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Physical review. B, Condensed matter

Spectral function of a hole in a Hubbard antiferromagnet

S. A. Trugman

Published Jan 1, 1990• 134 citations
Publisher

Abstract

The Green's function G(k,\ensuremath{\omega}) is calculated for a hole in a Hubbard antiferromagnet by a new nonperturbative, variational method for an infinite square lattice. The quasiparticle and excited-state wave functions show that the substantial structure in the ``incoherent'' spectrum is related to string states for large U. The strength of the quasiparticle pole and the anisotropic effective mass are obtained as a function of (U/t). The quasiparticle lifetime is infinite for most k.

DOI
10.1103/physrevb.41.892

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New Journal of Physics•Nov 16, 2021•20 citations
The Green's function G(k,\ensuremath{\omega}) is calculated for a hole in a Hubbard antiferromagnet by a new nonperturbative, variational method for an infinite square lattice. The quasiparticle and excited-state wave functions show that the substantial structure in the ``incoherent'' spectrum is related to string states for large U. The strength of the quasiparticle pole and the anisotropic effective mass are obtained as a function of (U/t). The quasiparticle lifetime is infinite for most k.
An analytic approach is presented of electronic one-particle spectra of the one-band Hubbard model at half filling in the antiferromagnetic phase. Starting from the strong-coupling regime U\ensuremath{\gg}:t, a projection technique is used to set up self-consistent coupled equations for the electron Green function, which are valid down to values U\ensuremath{\approx}t. The self-consistent equation for the hole propagator is a direct generalization of the one found from the t-J model. This gives further support to the ``string'' picture, where propagation of holes creates strings of overturned spins with which the holes interact. In the present work hopping of holes (or electrons) with up spin on the down sublattice is also taken into acount, as well as transitions between the lower and upper Hubbard bands. These are shown to change significantly the incoherent part of the t-J model spectra, by smearing out the shake-off peaks, reminiscent of higher bound string states due to multispin scattering. Coherent (quasiparticle) peaks exist at the band edges, on both sides of the insulating gap. We show that with decreasing U the quasiparticle concept loses its meaning for wave vectors at the center of the magnetic Brillouin zone (MBZ). For large values of U the dispersion of the quasiparticle is found to scale with its band width, which is of order J. Extrema are always found at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2). The weight of the quasiparticle at this k value decreases logarithmically with increasing U. In the strong-coupling limit the spectrum tends to be symmetric, i.e., to become an even function of the frequency around the chemical potential, for any wave vector. For small values of U the dispersion at the edge of the MBZ flattens away, as expected when approaching the noninteracting limit. The spectral function in this regime, for wave vectors away from the edge of the MBZ, is concentrated mainly on one side of the chemical potential.
The author calculates an exact expression for the Green function of a single hole in a Neel background, for finite Ising interaction J z identical to 4t 2 /U where t is the hopping (overlap) integral in the Hubbard model and U the usual on-site interaction. In 1D for fixed boundary conditions, loops are absent and thus the expression is exact and independent of the retraceable path approximation, while in 2D the expression is exact to within the exclusion of loops. The spectral function A( omega ) shows delta-function peaks at well defined energy spacings, with damping only at frequencies omega >5/2J z different from earlier results. These peaks are necessary for the existence of quasiparticles. The author presents an argument that, for sufficiently large U, the discrete spectrum in A( omega ) merges to a continuous 'band' as the weight varies as J z of the poles vanishes in the U= infinity limit and hence that use of the latter does not necessarily imply incoherent behaviour in the system.
The spectral functions of one hole in the t-J and one-band Hubbard models are calculated using exact diagonalization techniques on small lattices. Results for the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model are also presented. For the t-J model we found that there is a quasiparticle at the bottom of the hole spectrum with an energy well approximated by Eh{\mathit{E}}_{\mathit{h}}Eh​=-3.17+2.83J0.73{\mathit{J}}^{0.73}J0.73 (for 0.1\ensuremath{\le}J\ensuremath{\le}1.0, t=1) on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The rest of the spectrum is not incoherent: We identified at least two other peaks following a similar power-law behavior with J. We speculate that the J dependence of the results can be explained by a model where the hole is trapped in a confining potential as in the Ising limit. The bandwidth of the hole is linear in J in the region 0.1\ensuremath{\le}J\ensuremath{\le}0.4 although a power-law behavior is not excluded. The spectral weight of the quasiparticle grows like J0.5{\mathit{J}}^{0.5}J0.5 in the same region. We present new analytical results in the large J/t limit to understand the motion of the hole: In perturbation theory it can be shown that the momentum of the hole at large J/t is k=(\ensuremath{\pi},\ensuremath{\pi}) changing to k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) at intermediate J/t in agreement with numerical and spin-waves results. We show analytically and numerically that the bandwidth of the quasiparticle is of order t in the large J/t limit. This result corresponds to a spin-liquid state. The one-hole spectral function of the Hubbard model is obtained for lattices with 8 and 10 sites. A quasiparticle is also observed in this case. The bandwidth and the relation with the t-J model are discussed and a comparison with recent Monte Carlo results is made. We also review and extend previous results for the ground-state properties of the t-J model.
We calculated the spectral function of a hole in the t-J model using a numerical method on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. A sharp quasiparticle peak is observed with an energy that is well approximated by \ensuremath{\Delta}E=-3.17+2.83${J}^{\ensuremath{\alpha}}$ (for 0.1\ensuremath{\le}J\ensuremath{\le}1.0, t=1) with \ensuremath{\alpha}=0.73\ifmmode\pm\else\textpm\fi{}0.02. This result suggests that the picture of a hole attached to the origin by a string may be valid in the Heisenberg model. The rest of the spectrum is not completely incoherent but it has a nontrivial structure, stemming from the excited states of the string. The bandwidth of the quasiparticle is discussed. Results for the Ising model are also presented.
Abstract The dynamics of a hole motion in a quantum antiferromagnet has been studied in the past three decades because of its relationship to models related to superconductivity in cuprates. The same problem has received significant attention because of its connection to very recent experiments of the dynamics of ultra-cold atoms in optical lattices where models of strongly correlated electrons can be simulated. In this paper we apply the diagrammatic Monte Carlo method to calculate the single-hole Green’s function in the t–J model, where the J term is linearized, in a wide range of imaginary-time with the aim to examine the polaron formation and in particular the details of the contribution of the so-called string excitations found in such recent experiments. We calculate the single-hole spectral function by analytic continuation from imaginary to real time and study the various aspects that constitute the string picture, such as, the energy–momentum dependence of the main quasiparticle peak and its residue, the internal excitations of the string which appear as multiple peaks in the spectral function as well as their momentum dependence. We find that the earlier analysis of the spectral function based on a mobile-hole connected with a string of overturn spins and the contribution of the internal string excitations as obtained from the non-crossing approximation is accurate.