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arXiv:2312.14156 · Nature

Formation of individual stripes in a mixed-dimensional cold-atom Fermi–Hubbard system

Dominik Bourgund, Thomas Chalopin, Petar Bojović, Henning Schlömer, Si Wang, Titus Franz, Sarah Hirthe, Annabelle Bohrdt, Fabian Grusdt, Immanuel Bloch, Timon Hilker

Published Jan 1, 2025• 23 citations
arXivHTMLPDF

Abstract

Abstract The relation between d -wave superconductivity and stripes is fundamental to the understanding of ordered phases in high-temperature cuprate superconductors 1–6 . These phases can be strongly influenced by anisotropic couplings, leading to higher critical temperatures, as emphasized by the recent discovery of superconductivity in nickelates 7–10 . Quantum simulators with ultracold atoms provide a versatile platform to engineer such couplings and to observe emergent structures in real space with single-particle resolution. Here we show, to our knowledge, the first signatures of individual stripes in a cold-atom Fermi–Hubbard quantum simulator using mixed-dimensional (mixD) settings. Increasing the energy scale of hole–hole attraction to the spin exchange energy, we access the interesting crossover temperature regime in which stripes begin to form 11 . We observe extended, attractive correlations between hole dopants and find an increased probability of forming larger structures akin to individual stripes. In the spin sector, we study correlation functions up to the third order and find results consistent with stripe formation. These observations are interpreted as a precursor to the stripe phase, which is characterized by interleaved charge and spin density wave ordering with fluctuating lines of dopants separating domains of opposite antiferromagnetic order 12–14 .

DOI
10.1038/s41586-024-08270-7

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Abstract The relation between d -wave superconductivity and stripes is fundamental to the understanding of ordered phases in high-temperature cuprate superconductors 1–6 . These phases can be strongly influenced by anisotropic couplings, leading to higher critical temperatures, as emphasized by the recent discovery of superconductivity in nickelates 7–10 . Quantum simulators with ultracold atoms provide a versatile platform to engineer such couplings and to observe emergent structures in real space with single-particle resolution. Here we show, to our knowledge, the first signatures of individual stripes in a cold-atom Fermi–Hubbard quantum simulator using mixed-dimensional (mixD) settings. Increasing the energy scale of hole–hole attraction to the spin exchange energy, we access the interesting crossover temperature regime in which stripes begin to form 11 . We observe extended, attractive correlations between hole dopants and find an increased probability of forming larger structures akin to individual stripes. In the spin sector, we study correlation functions up to the third order and find results consistent with stripe formation. These observations are interpreted as a precursor to the stripe phase, which is characterized by interleaved charge and spin density wave ordering with fluctuating lines of dopants separating domains of opposite antiferromagnetic order 12–14 .
Studying the low-temperature phases of doped Hubbard models, including the emergence of collective order and exotic normal phases, is at the heart of strongly correlated physics. This thesis offers new theoretical insights into the low-energy physics of doped Hubbard models, with a particular emphasis on leveraging quantum simulation as a powerful investigative tool. We explore the emergence of collective order driven by many-body interactions, by predicting and observing stripe-like structures in quantum gas microscopes of engineered Hamiltonians. These fluctuating stripe patterns are developed into a theoretical framework for the pseudogap— a highly enigmatic phase in hole-doped cuprates—where we show how fluctuating domain walls yield effective toric code descriptions. Ultracold atoms in optical lattices provide a unique platform for directly probing and testing this theory. We further investigate unconventional bilayer nickelate superconductors, predicting an exceptionally stable superfluid state in a single-band Hubbard model at experimentally accessible temperatures. By proposing schemes to observe coherent pair-pair correlations, this advances the long-standing goal of realizing and observing long-range superconducting order in ultracold atomic systems in optical lattices. On non-bipartite lattices, we examine kinetic magnetism in Hubbard models and moiré heterostructures. Additionally, the low-temperature regime of doped Hubbard models with enhanced symmetries is analyzed, revealing exotic phenomena such as sub-dimensional polaronic particles. We develop machine learning techniques to extract key physical insights from many-body snapshots, offering a new avenue for understanding intricate quantum phases. Finally, we go beyond the Fermi-Hubbard paradigm and show how non-local strongly correlated models can be leveraged to address classical optimization problems through quantum annealing.
We study the electronic structure within a system of phase-decoupled one-dimensional superconductors coexisting with stripe spin and charge-density-wave order. This system has a nodal Fermi surface (Fermi arc) in the form of a hole pocket and an antinodal pseudogap. The spectral function in the antinodes is approximately particle-hole symmetric contrary to the gapped regions just outside the pocket. We find that states at the Fermi energy are extended whereas states near the pseudogap energy have localization lengths as short as the interstripe spacing. We consider pairing which has either local ddd-wave or sss-wave symmetry and find similar results in both cases, consistent with the pseudogap being an effect of local pair correlations. We suggest that this state is a stripe-ordered caricature of the pseudogap phase in underdoped cuprates with coexisting spin-, charge-, and pair-density-wave correlations. Lastly, we also model a superconducting state which (1) evolves smoothly from the pseudogap state, (2) has a signature subgap peak in the density of states, and (3) has the coherent pair density concentrated to the nodal region.
Abstract Several state-of-the-art numerical methods have observed static or fluctuating spin and charge stripes in doped two-dimensional Hubbard models, suggesting that these orders play a significant role in shaping the cuprate phase diagram. Many experiments, however, also indicate that the cuprates have strong electron-phonon (e-ph) coupling, and it is unclear how this interaction influences stripe correlations. We study static and fluctuating stripe orders in the doped single-band Hubbard-Holstein model using zero temperature variational Monte Carlo and finite temperature determinant quantum Monte Carlo. We find that the lattice couples more strongly with the charge component of the stripes, leading to an enhancement or suppression of stripe correlations, depending on model parameters like the next-nearest-neighbor hopping t′{t}^{{\prime} }t′ t′ or phonon energy Ω. Our results help elucidate how thee-ph interaction can tip the delicate balance between stripe and superconducting correlations in the Hubbard-Holstein model with implications for our understanding of the high-Tccuprates.
Connecting single-layer ttt-JJJ to Kondo lattice models: Exploration with cold atoms
The Kondo effect, a hallmark of many-body physics, emerges from the antiferromagnetic coupling between localized spins and conduction fermions, leading to a correlated many-body singlet state. Here we propose to use the mixed-dimensional (mixD) bilayer Hubbard geometry as a platform to study Kondo lattice physics with current ultracold atom experiments. At experimentally feasible temperatures, we predict that key features of the Kondo effect can be observed, including formation of the Kondo cloud around a single impurity and the competition of singlet formation with Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions for multiple impurities, summarized in the Doniach phase diagram. Moreover, we show that the mixD platform provides a natural bridge between the Doniach phase diagram of the Kondo lattice model, relevant to heavy-fermion materials, and the phase diagram of cuprate superconductors as described by a single-layer Zhang-Rice type ttt-JJJ model: It is possible to continuously tune between the two regimes by changing the interlayer Kondo coupling. Our findings demonstrate that the direct connection between high-temperature superconductivity and heavy-fermion physics can be experimentally studied using currently available quantum simulation platforms.
The simulation of high-temperature superconducting materials by implementing strongly correlated fermionic models in optical lattices is one of the major objectives in the field of analog quantum simulation. Here we show that local control and optical bilayer capabilities combined with spatially resolved measurements create a versatile toolbox to study fundamental properties of both nickelate and cuprate high-temperature superconductors. On the one hand, we present a scheme to implement a mixed-dimensional (mixD) bilayer model that has been proposed to capture the essential pairing physics of pressurized bilayer nickelates. This allows for the long-sought realization of a state with long-range superconducting order in current lattice quantum simulation machines. In particular, we show how coherent pairing correlations can be accessed in a partially particle-hole transformed and rotated basis. On the other hand, we demonstrate that control of local gates enables the observation of d -wave pairing order in the two-dimensional (single-layer) repulsive Fermi-Hubbard model through the simulation of a system with attractive interactions. Lastly, we introduce a scheme to measure momentum-resolved dopant densities, providing access to observables complementary to solid-state experiments—which is of particular interest for future studies of the enigmatic pseudogap phase appearing in cuprates. Published by the American Physical Society 2024
Competing orders represent a central challenge in understanding strongly correlated systems. In this work, we employ projector quantum Monte Carlo simulations to study a sign-problem-free bilayer extended Hubbard model. In this model, a charge stripe phase, characterized by a peak at (0±2πδ,0) is induced by highly anisotropic interlayer antiferromagnetic spin-exchange coupling Jz, and strongly suppressed by the decreasing spin-exchange anisotropy (i.e., introducing the spin-flip term J⊥); in contrast, the introduction of J⊥ favors the emergence of interlayer pairing superconductivity. We further demonstrate that the anisotropy of the interlayer spin-exchange directly governs the competition between these two phases, while the on-site interaction U plays a complex role in tuning both the charge stripe and superconductivity. Furthermore, against the background of charge stripes we discover the spin stripes, characterized by twice the period of charge stripes, indicating the profound internal connection between them. Our work identifies the key factors driving charge stripe formation, highlights the sensitivity of both the charge stripe and superconducting phases to interaction parameters, and thereby provides valuable insights into competing orders in strongly correlated systems.