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Physical review. B, Condensed matter

Interaction of holes in a Hubbard antiferromagnet and high-temperature superconductivity

S. A. Trugman

Published Feb 1, 1988• 502 citations
Publisher

Abstract

The ${\mathrm{CuO}}_{2}$ planes in the high-temperature superconductors are described by a two-dimensional Hubbard model. The model is investigated for an infinite system with one and two electrons less than half filling. The method used is to diagonalize the Hamiltonian exactly within a retained portion of the Hilbert space. A single hole is found not to be localized by a string potential that increases linearly with distance, although it does have a large effective mass. A pair of holes, which naively should be quite mobile, is found instead to be extremely heavy due to a previously unappreciated frustration effect that impedes their motion. This lack of mobility increases the energy of the pair so that they do not bind, contrary to some recently published results using mean field theory or intuitive arguments. The energy of one- and two-hole states is calculated as a function of wavevector k. The comparative energy of different angular momentum channels and magnetic polaron effects are discussed.

DOI
10.1103/physrevb.37.1597

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The CuO2{\mathrm{CuO}}_{2}CuO2​ planes in the high-temperature superconductors are described by a two-dimensional Hubbard model. The model is investigated for an infinite system with one and two electrons less than half filling. The method used is to diagonalize the Hamiltonian exactly within a retained portion of the Hilbert space. A single hole is found not to be localized by a string potential that increases linearly with distance, although it does have a large effective mass. A pair of holes, which naively should be quite mobile, is found instead to be extremely heavy due to a previously unappreciated frustration effect that impedes their motion. This lack of mobility increases the energy of the pair so that they do not bind, contrary to some recently published results using mean field theory or intuitive arguments. The energy of one- and two-hole states is calculated as a function of wavevector k. The comparative energy of different angular momentum channels and magnetic polaron effects are discussed.
The weak-coupling form of the effective interaction between two holes in the antiferromagnetic state of the two-dimensional Hubbard model near half-filling is derived. It is shown to predict a Cooper instability in the singlet ddd-wave channel, with binding temperature Tc>100{T}_{c}>100Tc​>100 K. Our results differ from those of the "spin-bag" theory of Schrieffer, Wen, and Zhang.
We introduce a Hubbard model on a particular class of geometries, and consider the effect of doping the highly spin-degenerate Mott-insulating state with a microscopic number of holes in the extreme strong-coupling limit. The geometry is quite general, with pairs of atomic sites at each superlattice vertex, and a highly frustrated inter-atomic connectivity: the one-dimensional realization is a chain of edge-sharing tetrahedra. The sole model parameter is the ratio of intra-pair to inter-pair hopping matrix elements. If the intra-pair hopping is negligible then introducing a microscopic number of holes results in a ferromagnetic Nagaoka groundstate. Conversely, if the intra-pair hopping is comparable with the inter-pair hopping then the groundstate is low spin with short-ranged spin correlations. We exactly solve the correlated motion of a pair of holes in such a state and find that, in 1d and 2d, they form a bound pair on a length scale that increases with diminishing binding energy. This result is pertinent to the long-standing problem of hole motion in the CuO 2 planes of the high-temperature superconductors: we have rigorously shown that, on our frustrated geometry, the holes pair up and a short-ranged low-spin state is generated by hole motion alone.
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.
Abstract. We introduce a Hubbard model on a particular class of geometries, and consider the effect of doping the highly spin-degenerate Mott-insulating state with a microscopic number of holes in the extreme strong-coupling limit. The geometry is quite general, with pairs of atomic sites at each superlattice vertex, and a highly frustrated inter-atomic connectivity: the one dimensional realization is a chain of edge-sharing tetrahedra. The sole model parameter is the ratio of intrapair to inter-pair hopping matrix elements. If the intra-pair hopping is negligible then introducing a microscopic number of holes results in a ferromagnetic Nagaoka groundstate. Conversely, if the intra-pair hopping is comparable with the inter-pair hopping then the groundstate is low spin with short-ranged spin correlations. We exactly solve the correlated motion of a pair of holes in such a state and find that, in 1-d and 2-d, they form a bound pair on a length scale that increases with diminishing binding energy. This result is pertinent to the long-standing problem of hole-motion in the CuO2 planes of the high-temperature superconductors: we have rigorously shown that, on our frustrated geometry, the holes pair up and a short-ranged low-spin state is generated by hole motion alone. Submitted to: J. Phys. A: Math. Gen. 1.
We study the propagation of a single hole in the spin-density-wave state of the half-filled two-dimensional Hubbard model. This state is insulating not because of strong electron correlations, but because of the single-particle gap caused by magnetic ordering. We calculate the self-energy of the hole in the one-loop approximation and solve the Dyson equation to extract quantities such as the quasihole spectral function, energy dispersion, and lifetime. At all values of U we find a quasiparticle peak in the hole spectral function. The strength of the peak decreases as t2{\mathit{t}}^{2}t2/U2{\mathit{U}}^{2}U2 for large U. We calculate the shape of the quasihole Fermi surface and find that it consists of four pockets around the points (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2). Finally, we discuss the large-U extrapolation of our theory and compare our results with those of recent mean-field theories of the strong-coupling antiferromagnet.
We study the ``t-J'' model with strongly correlated electrons (infinite-U Hubbard-model kinetic energy plus superexchange J) by exact diagonalization on small two-dimensional clusters of six, eight, and twelve sites on a square lattice with periodic boundary conditions. We find an attractive pairing potential for two holes when J/t\ensuremath{\ge}(J/t)c{)}_{\mathit{c}})c​\ensuremath{\simeq}0.2, with t the hopping matrix element. Ground-state calculations show that extended s-wave and d-wave pairing susceptibilities and equal-time pairing correlations grow weakly with increasing J and system size at fixed filling n. Pairing correlations may be enhanced for (J/t)>(J/t)c{)}_{\mathit{c}})c​\ensuremath{\approxeq}0.2. We compare equal-time pairing correlations to a mean-field theory for J=0 and note potential difficulties with interpreting numerical results for such small systems. The widths of one- and two-hole energy bands in low-spin states are of order J; the one-hole effective mass is larger than the two-hole effective mass in such states. Spin-charge correlation functions for a single hole in the eight- and twelve-site lattices are consistent with the spin-charge decoupling scenario for resonating-valence-bond states.