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arXiv:2402.14605

Observation of the antiferromagnetic phase transition in the fermionic Hubbard model

Hou-Ji Shao, Yu‐Xuan Wang, Dezhi Zhu, Zhu Yan-song, Haonan Sun, Chen, Si-Yuan, Chi Zhang, Zhi-Jie Fan, Youjin Deng, Xing-Can Yao, Yu-Ao Chen, Jian-Wei Pan

Published Feb 22, 2024• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2402.14605

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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.
Ultracold fermionic atoms in optical lattices offer pristine realizations of Hubbard models1, which are fundamental to modern condensed-matter physics2,3. Despite notable advancements4–6, the accessible temperatures in these optical lattice material analogues are still too high to address many open problems7–10. Here we demonstrate a several-fold reduction in temperature6,11–13, bringing large-scale quantum simulations of the Hubbard model into an entirely new regime. This is accomplished by transforming a low-entropy product state into strongly correlated states of interest via dynamic control of the model parameters14,15, which is extremely challenging to simulate classically10. At half-filling, the long-range antiferromagnetic order is close to saturation, leading to a temperature of T/t=0.05−0.05+0.06T/t=0.0{5}_{-0.05}^{+0.06}T/t=0.05−0.05+0.06​ based on comparisons with numerically exact simulations. Doped away from half-filling, it is exceedingly challenging to realize systematically accurate and predictive numerical simulations9. Importantly, we are able to use quantum simulation to identify a new pathway for achieving similarly low temperatures with doping. This is confirmed by comparing short-range spin correlations to state-of-the-art, but approximate, constrained-path auxiliary-field quantum Monte Carlo simulations16–18. Compared with the cuprates2,19,20, the reported temperatures correspond to a reduction from far above to below room temperature, at which physics such as the pseudogap and stripe phases may be expected3,19,21–24. Our work opens the door to quantum simulations that solve open questions in material science, develop synergies with numerical methods and theoretical studies, and lead to discoveries of new physics8,10. A several-fold reduction in temperature is accomplished using a neutral-atom Hubbard quantum simulator by transforming a low-entropy product state into strongly correlated states of interest via dynamic control of the model parameters.
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.
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.
Exotic phenomena in strongly correlated electron systems emerge from the interplay between spin and motional degrees of freedom. For example, doping an antiferromagnet is expected to give rise to pseudogap states and high-temperature superconductors. Quantum simulation with ultracold fermions in optical lattices offers the potential to answer open questions about the doped Hubbard Hamiltonian, and has recently been advanced by quantum gas microscopy. In order to take advantage of these possibilities, a stable, high-power, 2D square lattice system was developed and coupled with an image plane based spatial light modulator. The ability to finely tune the atomic potential has made it possible to realize an antiferromagnet in a repulsively interacting Fermi gas on a 2D square lattice. At our lowest temperatures of T/t=0.25(2)T/t=0.25(2)T/t=0.25(2), antiferromagnetic long-range order (LRO) manifests through the divergence of the correlation length that reaches the size of the system, the development of a peak in the spin structure factor and a value of the staggered magnetization approaching the ground state value. Similarly, by carefully shaping the confinement, we have produced ultra-low entropy band insulators, which promise to be a perfect starting point for more advanced cooling schemes. In addition to the production of new states, Fermi gas microscopy is superbly well suited to studies of many-body dynamics. To this end I report on preliminary measurements of the propagation of holes in a Mott insualtor. These results demonstrate that Fermi gas microscopy can address open questions on the low-temperature Hubbard model.
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.
We study the SU(3) symmetric Fermi-Hubbard model (FHM) in the square lattice at 1/31/31/3-filling using numerically exact determinant quantum Monte Carlo and numerical linked-cluster expansion techniques. We present the different regimes of the model in the $T\text{\ensuremath{-}}U$ plane, which are characterized by local and short-range correlations, and capture signatures of the metal-insulator transition and magnetic crossovers. These signatures are detected as the temperature scales characterizing the rise of the compressibility, and an interaction-dependent change in the sign of the diagonal spin-spin correlation function. The analysis of the compressibility estimates the location of the metal-insulator quantum critical point at ${U}_{c}/t\ensuremath{\sim}6$, and provides a temperature scale for observing Mott physics at finite TTT. Furthermore, from the analysis of the spin-spin correlation function we observe that for $U/t\ensuremath{\gtrsim}6$ and $T\ensuremath{\sim}J=4{t}^{2}/U$ there is a development of a short-range two-sublattice (2SL) antiferromagnetic structure, as well as an emerging three-sublattice (3SL) antiferromagnetic structure as the temperature is lowered below $T/J\ensuremath{\lesssim}0.57$. This crossover from 2SL to 3SL magnetic ordering agrees with Heisenberg limit predictions, and has observable effects on the density of on-site pairs. Finally, we describe how the features of the regimes in the $T\ensuremath{-}U$ plane can be explored with alkaline-earth-like atoms in optical lattices with currently achieved experimental techniques and temperatures. The results discussed in this paper provide a starting point for the exploration of the SU(3) FHM upon doping.