Rare region effects at classical, quantum and nonequilibrium phase transitions

Abstract

Rare regions, i.e., rare large spatial disorder fluctuations, can dramatically change the properties of a phase transition in a quenched disordered system. In generic classical equilibrium systems, they lead to an essential singularity, the so-called Griffiths singularity, of the free energy in the vicinity of the phase transition. Stronger effects can be observed at zero-temperature quantum phase transitions, at nonequilibrium phase transitions, and in systems with correlated disorder. In some cases, rare regions can actually completely destroy the sharp phase transition by smearing.This topical review presents a unifying framework for rare region effects at weakly disordered classical, quantum, and nonequilibrium phase transitions based on the effective dimensionality of the rare regions. Explicit examples include disordered classical Ising and Heisenberg models, insulating and metallic random quantum magnets, and the disordered contact process.

Rare regions of the susceptible-infected-susceptible model on Barabási-Albert networks

“…Furthermore, when k-dependent weighting was also applied, suppressing hubs or making the network disassortative, GP-like regions could be observed in the simulations. A systematic finite scaling study revealed that these power-laws saturate in the N → ∞ thermodynamic limit, suggesting smeared phase transitions [25]. This can be understood via the possibility of embedding even infinite dimensional, active RR-s in networks with d = ∞, leading to finite steady-state density: ρ(t → ∞,N → ∞).…”

“…An independent cause is related to arbitrarily large (l < N), active rare-regions (RR) with long lifetimes: τ ∝ exp(l) in the inactive phase (λ λ c ). In the region λ 0 c < λ < λ c , where λ 0 c is the critical point of the impure system, a so-called Griffiths phase (GP) [24,25] develops, with algebraic density decay ρ ∝ t −α , α being a nonuniversal exponent. At and below the λ 0 c rare clusters may still form, with an over-average mean degree, which are locally supercritical at a given λ and induce a slower-than-exponential but fasterthan-power-law decay of the global density.…”

Rare‐region effects and dynamics near the many‐body localization transition

“…Ref. [56] for a review. At some phase transitions, such as the quantum phase transition from a magnetically disordered to a magnetically ordered phase in the random transverse-field Ising spin chain, the leading divergences near the phase transition are due to such Griffiths effects [57].…”

Noisy branes

“…The action is given by the DBI for the probe D5-brane, and takes the form 6) where the tilde and the dot denote a partial derivative with respect to z and x respectively. The equations of motion for χ(z, x) and φ(z, x) follow readily from this action, and they have been written explicitly in the appendix of [24].…”

“…At the theoretical level it poses important questions as the existence and nature of disordered quantum critical points [5,6], and the possibility of disorder-induced metal to insulator phase transitions for strongly interacting systems.…”