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Physical review. B, Condensed matter

Hole motion in anS=1chain

Shoucheng Zhang, Daniel P. Arovas

Published Aug 1, 1989• 13 citations
Publisher

Abstract

We consider a simple t-J model of a spin-0 hole hopping in a spin-1 chain. The spin chain in the absence of the hole is governed by the Affleck-Kennedy-Lieb-Tasaki Hamiltonian; it has a disordered ground state and a gap to elementary excitations. In the presence of the hole we find some exact single- and multihole spin-singlet wave functions. We also discuss some properties of baglike excitations using a variational ansatz and kinematical analysis.

DOI
10.1103/physrevb.40.2708

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We consider a simple t-J model of a spin-0 hole hopping in a spin-1 chain. The spin chain in the absence of the hole is governed by the Affleck-Kennedy-Lieb-Tasaki Hamiltonian; it has a disordered ground state and a gap to elementary excitations. In the presence of the hole we find some exact single- and multihole spin-singlet wave functions. We also discuss some properties of baglike excitations using a variational ansatz and kinematical analysis.
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.
We present a controlled microscopic study of mobile holes in the spatially anisotropic (Abelian) gapped phase of the Kitaev honeycomb model. We address the properties of (i) a single hole (its internal degrees of freedom as well as its hopping properties); (ii) a pair of holes [their (relative) particle statistics and interactions]; and (iii) the collective state for a finite density of holes. We find that each hole in the doped model has an eight-dimensional internal space, characterized by three internal quantum numbers: the first two ``fractional'' quantum numbers describe the binding to the hole of the fractional excitations (fluxes and fermions) of the undoped model, while the third ``spin'' quantum number determines the local magnetization around the hole. The fractional quantum numbers also encode fundamentally distinct particle properties, topologically robust against small local perturbations: some holes are free to hop in two dimensions, while others are confined to hop in one dimension only; distinct hole types have different particle statistics, and in particular, some of them exhibit nontrivial (anyonic) relative statistics. These particle properties in turn determine the physical properties of the multihole ground state at finite doping, and we identify two distinct ground states with different hole types that are stable for different model parameters. The respective hopping dimensionalities manifest themselves in an electrical conductivity approximately isotropic in one ground state and extremely anisotropic in the other one. We also compare our microscopic study with related mean-field treatments, and discuss the main discrepancies between the two approaches, which in particular involve the possibility of binding fractional excitations as well as the particle statistics of the holes. On a technical level, we describe the hopping of mobile holes via a quasistationary approach, where effective hopping matrix elements are calculated between ground states with stationary holes at different positions. This approach relies on the fact that the model remains exactly solvable in the presence of stationary holes, and that the motion of sufficiently slow holes does not generate bulk excitations in a gapped phase. When the bare hopping amplitude is much smaller than the energy gap, many of our results, in particular those on the hopping properties and the particle statistics, are exact.
In contrast to the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) isotropic spin-1 Hamiltonian its elementary excitations are not known exactly in general. Based on a simple variational ansatz we construct approximate excitations in the form of dressed hidden domain walls. This allows us to give an improved upper bound for the gap of the AKLT Hamiltonian which is only 1% larger than the best currently available numerical result. We compare the oneand two-soliton dispersion curves with low-lying excitations for a chain of 16 sites as given by Fath and Solyom and find excellent agreement. Spin and string correlations are calculated and found to agree in leading order with the corresponding correlations for the bare kinks. Finally the weight of the string-ordered part of the lowest excited state is calculated and compared with the exact numerical result.
We present a spin-1 chain with a Hamiltonian which has three exactly solvable ground states. Two of these are fully dimerized, analogous to the Majumdar-Ghosh (MG) states of a spin-12\frac{1}{2}21​ chain, while the third is of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type. We use variational and numerical methods to study the low-energy excitations which interpolate between these ground states in different ways. In particular, there is a spin-12\frac{1}{2}21​ excitation which interpolates between the MG and AKLT ground states; this is the lowest excitation of the system and it has a surprisingly small gap. We discuss generalizations of our model of spin fractionalization to higher spin chains and higher dimensions.