We investigate the geometry of a critical system undergoing a second-order thermal phase transition. Using a local description for the dynamics characterizing the system at the critical point T=T(c), we reveal the formation of clusters with fractal geometry, where the term cluster is used to describe regions with a nonvanishing value of the order parameter. We show that, treating the cluster as an open subsystem of the entire system, new instanton-like configurations dominate the statistical mechanics of the cluster. We study the dependence of the resulting fractal dimension on the embedding dimension and the scaling properties (isothermal critical exponent) of the system. Taking into account the finite-size effects, we are able to calculate the size of the critical cluster in terms of the total size of the system, the critical temperature, and the effective coupling of the long wavelength interaction at the critical point. We also show that the size of the cluster has to be identified with the correlation length at criticality. Finally, within the framework of the mean field approximation, we extend our local considerations to obtain a global description of the system.We investigate the geometry of a critical system undergoing a second-order thermal phase transition. Using a local description for the dynamics characterizing the system at the critical point T=T(c), we reveal the formation of clusters with fractal geometry, where the term cluster is used to describe regions with a nonvanishing value of the order parameter. We show that, treating the cluster as an open subsystem of the entire system, new instanton-like configurations dominate the statistical mechanics of the cluster. We study the dependence of the resulting fractal dimension on the embedding dimension and the scaling properties (isothermal critical exponent) of the system. Taking into account the finite-size effects, we are able to calculate the size of the critical cluster in terms of the total size of the system, the critical temperature, and the effective coupling of the long wavelength interaction at the critical point. We also show that the size of the cluster has to be identified with the correlation length at criticality. Finally, within the framework of the mean field approximation, we extend our local considerations to obtain a global description of the system.

arXiv:hep-ph/0008047Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsSep 1, 200029 citations

Critical dynamics at the percolation threshold

Anthony C. Maggs, Robin B. Stinchcombe

Dilution-induced criticality is discussed using exact and scaling descriptions. A simple exact argument is used to obtain the scaling form for the density of states in dilute chains. For higher-dimensional systems it is shown that the condition for the existence of a 'fraction' edge is equivalent to the violation of a scaling relation between the fracton dimension of the infinite cluster, d and various static exponents. In addition the result dd/d f is obtained for the fracton dimensionality of the finite clusters near the percolation threshold; d f is the fractal dimension of the infinite cluster.Dilution-induced criticality is discussed using exact and scaling descriptions. A simple exact argument is used to obtain the scaling form for the density of states in dilute chains. For higher-dimensional systems it is shown that the condition for the existence of a 'fraction' edge is equivalent to the violation of a scaling relation between the fracton dimension of the infinite cluster, d and various static exponents. In addition the result dd/d f is obtained for the fracton dimensionality of the finite clusters near the percolation threshold; d f is the fractal dimension of the infinite cluster.

Journal of Physics A Mathematical and GeneralFeb 1, 19863 citations

Dynamical aspects of critical phenomena

M. P. Kawatra, J. I. Budnick

Jan 1, 197240 citations

Combination of Wavefunction and Density Functional Approximations for Describing Electronic Correlation

Alejandro Jesus Garza Gonzalez

Perhaps the most important approximations to the electronic structure problem in quantum chemistry are those based on coupled cluster and density functional theories. Coupled cluster theory has been called the ``gold standard'' of quantum chemistry due to the high accuracy that it achieves for weakly correlated systems. Kohn--Sham density functionals based on semilocal approximations are, without a doubt, the most widely used methods in chemistry and material science because of their high accuracy/cost ratio. The root of the success of coupled cluster and density functionals is their ability to efficiently describe the dynamic part of the electron correlation. However, both traditional coupled cluster and density functional approximations may fail catastrophically when substantial static correlation is present. This severely limits the applicability of these methods to a plethora of important chemical and physical problems such as, e.g., the description of bond breaking, transition states, transition metal-, lanthanide- and actinide-containing compounds, and superconductivity. In an attempt to tackle this problem, nonstandard (single-reference) coupled cluster-based techniques that aim to describe static correlation have been recently developed: pair coupled cluster doubles (pCCD) and singlet-paired coupled cluster doubles (CCD0). The ability to describe static correlation in pCCD and CCD0 comes, however, at the expense of important amounts of dynamic correlation so that the high accuracy of standard coupled cluster becomes unattainable. Thus, the reliable and efficient description of static and dynamic correlation in a simultaneous manner remains an open problem for quantum chemistry and many-body theory in general. In this thesis, different ways to combine pCCD and CCD0 with density functionals in order to describe static and dynamic correlation simultaneously (and efficiently) are explored. The combination of wavefunction and density functional methods has a long history in quantum chemistry (practical implementations have appeared in the literature since the 1970s). However, this kind of techniques have not achieved widespread use due to problems such as double counting of correlation and the symmetry dilemma---the fact that wavefunction methods respect the symmetries of Hamiltonian, while modern functionals are designed to work with broken symmetry densities. Here, particular mathematical features of pCCD and CCD0 are exploited to avoid these problems in an efficient manner. The two resulting families of approximations, denoted as pCCD+DFT and CCD0+DFT, are shown to be able to describe static and dynamic correlation in standard benchmark calculations. Furthermore, it is also shown that CCD0+DFT lends itself to combination with correlation from the direct random phase approximation (dRPA). Inclusion of dRPA in the long-range via the technique of range-separation allows for the description of dispersion correlation, the remaining part of the correlation. Thus, when combined with the dRPA, CCD0+DFT can account for all three-types of electron correlation that are necessary to accurately describe molecular systems. Lastly, applications of CCD0+DFT to actinide chemistry are considered in this work. The accuracy of CCD0+DFT for predicting equilibrium geometries and vibrational frequencies of actinide molecules and ions is assessed and compared to that of well-established quantum chemical methods. For this purpose, the $f^0$ actinyl series (UO $_2^{2+}$ , NpO $_ 2^{3+}$ , PuO $_2^{4+}$ ), the isoelectronic NUN, and Thorium (ThO, ThO $^{2+}$ ) and Nobelium (NoO, NoO $_2$ ) oxides are studied. It is shown that the CCD0+DFT description of these species agrees with available experimental data and is comparable with the results given by the highest-level calculations that are possible for such heavy compounds while being, at least, an order of magnitude lower in computational cost.Perhaps the most important approximations to the electronic structure problem in quantum chemistry are those based on coupled cluster and density functional theories. Coupled cluster theory has been called the ``gold standard'' of quantum chemistry due to the high accuracy that it achieves for weakly correlated systems. Kohn--Sham density functionals based on semilocal approximations are, without a doubt, the most widely used methods in chemistry and material science because of their high accuracy/cost ratio. The root of the success of coupled cluster and density functionals is their ability to efficiently describe the dynamic part of the electron correlation. However, both traditional coupled cluster and density functional approximations may fail catastrophically when substantial static correlation is present. This severely limits the applicability of these methods to a plethora of important chemical and physical problems such as, e.g., the description of bond breaking, transition states, transition metal-, lanthanide- and actinide-containing compounds, and superconductivity. In an attempt to tackle this problem, nonstandard (single-reference) coupled cluster-based techniques that aim to describe static correlation have been recently developed: pair coupled cluster doubles (pCCD) and singlet-paired coupled cluster doubles (CCD0). The ability to describe static correlation in pCCD and CCD0 comes, however, at the expense of important amounts of dynamic correlation so that the high accuracy of standard coupled cluster becomes unattainable. Thus, the reliable and efficient description of static and dynamic correlation in a simultaneous manner remains an open problem for quantum chemistry and many-body theory in general. In this thesis, different ways to combine pCCD and CCD0 with density functionals in order to describe static and dynamic correlation simultaneously (and efficiently) are explored. The combination of wavefunction and density functional methods has a long history in quantum chemistry (practical implementations have appeared in the literature since the 1970s). However, this kind of techniques have not achieved widespread use due to problems such as double counting of correlation and the symmetry dilemma---the fact that wavefunction methods respect the symmetries of Hamiltonian, while modern functionals are designed to work with broken symmetry densities. Here, particular mathematical features of pCCD and CCD0 are exploited to avoid these problems in an efficient manner. The two resulting families of approximations, denoted as pCCD+DFT and CCD0+DFT, are shown to be able to describe static and dynamic correlation in standard benchmark calculations. Furthermore, it is also shown that CCD0+DFT lends itself to combination with correlation from the direct random phase approximation (dRPA). Inclusion of dRPA in the long-range via the technique of range-separation allows for the description of dispersion correlation, the remaining part of the correlation. Thus, when combined with the dRPA, CCD0+DFT can account for all three-types of electron correlation that are necessary to accurately describe molecular systems. Lastly, applications of CCD0+DFT to actinide chemistry are considered in this work. The accuracy of CCD0+DFT for predicting equilibrium geometries and vibrational frequencies of actinide molecules and ions is assessed and compared to that of well-established quantum chemical methods. For this purpose, the $f^0$ actinyl series (UO$_2^{2+}$, NpO$_ 2^{3+}$, PuO$_2^{4+}$), the isoelectronic NUN, and Thorium (ThO, ThO$^{2+}$) and Nobelium (NoO, NoO$_2$) oxides are studied. It is shown that the CCD0+DFT description of these species agrees with available experimental data and is comparable with the results given by the highest-level calculations that are possible for such heavy compounds while being, at least, an order of magnitude lower in computational cost.

Jun 2, 20160 citations

f^0

actinyl series (UO

_2^{2+}

, NpO

_ 2^{3+}

, PuO

_2^{4+}

), the isoelectronic NUN, and Thorium (ThO, ThO

^{2+}

) and Nobelium (NoO, NoO

_2

) oxides are studied. It is shown that the CCD0+DFT description of these species agrees with available experimental data and is comparable with the results given by the highest-level calculations that are possible for such heavy compounds while being, at least, an order of magnitude lower in computational cost.