An Impurity Driven Phase Transition in the Antiferromagnetic Spin-1 Chain

Using an asymptotically exact real space renormalization procedure, we find that the Heisenberg antiferromagnetic spin-1 chain undergoes an impurity driven second order phase transition from the Haldane phase to the random singlet phase, as the bond distribution is broadened. In the Haldane phase and near the critical point, there is a Griffiths region in which the gap is filled and the susceptibility diverges in a non-universal manner. The correlation length critical exponent is ν ≈ 2.3. Pacs: 75.10.J, 75.30.H, 75.50.E Typeset using REVTEX 1 Recently, there has been tremendous interest in the antiferromagnetic (AF) spin-1 chain, inspired by the famous conjecture by Haldane [1] that integer-spin chains behave quite differently from half-odd-integer-spin chains. For example, in the absence of disorder, the spin-1 chain has short-range spin-spin correlations in the ground state and an excitation gap [1] whereas the spin-1/2 chain is critical. The ground state of the spin-1 chain also has a novel string-topological order [2]. Some of these results have been experimentally confirmed [3]. Randomness is always present in real materials. Theoretical work has demonstrated that randomness dramatically affects the physical properties of the AF spin-1/2 chain [4,5] and other random one-dimensional magnetic systems [6–9]. In this letter we report a systematic theoretical study on the effects of bond randomness on the AF spin-1 chain, based on the real space renormalization group scheme developed by Ma et al. [4] (see also Ref. [10]) and extended by Fisher [5]. Our main result is that in the presence of bond randomness, there are two distinct phases in the AF Heisenberg spin-1 chain, separated by a second order critical point. The nature of these two phases are described below. The ground state in the Haldane phase in the absence of randomness is well described by the valence bond solid (VBS) state [11]. In the VBS state each spin-1 is composed of two symetrized spin-1/2 objects. The spin-1/2 objects form singlets with spin-1/2 objects on neighboring sites. This state resembles the ground state of a dimerized spin-1/2 chain. The ground state is nondegenerate, there is an excitation gap, as well as a very stable topological structure. Thus we expect the Haldane phase and its topological structure to be stable against weak bond randomness [8]. On the other hand, when the randomness is strong and the distribution of bond strength is broad, the origonal spin-1 objects coupled by strong bonds form inert singlet pairs and generate effective further neighbor AF couplings. An asymptotically exact real space renormalization group (RG) analysis [4,5] shows that in this case the system flows toward a random singlet (RS) phase [5] with universal thermodynamic properties and power law behavior in averaged spin-spin correlations. In order to study the 2 transition from the Haldane phase to the random singlet (RS) phase, we have extended this RG scheme so that it may be used in both phases. We find the transition between these two phases is second order. The extended RG scheme becomes asymptotically exact in the low energy limit at the critical point, as well as in the RS phase. Thus we are able to extract exact information about the critical point. For examples, we find as the randomness strength approaches the critical point from the Haldane phase, the average spin-spin correlation length diverges in a power law manner with exponent ν = 6 √ 13−1 ≈ 2.3. The string-topological order parameter vanishes with a power law exponent γ = 2ν ≈ 4.6. Consider the Hamiltonian