Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9508038 · Physical review. B, Condensed matter

Single-hole properties in thet-Jand strong-coupling models

B. M. Elrick, Allan E. Jacobs

Published Oct 1, 1995• 6 citations
arXivHTMLPDF

Abstract

We report numerical results for the single-hole properties in the t-J model and the strong-coupling approximation to the Hubbard model in two dimensions. Using the hopping basis with over ${10}^{6}$ states we discuss (for an infinite system) the bandwidth, the leading Fourier coefficients in the dispersion, the band masses, and the spin-spin correlations near the hole. We compare our results with those obtained by other methods. The band minimum is found to be at (\ensuremath{\pi}/2,\ensuremath{\pi}/2) for the t-J model for 0.1\ensuremath{\le}t/J\ensuremath{\le}10, and for the strong-coupling model for 1\ensuremath{\le}t/J\ensuremath{\le}10. The bandwidth in both models is approximately 2J at large t/J, in rough agreement with loop-expansion results but in disagreement with other results. The strong-coupling bandwidth for t/J\ensuremath{\gtrsim}6 can be obtained from the t-J model by treating the three-site terms in first-order perturbation theory. The dispersion along the magnetic zone face is flat, giving a large parallel/perpendicular band mass ratio.

DOI
10.1103/physrevb.52.10369

Related papers

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.

Physical review. B, Condensed matter•May 1, 1990•315 citations

Superconductivity, hole motion, and spin-charge correlations in thet-Jmodel in two dimensions

K.-H. Luk, Daniel Lee Cox

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${)}_{\mathit{c}}$\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${)}_{\mathit{c}}$\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.

Physical review. B, Condensed matter•Mar 1, 1990•12 citations

Hubbard model in the strong-coupling approximation

B. M. Elrick, Mitchel D. Kovarik, Allan E. Jacobs +1 more

We report numerical results for the two-dimensional Hubbard model in the strong-coupling approximation, including the three-site terms omitted in the t-J model, for a 4\ifmmode\times\else\texttimes\fi{}4 system with periodic boundary conditions. We study the one- and two-hole band structures, the root-mean-square hole-hole separation, and the hole-hole binding energy for both the t-J and the strong-coupling models. In the physical parameter region (8U/t40) believed appropriate for the high-temperature superconductors, the three-site terms enhance the hole mobility, increasing the one- and two-hole bandwidths and decreasing the binding energy, but changing the hole-hole separation only slightly; quantitative differences between the models range up to 40%. The transition (at large U/t) to the ferromagnetic Nagaoka state is sharper in the strong-coupling model; our results do not support a transition through all intermediate S values in the bulk.

Physical review. B, Condensed matter•Mar 1, 1993•2 citations

Single-hole effective masses in thet-Jmodel

X. Zotos, P. Prelovek, Igor Sega

The ground state of a single hole in the one- and two-dimensional t-J model is studied for finite systems as a function of the phase in the kinetic energy term. The coherent effective hopping ${\mathit{t}}_{\mathit{c}}^{\mathrm{*}}$ and the incoherent hopping ${\mathit{t}}_{\mathit{i}}^{\mathrm{*}}$, evaluated in this way, are connected via the conductivity sum rule. It is shown that for even and odd chains ${\mathit{t}}_{\mathit{c}}^{\mathrm{*}}$\ensuremath{\sim}${\mathit{t}}_{\mathit{i}}^{\mathrm{*}}$, while their reduction from the free value t remains small even for large J/t. The Bethe ansatz solution, obtained for J=2t, is consistent with the latter result, yielding ${\mathit{t}}_{\mathit{c}}^{\mathrm{*}}$/t\ensuremath{\sim}0.938. In the 4\ifmmode\times\else\texttimes\fi{}4 system, the phase variation yields a nontrivial energy manifold. The minimum appears to be near the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},0) line in the Brillouin zone for J/t1, while moving towards the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},\ensuremath{\pi}) line for J/t\ensuremath{\gg}1.

Physical review. B, Condensed matter•Nov 1, 1990•53 citations

Loop-expansion study of the single-hole spectral function in thet-Jmodel

Zhiping Liu, Efstratios Manousakis

We consider a single hole in the t-J model where the Heisenberg and the hole-hopping terms are expanded in powers of the spin-deviation operators. The lowest order corresponds to the linear spin-wave approximation and leads to an effective Hamiltonian studied earlier by means of self-consistent perturbation theory. In this work we shall include all terms generating up to two-loop diagrams in the perturbative expansion of the single-hole Green's function. The hard-core constraints are partially respected and the overall effect gives a renormalization factor of 1.158 to the hopping parameter t. The vertex that describes the process where a spin wave is absorbed and emitted at the same time by the hole is far more important than others. We find that the numerical solutions of Dyson's equation including all two-loop diagrams lower considerably the single-hole ground-state energies over the previous one-loop results and our results on the 4\ifmmode\times\else\texttimes\fi{}4 lattice are in better agreement with those obtained by exact diagonalization in the physical regime (small J/t). Our method can provide numerical solutions to the hole Green's function on very large lattices. We find that with the inclusion of the two-loop diagrams the spectral function retains all its features found in the one-loop calculation and the bandwidth, the quasiparticle residues, and the ``string'' excitations feel very small corrections.

Physical review. B, Condensed matter•Feb 1, 1995•23 citations

Small-cluster study of a single-holet-Jmodel

Tao Xiang

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).

Physical review. B, Condensed matter•Aug 1, 1991•7 citations
We report numerical results for the single-hole properties in the t-J model and the strong-coupling approximation to the Hubbard model in two dimensions. Using the hopping basis with over 106{10}^{6}106 states we discuss (for an infinite system) the bandwidth, the leading Fourier coefficients in the dispersion, the band masses, and the spin-spin correlations near the hole. We compare our results with those obtained by other methods. The band minimum is found to be at (\ensuremath{\pi}/2,\ensuremath{\pi}/2) for the t-J model for 0.1\ensuremath{\le}t/J\ensuremath{\le}10, and for the strong-coupling model for 1\ensuremath{\le}t/J\ensuremath{\le}10. The bandwidth in both models is approximately 2J at large t/J, in rough agreement with loop-expansion results but in disagreement with other results. The strong-coupling bandwidth for t/J\ensuremath{\gtrsim}6 can be obtained from the t-J model by treating the three-site terms in first-order perturbation theory. The dispersion along the magnetic zone face is flat, giving a large parallel/perpendicular band mass ratio.
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.
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.
We report numerical results for the two-dimensional Hubbard model in the strong-coupling approximation, including the three-site terms omitted in the t-J model, for a 4\ifmmode\times\else\texttimes\fi{}4 system with periodic boundary conditions. We study the one- and two-hole band structures, the root-mean-square hole-hole separation, and the hole-hole binding energy for both the t-J and the strong-coupling models. In the physical parameter region (8U/t40) believed appropriate for the high-temperature superconductors, the three-site terms enhance the hole mobility, increasing the one- and two-hole bandwidths and decreasing the binding energy, but changing the hole-hole separation only slightly; quantitative differences between the models range up to 40%. The transition (at large U/t) to the ferromagnetic Nagaoka state is sharper in the strong-coupling model; our results do not support a transition through all intermediate S values in the bulk.
The ground state of a single hole in the one- and two-dimensional t-J model is studied for finite systems as a function of the phase in the kinetic energy term. The coherent effective hopping tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​ and the incoherent hopping ti∗{\mathit{t}}_{\mathit{i}}^{\mathrm{*}}ti∗​, evaluated in this way, are connected via the conductivity sum rule. It is shown that for even and odd chains tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​\ensuremath{\sim}ti∗{\mathit{t}}_{\mathit{i}}^{\mathrm{*}}ti∗​, while their reduction from the free value t remains small even for large J/t. The Bethe ansatz solution, obtained for J=2t, is consistent with the latter result, yielding tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​/t\ensuremath{\sim}0.938. In the 4\ifmmode\times\else\texttimes\fi{}4 system, the phase variation yields a nontrivial energy manifold. The minimum appears to be near the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},0) line in the Brillouin zone for J/t1, while moving towards the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},\ensuremath{\pi}) line for J/t\ensuremath{\gg}1.
We consider a single hole in the t-J model where the Heisenberg and the hole-hopping terms are expanded in powers of the spin-deviation operators. The lowest order corresponds to the linear spin-wave approximation and leads to an effective Hamiltonian studied earlier by means of self-consistent perturbation theory. In this work we shall include all terms generating up to two-loop diagrams in the perturbative expansion of the single-hole Green's function. The hard-core constraints are partially respected and the overall effect gives a renormalization factor of 1.158 to the hopping parameter t. The vertex that describes the process where a spin wave is absorbed and emitted at the same time by the hole is far more important than others. We find that the numerical solutions of Dyson's equation including all two-loop diagrams lower considerably the single-hole ground-state energies over the previous one-loop results and our results on the 4\ifmmode\times\else\texttimes\fi{}4 lattice are in better agreement with those obtained by exact diagonalization in the physical regime (small J/t). Our method can provide numerical solutions to the hole Green's function on very large lattices. We find that with the inclusion of the two-loop diagrams the spectral function retains all its features found in the one-loop calculation and the bandwidth, the quasiparticle residues, and the ``string'' excitations feel very small corrections.
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).