Application of a continuous time cluster algorithm to the two-dimensional random quantum Ising ferromagnet

A cluster algorithm formulated in continuous (imaginary) time is presented for Ising models in a transverse field. It works directly with an infinite number of time-slices in the imaginary time direction, avoiding the necessity to take this limit explicitly. The algorithm is tested at the zero-temperature critical point of the pure two-dimensional (2d) transverse Ising model. Then it is applied to the 2d Ising ferromagnet with random bonds and transverse fields, for which the phase diagram is determined. Finite size scaling at the quantum critical point as well as the study of the quantum Griffiths-McCoy phase indicate that the dynamical critical exponent is infinite as in 1d. PACS. 75.50.Lk Spin glasses and other random magnets – 05.30.-d Quantum statistical mechanics – 75.10.Nr Spin-glass and other random models Quantum phase transitions (QPT) in random transverse Ising models at and close to their quantum critical point at zero temperature have attracted a lot of interest recently [1]. In particular in one dimension many astonishing results have been obtained with powerful analytical [2,3] and numerical [4,5]) tools. Among the most important results is that at the critical point time scales diverge exponentially fast, implying that the dynamical exponent is zcrit = ∞. Such a behavior is reminiscent of thermally activated dynamics in classical random field systems [6], and has also been proposed for the Anderson-Mott transition of disordered electronic systems at the QPT [7]. The other important result for random transverse Ising models in any dimension is that close to the quantum critical point there is a whole region in which various susceptibilities diverge for T → 0 and that all these singularities, also called Griffiths-McCoy singularities [8,9], can be parameterized by a single dynamical exponent z(δ) that varies continuously with the distance δ from the critical point [1,10]. Recently it has been argued that experimentally observed non-Fermi liquid behavior in f -electron compounds can be nicely explained within such a scenario [11]. In one dimension z(δ) diverges for δ → 0. It is not clear in how far these properties, i.e. zcrit =∞ and z(δ → 0) = zcrit also apply in higher dimensions: in 2 and 3-dimensional transverse Ising spin glass models a finite value for zcrit has been reported [12] and for finite dimensional bond diluted ferromagnets it has been shown [13] that zcrit is infinite only at the percolation threshold. a e-mail: [email protected] b Present address: Department of Physics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachiohji, Tokyo, Japan. In this paper we therefore consider the twodimensional random ferromagnet (without dilution) in a transverse field with the help of a new Monte-Carlo cluster algorithm that is particularly suited to handle the inherent difficulties in the study of such a random quantum system: the origin of the Griffiths-McCoy singularities are strongly coupled clusters (with strong ferromagnetic bonds and weak transverse fields) which are extremely hard to equilibrate in a conventional Monte-Carlo algorithm, therefore we had to use a cluster method. Moreover, the exponent z(δ) is a non-universal quantity for which reason we really have to perform the so-called Trotter-limit explained below. A continuous time algorithm that incorporates this limit right from the beginning (in the spirit of Refs. [16, 17]) is the most efficient method we can think of. In the first part of this paper we present the method we use and apply it to the pure case in two dimensions, in the second part we present our results for the random case. In a related work [14] a short account of our results on the Griffiths-McCoy phase in the random system has been given and also a study of essentially the same model with a discrete time cluster algorithm, not performing the Trotter limit and therefore concentrating on the critical behavior, was presented. The system we are interested in is defined by the quantum mechanical Hamiltonian