The study by Shao et al.1 marks a critical advancement in quantum simulations by reporting the first experimental observation of the antiferromagnetic (AFM) phase transition in a 3D fermionic Hubbard model (FHM) using ultracold lithium-6 atoms. This experiment bridges a significant gap in our understanding of strongly correlated systems and provides a new platform to explore low-temperature quantum phases. This commentary will analyze the results, place them in the context of ongoing research, and discuss their broader implications for the field of quantum materials and technologies.
The study by Shao et al.1 marks a critical advancement in quantum simulations by reporting the first experimental observation of the antiferromagnetic (AFM) phase transition in a 3D fermionic Hubbard model (FHM) using ultracold lithium-6 atoms. This experiment bridges a significant gap in our understanding of strongly correlated systems and provides a new platform to explore low-temperature quantum phases. This commentary will analyze the results, place them in the context of ongoing research, and discuss their broader implications for the field of quantum materials and technologies.
Fermi-Hubbard model is a fundamental lattice model describing correlated electron systems in condensed matter physics and is closely related to high-temperature superconductivity. In recent years, cold-atom quantum simulations have become an important paradigm for studying the Fermi-Hubbard model, and advances in quantum many-body computations have contributed to our understanding of its fundamental properties. Notably, a recent ultracold-atom experiment achieving the well-known antiferromagnetic (AFM) phase transition in the three-dimensional (3D) Hubbard model represents a key step in quantum simulation, laying a foundation for exploring the link between the quantum magnetism and high-temperature superconductivity. In this paper, the experimental and theoretical research progress of Fermi-Hubbard model in 3D systems is reviewed, the development history and present status in this field are discussed, and the future development direction is also prospected. The paper is organized as follows. To begin with, recent progress of observing AFM phase transitions in the 3D Hubbard model is reviewed, focusing on an ultracold-atom experiment conducted by the research group at the University of Science and Technology of China (USTC). Next, a theoretical introduction to the fundamental properties of the 3D Hubbard model is provided, in which prior theoretical studies is summarized, the current research status is outlined, and some unresolved or under-explored problems are discussed. In Section 3, the quantum simulation of the Hubbard model using ultracold atoms in optical lattices is discussed, and the basic principle, historical developments and key challenges are outlined. The USTC team overcame these challenges through innovative techniques such as atom cooling, large-scale uniform box traps, and precise measurements of the AFM structure factor. Their work successfully confirms the AFM phase transition via the critical scaling analysis. Finally, the significance of this achievement is emphasized, and the future research prospects of the 3D Hubbard model are discussed, including experimental studies on the doped regions and related theoretical benchmarks.
The fermionic Hubbard model (FHM)[1], despite its simple form, captures essential features of strongly correlated electron physics. Ultracold fermions in optical lattices[2, 3] provide a clean and well-controlled platform for simulating FHM. Doping its antiferromagnetic ground state at half filling, various exotic phases are expected to arise in the FHM simulator, including stripe order[4], pseudogap[5], and d-wave superconductors[6], offering valuable insights into high-temperature superconductivity[7{9]. Although notable progress, such as the observation of antiferromagnetic correlations over short[10] and extended distances[11], has been obtained, the antiferromagnetic phase has yet to be realized due to the significant challenges of achieving low temperatures in a large and uniform quantum simulator. Here, we report the observation of the antiferromagnetic phase transition in a three-dimensional fermionic Hubbard system comprising lithium-6 atoms in a uniform optical lattice with approximately 800,000 sites. When the interaction strength, temperature, and doping concentration are finely tuned to approach their respective critical values, sharp increases in the spin structure factor (SSF) are observed. These observations can be well described by a power-law divergence, with a critical exponent of 1.396 from the Heisenberg universality class[12]. At half filling and with optimal interaction strength, the measured SSF reaches 123(8), signifying the establishment of an antiferromagnetic phase. Our results set the stage for exploring the low-temperature phase diagram of FHM.
Over the past few decades, quantum simulation with ultracold atoms in optical lattices has proven to be a useful tool for understanding many-body Hamiltonians.One of its primary directions is the experimental study of the Fermi Hubbard Model (FHM), which aims to capture the fundamental properties of cuprate superconductors and has produced a wide range of intriguing phenomena, including the metal-to-Mott insulator transition, antiferromagnetism, strange metallicity, pseudogap physics, and is frequently used in the study of superconductivity. In recent years, new kinds of quantum simulators have emerged, and in particular we focus on two of them: 1) quantum simulators that study high symmetry fermions in optical lattices, and 2) optical tweezer arrays which provide a bottom-up approach to engineering the FHM. The first class of emerging quantum simulators utilizes alkaline-earth-like atoms (AEA’s) – which have large nuclear spin I (such as 173Yb and 87Sr) – to engineer the SU(N = 2I +1) symmetric FHM, where N = 2,..., 10. The second class corresponds to quantum simulators based on optical tweezer arrays which combine deterministic single tweezer ground state loading and tunneling coupled sites. Both classes of quantum simulators are interesting on their own. For example, the SU(N) FHM is generating a lot of interest due to its connections to multiorbital solid state systems and theoretical predictions of rich phase diagrams which exhibit exotic phases of matter, including unusual ones like spin liquids. On the other hand, optical tweezer arrays have gained a lot of attention since they can be used to study arbitrary geometries and access very low-entropy samples. This thesis reports on the numerical study of both quantum simulation architectures. Results of the SU(N) FHM at finite temperature, for different fillings and interaction strengths that span the non-interacting to the strongly interacting limit are obtained using Determinant Quantum Monte Carlo, Exact Diagonalization, and Numerical Linked Cluster Expansion. In particular, we focus on how thermodynamic and magnetic observables, such as the number of on-site pairs, energy, entropy, spin correlations, and structure factors depend on N, the interaction strength U/t, temperature T/t, and the anisotropy of tunneling rates. Our results demonstrate that in a homogeneous square lattice with one particle per site on average, thermodynamic observables as a function of temperature obey a universal scaling with N, and that short-range antiferromagnetic correlations are stronger for larger N and in lower dimensions. In addition, we present theory-experiment comparisons where possible, where we perform thermometry and provide a precise characterization of the equation of state of the SU(N) FHM. Our results emerge as a tool to perform thermometry in experiments from theory-experiment comparison and also provide guidance for future experiments with AEAs in optical lattices. Additionally, we present calculations using numerically precise discrete variable representation methods for two-dimensional stroboscopic tweezer arrays and compare the outcomes with experimental data. We quantify the effects of stroboscopic potentials on Hubbard parameters like the interaction strength U and the tunneling t in optical tweezer arrays and illustrate how heating from the stroboscopic potential relies on strobe frequency. Our calculations enable evaluation and optimization of two-dimensional tweezer array experiments. An important milestone of the results of this thesis corresponds to the record temperatures and entropies achieved with these quantum simulation architectures: 1) the coldest fermions ever created in nature in absolute temperature and in cold atoms, and 2) the lowest entropy per particle fermions ever created, albeit in a so-far small system.
Ultracold atomic gases provide a fantastic platform to implement quantum simulators and investigate a variety of models initially introduced in condensed matter physics or other areas. One of the most promising applications of quantum simulation is the study of strongly correlated Fermi gases, for which exact theoretical results are not always possible with state-of-the-art approaches. Here, we review recent progress of the quantum simulation of the emblematic Fermi–Hubbard model with ultracold atoms. After introducing the Fermi–Hubbard model in the context of condensed matter, its implementation in ultracold atom systems, and its phase diagram, we review landmark experimental achievements, from the early observation of the onset of quantum degeneracy and superfluidity to the demonstration of the Mott insulator regime and the emergence of long-range anti-ferromagnetic order. We conclude by discussing future challenges, including the possible observation of high- T c superconductivity, transport properties, and the interplay of strong correlations and disorder or topology.
Abstract The celebrated antiferromagnetic (AFM) phase transition was realized in a most recent optical lattice experiment for the 3D fermionic Hubbard model [ Shao et al . Nature 632 267 (2024) ]. Despite this important progress, it was observed that the AFM structure factor (and also the critical entropy) reaches its maximum at an interaction strength U / t ≃ 11.75, which is significantly larger than the theoretical prediction of U / t ≃ 8. Here, we resolve this discrepancy by studying the interplay between the thermal entropy, density disorder, and antiferromagnetism in the half-filled 3D Hubbard model, using numerically exact auxiliary-field quantum Monte Carlo simulations. We have achieved an accurate entropy phase diagram, enabling us to simulate arbitrary entropy path on the temperature-interaction plane and track experimental parameters effectively. We find that above the discrepancy can be quantitatively explained by the entropy increase associated with increasing interaction strength in experiments, and together with the lattice density disorder present in the experimental setup. We further investigate the entropy dependence of double occupancy and predict universal behaviors that could serve as valuable probes in future optical lattice experiments.
Quantum simulation is emerging as an ambitious and active subfield of atomic physics. This thesis describes progress towards the goal of simulating condensed matter sys-tems, in particular the physics of the Fermi-Hubbard model, using ultracold Lithium atoms in an optical lattice. A major goal of the quantum simulation program is to observe phase transitions of the Hubbard model, into Néel antiferromagnetic phases and d-wave superfluid phases. Phase transitions are generally accompanied by a change in an underlying correlation in a physical system. Such correlations may be most amenable to probing by looking at fluctuations in the system. Experimental techniques for probing density and magnetization fluctuations in a variety of atomic Fermi systems are developed. The suppression of density fluctuations (or atom “shot noise”) in an ideal degenerate Fermi gas is observed by absorption imaging of time-of-flight expanded clouds. In-trap measurements of density and magnetization fluctuations are not easy to probe with absorption imaging, due to their extremely high attenuation. A method to probe