Physical review. B, Condensed matter

Hole dynamics in thet-Jmodel: An exact diagonalization study

Yukio Hasegawa, Didier Poilblanc

Published Nov 1, 1989• 169 citations

Abstract

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.

DOI: 10.1103/physrevb.40.9035

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Binding of holes and the pair spectral function in thet-Jmodel

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Hole binding in thet-Jmodel

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Hole dynamics in a strongly correlated two-dimensional spin background

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Physical review. B, Condensed matter

Hole dynamics in thet-Jmodel: An exact diagonalization study

Yukio Hasegawa, Didier Poilblanc

Published Nov 1, 1989• 169 citations

Publisher

Abstract

DOI: 10.1103/physrevb.40.9035

Related papers

Binding of holes and the pair spectral function in thet-Jmodel

Didier Poilblanc

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).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).

arXiv:cond-mat/9303029Physical review. B, Condensed matterAug 1, 199349 citations

Hole binding in thet-Jmodel

P. Wróbel, R. Eder

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 ${\mathit{d}}_{\mathit{x}}^{2}$ - ${\mathit{y}}^{2}$ . 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.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 ${\mathit{d}}_{\mathit{x}}^{2}$-${\mathit{y}}^{2}$. 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.

Physical review. B, Condensed matterJan 1, 199416 citations

Hole dynamics in a strongly correlated two-dimensional spin background

Holger Fehske, V. Waas, Heinrich Röder +1 more

In this paper we present results of finite-lattice studies of the t-J and the t-t'-J model for 18 (20) sites with up to 4 (2) holes. A modified Lanczos algorithm allowed for the classification of the ground state according to total spin S. In the one-hole sector we find evidence for the existence of hole pockets. We calculate the effective quasiparticle band and show qualitative differences between the t-J and t-t'-J model for J\ensuremath{\gtrsim}0.2. As U increases we notice a substantial band narrowing, which is maximal at the Nagaoka transition, accompanied by S-level crossings rendering a quasiparticle description difficult. In the high-doping regime we present hole binding energies for up to six holes for the t-J model. Evidence for phase separation is only found for J>t. We also studied the magnetic structure of the ground state on the 18-site lattice in the limit J=0. We present a complete ground-state classification for all numbers of holes ${\mathit{N}}_{\mathit{h}}$ and find S= ${\mathit{S}}_{\mathrm{max}}$ at quarter filling in addition to the Nagaoka case.In this paper we present results of finite-lattice studies of the t-J and the t-t'-J model for 18 (20) sites with up to 4 (2) holes. A modified Lanczos algorithm allowed for the classification of the ground state according to total spin S. In the one-hole sector we find evidence for the existence of hole pockets. We calculate the effective quasiparticle band and show qualitative differences between the t-J and t-t'-J model for J\ensuremath{\gtrsim}0.2. As U increases we notice a substantial band narrowing, which is maximal at the Nagaoka transition, accompanied by S-level crossings rendering a quasiparticle description difficult. In the high-doping regime we present hole binding energies for up to six holes for the t-J model. Evidence for phase separation is only found for J>t. We also studied the magnetic structure of the ground state on the 18-site lattice in the limit J=0. We present a complete ground-state classification for all numbers of holes ${\mathit{N}}_{\mathit{h}}$ and find S=${\mathit{S}}_{\mathrm{max}}$ at quarter filling in addition to the Nagaoka case.

Physical review. B, Condensed matterOct 15, 199143 citations

Hole pairing and clustering in the two-dimensionalt-Jmodel

Peter Prelovšek, X. Zotos

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> ${\mathit{J}}_{\mathit{c}}$ \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> ${\mathit{J}}_{\mathit{s}}$ \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> ${\mathit{J}}_{\mathit{s}}^{\mathrm{*}}$ \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.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>${\mathit{J}}_{\mathit{c}}$\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>${\mathit{J}}_{\mathit{s}}$\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>${\mathit{J}}_{\mathit{s}}^{\mathrm{*}}$\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.

Physical review. B, Condensed matterMar 1, 199381 citations

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

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

We have calculated S(q) and the single particle distribution function 〈 ${\mathit{n}}_{\mathbf{q}}$ 〉 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 ${\mathrm{CuO}}_{2}$ 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 have calculated S(q) and the single particle distribution function 〈${\mathit{n}}_{\mathbf{q}}$〉 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 ${\mathrm{CuO}}_{2}$ 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).

arXiv:cond-mat/9308010Physical review. B, Condensed matterFeb 1, 199427 citations

Strongly correlated electronic systems with one hole: Dynamical properties

Elbio R. Dagotto, Robert Joynt, Adriana Moreo +2 more

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- ${\mathit{J}}_{\mathit{z}}$ 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 ${\mathit{E}}_{\mathit{h}}$ =-3.17+2.83 ${\mathit{J}}^{0.73}$ (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 ${\mathit{J}}^{0.5}$ 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 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-${\mathit{J}}_{\mathit{z}}$ 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 ${\mathit{E}}_{\mathit{h}}$=-3.17+2.83${\mathit{J}}^{0.73}$ (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 ${\mathit{J}}^{0.5}$ 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.

Physical review. B, Condensed matterMay 1, 1990315 citations

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

{\mathit{d}}_{\mathit{x}}^{2}

{\mathit{y}}^{2}

. 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.

{\mathit{N}}_{\mathit{h}}

and find S=

{\mathit{S}}_{\mathrm{max}}

at quarter filling in addition to the Nagaoka case.

{\mathit{J}}_{\mathit{c}}

\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>

{\mathit{J}}_{\mathit{s}}

\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>

{\mathit{J}}_{\mathit{s}}^{\mathrm{*}}

\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.

We have calculated S(q) and the single particle distribution function 〈

{\mathit{n}}_{\mathbf{q}}

〉 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

{\mathrm{CuO}}_{2}

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).

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-

{\mathit{J}}_{\mathit{z}}

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

{\mathit{E}}_{\mathit{h}}

=-3.17+2.83

{\mathit{J}}^{0.73}

(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

{\mathit{J}}^{0.5}

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.