Observation of antiferromagnetic correlations in the Hubbard model with ultracold atoms

Abstract

Ultracold atoms in optical lattices have great potential to contribute to a better understanding of some of the most important issues in many-body physics, such as high-temperature (high-Tc) superconductivity [1]. Thirty years ago, Anderson suggested that the Hubbard model, a simplified representation of fermions moving on a periodic lattice, may contain the essence of copper oxide superconductivity [2]. The Hubbard model describes many of the features shared by the copper oxides, including an interaction-driven Mott insulating state and an antiferromagnetic (AFM) state. Optical lattices filled with a two-spin-component Fermi gas of ultracold atoms can faithfully realise the Hubbard model with readily tunable parameters, and thus provide a platform for the systematic exploration of its phase diagram [3,4]. Realisation of strongly correlated phases, however, has been hindered by the need to cool the atoms to temperatures as low as the magnetic exchange energy, and also by the lack of reliable thermometry [5]. Here we demonstrate spin-sensitive Bragg scattering of light to measure AFM spin correlations in a realisation of the three-dimensional (3D) Hubbard model at temperatures down to 1.4 times that of the AFM phase transition. This temperature regime is beyond the range of validity of a simple high-temperature series expansion, which brings our experiment close to the limit of the capabilities of current numerical techniques. We reach these low temperatures using a unique compensated optical lattice technique [6], in which the confinement of each lattice beam is compensated by a blue-detuned laser beam. The temperature of the atoms in the lattice is deduced by comparing the light scattering to determinantal quantum Monte Carlo [7] (DQMC) and numerical linked-cluster expansion [8] (NLCE) calculations. Further refinement of the compensated lattice may produce even lower temperatures which, along with light scattering thermometry, would open avenues for achieving and characterising other novel quantum states of matter, such as the pseudogap regime of the 2D Hubbard model.A two-spin-component Fermi gas in a simple cubic optical lattice may be described by a single-band Hubbard model with nearest-neighbour tunnelling t and on-site interaction U > 0. At a density n of one atom per site, and for sufficiently large U/t there is a crossover from a 'metallic' state to a Mott insulating regime [9] as the temperature T is reduced below U . The Mott regime has been demonstrated with ultracold atoms in an optical lattice by observing the reduction of doubly occupied sites [10] and the related reduction of the global compressibility [11]. For T below the Néel ordering temperature T N , which for U t is approximately equal to the exchange energy J = 4t 2 /U , the system undergoes a phase transition to an AFM state [12]. In the context of quantum simulations, AFM phases of Ising spins have been previously engineered with bosonic atoms in an optical lattice [13] and with spin-1 2 ions [14,15]. Also, nearest-neighbour AFM correlat...

A cold-atom Fermi–Hubbard antiferromagnet

“…Only at δ 0. 15 the high-temperature superconducting state [3]. But, unlike the cuprates, the system realized in our experiment is particle-hole symmetric and we expect the same critical value for particle doping.…”

“…In a square lattice, antiferromagnetic LRO manifests as a peak in the structure factor at q AFM = (π/a, π/a), whose amplitude is directly related to the staggered magnetization m z = S z (q AFM )/N . For cold atom systems the spin structure factor can be measured from noise correlations or Bragg scattering of light [15]. The site-resolved detection in our experiment allows for a direct measurement of the spin structure factor, obtained from averaging the squared Fourier transformation of individual single-spin images (Methods).…”

“…Ultracold fermions in optical lattices offer the potential to answer open questions on the lowtemperature regime of the doped Hubbard model [9][10][11], which is thought to capture essential aspects of the cuprate superconductor phase diagram but is numerically intractable in that parameter regime. Already, Mott-insulating phases and short-range antiferromagnetic correlations have been observed, but temperatures were too high to create an antiferromagnet [12][13][14][15]. A new perspective is afforded by quantum gas microscopy [16][17][18][19][20][21][22][23][24][25][26][27][28], which allows readout of magnetic correlations at the site-resolved level [25][26][27][28].…”

Spin and charge modulations in a single-hole-doped Hubbard ladder: Verification with optical lattice experiments

“…As a result, comparison between theory and experiment is not straightforward at times. On the other hand, Hubbard models can now be engineered in cold-atom experiments, with easy control of density and interaction [1][2][3][4][5][6]. In principle, one can obtain the solution of the Hubbard model by quantum simulation, i.e., finding the nature of the ground state directly from experiments.…”

Realizing exactly solvableSU(N)magnets with thermal atoms

“…While other approaches exist [9][10][11][12], the most common approach taken to implement a quantum spin model with ultracold atoms relies on preparing a Mott insulator in an optical lattice, where the internal states of atoms on each site define the effective spin [1,[13][14][15][16][17][18][19]. Virtual hopping processes to neighboring sites and back then give rise to effective superexchange spin-spin interactions.…”