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Physical review. B, Condensed matter

Quenched-impurity influence on quantum critical behavior

Gaetano Busiello, L. De Cesare, Ileana Rabuffo

Published Apr 1, 1984• 11 citations
Publisher

Abstract

The effects of short-range-correlated impurities on the critical behavior of some quantum models with a free-energy spectrum $\ensuremath{\sim}{k}^{\ensuremath{\sigma}}(0<\ensuremath{\sigma}<~2)$ are investigated by using the renormalization-group approach via the replica trick. At zero temperature, two nonsimultaneous expansion parameters ${\ensuremath{\epsilon}}_{q}={d}_{q}\ensuremath{-}d({d}_{q}=\ensuremath{\sigma}$ or $\frac{3}{2}\ensuremath{\sigma}$) and $\ensuremath{\epsilon}=2\ensuremath{\sigma}\ensuremath{-}d$ are involved in terms of which an unusual phase transition is predicted for $d<2\ensuremath{\sigma}$.

DOI
10.1103/physrevb.29.4189

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The effects of short-range-correlated impurities on the critical behavior of some quantum models with a free-energy spectrum \ensuremath∼k\ensuremathσ(0<\ensuremathσ< 2)\ensuremath{\sim}{k}^{\ensuremath{\sigma}}(0<\ensuremath{\sigma}<~2)\ensuremath∼k\ensuremathσ(0<\ensuremathσ< 2) are investigated by using the renormalization-group approach via the replica trick. At zero temperature, two nonsimultaneous expansion parameters \ensuremathϵq=dq\ensuremath−d(dq=\ensuremathσ{\ensuremath{\epsilon}}_{q}={d}_{q}\ensuremath{-}d({d}_{q}=\ensuremath{\sigma}\ensuremathϵq​=dq​\ensuremath−d(dq​=\ensuremathσ or 32\ensuremathσ\frac{3}{2}\ensuremath{\sigma}23​\ensuremathσ) and \ensuremathϵ=2\ensuremathσ\ensuremath−d\ensuremath{\epsilon}=2\ensuremath{\sigma}\ensuremath{-}d\ensuremathϵ=2\ensuremathσ\ensuremath−d are involved in terms of which an unusual phase transition is predicted for d<2\ensuremathσd<2\ensuremath{\sigma}d<2\ensuremathσ.
For random systems with quenched disorders distributed randomly in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions and perfectly correlated in the remaining ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ dimensions, the renormalization-group theory has been performed in the past in two different ways. In the single-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as finite and expands the perturbation terms only for \ensuremath{\epsilon}=dc{\mathit{d}}_{\mathit{c}}dc​-d, where dc{\mathit{d}}_{\mathit{c}}dc​ is the upper critical dimension. In the double-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as infinitesimal and performs an additional expansion for ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$. The former predicts smeared phase transitions, while the latter predicts sharp second-order phase transitions. We wish to find out which of the two predictions holds for a three-dimensional Ising model for which ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$=2 and ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$=1. We argue, using the replica method, that the spin correlation is isotropic for large distances as long as the fluctuation among the random bonds (in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions) is sufficiently smaller than the thermal energy kB{\mathit{k}}_{\mathit{B}}kB​T. With parameters chosen to satisfy this condition in the model, a Monte Carlo simulation has been performed. The results favor a sharp second-order phase transition.
The theoretic renormalization-group approach is applied to the study of the critical behavior of the ddimensional Ising model with long-range correlated quenched impurities, which has a power-like correlations r-(d-ρ).The asymptotic scaling law is studied in the framework of the expansion in e = 4 - d. In d ＜ 4, the dynamic exponent z .is calculated up to the second order in ρ with ρ = O(e1/2). The shape function is obtained in one-loop calculation.When d = 4, the logarithmic corrections to the critical behavior are found. The finite size effect on the order parameter relaxation rate is also studied.
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The critical properties of a ddd-dimensional Bose model in the presence of a random quenched field are investigated by renormalization-group techniques via the replica trick. To first order in $\ensuremath{\epsilon}=6\ensuremath{-}d$ it is found that both at nonzero- and zero-temperature regimes the grand-canonical critical exponents are the same as those of the corresponding classical model so that thermal and quantum fluctuations appear to be irrelevant. Furthermore, at zero temperature, the scaling laws already established for the pure model are retrieved with the replacement of ddd with $d\ensuremath{-}4$, but the random critical exponents are not derivable from the pure ones with the dimensional shift $d\ensuremath{\rightarrow}d\ensuremath{-}4$.
The theoretic renormalization-group approach is applied to the study of the critical behavior of the -dimensional Ising model with long-range correlated quenched impurities, which has a power-like correlations . The asymptotic scaling law is studied in the framework of the expansion in . In , the dynamic exponent is calculated up to the second order in with . The shape function is obtained in one-loop calculation. When , the logarithmic corrections to the critical behavior are found. The finite size effect on the order parameter relaxation rate is also studied.
The theoretical renormalization-group approach is applied to the study of the short-time critical behavior of the Ginzburg-Landau model with long-range random impurities which have a power-like correlation r—(d—ρ. The system initially at a high temperature is firstly quenched to the critical temperature Tc and then released to an evolution with a model A dynamics. The asymptotic scaling laws are studied in the frame of a double expansion in ϵ = 4 — d and δ = ϵ + ρ with δ of the order of ϵ. The initial slip exponent θ′ for the order parameter is calculated up to the first order in ϵ = 4 — d or in ϵ1/2 corresponding to different fixed points, respectively. For d > 4, the short-time behavior of the order parameter is investigated in one loop. Two different logarithmic and exponential-logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are also solved in d = 4 dimension. Crossover between the nonrandom behavior and quenched behavior is found for n = 4 (where n is the spin dimensionality) in d = 4 dimension.