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arXiv:cond-mat/0512074 · Physical Review B

Single-hole properties of the t−J model on the honeycomb lattice

Andreas Lüscher, Andreas Martin Läuchli, Weihong Zheng, Oleg P. Sushkov

Published Apr 25, 2006• 16 citations
arXivHTMLPDF

Abstract

Motivated by the possible charge ordering in the recently discovered superconductor ${\mathrm{Na}}_{x}\mathrm{Co}{\mathrm{O}}_{2}\phantom{\rule{0.2em}{0ex}}\mathrm{y}{\mathrm{H}}_{2}\mathrm{O}$, which at filling $x=1∕3$ would correspond to a half-filled honeycomb lattice, we investigate the single-hole dynamics of the $t\text{\ensuremath{-}}J$ model on this lattice. Using exact diagonalization, series expansion, and the self-consistent Born approximation, we calculate the quasiparticle dispersion, bandwidth, and residues and compare our findings to the well-established results for the square lattice. Given the similarities between both lattices, it is not surprising that we find qualitatively similar features for the honeycomb lattice, namely, an almost flat band along the edge of the magnetic Brillouin zone and well-defined hole pockets around the corners. However, we report a considerable disagreement between the three methods used concerning the bandwidth and discuss possible origins of this discrepancy.

DOI
10.1103/physrevb.73.155118

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Motivated by the possible charge ordering in the recently discovered superconductor NaxCoO2yH2O{\mathrm{Na}}_{x}\mathrm{Co}{\mathrm{O}}_{2}\phantom{\rule{0.2em}{0ex}}\mathrm{y}{\mathrm{H}}_{2}\mathrm{O}Nax​CoO2​yH2​O, which at filling x=1∕3x=1∕3x=1∕3 would correspond to a half-filled honeycomb lattice, we investigate the single-hole dynamics of the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model on this lattice. Using exact diagonalization, series expansion, and the self-consistent Born approximation, we calculate the quasiparticle dispersion, bandwidth, and residues and compare our findings to the well-established results for the square lattice. Given the similarities between both lattices, it is not surprising that we find qualitatively similar features for the honeycomb lattice, namely, an almost flat band along the edge of the magnetic Brillouin zone and well-defined hole pockets around the corners. However, we report a considerable disagreement between the three methods used concerning the bandwidth and discuss possible origins of this discrepancy.
We present quantum Monte Carlo (QMC) simulations for a single hole in a $t\ensuremath{-}J$ model from J=0.4tJ=0.4tJ=0.4t to J=4tJ=4tJ=4t on square lattices with up to $24\ifmmode\times\else\texttimes\fi{}24$ sites. The lower edge of the spectrum is directly extracted from the imaginary time Green's function. In agreement with earlier calculations, we find flat bands around $(0,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}),$ $(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi},0)$ and the minimum of the dispersion at $(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2).$ For small J, both self-consistent Born approximation and series expansions give a bandwidth for the lower edge of the spectrum in agreement with the simulations, whereas for J/t>1,J/t>1,J/t>1, only series expansions agree quantitatively with our QMC results. This band corresponds to a coherent quasiparticle. This is shown by a finite-size scaling of the quasiparticle weight $Z(\stackrel{\ensuremath{\rightarrow}}{k})$ that leads to a finite result in the thermodynamic limit for the considered values of J/t.J/t.J/t. The spectral function $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ is obtained from the imaginary time Green's function via the maximum entropy method. Resonances above the lowest edge of the spectrum are identified, whose J dependence is quantitatively described by string excitations up to J/t=2.J/t=2.J/t=2.
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.
We study one- and two-hole properties of the $t\ensuremath{-}J$ model at half-filling on the square lattice using series expansion methods at T=0.T=0.T=0. The dispersion curve for one-hole excitations is calculated and found to be qualitatively similar to that obtained by other methods, but the bandwidth for small t/Jt/Jt/J is some 20% larger than given previously. We also obtain the binding energy and dispersion relation for two-hole bound states. The lowest bound state as t/Jt/Jt/J increases is found to be first d wave, and then p wave, in accordance with predictions based upon the Kohn-Luttinger effect. We also carry out a similar study for the $t\ensuremath{-}{J}_{z}$ 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-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.
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We study the honeycomb lattice $t\text{\ensuremath{-}}J$ model using the fermionic tensor network approach. By examining the ansatz with various unit cells, we discover several different stripe states with different periods that compete strongly with uniform states. At very small doping $\ensuremath{\delta}<0.05$, we find almost degenerate uniform ddd-wave superconducting ground states coexisting with antiferromagnetic order. While at larger doping $\ensuremath{\delta}>0.05$, the ground state is an approximately half-filled stripe-ordered state, where the stripe period decreases with increasing hole doping $\ensuremath{\delta}$. Furthermore, the stripe states with the lowest variational energy always display ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing symmetry. The similarity between our results and those on the square lattice contributes to a more comprehensive understanding of doped Mott insulators.