The subtle interplay between kinetic energy, interactions, and dimensionality challenges our comprehension of strongly correlated physics observed, for example, in the solid state.In this quest, the Hubbard model has emerged as a conceptually simple, yet rich model describing such physics.Here we present an experimental determination of the equation of state of the repulsive two-dimensional Hubbard model over a broad range of interactions 0 ≲ U=t ≲ 20 and temperatures, down to k B T=t ¼ 0.63ð2Þ using highresolution imaging of ultracold fermionic atoms in optical lattices.We show density profiles, compressibilities, and double occupancies over the whole doping range, and, hence, our results constitute benchmarks for state-of-the-art theoretical approaches.The subtle interplay between kinetic energy, interactions, and dimensionality challenges our comprehension of strongly correlated physics observed, for example, in the solid state.In this quest, the Hubbard model has emerged as a conceptually simple, yet rich model describing such physics.Here we present an experimental determination of the equation of state of the repulsive two-dimensional Hubbard model over a broad range of interactions 0 ≲ U=t ≲ 20 and temperatures, down to k B T=t ¼ 0.63ð2Þ using highresolution imaging of ultracold fermionic atoms in optical lattices.We show density profiles, compressibilities, and double occupancies over the whole doping range, and, hence, our results constitute benchmarks for state-of-the-art theoretical approaches.

PhysicsApr 25, 20165 citations

Compressibility of a Fermionic Mott Insulator of Ultracold Atoms

Pedro M. Duarte, Russell Hart, Tsung‐Lin Yang +6 more

We characterize the Mott insulating regime of a repulsively interacting Fermi gas of ultracold atoms in a three-dimensional optical lattice. We use in situ imaging to extract the central density of the gas and to determine its local compressibility. For intermediate to strong interactions, we observe the emergence of a plateau in the density as a function of atom number, and a reduction of the compressibility at a density of one atom per site, indicating the formation of a Mott insulator. Comparisons to state-of-the-art numerical simulations of the Hubbard model over a wide range of interactions reveal that the temperature of the gas is of the order of, or below, the tunneling energy scale. Our results hold great promise for the exploration of many-body phenomena with ultracold atoms, where the local compressibility can be a useful tool to detect signatures of different phases or phase boundaries at specific values of the filling.We characterize the Mott insulating regime of a repulsively interacting Fermi gas of ultracold atoms in a three-dimensional optical lattice. We use in situ imaging to extract the central density of the gas and to determine its local compressibility. For intermediate to strong interactions, we observe the emergence of a plateau in the density as a function of atom number, and a reduction of the compressibility at a density of one atom per site, indicating the formation of a Mott insulator. Comparisons to state-of-the-art numerical simulations of the Hubbard model over a wide range of interactions reveal that the temperature of the gas is of the order of, or below, the tunneling energy scale. Our results hold great promise for the exploration of many-body phenomena with ultracold atoms, where the local compressibility can be a useful tool to detect signatures of different phases or phase boundaries at specific values of the filling.

arXiv:1409.8348Physical Review LettersFeb 20, 201571 citations

Equation of state of the fermionic two-dimensional Hubbard model

J. P. F. LeBlanc, Emanuel Gull

We present results for the equation of state of the two-dimensional Hubbard model on an isotropic square lattice as obtained from a controlled and numerically exact large-cluster dynamical mean field simulation. Our results are obtained for large but finite systems and are extrapolated to infinite system size using a known finite-size scaling relation, and are supplemented by reliable error bars accounting for all sources of errors. We establish the importance of examining the decay of spatial spin correlations to determine whether a sufficiently large cluster has been used and with this in mind we present the energy, entropy, double occupancy, and nearest-neighbor spin correlations extrapolated to the thermodynamic limit. We discuss the implications of these calculations on pseudogap physics of the 2D Hubbard model away from half filling, where we find a strong behavioral shift in energy below a temperature ${T}^{*}$ which becomes more pronounced for larger clusters. Finally, we provide reference calculations and tables for the equation of state for values of doping away from half filling which are of interest to cold-atom experiments.We present results for the equation of state of the two-dimensional Hubbard model on an isotropic square lattice as obtained from a controlled and numerically exact large-cluster dynamical mean field simulation. Our results are obtained for large but finite systems and are extrapolated to infinite system size using a known finite-size scaling relation, and are supplemented by reliable error bars accounting for all sources of errors. We establish the importance of examining the decay of spatial spin correlations to determine whether a sufficiently large cluster has been used and with this in mind we present the energy, entropy, double occupancy, and nearest-neighbor spin correlations extrapolated to the thermodynamic limit. We discuss the implications of these calculations on pseudogap physics of the 2D Hubbard model away from half filling, where we find a strong behavioral shift in energy below a temperature ${T}^{*}$ which becomes more pronounced for larger clusters. Finally, we provide reference calculations and tables for the equation of state for values of doping away from half filling which are of interest to cold-atom experiments.

arXiv:1305.6798Physical Review BOct 7, 201355 citations

Quantum simulation of the Hubbard model with ultracold fermions in optical lattices

Leticia Tarruell, Laurent Sanchez-Palencia

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.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.

arXiv:1809.00571Comptes Rendus PhysiqueSep 1, 2018176 citations

Two-dimensional Hubbard model at finite temperature: Weak, strong, and long correlation regimes

Fedor Šimkovic, Riccardo Rossi, Michel Ferrero

We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature, and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at temperatures as low as $T=0.067$ and interactions up to $U=7$ , while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature, and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at temperatures as low as $T=0.067$ and interactions up to $U=7$, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.

arXiv:2110.05863Physical Review ResearchDec 20, 202245 citations

Pair correlations in the attractive Hubbard model

Chan, C. F., Marcell Gall, Nicola Wurz +1 more

The mechanism of fermionic pairing is the key to understanding various phenomena such as high-temperature superconductivity and the pseudogap phase in cuprate materials. We study the pair correlations in the attractive Hubbard model using ultracold fermions in a two-dimensional optical lattice. By combining the fluctuation-dissipation theorem and the compressibility equation of state, we extract the interacting pair correlation functions and deduce a characteristic length scale of pairs as a function of interaction and density filling. At sufficiently low filling and weak on-site interaction, we observe that the pair correlations extend over a few lattice sites even at temperatures above the superfluid transition temperature.The mechanism of fermionic pairing is the key to understanding various phenomena such as high-temperature superconductivity and the pseudogap phase in cuprate materials. We study the pair correlations in the attractive Hubbard model using ultracold fermions in a two-dimensional optical lattice. By combining the fluctuation-dissipation theorem and the compressibility equation of state, we extract the interacting pair correlation functions and deduce a characteristic length scale of pairs as a function of interaction and density filling. At sufficiently low filling and weak on-site interaction, we observe that the pair correlations extend over a few lattice sites even at temperatures above the superfluid transition temperature.

arXiv:2005.09121May 18, 20201 citation

{T}^{*}

which becomes more pronounced for larger clusters. Finally, we provide reference calculations and tables for the equation of state for values of doping away from half filling which are of interest to cold-atom experiments.

T=0.067

and interactions up to

U=7

, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.