Comment on "Disorder induced quantum phase transition in random-exchange spin-1/2 chains".

In a recent Letter Hamacher, Stolze, and Wenzel [1] studied the disordered spin-1=2 antiferromagnetic XXZ chain H ˆ J X Lÿ1 iˆ1 ‰ i …S x i S x i‡1 ‡ S y i S y i‡1 † ‡ S z i S z i‡1 Š (1) with i independent identically distributed random variables uniformly distributed over the interval ‰1 ÿ W; 1 ‡ WŠ, with the parameter W controlling the strength of the disorder. The authors claim that for < 1 they found numerical evidence for nonuniversal behavior for weak disorder (W < 1) manifested in a continuously varying exponent …W† describing the asymptotic decay of the transverse spin correlations C xx …r† ˆ hS x i S x i‡r i / r ÿ…W† ; (2) where h i denotes the ground state expectation value averaged over the disorder and the sites i. They concluded that there is no universal infinite randomness fixed point (IRFP) as predicted by Fisher [2]. In this Comment we show that these conclusions are inadequate because the numerical data presented in [1] are not in the asymptotic regime. We demonstrate that the observed behavior is very compatible with the IRFP scenario due to the existence of a W-dependent crossover length scale W that describes the crossover from the pure fixed point to the only relevant IRFP: For L W one observes the critical behavior of the pure system (i ˆ const), and only for L W the true asymptotic critical behavior […W† ˆ 2, independent of W > 0] of the disordered chain becomes visible. Even for strong disorder (W ˆ 1:0) W is of the same order of magnitude as the system sizes considered in [1]. In order to be able to reach sufficiently large system sizes we restrict ourselves to ˆ 0 in which case (1) reduces to a free fermion model and the ground state computations are done following [3]. In Fig. 1 we show the averaged bulk correlation function C xx …L=2† for different strengths of the disorder. We observe that asymp-totically (i.e., for L ! 1) the data follow the behavior C xx …L† / L ÿ2 as predicted by the real space renormaliza-tion group [2]. Only for small L do the data appear to follow a nonuniversal (i.e., W-dependent) power law, and this is the region on which [1] reports. Because of the presence of a crossover length …