v 1 3 S ep 1 99 6 Higher dimensional realizations of activated dynamic scaling at random quantum transitions

We show that many of the unusual properties of the one-dimensional random quantum Ising model are shared also by dilute quantum Ising systems in the vicinity of a certain quantum transition in any dimension d > 1. Thus, while these properties are not an artifact of d = 1, they do require special circumstances in higher dimensions. PACS numbers:75.10.Nr, 05.50.+q, 75.10.Jm Typeset using REVTEX 1 There is considerable current interest in the properties of phase transitions in random quantum systems. Experimentally accessible quantum transitions, such as the transition from an insulator to a metal [1] or a superconductor [2], and various magnetic-nonmagnetic transitions in heavy fermion compounds [3], high-Tc cuprates [4], or insulating dipolar Ising magnets [5], often occur in situations with strong randomness, and are only poorly understood. Theoretically many authors [6–8] have analyzed the random Ising chain in a transverse field perhaps the simplest random quantum system. In particular, Fisher [8] used a real space renormalization group (RG) approach to obtain a detailed description of the thermodynamics and static correlation functions in the vicinity of the critical point. The properties of the system were found to be very unusual as compared to conventional quantum critical points. Specifically, length scales were found to diverge logarithmically with energy scales, implying a dynamic critical exponent z = ∞. The transition was shown to be flanked on either side by “Griffiths” regions with a susceptibility diverging due to contributions from statistically rare fluctuations. There are very few reliable results on other random quantum transitions, especially in finite dimensions d > 1; thus it is important to understand if the anomalous properties of the random quantum Ising chain are a specialty of d = 1, or if there exist higher dimensional quantum transitions which share these properties. Numerical work on higher dimensional transverse field Ising spin glasses has found evidence for the presence of Griffiths regions, but the dynamic scaling at the critical point seems conventional [9]. In this paper, we provide a simple example of such anomalous scaling in higher dimensional random quantum Ising systems. We consider bond or site diluted Ising models with short range interactions. As was first suggested by Harris [10], and as we argue below, these models have two quantum transitions (See Fig 1): At low dilution, below the percolation threshold, there is a phase transition when the long range ferromagnetic order is destroyed by increasing the transverse field. This is expected to be in the universality class of the generic random bond quantum Ising transition. Right at the percolation threshold, there is a finite range of transverse field strengths at which the system remains critical. There is thus another quantum transition, across the percolation threshold, at low but non-zero 2 transverse field strengths, which is potentially in a different universality class. These two critical lines meet at a multicritical point (M). The properties of the second transition, at the percolation threshold, are determined largely by the statistics and geometry of the percolating clusters about which much information is available. This permits us to make definitive statements about the scaling properties of this transition in any dimension. We show that the dynamic scaling is activated, with length scales diverging logarithmically with energy scales. We also demonstrate the existence of Griffiths phases on either side with diverging susceptibility. Our approach shows clearly the connection between these behaviors, and T = 0 phase transitions at which quantum fluctuations are “dangerously irrelevant” [8,11]; indeed, various exponents of the transition are given by those of the classical percolation transition, and are hence known either exactly or numerically. We also obtain bounds on exponents characterizing the multicritical point. For concreteness, we consider bond-diluted Ising models defined by the Hamiltonian