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arXiv:1706.07556 · Physical review. B./Physical review. B

Magnetic correlations in the two-dimensional repulsive Fermi-Hubbard model

Fedor Šimkovic, Youjin Deng, Nikolay V. Prokof'ev, Boris V. Svistunov, I. S. Tupitsyn, Evgeny Kozik

Published Aug 29, 2017• 17 citations
arXivHTMLPDF

Abstract

The repulsive Fermi-Hubbard model on a square lattice has a rich phase diagram near half-filling ($n=1$): at $n=1$ the ground state is an antiferromagnetic insulator, at $0.6<n\ensuremath{\lesssim}0.8$ the ground state is a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superfluid (at least for moderately strong interactions, $U\ensuremath{\lesssim}4$), and the region $1\ensuremath{-}n\ensuremath{\ll}1$ is likely subject to phase separation. Much less is known about the nature of strong magnetic fluctuations at finite temperature and how they change with the doping level. Recent experiments on ultracold atoms have now reached this interesting fluctuation regime. In this work we employ the skeleton diagrammatic method to quantify the characteristic temperature scale ${T}_{M}(n)$ for the onset of magnetic fluctuations with a large correlation length.

DOI
10.1103/physrevb.96.081117

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The repulsive Fermi-Hubbard model on a square lattice has a rich phase diagram near half-filling (n=1n=1n=1): at n=1n=1n=1 the ground state is an antiferromagnetic insulator, at 0.6<n\ensuremath≲0.80.6<n\ensuremath{\lesssim}0.80.6<n\ensuremath≲0.8 the ground state is a dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave superfluid (at least for moderately strong interactions, U\ensuremath≲4U\ensuremath{\lesssim}4U\ensuremath≲4), and the region 1\ensuremath−n\ensuremath≪11\ensuremath{-}n\ensuremath{\ll}11\ensuremath−n\ensuremath≪1 is likely subject to phase separation. Much less is known about the nature of strong magnetic fluctuations at finite temperature and how they change with the doping level. Recent experiments on ultracold atoms have now reached this interesting fluctuation regime. In this work we employ the skeleton diagrammatic method to quantify the characteristic temperature scale TM(n){T}_{M}(n)TM​(n) for the onset of magnetic fluctuations with a large correlation length.
The influence of spin and charge fluctuations on spectra of the two-dimensional fermionic Hubbard model is considered using the strong coupling diagram technique. Infinite sequences of diagrams containing ladder inserts, which describe the interaction of electrons with these fluctuations, are summed, and obtained equations are self-consistently solved for the ranges of Hubbard repulsions [Formula: see text], temperatures [Formula: see text] and electron concentrations [Formula: see text] with t the intersite hopping constant. For all considered U the system exhibits a transition to the long-range antiferromagnetic order at [Formula: see text]. At the same time no indication of charge ordering is observed. Obtained solutions agree satisfactorily with results of other approaches and obey moments sum rules. In the considered region of the U-T plane, the curve separating metallic solutions passes from [Formula: see text] at the highest temperatures to U = 2t at [Formula: see text] for half-filling. If only short-range fluctuations are allowed for the remaining part of this region is occupied by insulating solutions. Taking into account long-range fluctuations leads to strengthening of maxima tails, which transform a part of insulating solutions into bad-metal states. For low T, obtained results allow us to trace the gradual transition from the regime of strong correlations with the pronounced four-band structure and well-defined Mott gap for [Formula: see text] to the Slater regime of weak correlations with the spectral intensity having a dip along the boundary of the magnetic Brillouin zone due to an antiferromagnetic ordering for [Formula: see text]. For [Formula: see text] and [Formula: see text] doping leads to the occurrence of a pseudogap near the Fermi level, which is a consequence of the splitting out of a narrow band from a Hubbard subband. Obtained spectra feature waterfalls and Fermi arcs, which are similar to those observed in hole-doped cuprates.
The Kagome lattice Fermi-Hubbard model is one of the most physically rich, and at the same time most challenging, models to study in strongly-correlated physics. Among its special features are geometric frustration and a flat energy band that create conditions favorable to ferromagnetism near the band insulating limit. Here, we utilize two exact finite-temperature methods, the numerical linked-cluster expansion and the determinant quantum Monte Carlo, to study the extent as well as doping and interaction dependence of ferromagnetic correlations and other thermodynamic properties of the model. We find that repulsive interactions enhance ferromagnetic correlations at high electron densities and that increasing the interaction strength, extends the region with strong ferromagnetic correlations towards half filling, smoothly connecting it to Nagaoka ferromagnetism near the Mott insulating region. We also use the charge compressibility to obtain an accurate estimate for the critical interaction strength for the metal-insulator transition at half filling. These results improve our understanding of the magnetic tendencies of the model away from half filling and pave the way for further studies, including with ultracold atoms in optical lattices.
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Optical lattice experiments with ultracold fermion atoms and quantum gas microscopy have recently realized direct measurements of magnetic correlations at the site-resolved level. We calculate the short-range spin-correlation functions in the ground state of the two-dimensional repulsive Hubbard model with the auxiliary-field quantum Monte Carlo (AFQMC) method. The results are numerically exact at half filling where the fermion sign problem is absent. Away from half filling, we employ the constrained path AFQMC approach to eliminate the exponential computational scaling from the sign problem. The constraint employs unrestricted Hartree-Fock trial wave functions with an effective interaction strength UUU, which is optimized self-consistently within AFQMC. Large supercells are studied, with twist averaged boundary conditions as needed, to reach the thermodynamic limit. We find that the nearest-neighbor spin correlation always increases with the interaction strength UUU, contrary to the finite-temperature behavior where a maximum is reached at a finite UUU value. We also observe a change of sign in the next-nearest-neighbor spin correlation with increasing density, which is a consequence of the buildup of the long-range antiferromagnetic correlation. We expect the results presented in this paper to serve as a benchmark as lower temperatures are reached in ultracold atom experiments.
The cluster slave-spin method is employed to investigate systematically the ground state properties of the Hubbard model on a square lattice with doping $\ensuremath{\delta}$ and coupling strength UUU being its parameters. At half-filling, a relation between the staggered magnetization MMM and the antiferromagnetic (AFM) gap ${\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}$ is established in the small UUU limit to compare with that from the Hartree-Fock theory, and a first-order metal-insulator Mott transition in the paramagnetic state is substantiated, which is characterized by discontinuities and hystereses at UMott=10t{U}_{\text{Mott}}=10tUMott​=10t. The interaction Uc{U}_{c}Uc​ for the crossover in the AFM state, separating the weak- and strong-coupling regimes, is found to remain almost unchanged with large dopings, and smaller than UMott{U}_{\text{Mott}}UMott​ at half-filling because of long range AFM correlations. Finally, an overall phase diagram in the $U\ensuremath{-}\ensuremath{\delta}$ plane is presented, which is composed of four regimes: the AFM insulator at half-filling, the AFM metal with the compressibility $\ensuremath{\kappa}>0$ or $\ensuremath{\kappa}<0$, and the paramagnetic metal, as well as three phase transitions: (i) From the AFM metal to the paramagnetic metal, (ii) between the AFM metal phases with positive and negative $\ensuremath{\kappa}$, and (iii) separating the AFM insulating phase at $\ensuremath{\delta}=0$ from the AFM metal phase for $\ensuremath{\delta}>0$.
A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength (U=8t)(U=8t)(U=8t) and for moderate sizes (from $6\ifmmode\times\else\texttimes\fi{}6$ to $10\ifmmode\times\else\texttimes\fi{}10$). Our ansatz is superior to other widely used variational wave functions. The results for order parameters and correlation functions provide insight into the antiferromagnetic state at half-filling as well as strong evidence for a superconducting phase away from half-filling.