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arXiv:cond-mat/9609186 · Physical review. B, Condensed matter

Spectral properties of thet-Jmodel in the presence of hole-phonon interaction

Bumsoo Kyung, Sergei I. Mukhin, Vladimir N. Kostur, Richard A. Ferrell

Published Nov 1, 1996• 18 citations
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Abstract

We examine the effects of electron-phonon interaction on the dynamics of the charge carriers doped in two-dimensional (2D) Heisenberg antiferromagnet. The t-J model Hamiltonian with a Fr\"ohlich term which couples the holes to a dispersionless (optical) phonon mode is considered for low doping concentration. The evolution of the spectral density function, the density of states, and the momentum distribution function of the holes with an increase of the hole-phonon coupling constant g is studied numerically. As the coupling to a phonon mode increases the quasiparticle spectral weight decreases and a ``phonon satellite'' feature close to the quasiparticle peak becomes more pronounced. Furthermore, strong electron-phonon coupling smears the multimagnon resonances (``string states'') in the incoherent part of the spectral function. The jump in the momentum distribution function at the Fermi surface is reduced without changing the hole pocket volume, thereby providing a numerical verification of Luttinger theorem for this strongly interacting system. The vertex corrections due to electron-phonon interaction are negligible in spite of the fact that the ratio of the phonon frequency to the effective bandwidth is not small. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.13167

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Physical review. B, Condensed matter•Jul 1, 1996•7 citations
We examine the effects of electron-phonon interaction on the dynamics of the charge carriers doped in two-dimensional (2D) Heisenberg antiferromagnet. The t-J model Hamiltonian with a Fr"ohlich term which couples the holes to a dispersionless (optical) phonon mode is considered for low doping concentration. The evolution of the spectral density function, the density of states, and the momentum distribution function of the holes with an increase of the hole-phonon coupling constant g is studied numerically. As the coupling to a phonon mode increases the quasiparticle spectral weight decreases and a phonon satellite'' feature close to the quasiparticle peak becomes more pronounced. Furthermore, strong electron-phonon coupling smears the multimagnon resonances (string states'') in the incoherent part of the spectral function. The jump in the momentum distribution function at the Fermi surface is reduced without changing the hole pocket volume, thereby providing a numerical verification of Luttinger theorem for this strongly interacting system. The vertex corrections due to electron-phonon interaction are negligible in spite of the fact that the ratio of the phonon frequency to the effective bandwidth is not small. \textcopyright{} 1996 The American Physical Society.
The hole dynamics in a quantum antiferromagnetic background for a small doping concentration is studied within the t-J model description. The noncrossing approximation for the hole self-energy is used. The correct quasiparticle pole structure (without artificial broadening) in the spectral density function is employed. Dyson's equation for the hole Green's function is solved numerically by introducing a lattice in momentum space. The evolution of the density of states and the momentum-distribution function for holes is investigated as a function of hole-magnon coupling constant t at fixed J. The momentum distribution function shows a sharp drop at the Fermi surface for several different hole-magnon coupling constants. This indicates the validity of the Fermi liquid description for the t-J model in the low doping concentration. The volume (surface) enclosed by the Fermi surface (curve) is found to be invariant for all the coupling strengths we have studied. The incoherent spectrum below the chemical potential in the density of states becomes more significant for stronger interaction constants. These features numerically verify Luttinger's theorem for a small density of doped holes in the t-J model. The optical conductivity is computed using the lowest-order diagrams. As the hole-magnon coupling constant t increases, absorption becomes stronger at \ensuremath{\omega}\ensuremath{\simeq}2J, the characteristic frequency for spin waves. This feature may be associated with the experimentally observed midinfrared absorption band in underdoped cuprates.
Literature addressing the existence of ``hole pockets'' in experiments for the high-Tc{\mathrm{T}}_{\mathrm{c}}Tc​ cuprates and in theoretical analysis of electronic models of correlated electrons is reviewed. It is argued that the issue is not conclusively resolved, both in theory and experiments. The apparently large Fermi surface observed in numerical studies of the doped Hubbard and t-J models suggests the presence of 1-x carriers (with x the concentration of holes). However, this is in contradiction with results obtained in similar calculations for the Drude weight which scales with x at low doping. To address such a paradox, dressed operators are here used. Their spectral decomposition A(k,\ensuremath{\omega}) is analyzed specially using the t-J model on ladders, but considering also chains and two-dimensional (2D) clusters. The results are contrasted against those obtained with the standard bare operators. It is concluded that substantial changes in the spectral weight can occur by replacing the bare hole creation operator by its dressed version. Apparently large Fermi surfaces can turn into small ones by working with quasiparticle (qp) operators that represent accurately the state of one hole. Thus, large Fermi surfaces in angle-resolved photoemission (ARPES), obtained by the sudden removal of an electron, may not be in contradiction with a visualization of the normal state of lightly doped antiferromagnets as composed of a gas of spin polarons with energies approximately obtained from the rigid band doping of the half-filled dispersion. The coexistence of a large Fermi surface in ARPES with, e.g., a holelike Hall coefficient seems possible in systems with strong correlations. In this paper the expression ``hole pocket'' is used as representing a large accumulation of spectral weight centered at k=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) generated by antiferromagnetic correlations in 2D clusters, or in analogous positions for ladders and chains. The subtle issue of whether such hole pockets represent a true small Fermi surface or just large incoherent weight cannot be addressed with finite resolution techniques such as angle-resolved photoemission and numerical studies of electronic models. They only provide information about the location of the dominant weight in A(k,\ensuremath{\omega}). The ideas discussed here are very general and they can be applied to a variety of problems where quasiparticles are strongly dressed by low energy excitations of the medium in which they are immersed.
The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) is momentum dependent. Using Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.
We present an efficient numerical method for the description of ground state and spectral properties of a doped hole in generalized $t\text{\ensuremath{-}}J$ models coupled to optical phonons. In the weak-coupling regime, coherent hole motion is less affected by the electron-phonon (EP) interaction in the electron than in the hole-doped system. Increasing the strength of EP coupling leads to a vanishing low-energy spectral weight around $\ensuremath{\Gamma}$ point. Spectral properties are in the strong coupling regime consistent with recent high-resolution angle resolved photoemission spectra (ARPES) [K. M. Shen et al., Phys. Rev. Lett. 93, 267002 (2004)]. A waterfall-like feature appears connecting the dispersive low-energy band with the incoherent high-energy peak resembling ARPES data [F. Ronning et al., Phys. Rev. B 71, 094518 (2005)].
We investigate the effect of doping in the hole spectral function by exact diagonalization of a 16-site cluster with two holes. It is shown that for more than one hole the spectral function differs qualitatively from the one in the single-hole limit. In particular, we find no evidence for the formation of a Fermi surface having hole ``pockets'' localized around (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) in momentum space. On the contrary, a large low-energy peak appears in the spectral function of the hole propagator at k=(\ensuremath{\pi},0) or (0,\ensuremath{\pi}) which is then the best candidate for the existence of those pockets. A (\ensuremath{\pi}/2,\ensuremath{\pi}/2) quasihole state is obtained as a small energy excitation of the two-hole system. We discuss some consequences for photoemission and optical-conductivity experiments in high-Tc{\mathit{T}}_{\mathit{c}}Tc​ materials at intermediate doping.
The density of state for the single hole in the two-dimensional t-J model is studied in the presence of a constant and homogeneous magnetic field by means of exact calculations for small systems and spin-wave perturbative calculations for larger systems. As the field is turned on, the energy of the ground state is reduced by an amount proportional to parameter t and depending weakly on J. This result, together with the behavior of the Drude weight of the doped t-J model, yields strong support for the idea that the dynamics of charge carriers is governed by parameter t. This idea, confronted to the existence in the spectral density of the hole of a quasiparticle peak dispersing with J, constitutes evidence for the composite nature of quasiparticles. \textcopyright{} 1996 The American Physical Society.