The phase-diagram of the two-dimensional Blume-Capel model with a random crystal field is investigated within the framework of a real-space renormalization group approximation. Our results suggest that, for any amount of randomness, the model exhibits a line of Ising-like continuous transitions, as in the pure model, but no first-order transition. At zero temperature the transition is also continuous, but not in the same universality class as the Ising model. In this limit, the attractor (in the renormalization group sense) is the percolation fixed point of the site diluted spin-1/2 Ising model. The results we found are in qualitative agreement with general predictions made by Berker and Hui on the critical behaviour of random models. 75.10.Hk; 64.60.Ak; 64.60.Kw 30 kT J O D ∆ J 10 20 *

We study disordered spin-1 quantum chains with random exchange and biquadratic interactions using a real space renormalization group approach. We find that the dimerized phase of the pure biquadratic model is unstable and gives rise to a random singlet phase in the presence of weak disorder. In the Haldane region of the phase diagram we obtain a quite different behavior.

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 and as disorder decreases, the system shows a crossover from a Griffiths to a disordered Haldane phase.The study of the effects of disorder on quantum systems is an actual and important area of research [1]- [17]. Intensive work on the last decades has deepened our understanding of the phase transitions which occur in pure quantum systems [18]. Now the main effort is concentrated in understanding the role of randomness in these transitions. This gives rise to new and interesting phenomena as the existence of Griffiths phases [2,7]. In this connection random quantum spin chains have been intensive investigated. In the pure case their behavior is well known [19]. Also for spin−1/2 quantum antiferromagnetic chains a perturbative approach developed by Ma, Dasgupta and Hu (M DH) [1] and extended by Fisher [2] allows to obtain results which are essentially exact for this system. The picture which emerges for these chains is described by a random singlet phase (RSP ) where spins are coupled in pairs over arbitrary distances. In the renormalization group approach this random singlet phase is governed by an infinite randomness fixed point [2,7,17]. A straightforward extension of the M DH method for biquadratic spin−1 chains has shown that in the Heisenberg case the perturbative RG approach may fail even for the case of strong disorder [3]. The reason is that in the elimination process of strong interactions, in which consists the M DH approach, interactions stronger than those eliminated are generated. This failure is better demonstrated when the method is extended to finite temperatures where it gives rise to non-physical behavior as negative specific heat and so on [6]. Several proposals have been put forward to extend the M DH method for quantum spin chains, with S > 1/2 [11]-[14], without undisputed success. The challenge in the case of quantum integer spin chains is particularly exciting as it deals with the question of the fate of the Haldane phase [20] in the presence of disorder. The existence of a gap in the excitation spectrum is not sufficient to guarantee the robustness of pure chain behavior with respect to the effects of disorder. For gapped biquadratic chains, any amount of disorder drives the system to a random singlet phase or infinite randomness fixed point [3]. If this is not the case for Haldane chains there may be a unique property of integer chains which confers them a special stability with respect to the introduction of disorder. In fact Hida [16] using a density matrix renormalization group app...

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We use a modified perturbative renormalization group approach to study the random quantum antiferromagnetic spin-3/2 chain. We find that in the case of rectangular distributions there is a quantum Griffiths phase and we obtain the dynamical critical exponent Z as a function of disorder. Only in the case of extreme disorder, characterized by a power law distribution of exchange couplings, we find evidence that a random singlet phase could be reached. We discuss the differences between our results and those obtained by other approaches.