Griffiths-McCoy Singularities in Random Quantum Spin Chains: Exact Results through Renormalization

Abstract

The Ma-Dasgupta-Hu renormalization group (RG) scheme is used to study singular quantities in the Griffiths phase of random quantum spin chains. For the random transverse-field Ising spin chain we have extended Fisher's analytical solution to the off-critical region and calculated the dynamical exponent exactly. Concerning other random chains we argue by scaling considerations that the RG method generally becomes asymptotically exact for large times, both at the critical point and in the whole Griffiths phase. This statement is checked via numerical calculations on the random Heisenberg and quantum Potts models by the density matrix renormalization group method.In a random quantum system at zero temperature several physical quantities are singular not only at the critical point, but in a whole region, as well, which extends on both sides of the transition point [1]. In this Griffiths phase [2] a random quantum system is non-critical in the space direction (spatial correlations decay exponentially), whereas it is critical in the time direction and the corresponding behavior due to Griffiths-McCoy singularities [2,3] is controlled by a line of semi-critical fixed points characterized by the dynamical exponent z(δ), which depends on the value of the quantum control-parameter, δ. For example average autocorrelations decay in (imaginary) time, τ , as ∼ τ −z(δ) ; for a small magnetic field, H → 0, the magnetization behaves as ∼ H 1/z(δ) ; the low temperature susceptibility and specific heat are singular as ∼ T 1/z(δ)−1 and ∼ T 1/z(δ) , respectively. Let us mention that Griffiths-McCoy singularities are relevant in experiments on quantum spin glasses [4] and provides a theoretical explanation about the non-Fermi liquid behavior in U and Ce intermetallics [5].Among the theoretical methods developed to study random quantum systems the renormalization group (RG) scheme introduced by Ma, Dasgupta and Hu [6] plays a special rôle. For a class of systems, the critical behavior of those is controlled by an infiniterandomness fixed point [7] (IRFP), the RG method becomes asymptotically exact during iteration. For some one-dimensional models, random transverse-field Ising model [8] (RTIM) and the random XXZ-model [9], Fisher has obtained analytical solution of the RG equations and in this way many new exact results and new physical insight about the critical behavior of these models have been gained. Subsequent analytical [10] and numerical [11,10] investigations of the models are in agreement with Fisher's results. The RG-scheme has been numerically implemented in higher dimensions [7,12], as well, to study the critical behavior of the RTIM and reasonable agreement with the results of quantum MonteCarlo simulations [13] has been found. Considering the Griffiths-phase of random quantum spin chains here the RG-scheme has been rarely used [12], mainly due to the general belief that the method looses its asymptotically exact properties by leaving the vicinity of the scale invariant critical point.Our aim in the present Letter is to ...

Strong Griffiths singularities in random systems and their relation to extreme value statistics

“…At the fixed point the energy scale still goes to zero and the decimation equations are asymptotically exact leading to exact dynamical singularities, however the spatial correlations are short-ranged and correct only up to a range of ξ. The analytical solution of the RG equations by Fisher [4] is exact up to O(δ) which is then extended into the complete Griffiths phase [21,22]. The dynamical exponent, z, is found to be the positive root of the equation [22]:…”

“…[30] for the quantum critical point whereas in Ref. [21] for the Griffiths phase. The transformation rules for the bonds and external fields are of the form given in Eqs.…”

Strongly disordered spin ladders

“…However, a precise numerical calculation of a small ∆ by the DMRG method is very difficult, therefore we used another strategy, as described in details in Refs. 37,44. By this method one considers the equivalent AF chain with random first-and second-neighbor couplings (see Fig.…”

Random-bond Potts model in the large-qlimit

“…Similar, disorder dominated critical behavior occur in random quantum spin chains, where analytical results are available [21][22][23] , and also in 2d random quantum ferromagnets 24 . Whether exact results can be obtained also for the 2d RBPM in the large-q limit will be seen in future research.…”