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Europhysics Letters (EPL)

A self-consistent analytic treatment of two holes in the t - J model

H. Barentzen, V. S. Oudovenko

Published Jul 15, 1999• 8 citations
Publisher

Abstract

The behavior of two holes in the t - J model is analyzed on the infinite square lattice by an extension of the self-consistent Born approximation, which has proved to be very accurate in the single-hole problem. The main ingredient of our theory is the exact form of the eigenstates. Our results agree well with those of previous numerical studies of the problem. The d -wave bound state is found to have the lowest energy with a critical value J / t | c ≈ 0.4, and there is no indication for a crossover between the d -wave and the p -wave bound states.

DOI
10.1209/epl/i1999-00376-7

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We present results of an exact diagonalization calculation of the spectral function A(k,\ensuremath{\omega}) for a single hole described by the t-J model propagating on a 32-site square cluster. The minimum energy state is found at a crystal momentum k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2), consistent with theory, and our measured dispersion relation agrees well with that determined using the self-consistent Born approximation. In contrast to smaller cluster studies, our spectra show no evidence of string resonances. We also make a qualitative comparison of the variation of the spectral weight in various regions of the first Brillouin zone with recent angle-resolved-photoemission data.

arXiv:cond-mat/9509087•Physical review. B, Condensed matter•Dec 1, 1995•79 citations
The behavior of two holes in the t - J model is analyzed on the infinite square lattice by an extension of the self-consistent Born approximation, which has proved to be very accurate in the single-hole problem. The main ingredient of our theory is the exact form of the eigenstates. Our results agree well with those of previous numerical studies of the problem. The d -wave bound state is found to have the lowest energy with a critical value J / t | c ≈ 0.4, and there is no indication for a crossover between the d -wave and the p -wave bound states.
The spin bipolaron in the t–J model, i.e., two holes interacting with an antiferromagnetic spin background, is treated by an extension of the self-consistent Born approximation (SCBA), which has proved to be very accurate in the single-hole (spin polaron) problem. One of the main ingredients of our approach is the exact form of the bipolaron eigenstates in terms of a complete set of two-hole basis vectors. This enables us to eliminate the hole operators and to obtain the eigenvalue problem solely in terms of the boson (magnon) operators. The eigenvalue equation is then solved by a procedure similar to Reiter's construction of the single-polaron wave function in the SCBA. As in the latter case, the eigenvalue problem comprises a hierarchy of infinitely many coupled equations. These are brought into a soluble form by means of the SCBA and an additional decoupling approximation, whereupon the eigenvalue problem reduces to a linear integral equation involving the bipolaron self-energy. The numerical solutions of the integral equation are in quantitative agreement with the results of previous numerical studies of the problem. The d-wave bound state is found to have the lowest energy with a critical value J/t| c ≈ 0.4. In contrast to recent claims, we find no indication for a crossover between the d-wave and p-wave bound states.
The ttt-JJJ model in the spinless-fermion representation is studied. An effective Hamiltonian for the quasiparticles is derived using a canonical transformation approach. It is shown that the rather simple form of the transformation generator allows one to take into account the effect of hole interactions with the short-range spin waves and to describe the single-hole ground state. Obtained results are very close to ones of the self-consistent Born approximation. Further accounting of the long-range spin-wave interaction is possible on a perturbative basis. Spin-wave exchange and an effective interaction due to minimization of the number of broken antiferromagnetic bonds are included in the effective quasiparticle Hamiltonian. The two-hole bound state problem is solved using a Bethe-Salpeter equation. The only bound state found to exist in the region of 1<(t/J)<51<(t/J)<51<(t/J)<5 is the ddd wave. Both types of the hole-hole interaction are important for its formation. A discussion of the possible relation of the obtained results to the problem of superconductivity in real systems is presented.
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.
We report on a detailed examination of numerical results and analytical calculations devoted to a study of two holes doped into a two-dimensional, square lattice described by the $t\ensuremath{-}J$ model. Our exact diagonalization numerical results represent the first solution of the exact ground state of two holes in a 32-site lattice. Using this wave function, we have calculated several important correlation functions, notably the electron momentum distribution function and the hole-hole spatial correlation function. Further, by studying similar quantities on smaller lattices, we have managed to perform a finite-size scaling analysis. We have augmented this work by endeavouring to compare these results to the predictions of analytical work for two holes moving in an infinite lattice. This analysis relies on the canonical transformation approach formulated recently for the $t\ensuremath{-}J$ model. From this comparison we find excellent correspondence between our numerical data and our analytical calculations. We believe that this agreement is an important step helping to justify the quasiparticle Hamiltonian, and, in particular, the quasiparticle interactions that result from the canonical transformation approach. Also, the analytical work allows us to critique the finite-size scaling ansatzes used in our analysis of the numerical data. One important feature that we can infer from this successful comparison involves the role of higher harmonics in the two-particle, d-wave symmetry bound state---the conventional $[\mathrm{cos}{(k}_{x})\ensuremath{-}\mathrm{cos}{(k}_{y})]$ term is only one of many important contributions to the d-wave symmetry pair wave function.
The t-J model in the large-t/J and low-doping limit is approximated by an effective Hamiltonian that treats spin degrees of freedom within the linear-spin-wave approximation and the hole-hopping term with a hole-spin interaction vertex that is linear in the spin-wave operators. The Dyson's equation for the single-hole Green's function is solved numerically on finite- but sufficiently large-size square lattices. Our calculation is based on an approximate equation for the hole motion that has been obtained using a self-consistent perturbation approach where crossing diagrams are neglected. We have studied the contribution of vertex corrections and found that the leading two-loop crossing diagram is exactly zero. Thus, the leading nonzero contribution to the proper self-energy from crossing diagrams is a three-loop correction. In two dimensions, it is possible to solve Dyson's equation numerically including these three-loop corrections. The results demonstrate that the contribution of such vertex corrections is indeed small. Our solution for the hole spectral function, obtained by neglecting such small contributions from vertex corrections, shows that most features describing the hole motion are in close agreement with the results of the exact diagonalization on the 42{4}^{2}42 lattice in the region of J/t\ensuremath{\le}0.2. The results obtained on sufficiently large-size lattices suggest that certain important features of the spectral function remain while others change due to finite-size effects. In particular, we find well-defined peaks above the lowest-energy peak that survive in the thermodynamic limit. We analyze these higher-energy peaks of the spectral functions and find that they can be understood as ``string'' excitations, which are created by the hole moving in a nearly linear potential.
We present results of an exact diagonalization calculation of the spectral function A(k,\ensuremath{\omega}) for a single hole described by the t-J model propagating on a 32-site square cluster. The minimum energy state is found at a crystal momentum k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2), consistent with theory, and our measured dispersion relation agrees well with that determined using the self-consistent Born approximation. In contrast to smaller cluster studies, our spectra show no evidence of string resonances. We also make a qualitative comparison of the variation of the spectral weight in various regions of the first Brillouin zone with recent angle-resolved-photoemission data.