Rapid Note Critical exponents of random XX and XY chains: Exact results via random walks

We study random XY and (dimerized) XX spin-1/2 quantum spin chains at their quantum phase transition driven by the anisotropy and dimerization, respectively. Using exact expressions for magnetization, correlation functions and energy gap, obtained by the free fermion technique, the critical and off-critical (Griffiths-McCoy) singularities are related to persistence properties of random walks. In this way we determine exactly the decay exponents for surface and bulk transverse and longitudinal correlations, correlation length exponent and dynamical exponent. PACS. 75.50.Lk Spin glasses and other random magnets – 71.55.Jv Disordered structures; amorphous and glassy solids – 75.10.Jm Quantized spin models Disordered quantum spin chains have gained much interest recently [1–8]. It seems to be established right now that the critical properties in these one-dimensional system are governed by an infinite-disorder fixed-point [9] and the application of a renormalization group (RG) scheme à la Dasgupta and Ma [10] is a powerful tool to determine critical properties and static correlations of these new universality classes, either analytically — if possible — or numerically. Although the underlying renormalization scheme is extremely simple the analytical computations are sometimes tedious [1,2]. Therefore an alternative route to the exact determination of critical exponents and other quantities of interest is highly desirable, and this is what we are going to present in this letter. In doing so we follow a route on which we already traveled successfully for the random transverse Ising chain [11–13], and here we are going to do one step further studying random XX and XY models with the help of a straightforward and efficient mapping to random walk problems. This mapping is not only a short-cut to the results known from analytical RG calculations, it also gives new exact results in the off-critical region (the Griffiths-phase [14]) and provides a mean to study situations in which the RG procedure must fail, as for instance in the case of correlated disorder [15], where the distribution is not invariant under renormalization. Here we confine ourselves to a concise presentation of the basic ideas including the determination of various exponents for the first time. The technical details of the a e-mail: [email protected] derivations and further results are deferred to a subsequent publication [16]. The model that we consider is a spin-1/2 XY – quantum spin chain with L sites and open boundaries, defined by the Hamiltonian