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arXiv:cond-mat/0301067 · Physical Review B

Disorder-induced phases in higher-spin antiferromagnetic Heisenberg chains

Enrico Carlon, Péter Lajkó, Heiko Rieger, Ferenc Iglói

Published Apr 15, 2004• 16 citations
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Abstract

Extensive density-matrix renormalization-group calculations for spin $S=1/2$ and $S=3/2$ disordered antiferromagnetic Heisenberg chains show a rather distinct behavior in the two cases. While at sufficiently strong disorder both systems are in a random singlet phase, we show that weak disorder is an irrelevant perturbation for the $S=3/2$ chain, contrary to what expected from a naive application of the Harris criterion. The observed irrelevance is attributed to the presence of a new correlation length due to enhanced end-to-end correlations. This phenomenon is expected to occur for all half-integer $S>1/2$ chains. A possible phase diagram of the chain for generic S is also discussed.

DOI
10.1103/physrevb.69.144416

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Extensive density-matrix renormalization-group calculations for spin S=1/2S=1/2S=1/2 and S=3/2S=3/2S=3/2 disordered antiferromagnetic Heisenberg chains show a rather distinct behavior in the two cases. While at sufficiently strong disorder both systems are in a random singlet phase, we show that weak disorder is an irrelevant perturbation for the S=3/2S=3/2S=3/2 chain, contrary to what expected from a naive application of the Harris criterion. The observed irrelevance is attributed to the presence of a new correlation length due to enhanced end-to-end correlations. This phenomenon is expected to occur for all half-integer S>1/2S>1/2S>1/2 chains. A possible phase diagram of the chain for generic S is also discussed.
We use extensive density matrix renormalization group (DMRG) calculations to explore the phase diagram of the random S=1S=1S=1 antiferromagnetic Heisenberg chain with a power-law distribution of the exchange couplings. We use open chains and monitor the lowest gaps, the end-to-end correlation function and the string order parameter. For this distribution at weak disorder, the system is in the gapless Haldane phase with a disorder dependent dynamical exponent, zzz, and z=1z=1z=1 signals the border between the nonsingular and singular regions of the local susceptibility. For strong enough disorder, which approximately corresponds to a uniform distribution, a transition into the random singlet phase is detected, at which the string order parameter as well as the average end-to-end correlation function are vanishing and at the same time the dynamical exponent is divergent. Singularities of physical quantities are found to be somewhat different in the random singlet phase and in the critical point.
The zero-temperature phase diagram of the anisotropic, strongly disordered antiferromagnetic quantum spin-1 chain is investigated, within the framework of a real-space renormalization-group approach. In the presence of uniform anisotropy there occurs a phase transition from a random singlet phase to an Ising antiferromagnetic phase controlled by a fixed point that does not have Heisenberg symmetry. We find evidence of a fixed point due to randomness in the anisotropy similar to that which occurs for quantum Heisenberg spin-12\frac{1}{2}21​ chains.
We present a new perturbative real space renormalization group (RG) to study random quantum spin chains and other one-dimensional disordered quantum systems. The method overcomes problems of the original approach which fails for quantum random chains with spins larger than S=1/2. Since it works even for weak disorder, we are able to obtain the zero temperature phase diagram of the random antiferromagnetic Heisenberg spin-1 chain as a function of disorder. We find a random singlet phase for strong disorder. As the disorder decreases, the system shows a crossover from a Griffiths to a disordered Haldane phase.
We consider the properties of finite isotropic antiferromagnetic Heisenberg chains with S=1/2,1,3/2S=1/2,1,3/2S=1/2,1,3/2 spins when a weak magnetic field is applied on a few sites, using White's density-matrix renormalization-group method. For the S=1S=1S=1 chain there exists only one length scale in the system which determines the behavior of the one- and two-point correlation functions both around the local perturbation and near the free boundary. For the critical half-odd-integer spin cases the exponent of the spin-spin correlation function was found to be $\ensuremath{\eta}=1$ and the exponent of the decay of the site magnetization around the perturbed site is ${x}_{m}=\ensuremath{\eta}/2.$ Close to a free boundary, however, the behavior is completely different for S=1/2S=1/2S=1/2 and S>1/2.S>1/2.S>1/2.
Using an adaptive strategy which enables the study of quenched disordered system via the density-matrix renormalization-group method, we compute the various ground-state spin-spin correlation measures of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain with random coupling constants, namely, the mean values of the bulk and of the end-to-end correlations, the typical value of the bulk correlations, and the distribution of the bulk correlations. Our results are in agreement with the predictions of the strong-disorder renormalization-group method. We do not find any hint of logarithmic corrections either in the bulk average correlations, which were recently reported by Shu et al. [Phys. Rev. B 94, 174442 (2016)], or in the end-to-end average correlations. We report the existence of a logarithmic correction on the end-to-end correlations of the clean chain. Finally, we have determined that the distribution of the bulk correlations, when properly rescaled by an associated Lyapunov exponent, is a narrow and universal (disorder-independent) probability function.
The properties of spin-1/2 antiferromagnetic chains with various types of random exchange coupling are studied via an asymptotically exact decimation renormalization-group transformation, which is a generalization of that introduced by Dasgupta and Ma. Random-singlet phases occur in which each spin is paired with one other spin that may be very far away; more exotic phases also occur. The behavior of typical and mean correlation functions is analyzed and found to be very different, with very small sets of spins dominating the latter at long distances as well as the low-temperature thermodynamics. Some of the phase transitions that occur between antiferromagnetically ordered phases and random singlet or other antiferromagnetic phases are also analyzed. For example, if a small uniaxial anisotropy perturbation is added to a random Heisenberg antiferromagnetic chain, a transition occurs from a random-singlet phase to an Ising antiferromagnetic phase, as the anisotropy changes sign from easy plane to easy axis. The staggered magnetization vanishes at the transition with critical exponent \ensuremath{\beta}=8/(1+ \ensuremath{\surd}7 ). Possible implications for the properties of random quantum magnetic systems in higher dimensions are briefly discussed.