Perturbative Calculation of Self-Energy in the Hubbard Model on a Graphene Lattice at Finite Temperature

Core Concepts

This study demonstrates the effectiveness of a perturbative approach to calculate the self-energy of the Hubbard model on a graphene lattice at finite temperatures, showing good agreement with exact and numerical solutions for small lattices and moderate interaction strengths.

Abstract

• Bibliographic Information: Razmadze, L., & Luu, T. (2024). Hubbard interaction at finite T on a hexagonal lattice. Proceedings of Science.

• Research Objective: This research paper investigates the behavior of fermions on a hexagonal lattice (graphene) with Hubbard-type interaction at finite temperatures, focusing on calculating the self-energy and its effect on low-energy excitations.

• Methodology: The authors employ thermal field theory and perturbative calculations up to the leading non-trivial order (O(U^2)) to determine the self-energy. They validate their results by comparing them with exact solutions for small lattices (2-site and 4-site systems) and Hybrid Monte Carlo (HMC) simulations for a larger 2x3 graphene sheet.

• Key Findings:

  • The perturbative approach accurately predicts the zero-temperature energy shift, showing excellent agreement with exact solutions for the 2-site system and strong agreement for the 4-site system, even at large interaction strengths (U = 20).
  • The time evolution of correlators calculated perturbatively aligns well with exact results for both 2-site and 4-site models, even at relatively high interaction strengths.
  • For the 2x3 graphene sheet, the perturbative calculations of correlators show good consistency with HMC simulations for interaction strengths up to U = 3. Discrepancies arise at higher interaction strengths (U = 4).

• Main Conclusions: The study demonstrates that a perturbative approach can effectively calculate the self-energy and analyze the finite-temperature behavior of the Hubbard model on a graphene lattice, particularly for small lattices and moderate interaction strengths.

• Significance: This research contributes to the understanding of strongly correlated electron systems, particularly in two-dimensional materials like graphene. The developed perturbative method provides a computationally efficient alternative to exact diagonalization or HMC simulations, especially for exploring the finite-temperature regime.

• Limitations and Future Research: The accuracy of the perturbative approach decreases at larger interaction strengths and lattice sizes. Future research could explore higher-order perturbative corrections or develop more sophisticated analytical techniques to improve accuracy in these regimes. Additionally, extending the analysis to investigate temperature-dependent properties like specific heat and conductivity would be beneficial.

Stats

The leading order (O(U)) contributions to the self-energy cancel out, leading to non-trivial corrections starting at O(U^2).
For the 4-site system, the perturbative calculations remain accurate up to U = 20, which is well into the non-perturbative regime.
In the 2x3 graphene sheet, the off-diagonal terms in the self-energy matrix are an order of magnitude smaller than the diagonal terms, justifying an approximation that treats the matrix as diagonal.

Quotes

Key Insights Distilled From

Hubbard interaction at finite $T$ on a hexagonal lattice

by Lado Razmadz... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.03196.pdf

Deeper Inquiries

How does this perturbative approach compare to other methods for studying strongly correlated systems, such as dynamical mean-field theory (DMFT) or tensor network methods, in terms of accuracy and computational cost?

Perturbation theory, while computationally cheaper than other methods, has limitations, especially for strongly correlated systems where the interaction strength is significant. Here's a comparison:
Perturbation Theory:

Accuracy: Accurate for weak interactions (small U). As interaction strength increases, higher-order terms become relevant, making calculations complex and potentially inaccurate. This study shows good agreement for moderate U values but discrepancies arise at larger U, highlighting this limitation.
Computational Cost: Relatively inexpensive, especially for low-order calculations. This allows exploration of larger systems compared to more computationally intensive methods.
Applicability: Best suited for systems where interactions can be treated as a small perturbation from a non-interacting system.
Dynamical Mean-Field Theory (DMFT):

Accuracy: Can handle strong correlations better than perturbative methods. It maps the lattice problem onto a self-consistent impurity problem, capturing local correlations more effectively.
Computational Cost: More computationally demanding than perturbation theory, but less so than exact methods.
Applicability: Suitable for systems with strong local correlations, but may not fully capture long-range spatial correlations.
Tensor Network Methods:

Accuracy: Powerful numerical methods that can provide highly accurate results for both ground state and finite-temperature properties, even for strongly correlated systems.
Computational Cost: Computationally expensive, especially for large systems and at finite temperatures. The cost scales with the entanglement entropy of the system.
Applicability: Versatile and applicable to a wide range of strongly correlated systems, but computational cost can be a limiting factor.
Summary:
The choice of method depends on the specific problem and the desired balance between accuracy and computational cost. Perturbation theory is a valuable tool for gaining initial insights, especially for weakly correlated systems. However, for strongly correlated systems, DMFT or tensor network methods are often necessary to capture the essential physics.

Could the observed discrepancies between the perturbative calculations and HMC simulations at larger interaction strengths be attributed to the emergence of exotic phases or quantum critical phenomena not captured by the perturbative approach?

Yes, the discrepancies observed at larger interaction strengths could indeed signal the emergence of exotic phases or quantum critical phenomena that the perturbative approach fails to capture.
Here's why:

Breakdown of Perturbation Theory: Perturbation theory relies on the assumption that interactions are weak and can be treated as small disturbances to a non-interacting system. As the interaction strength increases, this assumption breaks down, and the perturbative expansion may no longer converge to the correct physical solution.
Emergent Phenomena: Strong correlations can give rise to collective behavior and emergent phenomena not present in the non-interacting system. These phenomena, such as Mott insulators, unconventional superconductivity, or spin liquids, are driven by strong interactions and cannot be described by simply adding small corrections to the non-interacting picture.
Quantum Criticality: The discrepancies could indicate the proximity to a quantum critical point. Near such points, quantum fluctuations become significant at all length scales, and the system's behavior deviates strongly from the perturbative regime.
In the context of the Hubbard model on a hexagonal lattice:

Mott Transition: The Hubbard model is known to exhibit a Mott metal-insulator transition at a critical interaction strength. This transition is a quintessential example of a phenomenon driven by strong correlations, where the system transitions from a metallic state to an insulating state due to strong electron-electron repulsion.
Other Exotic Phases: Depending on the specific parameters, the Hubbard model on a hexagonal lattice can host other exotic phases, such as spin density wave states or superconducting phases.
Further Investigation:
To determine the exact nature of the discrepancies observed in the study, further investigation is needed, potentially using non-perturbative methods like DMFT or tensor network simulations. These methods can provide insights into the ground state properties and the possibility of exotic phases or quantum critical behavior at larger interaction strengths.

Given the connection between the Hubbard model and high-temperature superconductivity, how could the insights from this study be applied to investigate the role of thermal fluctuations in these materials?

While this study focuses on the Hubbard model in the context of graphene, the insights gained are relevant for understanding high-temperature superconductivity, where the Hubbard model serves as a fundamental theoretical framework. Here's how:

Understanding Thermal Fluctuations: High-temperature superconductivity occurs at temperatures much higher than conventional superconductors, implying a significant role of thermal fluctuations. This study directly addresses the impact of finite temperatures on the behavior of the Hubbard model, providing a framework to analyze how thermal fluctuations affect the system's properties.
Competition with Superconductivity: Thermal fluctuations can disrupt the delicate pairing mechanism responsible for superconductivity. By studying the temperature dependence of the self-energy and correlators, one can gain insights into the energy scales associated with these fluctuations and how they compete with the formation of Cooper pairs.
Pseudogap Phase: High-temperature superconductors exhibit a mysterious "pseudogap" phase above the superconducting transition temperature. This phase is characterized by a partial suppression of electronic states and unusual transport properties. The study's focus on finite-temperature effects could help investigate the role of thermal fluctuations in the formation and properties of the pseudogap phase.
Exploring Parameter Space: The study's perturbative approach, while limited for strong interactions, allows for efficient exploration of the parameter space of the Hubbard model, including temperature, interaction strength, and doping. This can guide more computationally intensive simulations and experiments by identifying regions of interest where thermal fluctuations might play a crucial role in shaping the phase diagram.
Specific Applications:

Calculating Critical Temperature: The study's methodology could be extended to calculate the critical temperature (Tc) of the superconducting transition. By analyzing the temperature dependence of the superconducting order parameter, one can estimate Tc and study how it is affected by thermal fluctuations.
Probing Transport Properties: The calculated self-energy can be used to study the temperature dependence of transport properties like electrical conductivity and thermal conductivity. This can provide insights into the scattering mechanisms and the role of thermal fluctuations in the normal state of high-temperature superconductors.
Overall, this study's focus on finite-temperature effects in the Hubbard model offers valuable tools and insights that can be applied to investigate the complex interplay of thermal fluctuations and superconductivity in high-Tc materials.