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arXiv:cond-mat/0011479 · Physical review. B, Condensed matter

Breakdown of the perturbative renormalization group for S>~1 random antiferromagnetic spin chains

A. Saguia, B. Boechat, M. A. Contínentino, O. F. de Alcantara Bonfim

Published Jan 12, 2001• 7 citations
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Abstract

We investigate the application of a perturbative renormalization group (RG) method to random antiferromagnetic Heisenberg chains with arbitrary spin size. At zero temperature we observe that initial arbitrary probability distributions develop a singularity at $J=0,$ for all values of spin S. When the RG method is extended to finite temperatures, without any additional assumptions, we find anomalous results for $S>~1.$ These results lead us to conclude that the perturbative scheme is not adequate to study random chains with $S>~1.$ Therefore a random singlet phase in its more restrictive definition is only assured for spin-1/2 chains.

DOI
10.1103/physrevb.63.052414

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We investigate the application of a perturbative renormalization group (RG) method to random antiferromagnetic Heisenberg chains with arbitrary spin size. At zero temperature we observe that initial arbitrary probability distributions develop a singularity at J=0,J=0,J=0, for all values of spin S. When the RG method is extended to finite temperatures, without any additional assumptions, we find anomalous results for S> 1.S>~1.S> 1. These results lead us to conclude that the perturbative scheme is not adequate to study random chains with S> 1.S>~1.S> 1. Therefore a random singlet phase in its more restrictive definition is only assured for spin-1/2 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 introduce variants of the Ma-Dasgupta renormalization-group (RG) approach for random quantum spin chains, in which the energy-scale is reduced by decimation built on either perturbative or non-perturbative principles. In one non-perturbative version of the method, we require the exact invariance of the lowest gaps, while in a second class of perturbative Ma-Dasgupta techniques, different decimation rules are utilized. For the S = 1 random antiferromagnetic Heisenberg chain, both type of methods provide the same type of disorder dependent phase diagram, which is in agreement with density-matrix renormalization-group calculations and previous studies.
Combining quantum Monte Carlo simulations with the maximum entropy method, we study the dynamical structure factor S(k,ω)S(k,ω)S(k,ω) of spin-1/2 antiferromagnetic Heisenberg chains with various random bond distributions. We emphasize the crossover behavior in the dynamical properties from pure chain to disorder-dominated random singlet phase due to bond randomness. For the distribution corresponding to the infinite randomness fixed point, S(k,ω)S(k,ω)S(k,ω) develops broad non-spinon excitations as well as the random-singlet peak near (k,ω)=(π,0)(k, ω) = (π, 0)(k,ω)=(π,0), consistent with the known results obtained by the real-space renormalization group method. For weak disorder, however, we find clear signature of spinon excitations, reminiscent of pure spin chains, blurred by disorder. We discuss the implication for experiments on random-bond antiferromagnetic spin chains, realizable, e.g., in BaCu2_22​(Six_xx​Ge1−x_{1-x}1−x​)2_22​O7_77​.
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 study the random antiferromagnetic spin-1 chain following the evolution of the bond probability distributions under a renormalization group transformation. We use a mapping of the spin-1 chain into an effective spin-1/2 chain with both ferromagnetic (odd bonds) and antiferromagnetic (even and odd bonds) interactions. We obtain a recursion relation for the coupling constants, solving exactly up to a four-spin cluster. Our improved perturbation treatment on these larger clusters shows that the random singlet phase in the spin-1 chain, differently from previous results, is obtained only when 100100%100 of the odd bonds are strong ferromagnetic, i.e., larger than the even antiferromagnetic bonds. Otherwise the ground state is that of a dimerized spin-1/2 chain.
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.