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arXiv:1904.10782 · Physical review. B./Physical review. B

Magnetic susceptibility and simulated neutron signal in the two-dimensional Hubbard model

J. P. F. LeBlanc, Shaozhi Li, Xi Chen, Ryan Levy, Andrey E. Antipov, Andrew J. Millis, Emanuel Gull

Published Aug 12, 2019• 20 citations
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Abstract

We compute dynamic spin susceptibilities in the two-dimensional Hubbard model using the method of dual fermions, and we provide a comparison to lattice Monte Carlo and cluster dynamical mean-field theory. We examine the energy dispersion identified by peaks in $\mathrm{Im}\ensuremath{\chi}(\ensuremath{\omega},q)$, which define spin modes, and we compare the exchange scale and magnon dispersion to neutron experiments on the parent ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$ cuprate. We present the evolution of the spin excitations as a function of Hubbard interaction strengths and doping, and we explore the particle-hole asymmetry of the spin excitations. We also study the correlation lengths and the spin excitation dispersion peak structure, and we find a Y-shaped dispersion similar to neutron results on doped ${\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4+\ensuremath{\delta}}$.

DOI
10.1103/physrevb.100.075123

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We compute dynamic spin susceptibilities in the two-dimensional Hubbard model using the method of dual fermions, and we provide a comparison to lattice Monte Carlo and cluster dynamical mean-field theory. We examine the energy dispersion identified by peaks in Im\ensuremathχ(\ensuremathω,q)\mathrm{Im}\ensuremath{\chi}(\ensuremath{\omega},q)Im\ensuremathχ(\ensuremathω,q), which define spin modes, and we compare the exchange scale and magnon dispersion to neutron experiments on the parent La2CuO4{\mathrm{La}}_{2}{\mathrm{CuO}}_{4}La2​CuO4​ cuprate. We present the evolution of the spin excitations as a function of Hubbard interaction strengths and doping, and we explore the particle-hole asymmetry of the spin excitations. We also study the correlation lengths and the spin excitation dispersion peak structure, and we find a Y-shaped dispersion similar to neutron results on doped HgBa2CuO4+\ensuremathδ{\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4+\ensuremath{\delta}}HgBa2​CuO4+\ensuremathδ​.
We study the doping evolution of spin excitations in a one-dimensional (1D) Hubbard model and its downfolded spin Hamiltonians, by using exact diagonalization combined with cluster perturbation theory. In all models we observe hardening (softening) of spin excitations upon electron (hole) doping, which are reminiscent of recent experiments on two-dimensional (2D) cuprate materials. We also find that the three-site and even higher-order terms are crucial for the low-energy effective spin models to reproduce the magnetic spectra of doped Hubbard systems at a quantitative level. To interpret the numerical results, we further employ a strong coupling slave-boson mean-field theory. The mean-field theory provides an intuitive understanding of the overall compact support of dynamic spin structure factors, including the shift of zero-energy modes and change of spin excitation bandwidth with doping. Our results can serve as predictive benchmarks for future inelastic x-ray or neutron scattering experiments on doped 1D antiferromagnetic Mott insulators.
A dynamic cluster quantum Monte Carlo algorithm is used to study a spin-susceptibility representation of the pairing interaction for the two-dimensional Hubbard model with an on-site Coulomb interaction equal to the bandwidth for various doping levels. We find that the pairing interaction is well approximated by $\frac{3}{2}\overline{U}{(T)}^{2}\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$, with an effective temperature and doping dependent coupling U‾(T)\overline{U}(T)U(T) and the numerically calculated spin susceptibility $\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$. We show that at low temperatures, U‾\overline{U}U may be accurately determined from a corresponding spin-susceptibility based calculation of the single-particle self-energy. We conclude that the strength of the ddd-wave pairing interaction, characterized by the mean-field transition temperature, can be determined from a knowledge of the dressed spin susceptibility and the nodal quasiparticle spectral weight. This has important implications with respect to the questions of whether spin fluctuations are responsible for pairing in the high-Tc{T}_{c}Tc​ cuprates.
We investigate the properties of the frustrated underdoped Hubbard model on the square lattice using two complementary approaches, i.e., the dynamical cluster extension of the dynamical mean-field theory and the variational Monte Carlo simulations of Gutzwiller-Jastrow wave functions with backflow corrections. We compare and discuss data for the energy and the double occupancies, as obtained from both approaches. At small dopings, we observe a rapid crossover from a weakly correlated metal at low interaction strength UUU to a non-Fermi-liquid correlated state with strong local spin correlations. Furthermore, we investigate the stability of the correlated state against phase separation. We observe phase separation only for large values of UUU or very large frustration. No phase separation is present for the parameter range relevant for the cuprates.
Monte Carlo results for a two-dimensional Hubbard model in the intermediate-coupling regime U=4t are compared with a diagrammatic spin-fluctuation approximation. Simulations on an 8\ifmmode\times\else\texttimes\fi{}8 lattice doped away from half filling were carried out down to temperatures of order 140 of the bandwidth. Results for the spin susceptibility \ensuremath{\chi}(q,i\ensuremath{\omega}m{}_{\mathrm{m}}m​), the electron self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}n{}_{\mathrm{n}}n​), various pair-field susceptibilities, and the irreducible particle-particle scattering vertex \ensuremath{\Gamma}(p\ensuremath{'},i\ensuremath{\omega}${}_{\mathrm{n}\ensuremath{'}}$|p,i\ensuremath{\omega}n{}_{\mathrm{n}}n​) were obtained. A random-phase approximation for \ensuremath{\chi}(q,i\ensuremath{\omega}m{}_{\mathrm{m}}m​) with a renormalized Coulomb coupling U\ifmmode\bar\else\textasciimacron\fi{} is shown to provide a fit to the Monte Carlo data. A similar approximation for the Berk-Schrieffer spin-fluctuation interaction also provides a reasonable fit to the self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}n{}_{\mathrm{n}}n​) in the region explored by the Monte Carlo data. However, a similar approximation for the irreducible particle-particle interaction failed to reproduce the Monte Carlo results. Higher-order vertex corrections were calculated, but significant discrepancies with Monte Carlo results for \ensuremath{\Gamma} remain.
The momentum dispersion of the peak structure of the single-particle spectral weight A(p,\ensuremath{\omega}) of the two-dimensional Hubbard model is discussed. Using results obtained from Monte Carlo simulations on lattices up to 12\ifmmode\times\else\texttimes\fi{}12 in size, we determine the low-lying quasiparticle dispersion relation. This dispersion relation is anomalously flat near the (\ensuremath{\pi},0) and (0,\ensuremath{\pi}) points of the Brillouin zone, similar to the results of recent angular resolved photoemission measurements of the hole-doped cuprates. We argue that the generic nature of the quasiparticle dispersion relation observed in these materials arises from the strong Coulomb interaction and reflects the hole-spin correlations rather than the one-electron interactions which customarily determine the band structure.
A relaxation-function theory for the dynamic spin susceptibility in the $t\text{\ensuremath{-}}J$ model is presented. By a sum-rule-conserving generalized mean-field approximation (GMFA), the two-spin correlation functions of arbitrary range, the staggered magnetization, the uniform static susceptibility, and the antiferromagnetic correlation length are calculated in a wide region of hole doping and temperatures. A good agreement with available exact diagonalization (ED) data is found. The correlation length is in reasonable agreement with neutron-scattering experiments on ${\text{La}}_{2\ensuremath{-}\ensuremath{\delta}}{\text{Sr}}_{\ensuremath{\delta}}{\text{CuO}}_{4}$. Going beyond the GMFA, the self-energy is calculated in the mode-coupling approximation. The spin dynamics at arbitrary frequencies and wave vectors is studied for various temperatures and hole doping. At low doping a spin-wave-type behavior is found as in the Heisenberg model, while at higher doping a strong damping caused by hole hopping occurs, and a relaxation-type spin dynamics is observed in agreement with the ED results. The local spin susceptibility and its $\ensuremath{\omega}/T$ scaling behavior are calculated in a reasonable agreement with experimental and ED data.