Realizing the Emery Model in Optical Lattices for Quantum Simulation of Cuprates and Nickelates
Hannah Lange, Liyang Qiu, Robin Groth, Andreas von Haaren, Luca Muscarella, Titus Franz, Immanuel Bloch, Fabian Grusdt, Philipp M. Preiss, Annabelle Bohrdt
cond-mat.quant-gascond-mat.str-elphysics.atom-phquant-phSubmitted Mar 11, 2026• Announced Mar 12, 2026• 0 citations
The microscopic origin of high-temperature superconductivity in cuprates remains one of the central open questions in condensed matter physics. Growing experimental and theoretical evidence suggests that the bare single-band Fermi-Hubbard model may not fully capture properties of cuprates such as superconductivity, motivating us to revisit the canonical three-band model of the copper-oxide planes - the Emery model - from which the single-band counterpart was originally derived. Here, we propose and analyze a quantum simulation scheme for realizing the Emery model in regimes relevant to cuprates and infinite-layer nickelates with today's ultracold atom quantum simulation platforms, enabling the exploration of the three-band physics on system sizes that are challenging for current numerical methods. Specifically, we show that a two-dimensional optical lattice with a superimposed pattern of repulsive potentials can be designed to study low-temperature properties for variable parameter regimes of the Emery model relevant to cuprates as well as infinite-layer nickelates. Our results pave the way for real material simulations with ultracold atom quantum simulators and a better understanding of the physics of unconventional superconductors.
The microscopic origin of high-temperature superconductivity in cuprates remains one of the central open questions in condensed matter physics. Growing experimental and theoretical evidence suggests that the bare single-band Fermi-Hubbard model may not fully capture properties of cuprates such as superconductivity, motivating us to revisit the canonical three-band model of the copper-oxide planes - the Emery model - from which the single-band counterpart was originally derived. Here, we propose and analyze a quantum simulation scheme for realizing the Emery model in regimes relevant to cuprates and infinite-layer nickelates with today's ultracold atom quantum simulation platforms, enabling the exploration of the three-band physics on system sizes that are challenging for current numerical methods. Specifically, we show that a two-dimensional optical lattice with a superimposed pattern of repulsive potentials can be designed to study low-temperature properties for variable parameter regimes of the Emery model relevant to cuprates as well as infinite-layer nickelates. Our results pave the way for real material simulations with ultracold atom quantum simulators and a better understanding of the physics of unconventional superconductors.
The microscopic origin of high-temperature superconductivity in cuprates remains one of the central open questions in condensed matter physics. Growing experimental and theoretical evidence suggests that the bare single-band Fermi-Hubbard model may not fully capture properties of cuprates such as superconductivity, motivating us to revisit the canonical three-band model of the copper-oxide planes - the Emery model - from which the single-band counterpart was originally derived. Here, we propose and analyze a quantum simulation scheme for realizing the Emery model in regimes relevant to cuprates and infinite-layer nickelates with today's ultracold atom quantum simulation platforms, enabling the exploration of the three-band physics on system sizes that are challenging for current numerical methods. Specifically, we show that a two-dimensional optical lattice with a superimposed pattern of repulsive potentials can be designed to study low-temperature properties for variable parameter regimes of the Emery model relevant to cuprates as well as infinite-layer nickelates. Our results pave the way for real material simulations with ultracold atom quantum simulators and a better understanding of the physics of unconventional superconductors.
Strongly-correlated electrons in transition-metal oxides give rise to intriguing emergent phenomena, including high-temperature superconductivity in cuprates. While simplified one-band Hubbard models capture some aspects, explicitly describing the interplay of copper and oxygen orbitals -- as in the three-band Emery model -- is essential to capture the full phenomenology of cuprates. Quantum simulators based on ultracold atoms offer a promising route to study such systems in a controlled setting, but realizing realistic multi-orbital Hubbard models remains challenging. Here we propose an optical superlattice architecture that implements the three-band Emery model with ultracold fermions. By combining lattice beams with controllable interference, we engineer orbital degrees of freedom that reproduce key features of the cuprate band structure, while enabling independent control of orbital-dependent interactions and charge-transfer energy. We show that single-particle quantum walks can benchmark the resulting tight-binding model. Using determinant quantum Monte Carlo, we further investigate thermodynamic properties in the undoped regime and find a finite-temperature metal-insulator crossover accompanied by the onset of antiferromagnetic correlations accessible in current experiments. Finally, we apply a Hamiltonian learning protocol enabling to infer effective single-band Hubbard models from experimental realizations of Emery models. Our results provide a practical pathway to simulate multi-orbital Hubbard physics with quantum gas microscopes.
Understanding the physics underlying high-temperature superconductivity in cuprates and, more recently, infinite-layer nickelates has remained a central challenge in condensed-matter physics. We establish neural quantum states (NQS), specifically Hidden Fermion Determinant States (HFDS), as a scalable variational approach to the three-band Emery model of the copper- and nickel-oxide layers in these materials. After benchmarking HFDS against matrix product states on width-two geometries, we study ground states of fully two-dimensional (2D) systems of up to 10×10 unit cells (300 sites). We characterize the momentum-space distribution of dopants and find a pronounced electron-hole dichotomy similar to cuprate experiments. We further downfold the three-band model to effective single-band descriptions by constructing interacting Wannier functions. We consider a wide range of parameters -- from the charge-transfer regime relevant to cuprates to the Hubbard-Mott regime of nickelates, as well as systematic scans of the charge-transfer gap that has been demonstrated to impact the critical superconducting temperatures. Across all regimes, the effective model significantly deviates from the usual Fermi-Hubbard model: The typical ratio U/t is enhanced, some parameters experience a significant doping dependence, and sizable terms beyond the conventional Hubbard model are present, most notably a density-assisted hopping tn. Notably, in all effective models, tn has the largest contribution to particle-hole asymmetry, rather than next-nearest-neighbor hopping contributions. The effective parameters sensitively depend on the charge-transfer energy, doping, and interaction ratios. Our results establish HFDS as an efficient tool for studying the 2D Emery model and demonstrate that single-band descriptions can require interaction terms generated by the underlying multi-band models.
Strongly correlated materials exhibit complex emergent phenomena -- such as unconventional high-temperature superconductivity -- that cannot be explained within a single or weakly-interacting particle description, making understanding their physics a central challenge in condensed matter physics. This thesis develops and applies novel approaches to strongly correlated electron systems, focusing on neural quantum states (NQS) and quantum simulation with cold atoms in optical lattices or Rydberg tweezer arrays. A key contribution of this work is the advancement of neural-network-based representations of quantum many-body states. By employing state-of-the-art network architectures like Transformers and combining them with physical constructions such as determinants, Pfaffians and Gutzwiller-projection, NQS are extended to simulate large two-dimensional fermionic single- and multi-band systems with high accuracy and beyond the capabilities of conventional methods. In addition, the thesis explores how quantum simulators based on ultracold atoms in optical lattices can be leveraged to study models directly relevant to strongly correlated materials -- e.g. the multi-band model of cuprate superconductors -- as well as proposes strategies to probe superconductivity in this setting. Furthermore, hybrid approaches that combine NQS with data from quantum simulators are developed. This synergy allows for improved optimization and direct comparison between theory and experiment. These methodological developments are applied to investigate two fundamental questions in high-temperature superconductivity: On the one hand, the work proposes a microscopic mechanism based on Feshbach resonances and tests this hypothesis extensively in a model system relevant to cuprates, nickelates and heavy fermion materials. On the other hand, it develops systematic approaches to derive effective low-energy models from more complex systems like multi-band models or electron-phonon coupled systems, shining light on the timely question which single-band models capture the physics of high-temperature superconductors like cuprate or nickelate materials. Overall, the thesis demonstrates how combining machine learning and quantum simulation provides powerful tools for understanding strongly correlated quantum matter.
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
An outstanding problem in physics is how to understand strongly interacting quantum many-body systems such as the quark-gluon plasma, neutron stars, superfluid 4He, and the high-temperature superconducting cuprates. The physics approach to this problem is to reduce these complex systems to minimal models that are believed to retain relevant phenomenology. For example, the Hubbard model — the focus of this thesis — describes quantum particles tunneling between sites of a lattice with on-site interactions. The Hubbard model is conjectured to describe the low-energy charge and spin properties of high-temperature superconducting cuprates. Thus far, there are no analytic solutions to the Hubbard model, and numerical calculations are difficult and even impossible in some regimes (e.g., the Fermi-Hubbard model away from half-filling). Therefore, whether the Hubbard model is a minimal model for the cuprates remains unresolved. In the face of these difficulties, a new approach has emerged — quantum simulation. The premise of quan-tum simulation is to perform experiments on a quantum system that is well-described by the model we are trying to study, has tunable parameters, and is easily probed. Ultracold atoms trapped in optical lattices are an ideal candidate for quantum simulation of the Hubbard models. This thesis describes work on two such systems: a 87Rb (boson) optical lattice experiment in the group of Brian DeMarco at the University