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Physical review. B, Condensed matter

Dynamics of a single hole in the two-dimensionalt-Jmodel in the presence of a magnetic field and the composite nature of quasiparticles

P. Béran

Published Jul 1, 1996• 7 citations
Publisher

Abstract

The density of state for the single hole in the two-dimensional t-J model is studied in the presence of a constant and homogeneous magnetic field by means of exact calculations for small systems and spin-wave perturbative calculations for larger systems. As the field is turned on, the energy of the ground state is reduced by an amount proportional to parameter t and depending weakly on J. This result, together with the behavior of the Drude weight of the doped t-J model, yields strong support for the idea that the dynamics of charge carriers is governed by parameter t. This idea, confronted to the existence in the spectral density of the hole of a quasiparticle peak dispersing with J, constitutes evidence for the composite nature of quasiparticles. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.1391

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Numerical studies, from variational calculation to exact diagonalization, all indicate that the quasiparticle generated by introducing one hole into a two-dimensional quantum antiferromagnet has the same nature as a string state in the t-${\mathit{J}}_{\mathit{z}}$ model. Based on this observation, we attempt to visualize the quasiparticle formation and subsequent coherent propagation at low energy by studying the generalized t-${\mathit{J}}_{\mathit{z}}$-${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ model in which we first diagonalize the t-${\mathit{J}}_{\mathit{z}}$ model and then perform a degenerate perturbation in ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$. We construct the quasiparticle state and derive an effective Hamiltonian describing the coherent propagation of the quasiparticle and its interaction with the spin wave excitations in the presence of the N\'eel order. We expect that qualitative properties of the quasiparticle remain intact when analytically continuing ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ from the anisotropic ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$${\mathit{J}}_{\mathit{z}}$ to the isotropic ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=${\mathit{J}}_{\mathit{z}}$ limit, despite the fact that the spin wave excitations change from gapful to gapless. Extrapolating to ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=${\mathit{J}}_{\mathit{z}}$, our quasiparticle dispersion and spectral weight compare well with the exact numerical results for small clusters. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9507067•Physical review. B, Condensed matter•Jan 1, 1996•4 citations
The density of state for the single hole in the two-dimensional t-J model is studied in the presence of a constant and homogeneous magnetic field by means of exact calculations for small systems and spin-wave perturbative calculations for larger systems. As the field is turned on, the energy of the ground state is reduced by an amount proportional to parameter t and depending weakly on J. This result, together with the behavior of the Drude weight of the doped t-J model, yields strong support for the idea that the dynamics of charge carriers is governed by parameter t. This idea, confronted to the existence in the spectral density of the hole of a quasiparticle peak dispersing with J, constitutes evidence for the composite nature of quasiparticles. \textcopyright{} 1996 The American Physical Society.
We study the dynamics of one hole in the polarized t-J model. The model describes the physics of holes in polarized antiferromagnetic systems and the one-particle excitations in strong-coupling superconductors as well. The spectral densities of a single hole are calculated numerically using exact diagonalization of small clusters with toroidal boundary conditions. The quasiparticle spectrum is analyzed in detail. We show that a coherent quasiparticle peak is well defined for J/t>0.2. The dispersion relation of the quasiparticle is obtained for different magnetizations. The results are used to discuss the characteristics of one-particle excitations in small-coherence-length superconductors.
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).
We examine the effects of electron-phonon interaction on the dynamics of the charge carriers doped in two-dimensional (2D) Heisenberg antiferromagnet. The t-J model Hamiltonian with a Fr\"ohlich term which couples the holes to a dispersionless (optical) phonon mode is considered for low doping concentration. The evolution of the spectral density function, the density of states, and the momentum distribution function of the holes with an increase of the hole-phonon coupling constant g is studied numerically. As the coupling to a phonon mode increases the quasiparticle spectral weight decreases and a ``phonon satellite'' feature close to the quasiparticle peak becomes more pronounced. Furthermore, strong electron-phonon coupling smears the multimagnon resonances (``string states'') in the incoherent part of the spectral function. The jump in the momentum distribution function at the Fermi surface is reduced without changing the hole pocket volume, thereby providing a numerical verification of Luttinger theorem for this strongly interacting system. The vertex corrections due to electron-phonon interaction are negligible in spite of the fact that the ratio of the phonon frequency to the effective bandwidth is not small. \textcopyright{} 1996 The American Physical Society.
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 report a calculation of the dynamics of more than one hole in the t-J model using the Lanczos algorithm. The hole spectral weight develops a large quasiparticle peak that describes two holes in the same momentum state on the pseudo-Fermi-surface with antiparallel spins. We also show that doping leads to a narrowing of the bandwidth of the spin-wave dispersion. A larger number of higher-energy subpeaks in the spectral function (compared to half filling) are observed, although they are not predicted by mean-field theory.
Numerical studies, from variational calculation to exact diagonalization, all indicate that the quasiparticle generated by introducing one hole into a two-dimensional quantum antiferromagnet has the same nature as a string state in the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model. Based on this observation, we attempt to visualize the quasiparticle formation and subsequent coherent propagation at low energy by studying the generalized t-Jz{\mathit{J}}_{\mathit{z}}Jz​-${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ model in which we first diagonalize the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model and then perform a degenerate perturbation in ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$. We construct the quasiparticle state and derive an effective Hamiltonian describing the coherent propagation of the quasiparticle and its interaction with the spin wave excitations in the presence of the N\'eel order. We expect that qualitative properties of the quasiparticle remain intact when analytically continuing ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ from the anisotropic ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$Jz{\mathit{J}}_{\mathit{z}}Jz​ to the isotropic ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=Jz{\mathit{J}}_{\mathit{z}}Jz​ limit, despite the fact that the spin wave excitations change from gapful to gapless. Extrapolating to ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=Jz{\mathit{J}}_{\mathit{z}}Jz​, our quasiparticle dispersion and spectral weight compare well with the exact numerical results for small clusters. \textcopyright{} 1996 The American Physical Society.