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Science

Tackling the Hubbard model

Jelena Stajic

Published Aug 12, 2021• 0 citations
Publisher

Abstract

Physics The Hubbard model describes the physics of interacting particles on a lattice and is thought to contain elements essential to the superconductivity of the cuprates. Despite its apparent simplicity, solving the Hubbard model remains extremely challenging, especially at finite concentrations of hole dopants and intermediate temperatures. Wietek et al. used the minimally entangled typical thermal states (METTS) method to calculate the structural factors and thermodynamic properties for the two-dimensional Hubbard model with strong correlations at a finite doping and for a range of temperatures. The researchers found that a stripe order formed at low temperatures, whereas at higher temperatures, a phase resembling the experimentally observed pseudogap state took over. Phys. Rev. X. 11 , 0031007 (2021).

DOI
10.1126/science.373.6556.755-d

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The Hubbard model describes the physics of interacting particles on a lattice and is thought to contain elements essential to the superconductivity of the cuprates. Despite its apparent simplicity, solving the Hubbard model remains extremely challenging, especially at finite concentrations of hole dopants and intermediate temperatures. Wietek et al. used the minimally entangled typical thermal states (METTS) method to calculate the structural factors and thermodynamic properties for the two-dimensional Hubbard model with strong correlations at a finite doping and for a range of temperatures. The researchers found that a stripe order formed at low temperatures, whereas at higher temperatures, a phase resembling the experimentally observed pseudogap state took over.
Phys. Rev. X. 11 , 0031007 (2021).
The two-dimensional Hubbard model is one of the most fundamental models in condensed matter physics. It captures many properties of cuprate high-temperature superconductors. Despite its apparent simplicity, solving the Hubbard model is a challenging task. In this work, we employ the Hartree-Fock approximation, as well as functional renormalization group methods, to construct phase diagrams of the Hubbard model and to shed light on the zoo of magnetic, charge ordered, and superconducting states. The Hartree-Fock approximation can self-consistently determine the expectation values of local charge and magnetic order parameters. Mean-field theories like this can lead to a deeper understanding of the competition, coexistence and transition of different orders. We start by employing the Hartree-Fock approximation in a completely unrestricted form on finite-size lattices and complement the results with calculations in the thermodynamic limit. We construct mean-field phase diagrams of the two-dimensional Hubbard model in the thermodynamic limit for different values of the next-nearest neighbor hopping as a function of filling and temperature. We find that the dominant magnetic orders at moderate coupling strength are Néel order around half-filling, followed by spiral order, and stripe order as the hole doping is increased. We also clarify the nature of the transition from magnetic spiral to stripe order. We then extend the Hartree-Fock scheme by a functional renormalization group flow, which generates new effective interactions and allows us to also access the superconducting phase of the Hubbard model. We present an improved phase diagram, featuring Néel, spiral, and stripe order coexisting with d-wave superconductivity at low temperatures. We analyze the coexistence and competition between these orders. Finally, we perform unrestricted Hartree-Fock calculations of a Hubbard model on the graphene lattice and the Bernal bilayer graphene lattice. We also construct magnetic phase diagrams for these geometries, which feature Néel order around half-filling and various stripe orders in a secondary magnetic dome.
The Hubbard model is an iconic model in quantum many-body physics and has been intensely studied, especially since the discovery of high-temperature cuprate superconductors. Combining the complementary capabilities of two computational methods, we found superconductivity in both the electron- and hole-doped regimes of the two-dimensional Hubbard model with next-nearest-neighbor hopping. In the electron-doped regime, superconductivity was weaker and was accompanied by antiferromagnetic Néel correlations at low doping. The strong superconductivity on the hole-doped side coexisted with stripe order, which persisted into the overdoped region with weaker hole-density modulation. These stripe orders varied in fillings between 0.6 and 0.8. Our results suggest the applicability of the Hubbard model with next-nearest hopping for describing cuprate high–transition temperature ( T c ) superconductivity.
The relationship between the pseudogap and underlying ground-state phases has not yet been rigorously established. We investigated the doped two-dimensional Hubbard model at finite temperature using controlled diagrammatic Monte Carlo calculations, allowing for the computation of spectral properties in the infinite-size limit and with arbitrary momentum resolution. We found three distinct regimes as a function of doping and interaction strength: a weakly correlated metal, a correlated metal with strong interaction effects, and a pseudogap regime at low doping. We show that the pseudogap forms both at weak coupling, when the magnetic correlation length is large, and at strong coupling, when it is shorter. As the temperature goes to zero, the pseudogap regime extrapolates precisely to the ordered stripe phase found by ground-state methods.
We calculate the properties of the two-band Hubbard model using the dynamical cluster approximation. The phase diagram resembles the generic phase diagram of the cuprates, showing a strong asymmetry with respect to electron- and hole-doped regimes, in agreement with experiment. Asymmetric features are also seen in one-particle spectral functions and in the charge, spin, and ddd-wave pairing-susceptibility functions. We address the possible reduction of the two-band model to a low-energy single-band one, as it was suggested by Zhang and Rice. Comparing the two-band Hubbard model properties with the single-band Hubbard model ones, we have found similar low-energy physics provided that the next-nearest-neighbor hopping term ${t}^{\ensuremath{'}}$ has a significant value $({t}^{\ensuremath{'}}∕t\ensuremath{\approx}0.3)$. The parameter ${t}^{\ensuremath{'}}$ is the main culprit for the electron-hole asymmetry. However, a significant value of ${t}^{\ensuremath{'}}$ cannot be provided in a strict Zhang and Rice [Phys. Rev. B 37, R3759 (1988); 41, 7243 (1990)] picture where the extra holes added into the system bind to the existing Cu holes forming local singlets. We note that by considering approximate singlet states, such as plaquette states, reasonable values of ${t}^{\ensuremath{'}}$, which capture qualitatively the physics of the two-band model, can be obtained. We conclude that a single-band $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U$ Hubbard model captures the basic physics of the cuprates concerning superconductivity, antiferromagnetism, pseudogap, and electron-hole asymmetry, but is not suitable for a quantitative analysis or to describe physical properties involving energy scales larger than about 0.5 eV.
Abstract A microscopic understanding of the strongly correlated physics of the cuprates must account for the translational and rotational symmetry breaking that is present across all cuprate families, commonly in the form of stripes. Here we investigate emergence of stripes in the Hubbard model, a minimal model believed to be relevant to the cuprate superconductors, using determinant quantum Monte Carlo (DQMC) simulations at finite temperatures and density matrix renormalization group (DMRG) ground state calculations. By varying temperature, doping, and model parameters, we characterize the extent of stripes throughout the phase diagram of the Hubbard model. Our results show that including the often neglected next-nearest-neighbor hopping leads to the absence of spin incommensurability upon electron-doping and nearly half-filled stripes upon hole-doping. The similarities of these findings to experimental results on both electron and hole-doped cuprate families support a unified description across a large portion of the cuprate phase diagram.
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.