The Hubbard model: A computational perspective (2104.00064v1)

Abstract: The Hubbard model is the simplest model of interacting fermions on a lattice and is of similar importance to correlated electron physics as the Ising model is to statistical mechanics or the fruit fly to biomedical science. Despite its simplicity, the model exhibits an incredible wealth of phases, phase transitions, and exotic correlation phenomena. While analytical methods have provided a qualitative description of the model in certain limits, numerical tools have shown impressive progress in achieving quantitative accurate results over the last years. This article gives an introduction to the model, motivates common questions, and illustrates the progress that has been achieved over the last years in revealing various aspects of the correlation physics of the model.

• The paper establishes consensus on the Hubbard model's phase diagram, particularly confirming robust antiferromagnetic order at half-filling.
• It leverages diverse numerical methods like QMC, DMRG, and tensor networks to calculate ground state energies and correlation functions.
• The study identifies unresolved issues such as the pseudogap and stripe phases, setting the stage for future advances in correlated electron research.

Overview of "The Hubbard Model: A Computational Perspective"

The paper authored by Mingpu Qin, Thomas Schäfer, Sabine Andergassen, Philippe Corboz, and Emanuel Gull provides a comprehensive computational analysis of the Hubbard model, a cornerstone in the paper of strongly correlated electron systems. The Hubbard model, despite its simplicity, presents a rich phase diagram encompassing phenomena such as superconductivity, magnetism, and metal-insulator transitions, making it a fundamental problem in condensed matter physics.

Background and Methodological Developments

The Hubbard model is defined by its Hamiltonian, comprising a kinetic term represented by a hopping parameter $t$ and an interaction term characterized by the on-site interaction strength $U$. The model primarily addresses correlation effects within a single orbital, neglecting more complex interactions found in real materials. This simplified approach has offered significant insights into correlated electron systems, particularly in connection to high-temperature superconductivity and the physics of cuprates.

Over the years, a multitude of numerical methods have been employed to unravel the complexities of the Hubbard model. These include exact diagonalization, quantum Monte Carlo (QMC), density matrix renormalization group (DMRG), dynamical mean-field theory (DMFT), and tensor network techniques, among others. Each of these methods has its strengths and limitations, often necessitating the use of multiple approaches to gain consensus on the model's properties.

Achievements in Numerical Simulations

The convergence of results from different computational techniques has allowed researchers to reach agreement on several aspects of the model. Consensus has been particularly strong concerning the phase diagram of the Hubbard model at half-filling, where antiferromagnetic order is robust for all interaction strengths. Benchmark studies have facilitated this agreement by comparing results from various methods on specific observables, such as ground state energies and correlation functions.

One of the major unresolved issues that the research community has addressed is the nature of the pseudogap state and the role of stripe phases. At intermediate coupling strengths and certain doping levels, the model exhibits charge-ordered states (stripes), but these have been contentious. Recent studies using a combination of DMRG, quantum Monte Carlo, and other methods have shown that stripe phases without coexisting superconductivity are the ground state for a range of interaction strengths and doping levels.

Theoretical Implications and Future Directions

This paper underscores the complexities involved in connecting the predictions of the Hubbard model to experimental results, particularly in high-temperature superconductors. The presence of competing orders such as magnetism, charge density waves, and superconductivity poses challenges to understanding the phase diagram and the transitions between different states.

The authors highlight several open questions and future directions, suggesting that further computational advancements are needed to address issues like the precise location of phase boundaries, the nature of high-temperature metallic phases, and the full characterization of dynamical quantities. Additionally, the exploration of extensions to the Hubbard model, such as multi-orbital systems or models with non-local interactions, remains a significant research frontier.

Conclusion

The computational paper of the Hubbard model continues to be a dynamic and challenging field, with recent advances leading to significant consensus on its key features. While unresolved questions remain, the ongoing development of numerical methods promises further insights into this quintessential model of strongly correlated systems. As computational resources and techniques continue to evolve, the ability to address more complex models will undoubtedly enhance our understanding of correlated electron materials.