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arXiv:1006.1215 · Journal of Physics Condensed Matter

Thet–Jmodel on a semi-infinite lattice

A Sherman, Niina Voropajeva

Published Aug 31, 2010• 0 citations
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Abstract

The hole spectral function of the t-J model on a two-dimensional semi-infinite lattice is calculated using the spin-wave and non-crossing approximations. In the case of small hole concentrations and strong correlations, t >> J, several near-boundary site rows appear to be depleted of holes. The reason is a deformation of the magnon cloud, which surrounds the hole, near the boundary. The hole depletion leads to a more complicated spectral function in the boundary row in comparison with its bulk shape. This is connected with a maximum related to the underlying row, which reveals itself in the boundary spectrum.

DOI
10.1088/0953-8984/22/37/375603

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The hole spectral function of the t-J model on a two-dimensional semi-infinite lattice is calculated using the spin-wave and non-crossing approximations. In the case of small hole concentrations and strong correlations, t >> J, several near-boundary site rows appear to be depleted of holes. The reason is a deformation of the magnon cloud, which surrounds the hole, near the boundary. The hole depletion leads to a more complicated spectral function in the boundary row in comparison with its bulk shape. This is connected with a maximum related to the underlying row, which reveals itself in the boundary spectrum.
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).
Hole and magnon spectral functions of the 2D ¢-J model are self-consistently calculated from a set of Dyson's equations for the hole concentrations z < 0.3.For J < ¢, itis shown that near z = 0.05 the hole spectrum changes drastically, acquiring features of a weakly correlated spectrum.This change is accompanied by the entering of the chemical potential u into a sharp maximum of the density of states.With z increasing the maximum moves together with u upto z =~0.15 and then lags behind.This concentration is also distinctive in that the Fermi surface becomes a simply connected surface and the overdamped magnons which appear near z = 0.02 occupy the widest region in k space.Besides, a limiting frequency of usual magnons passes through a minimum here.Significant deviations from the normal Fermi liquid, which partly resemble the marginal Fermi liquid, are observed for x > 0.05.
In the spin-wave approximation the energy spectrum of one-hole states of the two-dimensional t-J model is considered in the parameter range 0.01\ensuremath{\le}J/t\ensuremath{\le}1, where J is the exchange constant and t the hopping-matrix element. Attention is focused on single-particle states (in contrast to hole-magnon continua) that form energy bands characterized by different values of the z component of the total spin Sz{\mathit{S}}_{\mathit{z}}Sz​, corresponding to ferromagnetically ordered regions of different size around the hole. For an infinite lattice, the shapes and relative positions of the lowest bands are computed by means of the Lanczos method. The formation process of the ferromagnetically ordered region in the one-hole ground state is discussed. It is connected with a characteristic dependence of the band energy on J/t for different Sz{\mathit{S}}_{\mathit{z}}Sz​. The shapes of the magnon cloud around the hole and the point symmetries of the eigenfunctions are determined for various points in the Brillouin zone and for different bands of interest.
The spectral functions of one hole in the t-J and one-band Hubbard models are calculated using exact diagonalization techniques on small lattices. Results for the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model are also presented. For the t-J model we found that there is a quasiparticle at the bottom of the hole spectrum with an energy well approximated by Eh{\mathit{E}}_{\mathit{h}}Eh​=-3.17+2.83J0.73{\mathit{J}}^{0.73}J0.73 (for 0.1\ensuremath{\le}J\ensuremath{\le}1.0, t=1) on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The rest of the spectrum is not incoherent: We identified at least two other peaks following a similar power-law behavior with J. We speculate that the J dependence of the results can be explained by a model where the hole is trapped in a confining potential as in the Ising limit. The bandwidth of the hole is linear in J in the region 0.1\ensuremath{\le}J\ensuremath{\le}0.4 although a power-law behavior is not excluded. The spectral weight of the quasiparticle grows like J0.5{\mathit{J}}^{0.5}J0.5 in the same region. We present new analytical results in the large J/t limit to understand the motion of the hole: In perturbation theory it can be shown that the momentum of the hole at large J/t is k=(\ensuremath{\pi},\ensuremath{\pi}) changing to k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) at intermediate J/t in agreement with numerical and spin-waves results. We show analytically and numerically that the bandwidth of the quasiparticle is of order t in the large J/t limit. This result corresponds to a spin-liquid state. The one-hole spectral function of the Hubbard model is obtained for lattices with 8 and 10 sites. A quasiparticle is also observed in this case. The bandwidth and the relation with the t-J model are discussed and a comparison with recent Monte Carlo results is made. We also review and extend previous results for the ground-state properties of the t-J model.
The momentum distribution function and electron spectral function for mobile holes in an antiferromagnet as described by the t-J model are discussed. The calculations are done using a trial wave function for the coherent motion of an individual hole based on a string picture. To discuss finite concentrations, the rigid-band approximation is assumed to be valid. Despite this both quantities considered are qualitatively similar to what one would expect for weakly interacting particles occupying the tight-binding band calculated from the hopping term in the Hamiltonian.
The self-consistent Born approximation for the self-energy and the wave function of a single hole moving in a two-dimensional quantum antiferromagnet is used to calculate various spin-correlation functions that characterize the perturbation induced by the hole. The results for these correlation functions agree with the available small-cluster calculations. The hole-spin correlation functions decay asymptotically with the distance from the hole as a power law and exhibit a dipolar distortion of the antiferromagnet. The quasiparticle spectral weight remains finite.