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Physical review. B, Condensed matter

Cumulant-expansion study of holes in quantum antiferromagnets

P. Prelovek, Igor Sega, J. Bona

Published Dec 1, 1990• 22 citations
Publisher

Abstract

The states of a single hole and of two holes in the planar quantum antiferromagnet are studied within the t-J model by performing the cumulant expansion in t/J and in the exchange anisotropy \ensuremath{\gamma}. Ground-state energies are calculated up to the n=12 order in t/J and to the m=6 order in \ensuremath{\gamma}. It is shown that results, which can be extended to smaller J/t, represent a very helpful supplement to exact diagonalization methods. In particular they yield finite-size corrections to previous results on the dispersion of single-hole states and to the binding energy of two holes, giving also more insight into the origin of binding.

DOI
10.1103/physrevb.42.10706

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arXiv:cond-mat/9404092•Physical review. B, Condensed matter•Nov 1, 1994•27 citations
The states of a single hole and of two holes in the planar quantum antiferromagnet are studied within the t-J model by performing the cumulant expansion in t/J and in the exchange anisotropy \ensuremath{\gamma}. Ground-state energies are calculated up to the n=12 order in t/J and to the m=6 order in \ensuremath{\gamma}. It is shown that results, which can be extended to smaller J/t, represent a very helpful supplement to exact diagonalization methods. In particular they yield finite-size corrections to previous results on the dispersion of single-hole states and to the binding energy of two holes, giving also more insight into the origin of binding.
We calculated the binding energy of two holes with antiparallel spin in the t-J model using a variational wave function based on the string or spin-bag picture. The symmetry of the bound state is dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2. Although the size of matrices to be diagonalized is only 3\ifmmode\times\else\texttimes\fi{}3 we could reproduce the results of various extensive numerical calculations with reasonable accuracy. We show that a sizeable contribution of the binding energy is due to the quantum fluctuations in the ground state of the Heisenberg antiferromagnet. In contrast to exact diagonalization results on small clusters in the thermodynamical limit the binding energy decreases rapidly with increasing t/J. A detailed discussion of the various finite-size effects in a small cluster is also included.
The effective single-band Hamiltonian for holes in the two-dimensional quantum antiferromagnet, relevant for the CuO2{\mathrm{CuO}}_{2}CuO2​ layers in copper-oxide superconductors, is studied by the exact diagonalization of a finite-size system with 16 cells. Single- and two-hole ground-state properties are calculated. Pair-binding-energy and hole-density correlation functions indicate that two holes bind for moderate exchange interactions, even in the case of the extreme anisotropic-Ising limit.
Exact diagonalizations of small clusters up to 26 sites are used to extrapolate the ground-state energy of a single hole in the N\'eel state of the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model or in the quantum antiferromagnetic (t-J model). A surprisingly rapid convergence with system size is observed for a wide range of parameters and for the largest clusters considered. In the Ising limit the energy dependence with Jz{\mathit{J}}_{\mathit{z}}Jz​ is remarkably close to a Jz2/3{\mathit{J}}_{\mathit{z}}^{2/3}Jz2/3​ law for intermediate coupling in agreement with Brinkman-Rice type of calculations. Nevertheless a small but significant bandwidth indicates that the hole can propagate coherently in this limit. Clearly the polaronic behavior \ensuremath{\sim}Jz1/2{\mathit{J}}_{\mathit{z}}^{1/2}Jz1/2​, for very small Jz{\mathit{J}}_{\mathit{z}}Jz​, is out of the range of the method. When quantum fluctuations are introduced (isotropic case) the ground-state energy behavior still remains close to that of the Ising limit although an overall increase of the energy by a factor 0.3t is observed. This suggests that the ``string scenario'' might have some relevance there.
Exact many-body states of square clusters up to 18 sites (with periodic boundary conditions) are investigated in order to study the dynamics of one or two holes in the t-J model. Our method takes full advantage of the translation, rotation, and reflection symmetries of the cluster. Using a modified Lanczos algorithm, we calculate the lowest state of all the different space-group representations and its total spin as well as its spin and hole correlations. For a single hole, at intermediate J/t values, the magnetic, kinetic energies, and staggered magnetization follow J/t power-law behaviors with size-dependent exponents. A binding energy of order J between two holes appears for sufficiently large J/t(\ensuremath{\ge}0.25). One-hole and two-hole calculations give different results regarding the stability of the Nagaoka state (against the singlet state) when J/t\ensuremath{\rightarrow}0. Energy expectation values of variational resonating-valence-bond states are calculated exactly and optimized. Significant overlaps with the exact states are found for any symmetry.
Clusters of the two-dimensional t-J model with two holes and up to 26 sites are diagonalized using a Lanczos algorithm. The behavior of the binding energy with system size suggests the existence of a finite critical value of J above which binding occurs in the bulk. Only the d-wave pair-field operator acting on the Heisenberg ground state has a finite overlap with the two-hole ground state for all the clusters considered. The related spectral function associated with the propagation of a d-wave (singlet) pair of holes in the antiferromagnetic background is calculated. The quasiparticle peak at the bottom of the spectrum as well as some structure appearing above the peak survive with increasing cluster size. Although no simple scaling law was found for the quasiparticle weight the data strongly suggest that this weight is finite in the bulk limit and is roughly proportional to the antiferromagnetic coupling J (for J1).
The effect of a magnetic (S=1/2) impurity coupled to a two-dimensional system of correlated electrons (described by the t-J model) is studied by exact diagonalizations. It is found that, if the exchange coupling of the impurity with the neighboring spins is ferromagnetic or weakly antiferromagnetic, an extra hole can form bound states of different spatial symmetries with the impurity extending to a few lattice spacings. The binding energy is maximum when the impurity is completely decoupled (vacancy) and vanishes for an antiferromagnetic coupling exceeding \ensuremath{\sim}0.3J. Several peaks appear in the single-hole spectral function below the lower edge of the quasiparticle band as signatures of the d-, s-, and p-wave bound states.