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arXiv:2006.02029 · Physical Review A

Thermodynamics and magnetism in the two-dimensional to three-dimensional crossover of the Hubbard model

Eduardo Ibarra-García-Padilla, Rick Mukherjee, Randall G. Hulet, Kaden R. A. Hazzard, Thereza Paiva, Richard T. Scalettar

Published Sep 29, 2020• 23 citations
arXivHTMLPDF

Abstract

The realization of antiferromagnetic (AF) correlations in ultracold fermionic atoms on an optical lattice is a significant achievement. Experiments have been carried out in one, two, and three dimensions, and have also studied anisotropic configurations with stronger tunneling in some lattice directions. Such anisotropy is relevant to the physics of cuprate superconductors and other strongly correlated materials. Moreover, this anisotropy might be harnessed to enhance AF order. Here we numerically investigate, using the determinant quantum Monte Carlo method, a simple realization of anisotropy in the three-dimensional (3D) Hubbard model in which the tunneling between planes, ${t}_{\ensuremath{\perp}}$, is unequal to the intraplane tunneling $t$. This model interpolates between the three-dimensional isotropic (${t}_{\ensuremath{\perp}}=t$) and two-dimensional (2D; ${t}_{\ensuremath{\perp}}=0$) systems. We show that at fixed interaction strength to tunneling ratio ($U/t$), anisotropy can enhance the magnetic structure factor relative to both 2D and 3D results. However, this enhancement occurs at interaction strengths below those for which the N\'eel temperature ${T}_{\mathrm{N}\stackrel{\ifmmode \acute{}\else \'{}\fi{}}{\mathrm{e}}\mathrm{el}}$ is largest, in such a way that the structure factor cannot be made to exceed its value in isotropic 3D systems at the optimal $U/t$. We characterize the 2D-3D crossover in terms of the magnetic structure factor, real space spin correlations, number of doubly occupied sites, and thermodynamic observables. An interesting implication of our results stems from the entropy's dependence on anisotropy. As the system evolves from 3D to 2D, the entropy at a fixed temperature increases. Correspondingly, at fixed entropy, the temperature will decrease going from 3D to 2D. This suggests a cooling protocol in which the dimensionality is adiabatically changed from 3D to 2D.

DOI
10.1103/physreva.102.033340

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We study the anisotropic 3D Hubbard model with increased nearest-neighbor tunneling amplitudes along one direction using the dynamical cluster approximation and compare the results to a quantum simulation experiment of ultracold fermions in an optical lattice. We find that the short-range spin correlations are significantly enhanced in the direction with stronger tunneling amplitudes. Our results agree with the experimental observations and show that the experimental temperature is lower than the strong tunneling amplitude. We characterize the system by examining the spin correlations beyond neighboring sites and determine the distribution of density, entropy, and spin correlation in the trapped system. We furthermore investigate the dependence of the critical entropy at the Néel transition on anisotropy.
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The fermionic Hubbard model (FHM)[1], despite its simple form, captures essential features of strongly correlated electron physics. Ultracold fermions in optical lattices[2, 3] provide a clean and well-controlled platform for simulating FHM. Doping its antiferromagnetic ground state at half filling, various exotic phases are expected to arise in the FHM simulator, including stripe order[4], pseudogap[5], and d-wave superconductors[6], offering valuable insights into high-temperature superconductivity[7{9]. Although notable progress, such as the observation of antiferromagnetic correlations over short[10] and extended distances[11], has been obtained, the antiferromagnetic phase has yet to be realized due to the significant challenges of achieving low temperatures in a large and uniform quantum simulator. Here, we report the observation of the antiferromagnetic phase transition in a three-dimensional fermionic Hubbard system comprising lithium-6 atoms in a uniform optical lattice with approximately 800,000 sites. When the interaction strength, temperature, and doping concentration are finely tuned to approach their respective critical values, sharp increases in the spin structure factor (SSF) are observed. These observations can be well described by a power-law divergence, with a critical exponent of 1.396 from the Heisenberg universality class[12]. At half filling and with optimal interaction strength, the measured SSF reaches 123(8), signifying the establishment of an antiferromagnetic phase. Our results set the stage for exploring the low-temperature phase diagram of FHM.
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