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arXiv:cond-mat/0402487 · Physical Review B

Self-energy corrections in an antiferromagnet: Interplay of classical and quantum effects on quasiparticle dispersion

Pooja Srivastava, Avinash Singh

Published Sep 13, 2004• 9 citations
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Abstract

Self-energy corrections due to to fermion-magnon interaction are studied in the antiferromagnetic state of the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}$ Hubbard model within the noncrossing approximation in the full $U$ range from weak to strong coupling. The role of classical (mean-field) features of fermion and magnon dispersion, associated with finite $U,{t}^{\ensuremath{'}},{t}^{\ensuremath{''}}$, are examined on quantum corrections to quasiparticle energy, weight, one-particle density of states, etc. A finite-$U$ induced classical dispersion term, absent in the $t\ensuremath{-}J$ model, is found to play an important role in suppressing the quasiparticle weight for states near $\mathbf{k}=(0,0)$, as seen in cuprates. For intermediate $U$, the renormalized antiferromagnetic (AF) band gap is found to be nearly half of the classical value, and the weak-coupling limit is quite nontrivial due to strongly suppressed magnon amplitude. For finite ${t}^{\ensuremath{'}}$, the renormalized AF band gap is shown to vanish at a critical interaction strength ${U}_{c}$, yielding a spin fluctuation driven first-order AF insulator--paramagnetic metal transition. Quasiparticle dispersion evaluated with the same set of Hubbard model cuprate parameters, as obtained from a recent magnon spectrum fit, provides excellent agreement with angle-resolved photoemission spectroscopy data for ${\mathrm{Sr}}_{2}\mathrm{Cu}{\mathrm{O}}_{2}{\mathrm{Cl}}_{2}$.

DOI
10.1103/physrevb.70.115103

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Self-energy corrections due to to fermion-magnon interaction are studied in the antiferromagnetic state of the t\ensuremath−t\ensuremath′\ensuremath−t\ensuremath′′t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}t\ensuremath−t\ensuremath′\ensuremath−t\ensuremath′′ Hubbard model within the noncrossing approximation in the full UUU range from weak to strong coupling. The role of classical (mean-field) features of fermion and magnon dispersion, associated with finite U,t\ensuremath′,t\ensuremath′′U,{t}^{\ensuremath{'}},{t}^{\ensuremath{''}}U,t\ensuremath′,t\ensuremath′′, are examined on quantum corrections to quasiparticle energy, weight, one-particle density of states, etc. A finite-UUU induced classical dispersion term, absent in the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model, is found to play an important role in suppressing the quasiparticle weight for states near k=(0,0)\mathbf{k}=(0,0)k=(0,0), as seen in cuprates. For intermediate UUU, the renormalized antiferromagnetic (AF) band gap is found to be nearly half of the classical value, and the weak-coupling limit is quite nontrivial due to strongly suppressed magnon amplitude. For finite t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′, the renormalized AF band gap is shown to vanish at a critical interaction strength Uc{U}_{c}Uc​, yielding a spin fluctuation driven first-order AF insulator--paramagnetic metal transition. Quasiparticle dispersion evaluated with the same set of Hubbard model cuprate parameters, as obtained from a recent magnon spectrum fit, provides excellent agreement with angle-resolved photoemission spectroscopy data for Sr2CuO2Cl2{\mathrm{Sr}}_{2}\mathrm{Cu}{\mathrm{O}}_{2}{\mathrm{Cl}}_{2}Sr2​CuO2​Cl2​.
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Ferromagnetism in the Hubbard model is investigated on sc, bcc, and fcc lattices using a systematic inverse-degeneracy (1∕N)(1∕\mathcal{N})(1∕N) expansion which incorporates self-energy and vertex corrections such that spin-rotation symmetry and the Goldstone mode are explicitly preserved. First-order quantum corrections to magnon energies are evaluated for several cases, providing a comprehensive picture of the interplay of lattice, band dispersion, and interaction effects on the stability of the ferromagnetic state with respect to both long- and short-wavelength fluctuations. Our results support the belief that ferromagnetism is a generic feature of the Hubbard model at intermediate and strong coupling provided the density of states (DOS) is sufficiently asymmetric and strongly peaked near the band edge, as for the fcc lattice with finite ${t}^{\ensuremath{'}}$. For short-wavelength modes, the behavior of a characteristic energy scale ${\ensuremath{\omega}}^{*}\ensuremath{\sim}{T}_{c}$ (magnon DOS peak energy) is in excellent agreement with the Tc{T}_{c}Tc​ vs nnn behavior within dynamical mean field theory, with respect to both the stable range of densities (0.20<n<0.85)(0.20<n<0.85)(0.20<n<0.85) as well as the optimal density n=0.65n=0.65n=0.65. However, our finding of vanishing spin stiffness near optimal density highlights the role of long-wavelength fluctuations in further reducing the stable range of densities.
Single-particle dynamics in the 120\ifmmode^\circ\else\textdegree\fi{} ordered antiferromagnetic state of the triangular-lattice Hubbard model is studied using a physically transparent fluctuation approach in terms of multiple magnon emission and absorption processes within the noncrossing approximation. Hole and electron spectral features are evaluated at intermediate UUU, and analyzed in terms of a competition between the frustration-induced direct hopping and the virtual hopping. Finite UUU-induced competing interactions and frustration effects contributing through the magnon dispersion are also discussed. Finite contribution to self-energy correction from long-wavelength (Goldstone) modes, together with the high density of electron scattering states in the narrow, sharp peak in the upper band, result in strong fermion-magnon scattering leading to pronounced incoherent behavior in the electron dynamics. The fluctuation-induced first-order metal-insulator transition due to vanishing band gap is also discussed.
Spin-only descriptions of the half-filled one-band Hubbard model are relevant for a wide range of Mott insulators. In addition to the usual Heisenberg exchange, many other types of interactions, including ring exchange, appear in the effective Hamiltonian in the intermediate coupling regime. In order to improve on the quantitative description of magnetic excitations in the insulating antiferromagnetic phase of copper-oxide (cuprate) materials, and to be consistent with band-structure calculations and photoemission experiments on these systems, we include second- and third-neighbor hopping parameters, ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, into the Hubbard Hamiltonian. A unitary transformation method is used to find systematically the effective Hamiltonian and any operator in the spin-only representation. The results include all closed four-hop electronic pathways in the canonical transformation. The method generates many ring exchange terms that play an important role in the comparison with experiments on La2CuO4{\text{La}}_{2}{\text{CuO}}_{4}La2​CuO4​. Performing a spin-wave analysis, we calculate the magnon dispersion as a function of UUU, ttt, ${t}^{\ensuremath{'}}$, and ${t}^{\ensuremath{''}}$. The four parameters are estimated by fitting the magnon dispersion to the experimental results of Coldea et al. [Phys. Rev. Lett. 86, 5377 (2001)] for La2CuO4{\text{La}}_{2}{\text{CuO}}_{4}La2​CuO4​. The ring exchange terms are found essential, in particular to determine the relative sign of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, with the values found in good agreement with independent theoretical and experimental estimates for other members of the cuprate family. The zero-temperature sublattice magnetization is calculated using these parameters and also found to be in good agreement with the experimental value estimated by Lee et al. [Phys. Rev. B 60, 3643 (1999)]. We find a value of the interaction strength $U\ensuremath{\simeq}8t$ consistent with Mott insulating behavior.
Quantum Monte Carlo (QMC) and maximum-entropy techniques are used to study the spectral function $A(\mathbf{p},\ensuremath{\omega})$ of the one-band Hubbard model with strong coupling including a next-nearest-neighbor electronic hopping with amplitude ${t}^{\ensuremath{'}}/t=\ensuremath{-}0.35$. These values of parameters are chosen to improve the comparison of the Hubbard model with angle-resolved photoemission (ARPES) data for Sr2CuO2Cl2{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{2}{\mathrm{Cl}}_{2}Sr2​CuO2​Cl2​. A narrow quasiparticle (qp) band is observed in the QMC analysis at the temperature of the simulation T=t/3T=t/3T=t/3, both at and away from half-filling. Such a narrow band produces a large accumulation of weight in the density of states at the top of the valence band. As the electronic density 〈n〉〈n〉〈n〉 decreases further away from half-filling, the chemical potential travels through this energy window with a large number of states, and by $〈n〉\ensuremath{\sim}0.70$ it has crossed it entirely. The region near momentum $(0,\ensuremath{\pi})$ and $(\ensuremath{\pi},0)$ in the spectral function is more sensitive to doping than momenta along the diagonal from (0,0)(0,0)(0,0) to $(\ensuremath{\pi},\ensuremath{\pi})$. The evolution with hole density of the quasiparticle dispersion contains some of the features observed in recent ARPES data in the underdoped regime. For sufficiently large hole densities the ``flat'' bands at $(\ensuremath{\pi},0)$ cross the Fermi energy, a prediction that could be tested with ARPES techniques applied to overdoped cuprates. The population of the qp band introduces a hidden density in the system which produces interesting consequences when the quasiparticles are assumed to interact through antiferromagnetic fluctuations and studied with the BCS gap-equation formalism. In particular, a region of extended sss-wave character is found to compete with the ddd wave in the overdoped regime, i.e., when the chemical potential has almost entirely crossed the qp band as 〈n〉〈n〉〈n〉 is reduced. The present study also shows that previous ``real-space'' pairing theories for the cuprates, such as the antiferromagnetic Van Hove scenario, originally constructed based on information gathered at half-filling, do not change their predictions if hole dispersions resembling noninteracting electrons with renormalized parameters are used.
Away from half filling and above its Neel temperature, the three-dimensional Hubbard model is a paramagnetic metal with antiferromagnetic spin fluctuations, where controlled momentum-resolved results have been lacking. Using connected-determinant diagrammatic Monte Carlo, we compute the momentum-resolved self-energy for n = 0.4-0.95, T/t = 0.05-0.33 and U/t = 4-8, and the full-zone spin susceptibility at twelve of these points, focusing on U/t = 4. At U/t = 4 and toward half filling the spin fluctuations become strong: for n = 0.7-0.9 at T/t = 0.14 they enhance the susceptibility peak up to sevenfold and raise the momentum-dependent part of the scattering rate to four to five times its second-order value. They are driven by an effective spin interaction screened to about 2U/3 and nearly uniform over the zone. Yet the peak height is set by the slightly smaller value of this interaction at the peak wave vector, and near magnetic order this small difference changes the height by a large factor. Two-particle self-consistent theory reproduces the peak because its sum rule fixes its single constant interaction near this value, but for n = 0.7-0.9 at T/t = 0.14 its self-energy overestimates the local scattering rate and underestimates the momentum-dependent part. Though strong, the fluctuations are short ranged, with a correlation length of at most about two lattice spacings, and for n = 0.7-0.9 and T/t >= 0.14 the scattering rate near the Fermi surface varies by at most about 1% at fixed band energy, without hot spots. At U/t = 4 dynamical mean-field theory (DMFT) reproduces the local scattering rate and quasiparticle weight within a few percent, and the nonlocal part it misses is approximately a function of the band energy. Strong spin fluctuations thus coexist with nearly isotropic quasiparticles, and the results provide a benchmark for extensions of DMFT and for spin-fluctuation theories.