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Physical Review Letters

Haldane gap in three dimensions: A rigorous example

Hal Tasaki

Published Apr 23, 1990• 21 citations
Publisher

Abstract

A quasi-one-dimensional S=1 antiferromagnet with XXZ-type intrachain couplings and Ising-type interchain couplings is studied. It has been suggested that when the interchain couplings are sufficiently small such a system has no long-range order, even at T=0, because of strong quantum fluctuation caused by the Haldane gap. Treating the problem in a restricted Hilbert space which describes the low-energy behavior of S=1 antiferromagnets rather accurately, we prove two theorems which establish the existence of a unique disordered ground state with a gap.

DOI
10.1103/physrevlett.64.2066

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A quasi-one-dimensional S=1 antiferromagnet with XXZ-type intrachain couplings and Ising-type interchain couplings is studied. It has been suggested that when the interchain couplings are sufficiently small such a system has no long-range order, even at T=0, because of strong quantum fluctuation caused by the Haldane gap. Treating the problem in a restricted Hilbert space which describes the low-energy behavior of S=1 antiferromagnets rather accurately, we prove two theorems which establish the existence of a unique disordered ground state with a gap.
We consider a quasi-one-dimensional system of spin-1 Heisenberg antiferromagnetic chains in two-dimensional and three-dimensional hypercubic lattices with interchain coupling J and uniaxial single-ion anisotropy D. Using large-scale numerical simulations, we map out the J-D phase diagram and investigate the low-lying excitations of the Haldane phase in the J≪1 limit. We also provide direct evidence that the Haldane phase remains a nontrivial symmetry-protected topological state for small but finite J.
The behavior of Haldane-gap antiferromagnets in a strong magnetic field is not universal. While the low-energy physics of the conventional one-dimensional spin-1 Heisenberg model in its magnetized regime is described by one incommensurate soft mode, other systems with somewhat perturbed coupling constants can possess two characteristic soft modes in a certain range of the field strength. Such a two-component Luttinger liquid phase is realized above the massive Haldane-gap phase, and in general above any massive nonmagnetic phase, when the ground state exhibits short-range incommensurate fluctuations already in the absence of the field.
We study the quenching of the Haldane gap in quasi-one-dimensional systems of weakly coupled spin-1 antiferromagnetic Heisenberg chains. The critical interchain coupling Jc required to stabilize long range magnetic order can be accurately determined from large scale quantum Monte Carlo calculations. Several different geometries of coupled chains are studied, illustrating the dependence of Jc on the coordination of chains. For bipartite geometries, ferromagnetically coupled chains yield similar magnitudes for Jc.
Recent experiments on the quasi-one-dimensional (1D) antiferromagnets R2BaNiO5(R{R}_{2}{\mathrm{BaNiO}}_{5} (RR2​BaNiO5​(R=rare earth) have shown the existence of purely 1D Haldane gap excitations propagating on the Ni chains. Below an ordering temperature, the gap excitations survive and coexist with the conventional spin waves in the ordered phase. We construct a model mixed-spin system in 2D for which the ground state can be exactly specified. Using the matrix product method, we show the existence of Haldane gap excitations in the ordered phase. We consider different cases of ordering to study the effect of ordering on the degeneracy of the Haldane gap excitations.
We study the quenching of the Haldane gap in quasi-one-dimensional systems of weakly coupled spin-1 antiferromagnetic Heisenberg chains. The critical interchain coupling Jc required to stabilize long range magnetic order can be accurately determined from large scale quantum Monte Carlo calculations. Several different geometries of coupled chains are studied, illustrating the dependence of Jc on the coordination of chains. For bipartite geometries, ferromagnetically coupled chains yield similar magnitudes for Jc.
Since Haldane conjectured that ground state of one-dimensional Heisenberg antiferromagnet has a finite spin gap for integer spins, while gapless excitations for half-odd integer spins, it has inspired lots of theoretical and experimental studies on low-dimensional quantum magnets. Several one-dimensional chain systems of integer spins, so called Haldane chain, have been discovered and studied, such as CsNiCl3, Y2BaNiO5, PbNi2V2O8, SrNi2V2O8, AgVP2S6. It has been known that most of the Haldane chain has antiferromagnetic coupling for the intrachain exchange interaction. Recently, we have found a Haldane chain compound in which antiferromagnetic order is accompanied by antiferromagnetic interchain coupling between one-dimensional ferromagnetic chains below 30.5 Kelvin. In this presentation, we present comprehensive study of its magnetic and thermal properties.