Absence of logarithmic scaling in the ageing behaviour of the 4D spherical model

Abstract

The non-equilibrium dynamics of the kinetic spherical model with a non-conserved orderparameter, quenched to T ≤ T c from a fully disordered initial state, is studied at its upper critical dimension d = d * = 4. In the scaling limit where both the waiting time s and the observation time t are large and the ratio y = t/s > 1 is fixed, the scaling functions of the two-time autocorrelation and autoresponse functions do not contain any logarithmic correction factors and the typical size of correlated domains scales for large times as L(t) ∼ t 1/2 .

Non-Local Meta-Conformal Invariance, Diffusion-Limited Erosion and the XXZ Chain

“…2. The explicit response function (6b) of the dle process is distinct from the predictions (12,14,16), see also table 2. The form of the meta-1 conformal two-point function (14), is clearly different for finite values of the scaling variable v = (r 1 − r 2 )/(t 1 − t 2 ), and similarly for the galilean conformal case (16).…”

“…In contrast to the interface width w(t), which shows a logarithmic growth at d = d * = 1, logarithms cancel in the two-time correlator C and response R, up to additive logarithmic corrections to scaling. This is well-known in the physical ageing at d = d * of simple magnets or of the Arcetri model [20,14,32].…”

“…Example 5: Taking the limit µ → 0 in the meta-conformal representation (13) produces the generators Hence the cga is not a meta-conformal algebra, although z = 1. The co-variant two-point correlator can either be obtained from the generators (15), using techniques similar to those applied in the above example of meta-conformal invariance [31,33], or else by letting µ → 0 in (14). Both approaches give…”

“…The non-analyticity of the correlators (14), and especially (16), in general overlooked in the literature, is required in order to achieve C 12 → 0 for large time-or space-separations, viz. All two-point functions (12,14,16) have indeed the symmetries C 12 (t 1 , t 2 ; r, r) = C 21 (t 2 , t 1 ; r, r) and C 12 (t, t; r 1 , r 2 ) = C 21 (t, t; r 2 , r 1 ), under permutation ϕ 1 ↔ ϕ 2 of the two scaling operators, as physically required for a correlator. The shape of the scaling functions of these three two-point functions is compared in figure 3.…”

Axiomatic construction of quantum Langevin equations

“…First, for a → ∞, one can straightforwardly expand in 1/a, with the result, using [ where again [32, p. 174, eqs. (17,19,21)] was used (the prime indicates the derivative). We also see that the terms of order 1/τ , which arose in (B.7) as distributions, cancel.…”