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Physical review. B, Condensed matter

Exotic quantum phase transition in systems with quenched disorder

L. De Cesare, Maria Teresa Mercaldo

Published Dec 17, 2001• 8 citations
Publisher

Abstract

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 $T=0$ 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 ${\ensuremath{\varepsilon}}_{\ensuremath{\tau}}.$

DOI
10.1103/physrevb.65.024202

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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 \ensuremathε\ensuremathτ{\ensuremath{\varepsilon}}_{\ensuremath{\tau}}\ensuremathε\ensuremathτ​ of the imaginary time direction \ensuremathτ\ensuremath{\tau}\ensuremathτ as an additional small expansion parameter. We work at the fixed physical value \ensuremathε\ensuremathτ=1{\ensuremath{\varepsilon}}_{\ensuremath{\tau}}=1\ensuremathε\ensuremathτ​=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 \ensuremathτ\ensuremath{\tau}\ensuremathτ direction rather than the artificial and doubtful expansion in small \ensuremathε\ensuremathτ.{\ensuremath{\varepsilon}}_{\ensuremath{\tau}}.\ensuremathε\ensuremathτ​.
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.
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 critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute mmm-vector model, the n=0n=0n=0 limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic mmm-component fixed point, which characterizes the critical behavior of the pure system, is stable. For m<mcm<{m}_{c}m<mc​, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for m=1m=1m=1. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.
The nonanalyticity of the Loschmidt echo at critical times in quantum quenched systems is termed as the dynamical quantum phase transition, extending the notion of quantum criticality to a nonequilibrium scenario. In this paper, we establish a new paradigm of dynamical phase transitions driven by a sudden change in the internal spatial correlations of the disorder potential in a low-dimensional disordered system. The quench dynamics between prequenched pure and postquenched random system Hamiltonian reveals an anomalous dynamical quantum phase transition triggered by an infinite disorder correlation in the modulation potential. The physical origin of the anomalous phenomenon is associated with the overlap between the two distinctly different extended states. Furthermore, we explore the quench dynamics between the prequenched random and postquenched pure system Hamiltonian. Interestingly, the quenched system undergoes dynamical quantum phase transitions for the prequench white-noise potential in the thermodynamic limit. In addition, the quench dynamics also shows a clear signature of the delocalization phase transition in the correlated Anderson model.
The theoretic renormalization-group approach is applied to the study of short-time critical behavior of the ddd-dimensional spin systems (model A) in the presence of quenched impurities with a long-range correlations decaying as ${r}^{\ensuremath{-}(d\ensuremath{-}\ensuremath{\rho})}$. The asymptotic scaling laws are studied in the frame of a double expansion in $ϵ=4\ensuremath{-}d$ and $\ensuremath{\rho}$ with $\ensuremath{\rho}$ of order ϵϵϵ. In d<4d<4d<4, the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ of the magnetization and $\ensuremath{\theta}$ of the response function, are calculated up to two-loop order. The crossover between fixed points is obtained. The long-time limit of the fluctuation-dissipation ratio is found in the aging regime, and its connection to equilibrium quantities is discussed. The comparison of our results with those of other systems without long-range correlated quenched impurities is also investigated.
In quenched random-spin systems, the renormalization group can be used to develop recursion relations for the probability distribution for random potentials. Alternatively, recursion relations for the average values of potentials and their higher cumulants can be obtained. In this paper, the above technique is used to study phase transitions in quenched random nnn-component classical spin systems using the $\ensuremath{\epsilon}$ expansion to second order in $\ensuremath{\epsilon}$. It there are long-range correlations in the random potentials (e.g., all potentials along a line are equal), there are no stable physical fixed points within the $\ensuremath{\epsilon}$ expansion. This is interpreted as a smeared transition. If there are no long-range correlations in the random potentials, there is a sharp transition with pure system exponents if the specific-heat exponent ${\ensuremath{\alpha}}^{H}$ of the pure system is negative. If ${\ensuremath{\alpha}}^{H}>0$ and n>1n>1n>1, there is a sharp transition with new exponents $\ensuremath{\eta}=[\frac{(5{n}^{2}\ensuremath{-}8n)}{256{(n\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}$ and $\ensuremath{\nu}=\frac{1}{2}+[\frac{3n}{32(n\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{n(127{n}^{2}\ensuremath{-}572n\ensuremath{-}32)}{4096{(n\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}$. For n=1n=1n=1, there is no stable fixed point, which is again interpreted as a smeared transition.