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Physical review. B, Condensed matter

Motion of a single hole in a resonating-valence-bond background: Some exact results

Indrani Bose

Published Jun 1, 1991• 5 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 (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 ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$ is also highlighted.

DOI
10.1103/physrevb.43.13602

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Physical review. B, Condensed matter•Apr 1, 1989•16 citations
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 {\mathrm{La}}_{2}$${\mathrm{CuO}}_{4} is also highlighted.
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.
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.
We formulate a quasiparticle theory for a single hole in a quantum antiferromagnet in the limit that the Heisenberg exchange energy is much less than the hopping matrix element, J\ensuremath{\ll}t. We consider the ground state of the spins to be either a quantum N\'eel state or a d-wave resonating-valence-bond (RVB) state. We show in a self-consistent perturbation theory that the hole spectrum is strongly renormalized by the interactions with spin excitations. The hole can be described by a narrow quasiparticle band located at an energy of order -t with a quasiparticle residue of order J/t and a bandwidth of order J. Above the quasiparticle band is an incoherent band of width of order t. Our results indicate that the energy scale for any coherent phenomenon involving the holes is \ensuremath{\delta}J, where \ensuremath{\delta} is the doping concentration. In the N\'eel state we perform a spin-wave expansion on an anisotropic Heisenberg model. In the Ising limit we reproduce previously known results and then expand perturbatively about that limit. In this expansion we find that the holes have a quasiparticle residue of Jz{J}_{z}Jz​/t and a bandwidth of ${J}_{\ensuremath{\perp}}$. In the Heisenberg limit we employ a ``dominant pole'' approximation in which we ignore contributions to the self-energy from the incoherent part of the hole spectrum. A similar technique is used to study the d-wave RVB state. The relevance of our results to recent optical experiments is discussed.
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.