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arXiv:cond-mat/9310023 · Physical review. B, Condensed matter

Hole dynamics in a quantum antiferromagnet: Extension of the retraceable-path approximation

Q. F. Zhong, Sandro Sorella, Alberto Parola

Published Mar 1, 1994• 9 citations
arXivHTMLPDF

Abstract

The one-hole spectral weight for two chains and two-dimensional lattices is studied numerically using a method of analysis of the spectral function within the Lanczos iteration scheme: the Lanczos spectral decoding method. This technique is applied to the t-${\mathit{J}}_{\mathit{z}}$ model for ${\mathit{J}}_{\mathit{z}}$\ensuremath{\rightarrow}0, directly on an infinite-size lattice. By a careful investigation of the first 13 Lanczos steps and the first 26 ones for the two-dimensional and the two-chain cases, respectively, we find several interesting features of the one-hole spectral weight. A sharp incoherent peak with a clear momentum dispersion is identified, together with a second broad peak at higher energy. The spectral weight is finite up to the Nagaoka energy where it vanishes in a nonanalytic way. Thus the lowest energy of one hole in a quantum antiferromagnet is degenerate with the Nagaoka energy in the thermodynamic limit.

DOI
10.1103/physrevb.49.6408

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Dynamical properties of a single hole in an antiferromagnet

Didier Poilblanc, Timothy A. L. Ziman, Heinz J. Schulz +1 more

A finite-size scaling analysis of the spectral function and of the optical conductivity of a single hole moving in an antiferromagnetic background is performed. It is shown that both the low-energy quasiparticle peak and the broad higher-energy structure are robust with increasing cluster size from 4\ifmmode\times\else\texttimes\fi{}4 to \ensuremath{\surd}26 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}26 sites. In the absence of spin fluctuations, for most static or dynamical quantities saturation occurs when the size exceeds a characteristic size ${\mathit{N}}_{\mathit{c}}$(${\mathit{J}}_{\mathit{z}}$). Typically, 16- and 26-site clusters give reliable results for ${\mathit{J}}_{\mathit{z}}$>0.75 and ${\mathit{J}}_{\mathit{z}}$>0.3, respectively. The hole optical mass is shown to be very large (>20) in agreement with the small bandwidth. Due to the energy gap to flip a spin in the vicinity of a hole, a small gap \ensuremath{\propto}${\mathit{J}}_{\mathit{z}}$ separates the low-energy \ensuremath{\delta} function from the rest of the spectrum in the dynamical correlation functions. On the other hand, with ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ this gap seems to disappear with increasing system size as one would expect since the spin waves are gapless in the thermodynamic limit. The large momentum dependence of the quasiparticle weight in the isotropic case is inconsistent with a string picture but agrees well with the self-consistent Born approximation.An accurate estimation of the higher-energy part of the spectral functions of the t-J model can be made for momenta close to (0,0) or (\ensuremath{\pi},\ensuremath{\pi}); it consists of a broad featureless band. For momenta close to the Fermi surface, no evidence for excited string states is found. This may be either because the string scenario breaks down or because exact calculations at momenta near (\ensuremath{\pi}/2,\ensuremath{\pi}/2) are limited by the differing geometries of the finite clusters. In the Ising limit, the spectral function also depends crucially on the momentum, but a feature resembling a string excitation is seen for momenta close to (0,0). Our data on the finite-frequency optical conductivity of a single hole in the t-J model reveal that the total spectral weight at finite frequency has a strong size dependence and decreases with increasing size (for J not too small). This suggests that the optical mass and the optical absorption of the t-J model may have been overestimated in the past.

arXiv:cond-mat/9212010•Physical review. B, Condensed matter•Jun 1, 1993•54 citations
The one-hole spectral weight for two chains and two-dimensional lattices is studied numerically using a method of analysis of the spectral function within the Lanczos iteration scheme: the Lanczos spectral decoding method. This technique is applied to the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model for Jz{\mathit{J}}_{\mathit{z}}Jz​\ensuremath{\rightarrow}0, directly on an infinite-size lattice. By a careful investigation of the first 13 Lanczos steps and the first 26 ones for the two-dimensional and the two-chain cases, respectively, we find several interesting features of the one-hole spectral weight. A sharp incoherent peak with a clear momentum dispersion is identified, together with a second broad peak at higher energy. The spectral weight is finite up to the Nagaoka energy where it vanishes in a nonanalytic way. Thus the lowest energy of one hole in a quantum antiferromagnet is degenerate with the Nagaoka energy in the thermodynamic limit.
We review in detail a recently proposed technique to extract information about dynamical correlation functions of many-body Hamiltonians with a few Lanczos iterations and without the limitation of finite size. We apply this technique to understand the low-energy properties and the dynamical spectral weight of a simple model describing the motion of a single hole in a quantum antiferromagnet: the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model in two spatial dimension and for a double-chain lattice. The simplicity of the model allows a well controlled numerical solution, especially for the two-chain case. Contrary to previous approximations, we have found that the single-hole ground state in the infinite system is continuously connected with the Nagaoka fully polarized state for Jz{\mathit{J}}_{\mathit{z}}Jz​\ensuremath{\rightarrow}0. Analogously, we have obtained an accurate determination of the dynamical spectral weight relevant for photoemission experiments. For Jz{\mathit{J}}_{\mathit{z}}Jz​=0 an argument is given that the spectral weight vanishes at the Nagaoka energy faster than any power law, as supported also by clear numerical evidence. It is also shown that spin-charge decoupling is an exact property for a single hole in the Bethe lattice but does not apply to the more realistic lattices where the hole can describe closed-loop paths.
We have investigated one-hole motion in the anisotropic three-dimensional (A3D) quantum antiferromagnet by use of the A3D t-J model in a simple cubic lattice. The hopping matrix element and the Heisenberg exchange energy are t and J in the xy plane. They are ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ and ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ in the z direction. Within the self-consistent mean-field approximation, the hole spectral function has been calculated by use of an iterative technique. We find that the quasiparticle picture still holds and the lowest-energy quasiparticle state has momentum either k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2,\ensuremath{\pi}/2) or k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2,0) depending on the parameters of the model. Holding J and ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ constant and allowing ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ to increase, the spectral weight and the peak height of the quasiparticle decreases. In the quasi-two-dimensional cases with small ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ and ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, the spectral weight and the peak height of the quasiparticle are smaller than those in the exact two-dimensional case.
We report a calculation of the dynamics of more than one hole in the t-J model using the Lanczos algorithm. The hole spectral weight develops a large quasiparticle peak that describes two holes in the same momentum state on the pseudo-Fermi-surface with antiparallel spins. We also show that doping leads to a narrowing of the bandwidth of the spin-wave dispersion. A larger number of higher-energy subpeaks in the spectral function (compared to half filling) are observed, although they are not predicted by mean-field theory.
We solved numerically the integral equation for the selfenergy which describes the motion of a single hole in a two-dimensional quantum antiferromagnet (AF) within the selfconsistent Born approximation. This formulation stresses the similarity of the AF-spin polaron with the standard polaron problem. We confine our calculation to finite cluster geometries and compare with results from previous exact diagonalization studies. The spectral function is characterized by a narrow quasiparticle (qp) peak at the low energy side of the spectra, which appears to be well separated from the incoherent band part for large enough J values. For small J we find a reduced width of ~7t for the incoherent band. The bottom of the coherent qp band always occurs at (±π/2, ±π/2). Its bandwidth initially increases with J until J≃t and then decreases as 2t2/J. A complete characterization of our solution is given, including the dispersion relation and effective masses of this quasiparticle. The comparison with exact diagonalization studies for a 4×4 cluster is remarkably good. From our results we see that a lattice of 16×16 sites describes adequately the thermodynamic limit. We conclude that the simple Born approximation is a valuable scheme to characterize the dynamics of one hole in the t-J model. both in the perturbative and the strong coupling regimes.
A finite-size scaling analysis of the spectral function and of the optical conductivity of a single hole moving in an antiferromagnetic background is performed. It is shown that both the low-energy quasiparticle peak and the broad higher-energy structure are robust with increasing cluster size from 4\ifmmode\times\else\texttimes\fi{}4 to \ensuremath{\surd}26 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}26 sites. In the absence of spin fluctuations, for most static or dynamical quantities saturation occurs when the size exceeds a characteristic size Nc{\mathit{N}}_{\mathit{c}}Nc​(Jz{\mathit{J}}_{\mathit{z}}Jz​). Typically, 16- and 26-site clusters give reliable results for Jz{\mathit{J}}_{\mathit{z}}Jz​>0.75 and Jz{\mathit{J}}_{\mathit{z}}Jz​>0.3, respectively. The hole optical mass is shown to be very large (>20) in agreement with the small bandwidth. Due to the energy gap to flip a spin in the vicinity of a hole, a small gap \ensuremath{\propto}Jz{\mathit{J}}_{\mathit{z}}Jz​ separates the low-energy \ensuremath{\delta} function from the rest of the spectrum in the dynamical correlation functions. On the other hand, with ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ this gap seems to disappear with increasing system size as one would expect since the spin waves are gapless in the thermodynamic limit. The large momentum dependence of the quasiparticle weight in the isotropic case is inconsistent with a string picture but agrees well with the self-consistent Born approximation.An accurate estimation of the higher-energy part of the spectral functions of the t-J model can be made for momenta close to (0,0) or (\ensuremath{\pi},\ensuremath{\pi}); it consists of a broad featureless band. For momenta close to the Fermi surface, no evidence for excited string states is found. This may be either because the string scenario breaks down or because exact calculations at momenta near (\ensuremath{\pi}/2,\ensuremath{\pi}/2) are limited by the differing geometries of the finite clusters. In the Ising limit, the spectral function also depends crucially on the momentum, but a feature resembling a string excitation is seen for momenta close to (0,0). Our data on the finite-frequency optical conductivity of a single hole in the t-J model reveal that the total spectral weight at finite frequency has a strong size dependence and decreases with increasing size (for J not too small). This suggests that the optical mass and the optical absorption of the t-J model may have been overestimated in the past.