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arXiv:cond-mat/0208554 · Physical review. B, Condensed matter

Single-hole dynamics in the t−J model

W. Apel, H. U. Everts, U. Körner

Published Nov 15, 2002• 0 citations
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Abstract

The quasiparticle weight of a single hole in an antiferromagnetic background is studied in the semiclassical approximation. We start from the $t\ensuremath{-}J$ model, generalize it to arbitrary spin S by employing an appropriate coherent state representation for the hole, and derive an effective action for the dynamics in the long-wavelength low-energy limit. In the same limit, we find an expression for the single-hole Green's function which we evaluate in an $1/S$ expansion. Our approach has the advantage of being applicable in one and in two dimensions. We find two qualitatively different results in these two cases: while in one dimension our results are compatible with a vanishing quasiparticle weight, this weight is found to be finite in two dimensions, indicating normal quasiparticle behavior of the hole in this last case.

DOI
10.1103/physrevb.66.174423

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Analytical results for a hole in an antiferromagnet

No author provided

The Green's function for a hole moving in an antiferromagnet is derived analytically in the long-wavelength limit. We find that the infrared divergence is eliminated in two and higher dimensions so that the quasiparticle weight is finite. Our results also suggest that the hole motion is polaronic in nature with a bandwidth proportional to t(2)/J exp[-c(t/J)(2)] (c is a constant) for J/t greater than or equal to 0.5. The connection of the long-wavelength approximation to the first-order approximation in the cumulant expansion is also clarified.

•Sep 10, 2026•0 citations
The quasiparticle weight of a single hole in an antiferromagnetic background is studied in the semiclassical approximation. We start from the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model, generalize it to arbitrary spin S by employing an appropriate coherent state representation for the hole, and derive an effective action for the dynamics in the long-wavelength low-energy limit. In the same limit, we find an expression for the single-hole Green's function which we evaluate in an 1/S1/S1/S expansion. Our approach has the advantage of being applicable in one and in two dimensions. We find two qualitatively different results in these two cases: while in one dimension our results are compatible with a vanishing quasiparticle weight, this weight is found to be finite in two dimensions, indicating normal quasiparticle behavior of the hole in this last case.
The motion of a single hole in an arbitrary magnetic background is investigated for the two-dimensional $t\ensuremath{-}J$ model. The wave function of the hole is described within a generalized string picture which leads to a modified concept of spin polarons. We calculate the one-hole spectral function using a large-string basis for the limits of a N\'eel ordered and a completely disordered background. In addition we use a simple approximation to interpolate between these cases. For the antiferromagnetic background we reproduce the well-known quasiparticle band. In the disordered case the shape of the spectral function is found to be strongly momentum dependent, the quasiparticle weight vanishes for all hole momenta. Finally, we discuss the relevance of results for the lowest energy eigenvalue and its dispersion obtained from calculations using a polaron of minimal size as found in the literature.
The motion of a single hole in a two-dimensional Heisenberg antiferromagnet (hAF) is studied in a representation where holes are described as spinless fermions (holons) and spins as normal bosons. Assuming long-range AF order the spin dynamics is treated in linear spin-wave theory. The formulation highlights the close relation with the conventional polaron problem. The holon Green's function is calculated self-consistently within the Born approximation using finite-cluster geometries for the numerical solution. As a remarkable result we find close agreement with the spectral function A(k,\ensuremath{\omega}) of a hole calculated by exact diagonalization methods. A(k,\ensuremath{\omega}) is characterized by a narrow quasiparticle (QP) peak at the low-energy side of the spectrum, which is well separated from the incoherent part for large enough J values. A complete characterization of our solution is given, including the spectral weight, the dispersion relation, and effective masses of the QP state. A finite-size-scaling study gives a nonvanishing spectral weight of the QP in the thermodynamic limit for values J/t typical for copper oxide superconductors. Our calculations indicate that the self-consistent Born approximation is a valuable scheme for characterizing the dynamics of a hole in the t-J model, even in the strong-coupling regime.
We develop an efficient numerical method for the description of a single-hole motion in the antiferromagnetic background. The method is free of finite-size effects and allows the calculation of physical properties at an arbitrary wave vector. A methodical increase of the functional space leads to results that are valid in the thermodynamic limit. We found good agreement with cumulant expansion, exact-diagonalization approaches on finite lattices as well as self-consistent Born approximations. The method allows a straightforward addition of other inelastic degrees of freedom, such as lattice effects. Our results confirm the existence of a finite quasiparticle weight near the band minimum for a single hole and the existence of stringlike peaks in the single-hole spectral function.
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.
The Green's function for a hole moving in an antiferromagnet is derived analytically in the long-wavelength limit. We find that the infrared divergence is eliminated in two and higher dimensions so that the quasiparticle weight is finite. Our results also suggest that the hole motion is polaronic in nature with a bandwidth proportional to $\frac{{t}^{2}}{J}\mathrm{exp}[\ensuremath{-}c{(\frac{t}{J})}^{2}]$ (ccc is a constant) for $\frac{J}{t}\ensuremath{\gtrsim}0.5$. The connection of the long-wavelength approximation to the first-order approximation in the cumulant expansion is also clarified.
The Green's function for a hole moving in an antiferromagnet is derived analytically in the long-wavelength limit. We find that the infrared divergence is eliminated in two and higher dimensions so that the quasiparticle weight is finite. Our results also suggest that the hole motion is polaronic in nature with a bandwidth proportional to t(2)/J exp[-c(t/J)(2)] (c is a constant) for J/t greater than or equal to 0.5. The connection of the long-wavelength approximation to the first-order approximation in the cumulant expansion is also clarified.