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arXiv:cond-mat/9611001 · Physical review. B, Condensed matter

Two-hole problem in the t - J model: A canonical transformation approach

V. I. Belinicher, A. L. Chernyshev, V. A. Shubin

Published Aug 1, 1997• 54 citations
arXivHTMLPDF

Abstract

The $t$-$J$ 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 $1<(t/J)<5$ is the $d$ 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.

DOI
10.1103/physrevb.56.3381

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A SELF-CONSISTENT ANALYTIC THEORY OF THE SPIN BIPOLARON IN THE t–J MODEL

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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.

International Journal of Modern Physics B•Mar 30, 2000•0 citations
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.
The report discusses the slave-fermion representations of the t-J model and describes another representation, in which fermions and bosons are completely commuting and in which the properties of fermions are directly related to the properties of physical holes. For a study of the system in the new representation at half-filling, interaction of fermions with two magnons is treated in mean-field theory. The obtained effective model, in comparison to that of the usual slave-fermion representation, has an additional bare hole dispersion due to the hole moving by using quantum spin fluctuations present in the undoped antiferromagnetic ground state. The single-hole Green's function at half-filling is then found numerically using the self-consistent Born approximation. For all studied quantities good or excellent agreement with numerical data is observed in the entire parameter range, noticeably better than in the studies with the slave-fermion representation. Using the same effective model, the two-hole problem is also studied by solving numerically the Bethe-Salpeter equation with noncrossing diagrams.
The motion of a single hole in a two-dimensional Heisenberg antiferromagnet (hAF) is studied in a representation where holes are described as spinless fermions (holons) and spins as normal bosons. Assuming long-range AF order the spin dynamics is treated in linear spin-wave theory. The formulation highlights the close relation with the conventional polaron problem. The holon Green's function is calculated self-consistently within the Born approximation using finite-cluster geometries for the numerical solution. As a remarkable result we find close agreement with the spectral function A(k,\ensuremath{\omega}) of a hole calculated by exact diagonalization methods. A(k,\ensuremath{\omega}) is characterized by a narrow quasiparticle (QP) peak at the low-energy side of the spectrum, which is well separated from the incoherent part for large enough J values. A complete characterization of our solution is given, including the spectral weight, the dispersion relation, and effective masses of the QP state. A finite-size-scaling study gives a nonvanishing spectral weight of the QP in the thermodynamic limit for values J/t typical for copper oxide superconductors. Our calculations indicate that the self-consistent Born approximation is a valuable scheme for characterizing the dynamics of a hole in the t-J model, even in the strong-coupling regime.
The report discusses the slave-fermion representations of the t-J model and describes another representation, in which fermions and bosons are completely commuting and in which the properties of fermions are directly related to the properties of physical holes. For a study of the system in the new representation at half-filling, interaction of fermions with two magnons is treated in mean-field theory. The obtained effective model, in comparison to that of the usual slave-fermion representation, has an additional bare hole dispersion due to the hole moving by using quantum spin fluctuations present in the undoped antiferromagnetic ground state. The single-hole Green's function at half-filling is then found numerically using the self-consistent Born approximation. For all studied quantities good or excellent agreement with numerical data is observed in the entire parameter range, noticeably better than in the studies with the slave-fermion representation. Using the same effective model, the two-hole problem is also studied by solving numerically the Bethe-Salpeter equation with noncrossing diagrams.
This report discusses the slave-fermion representations of the $t\ensuremath{-}J$ model and describes another representation in which fermions and bosons are completely commuting. For a study of the system in the new representation at half-filling, the interaction of fermions with two magnons is treated in mean-field theory. The obtained effective model, in comparison to that of the usual slave-fermion representation, has an additional bare hole dispersion due to the hole moving by using quantum spin fluctuations present in the undoped antiferromagnetic ground state. The single-hole Green's function at half-filling is then found numerically using the self-consistent Born approximation. For all studied quantities good or excellent agreement with numerical data is observed in the entire parameter range, noticeably better than in the studies with the slave-fermion representation. Using the same effective model, the two-hole problem is also studied by solving numerically the Bethe-Salpeter equation with noncrossing diagrams.
The formation of bound states of holes in an antiferromagnetic spin-1/2 background is studied using numerical techniques applied to the t-J Hamiltonian on clusters with up to 26 sites. An analysis of the binding energy as a function of cluster size suggests that a two-hole bound state is formed for couplings larger than a ``critical'' value J/t${\mathrm{\ensuremath{\Vert}}}_{\mathit{c}}$ The symmetry of the bound state is dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2. We also observed that its ``quasiparticle'' weight Z2h{\mathit{Z}}_{2\mathit{h}}Z2h​ (defined in the text), is finite for coupling J/t\ensuremath{\ge}J/t${\mathrm{\ensuremath{\Vert}}}_{\mathit{c}}$. Thus, in the region J/t\ensuremath{\ge}J/t${\mathrm{\ensuremath{\Vert}}}_{\mathit{c}}$ the bound state of two holes behaves like a quasiparticle with charge Q=2e, spin S=0, and dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 internal symmetry. The relation with recent ideas that have suggested the possibility of d-wave pairing in theories of the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprates is briefly discussed.
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.