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arXiv:cond-mat/9507067 · Physical review. B, Condensed matter

Quasiparticle structure and coherent propagation in thet- Jz - J⊥ model

Junwu Gan, Per Hedegård

Published Jan 1, 1996• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.53.911

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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-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​-J\ensuremath⊥{\mathit{J}}_{\mathrm{\ensuremath{\perp}}}J\ensuremath⊥​ model in which we first diagonalize the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model and then perform a degenerate perturbation in J\ensuremath⊥{\mathit{J}}_{\mathrm{\ensuremath{\perp}}}J\ensuremath⊥​. 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 J\ensuremath⊥{\mathit{J}}_{\mathrm{\ensuremath{\perp}}}J\ensuremath⊥​ from the anisotropic {\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$${\mathit{J}}_{\mathit{z}} to the isotropic J\ensuremath⊥{\mathit{J}}_{\mathrm{\ensuremath{\perp}}}J\ensuremath⊥​=Jz{\mathit{J}}_{\mathit{z}}Jz​ limit, despite the fact that the spin wave excitations change from gapful to gapless. Extrapolating to J\ensuremath⊥{\mathit{J}}_{\mathrm{\ensuremath{\perp}}}J\ensuremath⊥​=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.
We explore with exact diagonalization the propagation of a single hole in four magnetic phases of the $t\text{\ensuremath{-}}J$-like Kitaev-Heisenberg model on a honeycomb lattice: the N\'eel antiferromagnetic, stripe, zigzag, and Kitaev spin-liquid phases. We find coherent propagation of spin-polaron quasiparticles in the antiferromagnetic phase by a mechanism similar to that in the $t\text{\ensuremath{-}}J$ model for high-Tc{T}_{c}Tc​ cuprates. In the stripe and zigzag phases clear quasiparticles features appear in spectral functions of those propagators where holes are created and annihilated on one sublattice, while they remain largely hidden in those spectral functions that correspond to photoemission experiments. As the most surprising result, we find a totally incoherent spectral weight distribution for the spectral function of a hole moving in the Kitaev spin-liquid phase in the strong-coupling regime $t\ensuremath{\gg}J$ relevant for iridates. At intermediate coupling the finite systems calculation reveals a well-defined quasiparticle at the $\ensuremath{\Gamma}$ point; however, we find that the gapless spin excitations wipe out quasiparticles at finite momenta. Also for this more subtle case we conclude that in the thermodynamic limit the lightly doped Kitaev liquid phase does not support quasiparticle states in the neighborhood of $\ensuremath{\Gamma}$, and therefore yields a non-Fermi liquid, contrary to earlier suggestions based on slave-boson studies. These observations are supported by the presented study of the dynamic spin-structure factor in the Kitaev spin liquid regime.
The $t\ensuremath{-}{J}_{z}$ model is the strongly anisotropic limit of the $t\ensuremath{-}J$ model which captures some general properties of doped antiferromagnets (AF's). The absence of spin fluctuations simplifies the analytical treatment of hole motion in an AF background, and allows us to calculate single- and two-hole spectra with a high accuracy using a regular diagram technique combined with a real-space approach. At the same time, numerical studies of this model via exact diagonalization on small clusters show negligible finite-size effects for a number of quantities, thus allowing a direct comparison between analytical and numerical results. Both approaches demonstrate that the holes have a tendency to pair in p- and d-wave channels at realistic values of t/J.t/J.t/J. Interactions leading to pairing and effects selecting p and d waves are thoroughly investigated. The role of transverse spin fluctuations is considered using perturbation theory. Based on the results of the present study, we discuss the pairing problem in the realistic $t\ensuremath{-}J$-like model. Possible implications for preformed pairs formation and phase separation are drawn.
We solved numerically the integral equation for the selfenergy which describes the motion of a single hole in a two-dimensional quantum antiferromagnet (AF) within the selfconsistent Born approximation. This formulation stresses the similarity of the AF-spin polaron with the standard polaron problem. We confine our calculation to finite cluster geometries and compare with results from previous exact diagonalization studies. The spectral function is characterized by a narrow quasiparticle (qp) peak at the low energy side of the spectra, which appears to be well separated from the incoherent band part for large enough J values. For small J we find a reduced width of ~7t for the incoherent band. The bottom of the coherent qp band always occurs at (±π/2, ±π/2). Its bandwidth initially increases with J until J≃t and then decreases as 2t2/J. A complete characterization of our solution is given, including the dispersion relation and effective masses of this quasiparticle. The comparison with exact diagonalization studies for a 4×4 cluster is remarkably good. From our results we see that a lattice of 16×16 sites describes adequately the thermodynamic limit. We conclude that the simple Born approximation is a valuable scheme to characterize the dynamics of one hole in the t-J model. both in the perturbative and the strong coupling regimes.
Solid state physics, a very broad field in natural science, aims to explain physical properties of materials modeled as a many-body ensemble of the order of ∼ 10^23 interacting particles. For interacting metallic systems, one of the most powerful tools for unraveling the consequences of the collective interaction is the Fermi liquid theory. While still being an approximation, it excellently models many materials especially in the low-temperature and low-energy regime by treating themany-body problem as being equivalent to effectively non-interacting ”particles”, which is done by introducing the concept of quasiparticles. During the last few decades, several exotic phases of matter have been discovered, which originate from strong Coulomb interactions between the electrons, whose description is beyond the Fermi liquid theory. This thesis addresses the question of the validity of Fermi liquid theory once a single hole is introduced into an antiferromagnet with completely localized spins due to the strong interactions. Here, the most-studied case concerns the so-called 2D t − J model, which hosts a well-defined quasiparticle, the spin polaron. Focusing on a more tractable case, the t − Jz model without quantum spin fluctuations, we study the stability of the spin polaron solution once the nextnearest neighbor hopping t′ is introduced into the system. We focus on observations at the Γ-point and the X-point of systems with fixed Jz = 0.4t and varying t′ ∈ [−0.5t, 0.5t]. The model is numerically solved using exact diagonalization on finite-size lattices of up to 32 sites. A study of the quasiparticle weight under finite-size scaling reveals an “anomalous” regime at the Γ-point fort′ ≤ −0.3t, indicated by a vanishing quasiparticle weight. In order to characterize the low-energy dynamics within this “anomalous” regime, we investigated the spin correlation centered around the hole, as well as the magnon number distribution of a single-doped 20-site system. Based on the results, we present a preliminary picture of the antiferromagnetic ground state doped with one hole within that “anomalous” regime (t′ ≤ −0.3t) that deviates from the standard spin polaron as the cloud of magnetic fluctuations, typically localized around the hole, delocalizes over the whole finite-size lattice. Consequently, such systems show a weakly suppressed antiferromagnetic configuration under single-hole doping.
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.
Based on a variational approach, we propose that there are two kinds of low-energy states in the t-J-type models at low doping. In a quasiparticle state an unpaired spin bound to a hole with a well-defined momentum can be excited with spin waves. The resulting state shows a suppression of antiferromagnetic order around the hole with the profile of a spin bag. These spin-bag states with spin and charge or hole separated form a continuum of low-energy excitations. Very different properties predicted by these two kinds of states explain a number of anomalous results observed in the exact diagonalization studies on small clusters up to 32 sites.