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Physical review. B, Condensed matter

Spatial distribution of mobile holes in thet-Jmodel

K. J. E. Vos, Robert J Gooding, P. W. Leung, Liang Chen

Published May 1, 1995• 0 citations
Publisher

Abstract

We have calculated the hole-hole correlation function for two holes doped into a S=1/2 antiferromagnetic insulator described by the t-J model on a 16-site \ensuremath{\surd}8\ifmmode\times\else\texttimes\fi{}\ensuremath{\surd}32 lattice and a 24-site \ensuremath{\surd}18\ifmmode\times\else\texttimes\fi{}\ensuremath{\surd}32 lattice using an exact diagonalization technique. We predict that the carriers described by this model, in the parameter range relevant to ${\mathrm{CuO}}_{2}$ planes, tend to stay as far apart as possible. This result is in direct contradiction to previously published exact diagonalization data [e.g., D. Poilblanc, Phys. Rev. B 49, 1477 (1994)]. We suggest that if the carriers do tend to stay as far apart from one another as possible, then the electron momentum distribution function shows evidence for hole pockets at k\ensuremath{\rightarrow}=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2), consistent with many recent studies of these models, including recent photoemission work.

DOI
10.1103/physrevb.51.12034

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Exact-diagonalization demonstration of incommensurability and the associated Fermi surface forNholes in thet-Jmodel

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Physical review. B, Condensed matter•Mar 1, 1993•81 citations
We have calculated the hole-hole correlation function for two holes doped into a S=1/2 antiferromagnetic insulator described by the t-J model on a 16-site \ensuremath{\surd}8\ifmmode\times\else\texttimes\fi{}\ensuremath{\surd}32 lattice and a 24-site \ensuremath{\surd}18\ifmmode\times\else\texttimes\fi{}\ensuremath{\surd}32 lattice using an exact diagonalization technique. We predict that the carriers described by this model, in the parameter range relevant to CuO2{\mathrm{CuO}}_{2}CuO2​ planes, tend to stay as far apart as possible. This result is in direct contradiction to previously published exact diagonalization data [e.g., D. Poilblanc, Phys. Rev. B 49, 1477 (1994)]. We suggest that if the carriers do tend to stay as far apart from one another as possible, then the electron momentum distribution function shows evidence for hole pockets at k\ensuremath{\rightarrow}=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2), consistent with many recent studies of these models, including recent photoemission work.
We have calculated S(q) and the single particle distribution function 〈nq{\mathit{n}}_{\mathbf{q}}nq​〉 for N holes in the t-J model on a nonsquare \ensuremath{\surd}8 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}32 16-site lattice with periodic boundary conditions. We justify the use of this lattice by appealing to results obtained from the conventional 4\ifmmode\times\else\texttimes\fi{}4 16-site cluster, and an undoped 32-site system, each having the full square symmetry of the bulk. This new cluster has a high density of k points along the diagonal of reciprocal space, viz. along k=k(1,1). The results clearly demonstrate that when the single-hole problem has a ground state with a system momentum of k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2), the resulting ground state for N holes involves a shift of the peak of the system's structure factor away from the antiferromagnetic state q=(\ensuremath{\pi},\ensuremath{\pi}). This shift effectively increases continuously with N. When the single hole problem has a ground state with a momentum that is not equal to k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2), something that may easily be accomplished through the use of the t-t'-J model with t'/t small and positive, then the above-mentioned incommensurability for N holes is not found---the maximum of S(q) remains at q=(\ensuremath{\pi},\ensuremath{\pi}) for all N.The existence of the incommensurate ground states are presented in conjunction with the electron and hole momentum distribution functions. Our results may be interpreted as evidence for rigid-band filling of the dipolar spin distortion states of Shraiman and Siggia---by studying the hole-hole correlation function we demonstrate that the holes on our cluster tend to stay as far apart as is possible, and thus rigid-band filling is not unexpected. Thus, these results demonstrate that in some instances important results for moderately doped CuO2{\mathrm{CuO}}_{2}CuO2​ planes can be predicted from a knowledge of the properties of weakly doped planes. However, the electron Fermi surface that we obtain also satisfies Luttinger's theorem, and has dimples at the wave vectors of the single hole ground state, k=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2). Finally, we comment on the relation of all of these results to the incommensurability found in S(q).
We solve the t-J model with four holes on a 32-site lattice using exact diagonalization. This is the largest scale calculation ever reported for this model. The size of the matrix involved in this problem is about 3×109 by 3×109. We find that at the realistic parameter J/t = 0.3, holes in the ground state of this system are unbound. They have short range repulsion due to lowering of kinetic energy. The electron momentum distribution function resembles hole pockets. We show that in case antiferromagnetic order is strong, holes form d-wave bound pairs and there is mutual repulsion among the holes. This presumably will occur at low doping level.
We study the $t\text{\ensuremath{-}}J$ model with four holes on a 32-site square lattice using exact diagonalization. This system corresponds to doping level x=18x=\frac{1}{8}x=81​. At the ``realistic'' parameter J∕t=0.3J∕t=0.3J∕t=0.3, holes in the ground state of this system are unbound. They have short range repulsion due to the lowering of the kinetic energy. There is no antiferromagnetic spin order and the electron momentum distribution function resembles hole pockets. Furthermore, we show evidence that in case antiferromagnetic order exists, holes form ddd-wave bound pairs and there is mutual repulsion among hole pairs. This presumably will occur at low doping levels. This scenario is compatible with a checkerboard-type charge density state proposed to explain the ``18\frac{1}{8}81​ anomaly'' in the lanthanum cuprate family, except that it is the ground state only when the system possesses strong antiferromagnetic order.
The evolution of the Fermi surface upon hole doping is studied in the t-J model of correlated electrons by exact diagonalization of chains and planes. In one dimension at low hole doping, the momentum distribution function n(k) indicates the presence of pocketlike features at the (noninteracting) Fermi momentum, while increasing the density of holes a large Fermi surface is observed. Although the results in two dimensions are consistent with this picture, conclusive evidence for the existence of hole pockets (at intermediate doping levels) cannot be provided in the present study of 4\ifmmode\times\else\texttimes\fi{}4 and \ensuremath{\surd}18 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}18 square lattices. In order to improve the resolution in momentum space, twisted boundary conditions are used for the two-dimensional clusters.
We investigate the effect of doping in the hole spectral function by exact diagonalization of a 16-site cluster with two holes. It is shown that for more than one hole the spectral function differs qualitatively from the one in the single-hole limit. In particular, we find no evidence for the formation of a Fermi surface having hole ``pockets'' localized around (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) in momentum space. On the contrary, a large low-energy peak appears in the spectral function of the hole propagator at k=(\ensuremath{\pi},0) or (0,\ensuremath{\pi}) which is then the best candidate for the existence of those pockets. A (\ensuremath{\pi}/2,\ensuremath{\pi}/2) quasihole state is obtained as a small energy excitation of the two-hole system. We discuss some consequences for photoemission and optical-conductivity experiments in high-Tc{\mathit{T}}_{\mathit{c}}Tc​ materials at intermediate doping.
Hole-density correlation functions in the t-J model on lattices with square geometry are studied by the method of exact diagonalization for systems of various sizes with up to 26 sites. For the largest sizes we employ only a restricted basis, whereby for the extrapolation we introduce and test a scaling procedure. For two holes in an antiferromagnet our results confirm their binding for J>Jc{\mathit{J}}_{\mathit{c}}Jc​\ensuremath{\sim}0.2t. For four holes the two-point and four-point density correlations, calculated for N=18,20 sites, show clear evidence for hole-clustering instabilities for J>Js{\mathit{J}}_{\mathit{s}}Js​\ensuremath{\sim}1.5t; a first instability is likely to be towards a striped phase, with holes forming domain walls along the (0,1) or (1,0) direction. Signs for the phase separation into a hole-rich and a hole-free phase appear only at J>Js∗{\mathit{J}}_{\mathit{s}}^{\mathrm{*}}Js∗​\ensuremath{\sim}2.5t. Results in the intermediate regime 0.4J/t1.2 are consistent with the pairing picture, although there are also indications for a large weight of configurations representing domain walls along the (1,1) direction.