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Physical review. B, Condensed matter

Motion of two holes in an interacting spin background

Indrani Bose

Published May 1, 1993• 0 citations
Publisher

Abstract

An antiferromagnetic spin model (S=1/2) consisting of a chain of octahedra is considered. In a certain parameter regime of the Hamiltonian the exact ground state is a local resonating-valence-bond state. We start with the t-J Hamiltonian for this model describing two holes in the same octahedron, one in the base and the other at one of the vertex sites of the octahedron. The t-J Hamiltonian consists of three parts: ${\mathit{H}}_{\mathit{t}1}$, describing the motion of holes along the chain, ${\mathit{H}}_{\mathit{t}2}$, describing the motion of holes in the bases of the octahedra, and ${\mathit{H}}_{\mathit{J}}$, describing the antiferromagnetic spin-spin interaction. Some exact eigenstates of ${\mathit{H}}_{\mathit{t}2}$ and ${\mathit{H}}_{\mathit{J}}$ are constructed. Successive applications of ${\mathit{H}}_{\mathit{t}1}$ on some of these states show two types of localization for one of the holes. In one type of localization one of the holes is localized at its original location while the other hole moves. In the other type of localization, a hole, located in the base of an octahedron, cannot move away from the other hole sitting at a vertex site of the same octahedron. The localization in both cases is produced by the topology of the lattice on which the t-J model is defined.

DOI
10.1103/physrevb.47.11537

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An antiferromagnetic spin model (S=1/2) consisting of a chain of octahedra is considered. In a certain parameter regime of the Hamiltonian the exact ground state is a local resonating-valence-bond state. We start with the t-J Hamiltonian for this model describing two holes in the same octahedron, one in the base and the other at one of the vertex sites of the octahedron. The t-J Hamiltonian consists of three parts: Ht1{\mathit{H}}_{\mathit{t}1}Ht1​, describing the motion of holes along the chain, Ht2{\mathit{H}}_{\mathit{t}2}Ht2​, describing the motion of holes in the bases of the octahedra, and HJ{\mathit{H}}_{\mathit{J}}HJ​, describing the antiferromagnetic spin-spin interaction. Some exact eigenstates of Ht2{\mathit{H}}_{\mathit{t}2}Ht2​ and HJ{\mathit{H}}_{\mathit{J}}HJ​ are constructed. Successive applications of Ht1{\mathit{H}}_{\mathit{t}1}Ht1​ on some of these states show two types of localization for one of the holes. In one type of localization one of the holes is localized at its original location while the other hole moves. In the other type of localization, a hole, located in the base of an octahedron, cannot move away from the other hole sitting at a vertex site of the same octahedron. The localization in both cases is produced by the topology of the lattice on which the t-J model is defined.
An antiferromagnetic spin model (S=1/2) consisting of a chain of octahedra is considered. In a certain parameter regime of the Hamiltonian the exact ground state is a local resonating-valence-bond (RVB) state. We start with the t-J Hamiltonian for this model describing a single hole in the system and derive exact ground and excited states of the Hamiltonian when the hole is static. The local spin distortion around a hole is termed a spin polaron. The dynamics of the hole is studied through construction of exact eigenstates describing spin polarons in extended states. Coherent states for all momentum wave vectors in the chain direction exist for certain spin polarons. In the Nagaoka limit, the RVB polaronic states have energy greater than that of the ferromagnetic state. For exchange interaction J\ensuremath{\ne}0, the RVB states have lower energy. Other exact states obtained include states in which the hole motion is localized. Structural similarity of the present model to the arrangement of Cu spins in La2{\mathrm{La}}_{2}La2​CuO4{\mathrm{CuO}}_{4}CuO4​ is also highlighted.
A quasi-one-dimensional spin model that consists of a chain of octahedra of spins has been suggested, for which in a certain parameter regime of the Hamiltonian the ground state can be written down exactly. The ground state is highly degenerate and can be other than a singlet. Also, several excited states can be constructed exactly. The ground state is a local resonating valence bond state for which resonance is confined to rings of spins. Some exact numerical results for an octahedron of spins have also been reported.
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.
A two-chain ladder model is considered that is described by the strong-coupling t - t ´- J - J ´ Hamiltonian. For the case of two holes moving in a background of antiferromagnetically interacting spins, exact, analytical results are derived for the ground-state energy and low-lying excitation spectrum. The ground state is a bound state of two holes with total spin S = 0. The charge excitation is gapless and the spin excitation has a gap. The corresponding wave functions are also exactly determined. The bound hole pair is found to have symmetry of the d-wave type. In the limit of strong rung coupling, the model maps onto an effective hard-core-boson model which exhibits dominant superconducting pairing correlations.
Two spin models in two dimensions are constructed for which the exact ground states can be written down in a certain parameter regime of the Hamiltonian. The ground state of the first model is a local resonating valence bond (RVB) state in which not all the spins participate in resonance. The ground state is highly degenerate and there is a gap in the excitation spectrum. The second model is shown to have a ground state which is a quantum spin liquid with all the spins taking part in resonance. The ground state obtained is non-degenerate and there is no gap in the excitation spectrum.
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.