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arXiv:cond-mat/9806367 · Physical review. B, Condensed matter

One- and two-hole states in the two-dimensional t−J model via series expansions

Chris J. Hamer, Zheng Weihong, J. Oitmaa

Published Dec 15, 1998• 27 citations
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Abstract

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 $T=0.$ 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 $t/J$ 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 $t/J$ 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.

DOI
10.1103/physrevb.58.15508

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We study one- and two-hole properties of the t\ensuremath−Jt\ensuremath{-}Jt\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−Jzt\ensuremath{-}{J}_{z}t\ensuremath−Jz​ model.
Series expansions at T=0T=0T=0 are used to study properties of the half-filled $t\ensuremath{-}J$ ladder doped with one or two holes, and at quarter filling. Dispersion curves are obtained for one-hole symmetric and antisymmetric (bonding and antibonding) excitations and for the two-hole bound state. The line in the phase diagram that separates bound and unbound states is determined. For quarter filling we compute the ground-state energy and estimate the location of the phase separation line. Comparisons with other numerical and analytical results are presented.
The two-hole excitation spectrum of the $t\ensuremath{-}J$ ladder at half-filling is studied using linked-cluster series expansion methods. A rich spectrum of bound states emerges, particularly at small t/J.t/J.t/J. Their dispersion relations and coherence lengths are computed, along with the threshold behavior as the bound states merge into the continuum. A class of four-hole bound states is also studied, leading to the conclusion that phase separation occurs for $t/J\ensuremath{\lesssim}0.5,$ in agreement with other studies.
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 $t\text{\ensuremath{-}}J$ 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.
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.
The ground state of the two-dimensional Hubbard model on a square lattice is studied in the large-U limit in a half-filled band with a dynamical hole. For a 4\ifmmode\times\else\texttimes\fi{}4 lattice with hopping t=1 and coupling JJc{J}_{c}Jc​\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}Jc{J}_{c}Jc​ the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.
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.