes, exploring alternative parameterizations of the effective Hamiltonian holds significant potential for enhancing the accuracy and applicability of this extended MBPT method. Here's a breakdown of strategies and considerations:
1. Beyond Simple Rational Functions:

The paper primarily focuses on a rational function form for the effective interaction (Equation 4). Investigating other functional forms, such as exponential functions, trigonometric functions, or piecewise functions, could offer advantages in specific physical contexts.
2. Density Dependence:

Incorporating density dependence into the effective interaction parameters could be particularly relevant for systems with strong spatial correlations or inhomogeneous densities.
3. Symmetry Considerations:

Leveraging the symmetries of the system to guide the choice of parameterization can lead to more physically motivated and efficient representations.
4. Constraints from Higher-Order Correlations:

Imposing constraints derived from knowledge of higher-order correlations (e.g., three-body correlations) could further refine the effective Hamiltonian.
5. Optimization Strategies:

Developing systematic optimization strategies for determining the parameters of the effective Hamiltonian, potentially using machine learning techniques, could enhance accuracy.
Benefits of Exploration:

Improved Accuracy: Alternative parameterizations might better capture the underlying physics in certain regimes, leading to more accurate results.
Extended Applicability: Different parameterizations could extend the method's applicability to a broader class of strongly correlated systems.
Deeper Physical Insights: Exploring the performance of various parameterizations could provide insights into the essential correlations governing the system's behavior.

What are the potential implications of interpreting the results of this extended MBPT method within the framework of effective field theory for our understanding of emergent phenomena in strongly correlated systems?

Interpreting the extended MBPT method through the lens of effective field theory (EFT) offers a powerful framework for understanding emergent phenomena in strongly correlated systems. Here's a look at the potential implications:
1. Separation of Scales:

EFT emphasizes the separation of energy scales. In this context, the extended MBPT method explicitly separates the low-energy, strongly correlated physics (captured by the effective Hamiltonian) from the high-energy details (integrated out or renormalized).
2. Universality and Model Independence:

EFTs often exhibit universality, meaning that the low-energy physics becomes insensitive to the microscopic details of the underlying theory. The extended MBPT method, by focusing on low-energy effective Hamiltonians, could help uncover universal behavior in strongly correlated systems.
3. Systematic Power Counting:

EFTs provide a systematic way to organize calculations in terms of a small expansion parameter. The extended MBPT method's ability to match low- and high-scale limits suggests a potential connection to a well-defined power counting scheme.
4. Emergent Collective Excitations:

Strongly correlated systems often exhibit emergent collective excitations, such as Cooper pairs in superconductors or spin waves in magnets. The extended MBPT method, by incorporating correlations beyond mean-field, could provide a theoretical handle on describing these emergent degrees of freedom.
5. Bridging Scales:

One of the key strengths of EFT is its ability to bridge different energy scales. The extended MBPT method, by connecting to the high-scale limit, could offer a way to relate low-energy observables to high-energy parameters or constraints.
Implications for Understanding Emergent Phenomena:

Predictive Power: This approach could lead to more accurate and predictive theories for strongly correlated systems, even in regimes where traditional perturbative methods fail.
Conceptual Unification: It could provide a unifying conceptual framework for understanding a wide range of emergent phenomena across condensed matter physics, nuclear physics, and ultracold atom systems.
New Material Design: By elucidating the connection between microscopic interactions and emergent properties, it could aid in the design of new materials with exotic properties.