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Journal of Physics Condensed Matter

Bound and antibound states of holes in a coupled-chain model

Saurabh Gayen, Indrani Bose

Published Jul 17, 1995• 6 citations
Publisher

Abstract

The binding of two holes in a coupled-chain model described by the t-t'-J Hamiltonian is studied. Using the Bethe Ansatz analysis, it is shown that two holes form antibound states in a triplet spin state. This result can be extended to the case of more than two holes. The eigenvalue problem when the two holes are in a singlet state is solved exactly and one finds that the two holes can bind for all values of J. In this case, antibound states are found for sufficiently large values of J.

DOI
10.1088/0953-8984/7/29/014

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arXiv:cond-mat/9605143•Physical review. B, Condensed matter•Mar 1, 1997•165 citations
The binding of two holes in a coupled-chain model described by the t-t'-J Hamiltonian is studied. Using the Bethe Ansatz analysis, it is shown that two holes form antibound states in a triplet spin state. This result can be extended to the case of more than two holes. The eigenvalue problem when the two holes are in a singlet state is solved exactly and one finds that the two holes can bind for all values of J. In this case, antibound states are found for sufficiently large values of J.
Some exact results on single-hole dynamics in a coupled-chain model are derived. The hole moves in the background of a dimer state which is the exact ground state of the model in the absence of the hole. Three different types of exact eigenfunctions have been determined describing (1) the motion of a bound object of the hole and a free spin 1/2 in a sea of singlets, (2) a state in which the bound object of hole and free spin 1/2 stays close to a localized triplet excitation, and (3) a state in which the bound object moves away from the triplet excitation. The energy eigenvalues corresponding to the three eigenfunctions have been exactly determined.
Using the Bethe ansatz, we calculate the binding energies of two holes just off half-filling in one-dimensional Hubbard chains. Somewhat surprisingly, we find binding comparable to that seen in two-dimensional systems in chains that have a degeneracy at the Fermi level in the noninteracting (U=0) limit. The binding initially IincreasesR with lattice size before beginning a slow decay to zero in the thermodynamic limit. We discuss the general implications of our results for binding energy and other numerical calculations of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ models.
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.
Exact many-body states of square clusters up to 18 sites (with periodic boundary conditions) are investigated in order to study the dynamics of one or two holes in the t-J model. Our method takes full advantage of the translation, rotation, and reflection symmetries of the cluster. Using a modified Lanczos algorithm, we calculate the lowest state of all the different space-group representations and its total spin as well as its spin and hole correlations. For a single hole, at intermediate J/t values, the magnetic, kinetic energies, and staggered magnetization follow J/t power-law behaviors with size-dependent exponents. A binding energy of order J between two holes appears for sufficiently large J/t(\ensuremath{\ge}0.25). One-hole and two-hole calculations give different results regarding the stability of the Nagaoka state (against the singlet state) when J/t\ensuremath{\rightarrow}0. Energy expectation values of variational resonating-valence-bond states are calculated exactly and optimized. Significant overlaps with the exact states are found for any symmetry.
Using numerical results from density matrix renormalization group (DMRG) calculations for the t-J model, on systems as large as 10\ifmmode\times\else\texttimes\fi{}7, we examine the structure of the one and two hole ground states in ladder systems and in two dimensional clusters. A simple theoretical framework is used to explain why holes bind in pairs in two-dimensional antiferromagnets. For the case J/t=0.5, which we have studied, the hole pairs reside predominantly on a 2\ifmmode\times\else\texttimes\fi{}2 core plaquette with the probability that the holes are on diagonal sites greater than nearest-neighbor sites. There is a strong singlet bond connecting the spins on the two remaining sites of the plaquette. We find that a general characteristic of dynamic holes in an antiferromagnet is the presence of frustrating antiferromagnetic bonds connecting next-nearest-neighbor sites across the holes. Pairs of holes bind in order to share the frustrating bonds. At low doping, in addition to hole pairs, there are two additional low-energy structures which spontaneously form on certain finite systems. The first is an undoped L\ifmmode\times\else\texttimes\fi{}2 spin-liquid region, or ladder. The second is a hole moving along a one dimensional chain of sites. At higher doping we expect that hole pairing is always favored.