Spin and Charge Correlations across the Metal-to-Insulator Crossover in the Half-Filled 2D Hubbard Model

Abstract

The 2d Hubbard model with nearest-neighbour hopping on the square lattice and an average of one electron per site is known to undergo an extended crossover from metallic to insulating behavior driven by proliferating antiferromagnetic correlations. We study signatures of this crossover in spin and charge correlation functions and present results obtained with controlled accuracy using diagrammatic Monte Carlo in the range of parameters amenable to experimental verification with ultracold atoms in optical lattices. The qualitative changes in charge and spin correlations associated with the crossover are observed at well-separated temperature scales, which encase the intermediary regime of non-Fermi-liquid character, where local magnetic moments are formed and non-local fluctuations in both channels are essential.Recent developments of quantum emulators based on ultra-cold atoms loaded in an optical lattice [1-8] have enabled accurate experimental realization and probing of the quintessential single-band 2d Hubbard model of correlated electrons in solids:(1) where c xσ annihilates a fermion with spin σ on the site x, xy implies nearest-neighbour sites, n σ (x) = c † xσ c xσ is the corresponding number operator, t is the hopping amplitude (set to unity) between nearest-neighbor sites, U the on-site repulsion, and µ the chemical potential. Despite seeming simplicity, the model harbors extremely rich physics, including, e.g., unconventional [9] and possibly high-temperature superconductivity [10], while a priori accurate theoretical results in the thermodynamic limit are remarkably scarce [11].Central among properties of the Hubbard model is the state of the interaction-induced insulator at half-filling ( n ↑ + n ↓ = 1), when the non-interacting system is a metal. Here, an important ingredient is the tendency toward antiferromagnetic (AFM) ordering due to nesting of the Fermi surface (FS), i.e. the existence of a single wavevector Q = (π, π) that connects any point on the FS to another point on the FS. At U/t 1, an exponentially small ∼ t exp(−2π t/U ) energy gap in charge excitations emerges due to an exponential increase of the AFM correlation length [12], despite the absence of longrange order at any T > 0 [13,14]. At U/t 1, the charge gap ∼ U/2 is due to on-site repulsion, while AFM correlations develop at much smaller scales ∼ 4t 2 /U and are irrelevant for the insulator. This drastic qualitative difference between the limiting cases-a local scenario at strong coupling versus that local in the momentum space at weak coupling-makes physics at intermediate U ∼ t particularly intriguing and challenging to describe reliably.When extended AFM correlations are explicitly suppressed, a Mott insulator is expected to emerge by a first-order metal-to-insulator transition [15][16][17][18][19][20][21][22][23][24][25]. In the 2d Hubbard model (1) currently realized in experiments, extending correlations make the insulator develop in a smooth crossover [26][27][28][29]. Recent controlled results [29] by diagrammatic determinan...

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