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Physical review. B, Condensed matter

One-hole motion in the anisotropic three-dimensional antiferromagnet

Weijie Zhang, Jue-lian Shen

Published May 1, 1993• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physrevb.47.12300

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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 t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ and J\ensuremath⊥{\mathit{J}}_{\mathrm{\ensuremath{\perp}}}J\ensuremath⊥​ 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 J\ensuremath⊥{\mathit{J}}_{\mathrm{\ensuremath{\perp}}}J\ensuremath⊥​ constant and allowing t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ to increase, the spectral weight and the peak height of the quasiparticle decreases. In the quasi-two-dimensional cases with small t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ and J\ensuremath⊥{\mathit{J}}_{\mathrm{\ensuremath{\perp}}}J\ensuremath⊥​, the spectral weight and the peak height of the quasiparticle are smaller than those in the exact two-dimensional case.
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 formulate a quasiparticle theory for a single hole in a quantum antiferromagnet in the limit that the Heisenberg exchange energy is much less than the hopping matrix element, J\ensuremath{\ll}t. We consider the ground state of the spins to be either a quantum N\'eel state or a d-wave resonating-valence-bond (RVB) state. We show in a self-consistent perturbation theory that the hole spectrum is strongly renormalized by the interactions with spin excitations. The hole can be described by a narrow quasiparticle band located at an energy of order -t with a quasiparticle residue of order J/t and a bandwidth of order J. Above the quasiparticle band is an incoherent band of width of order t. Our results indicate that the energy scale for any coherent phenomenon involving the holes is \ensuremath{\delta}J, where \ensuremath{\delta} is the doping concentration. In the N\'eel state we perform a spin-wave expansion on an anisotropic Heisenberg model. In the Ising limit we reproduce previously known results and then expand perturbatively about that limit. In this expansion we find that the holes have a quasiparticle residue of Jz{J}_{z}Jz​/t and a bandwidth of ${J}_{\ensuremath{\perp}}$. In the Heisenberg limit we employ a ``dominant pole'' approximation in which we ignore contributions to the self-energy from the incoherent part of the hole spectrum. A similar technique is used to study the d-wave RVB state. The relevance of our results to recent optical experiments is discussed.
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.
We have studied hole states on square lattices with 8, 10, 16, 18, and 20 sites by exactly diagonalizing the single-hole t-J model with twisted boundary conditions in the whole Brillouin zone. The bottom of the hole band is found to be extremely flat and anisotropic. The hole effective mass is highly anisotropic and enhanced in a quantum antiferromagnet. The momentum of a hole in the ground state is not located at high-symmetry points such as (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2), (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi},0), or (0,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}). They are strongly lattice-size dependent and weakly parameter J/t (t is the hopping constant and J is the exchange constant) dependent. As the lattice size increases, the momentum of the ground state approaches the zone face center (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2).
Using exact diagonalization techniques, we study one dynamical hole in the two-dimensional t-J and Hubbard models on a square lattice including a next-nearest-neighbor hopping t'. We present the phase diagram in the parameter space (J/t,t'/t), discussing the ground-state properties of the hole. At J=0, a crossing of levels exists at some value of t' separating a ferromagnetic from an antiferromagnetic ground state. For nonzero J, at least four different regions appear where the system behaves like an antiferromagnet or a (not fully saturated) ferromagnet. We study the quasiparticle behavior of the hole, showing that for small values of \ensuremath{\Vert}t'\ensuremath{\Vert} the previously presented string picture is still valid. We also find that, for a realistic set of parameters derived from the Cu-O Hamiltonian, the hole has momentum (\ensuremath{\pi}/2,\ensuremath{\pi}/2), suggesting an enhancement of the p-wave superconducting mode due to the second-neighbor interactions in the spin-bag picture. Results for the t-t'-U model are also discussed with conclusions similar to those of the t-t'-J model. In general we found that t'=0 is not a singular point of these models.