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Frontiers of Physics

Quasiparticle scattering interference in the renormalized Hubbard model

Shuhua Wang, Huaisong Zhao, Feng Yuan

Published Sep 27, 2014• 3 citations
Publisher

Abstract

In this paper, we study the quasiparticle scattering interference phenomenon in the presence of a single impurity within the renormalized Hubbard model. By calculating the energy and momentum dependence of the Fourier-transformed local density of states in the full Brillouin zone, we can qualitatively describe the main features of the quasiparticle scattering interference phenomenon in cuprate superconductors using a single point-like impurity. In particular, we show that with increasing energy, the position of the peak along the nodal ([0, 0] → [ π, π ]) direction moves steadily to a large momentum region, while the position of the peak along the antinodal ([0, 0] → [ π , 0]) direction moves toward the center of the Brillouin zone.

DOI
10.1007/s11467-014-0446-9

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In this paper, we study the quasiparticle scattering interference phenomenon in the presence of a single impurity within the renormalized Hubbard model. By calculating the energy and momentum dependence of the Fourier-transformed local density of states in the full Brillouin zone, we can qualitatively describe the main features of the quasiparticle scattering interference phenomenon in cuprate superconductors using a single point-like impurity. In particular, we show that with increasing energy, the position of the peak along the nodal ([0, 0] → [ π, π ]) direction moves steadily to a large momentum region, while the position of the peak along the antinodal ([0, 0] → [ π , 0]) direction moves toward the center of the Brillouin zone.
We revisit the interpretation of quasiparticle scattering interference in cuprate high-Tc{T}_{c}Tc​ superconductors. This phenomenon has been very successful in reconstructing the dispersions of ddd-wave Bogoliubov excitations, but the successful identification and interpretation of quasiparticle interference (QPI) in scanning tunneling spectroscopy (STS) experiments rely on theoretical results obtained for the case of isolated impurities. We introduce a highly flexible technique to simulate STS measurements by computing the local density of states using real-space Green's functions defined on two-dimensional lattices with as many as 100 000 sites. We focus on the following question: to what extent can the experimental results be reproduced when various forms of distributed disorder are present? We consider randomly distributed pointlike impurities, smooth ``Coulombic'' disorder, and disorder arising from random on-site energies and superconducting gaps. We find an apparent paradox: the QPI peaks in the Fourier-transformed local density of states appear to be sharper and better defined in experiment than those seen in our simulations. We arrive at a no-go result for smooth-potential disorder since this does not reproduce the QPI peaks associated with large-momentum scattering. An ensemble of pointlike impurities gets closest to experiment, but this goes hand in hand with impurity cores that are not seen in experiment. We also study the effects of possible measurement artifacts (the ``fork mechanism''), which turn out to be of relatively minor consequence. It appears that a more microscopic model of the tunneling process needs to be incorporated in order to account for the sharpness of the experimentally obtained QPI peaks.
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.
We investigate the temperature dependence of quasiparticle interference in the high TcT_cTc​-cuprates using an Exact-Diagonalization + Monte-Carlo based scheme to simulate the ddd-wave superconducting order parameter. The quasiparticle interference patterns have features largely resulting from the scattering vectors of the octet model at lower temperature. Our findings suggest that the features of quasiparticle interference in the pseudogap region of the phase diagram are also dominated by the set of scattering vectors belonging to the octet model because of the persisting antinodal gap beyond the superconducting transition TcT_cTc​. However, beyond a temperature when the antinodal gap becomes very small, a set of scattering vectors different from those belonging to the octet model are responsible for the quasiparticle interference patterns. With a rise in temperature, the patterns are increasingly broadened.
We investigate the temperature dependence of quasiparticle interference in the high T_c T c -cuprates using an exact-diagonalization + Monte-Carlo based scheme to simulate the d d -wave superconducting order parameter. The quasiparticle interference patterns have features largely resulting from the scattering vectors of the octet model at lower temperature. Our findings suggest that the features of quasiparticle interference in the pseudogap region of the phase diagram are also dominated by the set of scattering vectors belonging to the octet model because of the persisting antinodal gap beyond the superconducting transition T_c T c . However, beyond a temperature when the antinodal gap becomes very small, a set of scattering vectors different from those belonging to the octet model are responsible for the quasiparticle interference patterns. With a rise in temperature, the patterns are increasingly broadened.
We examine the quasiparticle lifetime and spectral weight near the Fermi surface in the two-dimensional Hubbard model. We use the FLEX approximation to self-consistently generate the Matsubara Green's functions and then we analytically continue to the real axis to obtain the quasiparticle spectral functions. We compare the spectral functions found in the nearest neighbor hopping only Hubbard model with those found when second neighbor hopping is included. This separates the effects of nesting, the Van Hove singularity and the short-ranged antiferromagnetic correlations. The quasiparticle scattering rate is enhanced along the (0,pi) to (pi,pi) Brillouin zone diagonal. When the density is close to half-filling these 'hot-spots' lie on the Fermi surface and the scattering rate increases with decreasing temperature. For the next-nearest neighbor hopping scenario we observe a large range of doping where there is no antiferromagnetism but the scattering rate has a linear temperature dependence. On decreasing the interaction this non-Fermi liquid behavior is confined to doping levels where the Fermi energy lies near to the Van Hove singularity. We conclude that the 'hot-spots' are associated with the antiferromagnetic phase transition while the linear temperature dependence of the scattering rate is associated with the Van Hove singularity.
We generalize the numerical renormalization group (RG) scheme of Honerkamp et al. ( Phys. Rev. B , 63 (2001) 035109), Zanchi et al. ( Europhys. Lett. , 44 (1997) 235; Phys. Rev. B , 61 (2000) 13609) and Halboth et al. ( Phys. Rev. Lett. , 85 (2000) 5162) to study the phonon-mediated retarded interactions in the high- T c cuprates. We find that three sets of phonon-mediated retarded quasiparticle scatterings grow under RG flow. These scatterings share the following common features: 1) the initial and final quasiparticle momenta are in the antinodal regions, and 2) the scattering amplitudes have a x 2 - y 2 symmetry. All three sets of retarded interaction are driven to strong coupling by the magnetic fluctuations around (π/ a ,π/ a ). After growing strong, these retarded interaction will trigger density wave orders with d -wave symmetry. However, due to the d -wave form factor they will leave the nodal quasiparticle unaffected. We conclude that the main effect of electron-phonon coupling in the cuprates is to promote these density wave orders.