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Journal of Physics A Mathematical and General

Self-consistent treatment of impurity influence in quantum systems at T=0

Gaetano Busiello, L. De Cesare, Ileana Rabuffo

Published Sep 11, 1985• 5 citations
Publisher

Abstract

The zero-temperature properties of quantum systems in the presence of quenched impurities which coupled quadratically to the order parameter are investigated by means of a self-consistent treatment. A physical interpretation of the renormalisation group fixed point instability recently discovered is suggested and a comparison with some experiments on doped quantum incipient ferroelectrics is also made.

DOI
10.1088/0305-4470/18/13/003

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The zero-temperature properties of quantum systems in the presence of quenched impurities which coupled quadratically to the order parameter are investigated by means of a self-consistent treatment. A physical interpretation of the renormalisation group fixed point instability recently discovered is suggested and a comparison with some experiments on doped quantum incipient ferroelectrics is also made.
Double expansion renormalization group treatments are still employed in literature to describe the effects of symmetry-conserving quenched disorder on quantum phase transitions as an alternative to more or less phenomenological models. If one assumes = 4 – d and the dimensionality τ of the imaginary-time space involved in a generalized quantum action as simultaneously small parameters, a stable random fixed point is found which governs a new disorder-induced quantum criticality usually considered meaningful also after extrapolation to the physical time-like dimension τ = 1. In contrast, by considering τ = 1 at the beginning, a runaway takes place in the parameter space for dimensionalities d < 4. To give some insight into these contrasting results, we employ here a simple self-consistent approach for case of short-range correlated quenched disorder which allows us to analyze analytically what happens by continuous variation of the fictitious time-like dimension τ We find that a τ *( n ) < 1 exists, depending on the symmetry index n , above which the disorder inhibits the occurrence of a conventional quantum phase transition. This suggests that the usual procedure to extrapolate the small τ -predictions to the value of interest τ = 1 may have no real physical meaning.
Zero temperature phase transitions not only occur in the bulk of quantum systems, but also at boundaries or impurities. We review recent work on quantum phase transitions in impurity models that are generalizations of the standard Kondo model describing the interaction of a localized magnetic moment with a metallic fermionic host. Whereas in the standard case the moment is screened for any antiferromagnetic Kondo coupling as T to 0, the common feature of all systems considered here is that Kondo screening is suppressed due to the competition with other processes. This competition can generate unstable fixed points associated with phase transitions, where the impurity properties undergo qualitative changes. In particular, we discuss the coupling to both non-trivial fermionic and bosonic baths as well as two-impurity models, and make connections to recent experiments.
The possibility of a quantum phase transition in a d-dimensional model with quenched disorder is analyzed via a renormalization group treatment without using the dimensionality ${\ensuremath{\varepsilon}}_{\ensuremath{\tau}}$ of the imaginary time direction $\ensuremath{\tau}$ as an additional small expansion parameter. We work at the fixed physical value ${\ensuremath{\varepsilon}}_{\ensuremath{\tau}}=1$ from the beginning, assuming the infinite correlation length in the timelike direction at temperature T=0T=0T=0 as generated by an appropriate limit of a time-dependent random potential with long-range correlation in time. A stable fixed point is found to occur for realistic dimensionalities which, in the time-totally-correlated limit, is assumed to govern an exotic second order quantum phase transition in the original physical model. Also in the present approach a double expansion procedure is used but, more physically, we involve the extension of the random correlation in the $\ensuremath{\tau}$ direction rather than the artificial and doubtful expansion in small ${\ensuremath{\varepsilon}}_{\ensuremath{\tau}}.$
Recent work is reviewed on continuous quantum phase transitions in impurity models, both with fermionic and bosonic baths – these transitions are interesting realizations of boundary critical phenomena at zero temperature. The models with fermion bath are generalizations of the standard Kondo model, with the common feature that Kondo screening of the localized spin can be suppressed due to competing processes. The models with boson bath are related to the spin–boson model of dissipative two-level systems, where the interplay between tunneling and friction results in multiple phases. The competition inherent to all models can generate unstable fixed points associated with quantum phase transitions, where the impurity properties undergo qualitative changes. Interestingly, certain impurity transitions feature both lower-critical and upper-critical “dimensions” and allow for epsilon-type expansions. Results are presented for a number of observables, obtained by both analytical and numerical renormalization group techniques, and connections made to experiments.